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B. G.MIRKIN .
. ANALYSIS OF QUALITATIVE
iATTIDBUTESANDSTRUCTURES
, .
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Moscow
" · STATISTIKA.
1980
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17
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u "
KOB B COOTBeTCTBHH co CTPYKTYPOH OTHOIlleHHH Ha MHO>KeCTBe
3HaqeHHH OCHOBbIBaeTCH HenocpeJ{cTBeHHo Ha onpe,ne.neHHH
npH3HaKa KaK OTOOpa:>KeHHH MHO:>KeCTBa OO'beKTOB B MH01KeCT-
BO 3HaqeHHH. HecMoTpSl Ha pacnpocTpaHeHHocTb H3J10>KeHHOH
TOqKH 3peHH5I, OHa 3HaqHTeJIbHO YCTynaeT no rnYOHHe H TOq..
HOCTH pa3paOOTaHHOMY K HaCTOSlIneMY BpeMeHH oonee IIIHpO-
KOMY H YHHBepCaJIbHOMY B3rJlH,!{Y H8 npOOJIeMY THnH3anHH
IlIKaJI npH3HaKOB H OCMbICJleHjIOCTH cpaBHeHHH HX 3HaQeHHH,
OCHOBaHHOMY Ha npe,!{CTaBJIeHHH npH3HaKa CeMeHCTBOM OTO..
6pajl{eHHH (a He e,!{HHCTBeHHhIM OTOOpa>KeHHeM) MHO>KeCTBa
OO'beKTOB B MHO:>KeCTBO BexnecTBeHHbIX tlHCeJl. B aToA TeopHH
CtlHTaeTcSI, tlTO 3HaqeHHH npR3HaKoB - qHCJlOBhIe, T. e. AJln
nI06oro x: in -+ B x MHO:>KeCTBO B % - tlHCJIOBoe. OrpaHHQeHHjJ
06 HOCTH He npOHCXO,!{HT, TaK KaK 9J1eMeHThI MHO>KeCTBa B x,
19
... -
__ Z--
'111-
- "III
"'1'
HMelOInerO MOInHOCTb, He OOJIblIlYIO KOHTHHYYMa, MO>KHO 3a-
KOAHpOBaTb lIHCJIaMH. TIpH 3TOM oTHorueHH5I Me>KAY 3HalleHH5I-
u
MH npH3HaKa AOJI>KHbI nepeHTH B COOTBeTCTBYIOInHe tIHCJIOBbIe
OTHOlIleHH5I: «x paBHO Y» - B OTHOlIleHHe paBeHCTBa, i xY>Ke
j - B OTHOlIleHHe «MeHblIle» H T. n. IlpH3HaK KaK OTOOpa>KeHHe
3aAaeTC5I BMeCTe C MHO>KeCTBOM CBOHX oonycmuMblx npeo6pa-
30eaHUU. A HMeHHO cpHKcHpyeTc5I MHO>KeCTBO <D tIHCJIOBbIX npe-
oopa30BaHHH, He MeH5IIOIlJJIX STOT npH3HaK. TOQHee rOBOp5l,
npH3HaK 3aAaeTCSI He TOJIbKO OTOOpa>KeHHeM x: m -+ B x, HO
II BceB03MO>KHbIMH oTo6pa>KeHH5IMH BH.L{a cpx, r.L{e cp E <D.
HanOMHHM, 1JTO AJISI JIIOOoro 06'beKTa i E m 3HatIeHHe cpx (i)
onpeAeJIHeTCSI KaK pe3YJIbTaT nOCJIe.L{OBaTeJIbHOrO npHMeHeHH5I
OOOHX OTOOpa1KeHHH: cpx (i) = cp (x (i)).
HanpHMep, 3HalleHH5I HOMHHaJIbHOrO npH3HaKa «nOJI» MO}l{-
HO 3aKO.L{HpoBaTb JIIOObIMH .L{BYMSI HepaBHbIMH lIHCJIaMH: 0 HI,
20 H 18 H T. 'A. Ba>KeH TOJIbKO <paKT pa3JIHtIHSI 3THX tIHCeJI, HO
He HX COOTHOllIeHHe - rpynnHpoBKa JIIO.L{eH no nOJIY OCTa-
eTCSI HeH3MeHHOH. TI03TOMY AJI5I HOMHHaJIbHOrO npH3HaKa
BMeCTO OTOOpa>KeHHSI x : m -+ B x MO>KHO B3S1Tb JIIOOoe APY-
roe, nOJIYlIeHHoe H3 Hero B3aHMHO-0.L{H03HallHbIM npeoopa30Ba-
HHeM. B3aHMHa51 0.L{H03HallHOCTb HeOOXO.L{HMa, 11 TOObI HepaB-
HbIe 3HalleHH5I OCTaBaJIHCb HepaBHbIMH.
JISI paHrOBoro npH3HaKa Ba>KHO He TOJIbKO pa3JIHtIHe B
tIHCJIOBbIX 3HalleHHSlX, HO H HX ynop5l.L{OtIeHHe. PaHrOBbIH npH-
3HaK He H3MeHHTC5I npH JIIOOOM MOHOTOHHO B03paCTalO eM npe-
oopa30BaHHH ero 3HaqeHHH; .L{PyrHe npe06pa30BaHH5I MorYT
HapYlIlHTb nOp5l.L{OK 3HalleHHH npH3HaKa.
MHO}l{eCTBO .L{OnYCTHMbIX npeoopa30BaHHH cI> xapaKTepH-
3yeT mun UlKallbl npII3HaKa. JISI tIaCTO BCTpetIalOIlJ,HXC5I MHO-
:meCTB <I> npHHSlTbI Cne HaJIbHbIe HaHMeHOBaHHSI. ECJIH <I> CO-
CTOHT H3 Bcex B3aHMHO-0.L{H03HallHbIX npeoopa30BaHHH, lIlKaJIa
Ha3bIBaeTC5I HOMUflallbflOU, eCJIH cI> COCTOHT H3 Bcex MOHOTOHHO
B03paCTalO HX npeoopa30BaHHH, llIKaJIa Ha3bIBaeTCSI pafl-
20eou. CpeAH KOJIHlIeCTBeHHbIX IllKaJI eCTeCTBeHHO BbI.L{eJI5IIOTCH
CJIeAYIO He TPH THna. IlIKaJI3 Ha3bIBaeTC5I UHmepeallbHOU.
eCJIH <P COCTOHT H3 BceB03MO>KHbIX nOAOJICUmeAbflblX AUHeUHblX
t/JYflKl{UU BHAa cp (x) = ax + (a > 0, - Juoooe); lIlKaJIOH
omflOtueflUU HJIH n0006UJl, eCJIH.<D COCTOHT H3 npeoopa30BaHHH
nOa06UJl cp (x) = ax; a6cOAlOmHOU, eCJIH <D COCTOHT H3 e,nHHCT-
BeHHoro TO>K.l(eCTBeHHoro npeoopa30BaHHH cp (x) = X.
KpaTKO 00CYAHM Ha3BaHHbIe THnbI lIlKaJI. KOHKpeTHoe
JIHHeHHOe npeoopa30BaHIIe cp (x) = ax + npH3HaKa x onpe-
.L{eJISleTCH BbIOOpOM ,Z(BYX KOHCTaHT: ex II . OAHa H3 HHX,
= cp...(O), xapaKTepII3yeT HOBoe MeCTOnOJIOJKeHHe HYJI5I, T. e.
20
t
,
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"
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t
'I1II1IIIII1I11111111
T01JKY OTCqeTa, .L{pyra51 - MaCIIlTaO. BC5IKHH tJHCJIOBOM HH-
TepBaJI (X, y) npeoopa3yeTC5I B a pa3 npeBOCXO.L{51InHH ero HH-
TepBaJI (cp (x), cp (y)); cp (y) - cp (x) = a (y - x). B IIlKaJIe
HHTepBaJIOB OTHOllleHHe HHTepBaJIOB Me:>K.L{Y JIIOObIMH 3Haqe-
HH5IMH OCTaeTC5I HeH3MeHHbIM npH JIIOOOM .L{OnYCTHMOM npe-
00pa30BaHHH: Kam.L{bIM MeH5IeTC5I B O)1.HO H TO }Ke qHCJIO a pa3.
npHMep IIlKaJIhI HHTepBaJIOB - IllKaJIa H3MepeHH5I BpeMeHH.
B IlIKaJIe nO.L{OOH5I HaqaJIO OTCqeTa He MeH5IeTC5I: cp (x) = ax,
TaK qTO = o. npH3HaK, H3MepeHHbIM B 3TOM IllKaJIe, onpe-
,neJIeH C TOqHOCTblO .L{O MaCIIlTaoa, KaK, Hanp HMep, «Macca»,
«.L{JIHHa» H T. n.
B aOCOJIIOTHOH IllKaJIe H3Mep5lIOTC5I KOJIHqeCTBa OO'beKTOB.
BCTpeQaIOTC5I H .L{PyrHe THnbI KOJIHqeCTBeHHbIX IlIKaJI, Ha-
npHMep c cp (x) = eax(a> 0); JIOrapHcpMHpOBaHHeM 3Ta
IIlKaJIa CBO.L{HTC5I K IllKaJIe OTHOIlleHHM. K KOIlU4,eCm8eHHbtM
tuKallaM eCTeCTBeHHO OTHOCHTb TaKHe, B KOTOpbIX .L{OnYCTHMbIe
npeoopa30BaHHH onpe.L{eJI5IIOTC5I KOHeqHbIM qHCJIOM KOJIH1Je-
CTBeHHbIX napaMeTpOB. BnpOqeM, CYIneCTBOBaHHe IllKaJI, onpe-
.L{eJI5IeMhIX oOJIee qeM .L{BYM5I napaMeTpaMH, KOTopbIe TeM HJIH
HHbIM cnocoooM MO:>KHO CB5I3aTb C xapaKTepHcTHKaMH MaCIlITa-
oa H TOqKH OTCqeTa, BeCbMa COMHHTeJIbHO.
B TepMHHax .L{OnYCTHMbIX npeoopa30BaHHH Bonpoc 0 IllKa-
JIax .L{OMHHHpOBaHH5I H nOXO:>KeCTH, 0 HanpaBJIeHHbIX H He-
HanpaBJIeHHbIX IllKaJIaX YTOqH5IeTC5I CJIe.L{YIOInHM 00pa30M.
PaHrOBa51 IlIKaJIa, onpe.L{eJIeHHa51 BbIIlIe, - HanpaBJIeHHa5l. npH-
3HaK no 3TOM IlIKaJIe onpe.L{eJIeH C TOqHOCTblO .L{O MOHOTOHHO
B03paCTalOInerO npeoopa30BaHH5I. HeHanpaBJIeHHa51 nop5l.L{-
KOBa51 IllKaJIa xapaKTepH3yeTC5I MHO}KeCTBOM <D, COCT05l HM
H3 Bcex MOHOTOHHbIX npeoopa30BaHHM - H B03pacTalO Hx,
H YObIBaIOInHx. 9TO MHO}KeCTBO <D COXpaH5IeT nOp5l.L{OK 3H3-
u
qeHHH C TOqHOCTblO .L{O HanpaBJIeHH5I, OCTaBJI5I5I HeH3MeHHhIM
OTHOIlleHHe B3aHMHoro COCe.L{CTBa OO'beKTOB, TaK tlTO IlIKaJIa
xapaKTepH3yeT He .L{OMHHHpOBaHHe, a JIHIllb nOXO:>KeCTb. AHa-
JIOfHqHhIM 00pa30M BbIllle ObIJIH onpe.L{eJIeHbI IlIKaJIbI HHTepBa-
JIOB H OTHOIIleHHH, KOTopbIe 51BJI5IIOTC5I HanpaBJIeHHbIMH, TaK
KaK npH nOJIO:>KHTeJIbHOM a OOJIbIlIHe 3HaqeHH5I nepeXO,Z(51T B
OOJIhlIIHe. HeHanpaBJIeHHbIe IllKaJIbI HHTepBaJIOB H OTHOIlleHHH
nOpO>K.L{aIOTC5I MHomeCTBaMH .L{OnYCTHMbIX npeoopa30BaHHH BH-
,na cp (x) = ax + H cp (x) = ax COOTBeTCTBeHHO, r.L{e a -
rrpOH3BOJIbHOe He paBHoe HYJIIO BeIneCTBeHHoe tlHCJIO. .aJI5I
3THX HeHanpaBJIeHHbIX IllKaJI HHBapHaHTHbIMH 51BJI5IIOTCH He
OTHOlIIeHH5I HHTepBaJIOB C yqeTOM 3HaKa, HO JIHIllb OTHOlIIeHH5I
HX .L{JIHH. MHorHe nCHXOJIOrHqeCKHe IllKaJIbI THna «HHTpaBepT-
HOCTb--SKCTpaBepTHOCTh», «JIOrHqeCKHM--00pa3H@M» (CTHJIb
21
"'- '" '-
-'lilll
MbIIIlJIeHH5I) OneHHBalOT JIHIllb nOXO:meCTb CYO'beKTOB no BOC-
npHHTHIO, HO He npHnHCbIBalOT HanpaBJIeHH5I, He3aBHCHMO OT
Toro, B KOJIHqeCTBeHHbIX HJIH KaqeCTBeHHbIX TepMHHax H3Me-
p5lIOTC5I 3TH npH3HaKH.
nOH5ITHe ,l{OnYCTHMoro npeoopa30BaHHSI n03BOJI5IeT TpaK-
TOBaTb Bonpoc 00 OCMbICJIeHHOCTH pa3JIHqHbIX cpaBHeHHH 3Ha-
qeHHH npH3HaKOB KaK qaCTb OOIneR npOOJIeMbI OCMbICJIeHHOCTH
YTBep)K,l{eHHH 0 pe3YJIbTaTaX H3MepeHHH. PaCCMOTpHM KaKoe-
2+6
JIHOO qHCJIOBOe YTBep)K,l{eHHe, HanpHMep = 3 HJIH 2 +
+ 3 = 3 + 2. nepBoe, OqeBH.L{HO, JIO)KHO, TOr.L{a KaK BTO-
poe - HCTHHHO. .aonycTHM, qTO H3MepeHHSI npOBe.L{eHbI B IllKa-
JIe oTHolIleHHH, T. e. .L{OnYCTHMO JIlOOoe npeoopa30BaHHe BH,l{a
cp (x) = ax. nOCMOTpHM, He H3MeHHTCSI JIH HCTHHHOCTb YTBep-
>K,l{eHHH nOCJIe npHMeHeHHSI ,l{onYCTHMoro npeoopa30BaHH5I.
I10JIYQaeM cp(2) 1 cp(6) = fJJ (3), a2 + a3 = a3 + a2. 3Ha-
qeHHe HCTHHHOCTH 3THX paBeHCTB He HaMeHHTC5I npH cOKpaw.e-
HHH Ha a, B pe3YJIbTaTe qero nOJIyqalOTCSI HCXO.L{HbIe paBeHCT-
Ba. 06a YTBep)K,l{eHHSI OCMbICJleHHbI B IllKaJIe OTHOlIleHHH, TaK
KaK nepBoe OCTaeTC5I JIOLKHhIM, a BTopoe HCTHHHbIM npH JIIO-
OOM ,nonYCTHMOM npe06pa30BaHHH. MOLKHO npOBepHTb, qTO
v
3TH YTBep)K,l{eHHH He MeH5IIOT CBOHX 3HaqeHHH HCTHHHOCTH H B
IIlKaJIe HHTepBaJIOB. Ho B nopSl,l{KoBOH IllKaJIe nepBoe YTBep)K-
,l{eHHe oeCCMbICJleHHO, TaK KaK OHO MO)KeT CTaTb HCTHHHhIM npH
nO;tXO,l{SlIneM MOHOTOHHO B03paCTalO eM npe06pa30BaHHH cpo
HanpHMep, npH cp (2) = 10, cp (3) = 15, cp (6) = 20 YTBep)K-
1
,neHHe"2 (cp (2) + cp (6») = cp (3) HCTHHHO. B TO >Ke BpeMS! BTO-
poe YTBep)K,neHHe OCTaeTC5I CnpaBe,l{JIHBbIM H, CJIe,nOBaTeJIbHO,
OCMbICJIeHHbIM npH JIIOOOM npeoopa30BaHHH cp, B TOM qHCJIe H B
u
HOMHHaJIbHOH IIlKaJIe.
TaKHM 06pa30M, qHCJIOBOe YTBep)K,neHHe SlBJISleTC5I oc-
MbLC/leflflblM (B oo eH TeopHH H3MepeHHH HCnOJIb3yeTCS! Tep-
MHH aaeK8amflocmb [94, 83]), eCJIH ero HCTHHHOCTb (HJIH JIO)K-
HOCTb) He H3MeHSleTC5I npH JII060M ,nonYCTHMOM npeoopa30Ba-
HHH BXO,nSl HX B Hero qHCeJI. 06paTHM BHHMaHHe, qTO 3.L{eCb
petlb H,neT 00 OCMbICJIeHHOCTH HMeHHO qHCJIOBbIX YTBep:>K,neHHH,
a He qHCJIOBbIX onepanHH, KaK 3TO HHor,na ,l{eJIaeTC5I. npHMe-
HeHHe onepaUHH OCMbICJIeHHO B O,l{HHX YTBep>K,neHHHX H oec-
CMbICJIeHHO B ,npyrHX. TaK, B HHTepBaJIbHoit IlIKaJIe YTBep)K.L{e-
HHe 2 + 3 = 5 6eCCMbICJIeHHO (a2 + b + a3 + b =1= as + b
npM b =1= 0), a YTBep}K,Z(eHHe 2 + 3 ' 4 + 1 OCMbICJIeHHO,
XOTH ooa cO)(ep>KaT onepanHIO CJIOJKeHHSI. PaBeHCTBO 2 + 3 =
22
!
""
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/
1IIIIIIIIIIIIIIIIIIIII11JIIIIIII
r
= 3 + 2 OCMbICJIeHHO .L{ame B HOMHHaJIbHOM III KaJ1e. OCMbIC-
JIeHHbIM B HHTepBaJIbHOM IlIKaJIe 51BJI5IeTC5I cpaBHeHHe cpeJ(-
HlfX apH<pMeTHqeCKlfX ,l{Jl5I .L{BYX cOBoKynHocTefI 06'beKToB.
1 P I q
.IT.eHCTBHTeJIbHO, BbICKa3bIBaHHe - Xi >- Yj 3KBHBa-
P i= I q i = 1
JIeHTHO YTBep*,l{eHHIO 0 TOM, tlTO
I P ex P 1 q .
- (aXi+ )=- xd- >- (aYrt- )=
p. 1 P i 1 q. I
t= = 1 =
ex q
==- Yj + .
q . 1
1=
Bonpoc 0 MHO}KeCTBe ,nOnYCTHMbIX npe06pa30BaHHH ,naH-
HOrO npH3HaKa, T. e. 0 THne ,l{aHHOH lIIKaJIbl, ,l{OBOJIbHO CJIo»<eH.
no HameMY MHeHHIO, ,nonYCTHMOCTb npe06pa30BaHH5I npH3Ha-
K8 onpe,l{eJI5IeTC5I TeMH TeOpeTHqeCKHMH 3aKOHOMepHOCT5IMU,
B KOTOphIX OH YtlaCTByeT. npeo6pa30B8HHe ,l{OnYCTHMO, eCJIU
3aKOHOMepHOCTH He H8pyrnaIOTC5I. B COOTBeTCTBHH C MeTO,l{O-
JIOrHeH aHaJIH3a ,l{aHHbIX, H3JIO:>KeHHOH BO BBe,l{eHHH, M01KHO
CKa3aTb, qTO npe06pa30BaHHe npH3HaKa (HJIH, 60JIee o6mo,
BceH Ta6JIHnbl ,naHHbIX) ,nonYCTHMO, eCJIH OHO He MeH5IeT no-
JIyqeHHhIX no STOH Ta6JIHne onHcaHHH.
B pa3BHTbIX <pH3HqeCKHX TeOpH5IX npH3HaKH, xapaKTepu-
3Y lO rnHe 06'beKTbl, CB5I3aHbl KOJIHqeCTBeHHbIMH COOTHorneHUSI-
MH, BbIpa}l{alOrnHMH onHCaHH5I O,l{HHX npH3HaKOB Qepe3 ,npy-
rife, - TeOpeTHqeCKHMH H SMnHpHqeCKHMH 3aKOHOMepHOCTSI-
MH. .aonYCTHMbI JII06bIe npe06pa30BaHH5I, He MeH5IlOrnHe STUX
COOTHOIIleHHH. TaK, 3aKOH KJIaneHpOHa PV = const, CBH3bl-
T
BalOrnHM a6COJIIOTHYIO TeMnepaTYPY T, 06'beM V H ,l{aBJIeHHe P
H.L{eaJIbHOro ra3a, HHBapHaHTeH OTHOCHTeJIbHO npeo6pa30Ba-
HHM nO,l{06H5I cp "(x) = ax STHX napaMeTpOB. 3aKoH KJIaneH-
pORa n03BOJISleT nporH03HpOBaTb no KOJIHtleCTBeHHOMY H3-
MeHeHHIO O,l{HHX napaMeTpOB KOJIHqeCTBeHHbIe H3MeHeHHSI
.npyrHX napaMeTpOB. B TO }Ke BpeM5I BO MHorHX SMnHpHqeCKUX
HCCJIe,l{OBaHH5IX ,name npH3HaKH, H3MepeHHbIe B KOJIHqeCTBeH-
HhIX «<plf3HqeCKHX» e,l{HHHnax, He CBSl3aHbI C ,nPyrHMH nOKa3a-
TeJISlMH KOJIHqeCTBeHHbIMH COOTHOrneHH5IMH, nOSTOMY HX KO-
JIHQeCTBeHHoe H3MeHeHHe He Mo*eT npHBO,l{HTb K KOJIHqeCT-
23
'oir.:"
.
'II
BeHHOMY npOrH03Y M3MeHeHHH .L{PyrHx napaMeTpOB. B TaKHX
CJIyqaHX MHOLKeCTBO .L{OnYCTHMbIX npe06pa30BaHllH eCTeCTBeH-
HO paCIllMp5IeTCH, BnJIOTb .L{O MHOlKeCTB3 BCeX B3aMf\lHO-0 HO-
3HallHbIX npe06pa30BaHHH, TaK liTO npH3HaK MOmeT paCCMa-
TpHBaTbC5I KaK HOMHHaJIbHbIiL HanpMMep, BO MHOrHX CO[(HO-
JIOrHlIeCKHX aHKeTaX cpHrYPHpyeT TaKOH Ha nepBbUi B3fJI5I.L{
KOJIHlIeCTBeHHbUi npH3HaK, KaK «B03paCT». TIpH OTCYTCTBHM
KOJIHlIeCTBeHHbIX COOTHOIlIeHHH, CB5I3bIBalO MX B03paCT C .L{Py-
fHMH XapaKTepHCTHKaMM, STOT nOKa3aTeJIb MO:>KeT 6bITb HC-
nOJIb30BaH JIHlIlh .L{JI5I }(aqeCTBeHHoro nporH03a TMna «qeM
60JIbille B03paCT, TeM BbIllie CTaJK pa60TbI» (nOp5l.L{KOBOe onM-
caHMe) HJIH .L{ame «pa3HbIe B03paCTHbIe fpynnbI no-pa3HOMY
OTHOC5ITC5I K TypH3MY» (HOMHHaJIbHOe onHcaHHe). I1MeHHo OT-
CYTCTBHe cO.L{ep)KaTeJIbHbIX KOJIHlIeCTBeHHbIX COOTHOIlleHHH npH-
BO.L{HT K He06xo,1l.HMOCTH nOp5l.L{KOBOro HJIH HOMHHaJIbHOfO npe.L{-
CTaBJIeHMH KOJIMlIeCTBeHHbIX npH3HaKOB C nOMO blO pa36He-
HH5I 06JIaCTH HX 3HalleHHH, liTO M 06'b5lCHHeT CTOJIb 60JIbIllYIO>
pOJIb rpynnHpoBOK B 3MnHpHlIeCKHX HaYKax.
B TO :>Ke BpeM5I HaJIHlIHe a.L{eKBaTHbIX KOJIHtJeCTBeHHbIX onH--
caHHH, BKJIlOlIalO HX lIHCJIOBbIe npe.L{CTaBJIeHM5I KalleCTBeHHbIX.
npH3HaKOB, nepeBO.L{HT 3TH npH3HaKH B KJIaCC KOJIHlIeCTBeH--
HbIX. TaK, paHrOBbIH npM3HaK «KBaJIH<pMKa[(HH» MOjl{eT OKa--
3aTbC5I BKJIlOlIeHHbIM B lIHCJIO aprYMeHTOB KOJIHlIeCTBeHHoit
<PYHK[(HH perpeCCMH, TaK liTO no ero 3HalleHM5IM MO)KHO C .L{O-
u
CTaTOllHOH TOllHOCTblO npe.L{CKa3bIBaTb, CKa:meM, ypOBeHb 3a...
pa60THOH nJIaTbI.
Heo6xo.L{MMO OTMeTHTb, liTO lIaCTO KOJIHlIeCTBeHHbIe COOT-
HOIlleHH5I npe.L{CTaBJI5IIOT C060H npOCTO cnoc06 BBe.L{eHM5I HO-
Boro TepMHHa (npM3HaKa), KOTOpbIH lIepe3 3TO COOTHoweHHe
onHCbIBaeTC5I HCXO.L{HbIMH npH3HaKaMH. Bce BeJIHlIHHbI TeopHH
HeKOToporo KOHKpeTHoro 51BJIeHH5I pa3.L{eJIHIOTC5I Ha ,l{Ba KJIaC-
ca: nepBHlIHbIe, <PYH.L{aMeHTaJIbHbIe npH3HaKH II BTOpHlIHbIe,
npOH3BO.L{HbIe npH3HaKH, CKOHCTpYHpOBaHHbIe Ha OCHOBe nep-
BHlIHbIX. I1cTopH5I HaYKH nOKa3bIBaeT, liTO pa3BHTHe HaYlIHbIX
,nHC[(HnJIHH npe.L{CTaBJI5IeT cODoH [(enb peBOJIIO[(HOHHqIX nepe-
BOpOTO , CB5I3aHHbIX C H3MeHeHHeM cnHCKa nepBHlIHbIX npH-
3HaKOB (a TaKJKe 06'beKToB) H nepeBO.L{OM cTapbIx nepBHlIHbIX
nOKa3aTeJIeH B npOH3BO.L{Hble. 3MnHpHlIecKMe COOTHOlIleHH5I
Me:>K.L{Y cTapbIMH nepBHlIHbIMH npH3HaKaMH npeBpa aIOTC5I
npH 3TOM B COOTHOlIleHHH, BbIBO.L{HMbIe H3 onpe.L{eJIeHH5I 3THX
npH3HaKoB lIepe3 HOBbIe nepBMlIHbIe npM3HaKH. POJIb aHaJIH3a
.L{aHHbIX npH 3TOM COCTOMT B TOM, lITOObI HaKonHTb .L{OCTaTOll-
HbIH 3anac onHcaHMH, HeOOXO,l{HMbIH .L{JISI TBopIIeCKoro nepexo.L{(l.I
K HOBOMY 3Tany pa3BHTHH TeopMH.
24
t
..
i
.
1111"' 1
1.1.4. JI. UXOTOMU"leCK,Ue npU3flaK,U
PaCCMOTpHM YHHKaJIbHbIH KJIaCC npH3HaKoB, MHO-
u
Jl{eCTBO 3HaqeHHH Ka:>K.L{OrO H3 KOTOpbIX COCTOMT Bcero H3 .L{BYX
9JIeMeHToB. TaKHe npH3HaKH Ha3bIBaIOTC5I ouXOmOMUtLeCKU-
.MU, 6YlleBcKuMu, 6uHapHbl./vtu. u
MHomecTBo Bcex B3aHMHO-O.L{H03HaqHbIX npeoopa30BaHHH
,.L{.n51 ,nHXoTOMHqeCKHX npH3HaKoB COBna.L{aeT C MHO}KeCTBOM
.JIHHeHHbIx npeoopa30BaHHH. ,[{eHcTBMTeJIbHO, B3aHMHO-0,l{HO-
3HaqHOe COOTBeTCTBMe ,l{BYX KOHKpeTHblx 3HaqeHHH Xl' X2, BO-
oome rOBOp5l, .L{PyrMM ,l{BYM 3HaqeHH5IM - Yl, Y2' MO:>KeT ObITb
3a.L{aHO KaK JIHHeHHOe C napal'vleTpaMH ex M , Y.L{OBJIeTBOp5l1O-
mHMH COOTHOlIleHH5IM Yi = exXi + (i = 1, 2), KOTopbIe npH
Xl =F X 2 0,l{H03Ha1JHO pa3pelIlHMbI OTHOCI1TeJIbHO ex H ; npHqeM
113 a =1= 0 CJIe.L{yeT Yl =1= Y2. aTO 03HaqaeT, qTO HOMHHaJIbHbIH
v
,IlMXOTOMl1qeCKI1H npH3HaK Ha CaMOM ,l{eJIe H3MepeH B HeHa-
npaBJIeHHOH IIlKaJIe HHTepBaJIOB. AHaJIOrHqHO ,l{OKa3bIBaeTC5I,
1JTO paHrOBbIM (no ,l{OMHHHpOBaHIIlO) ,l{HXOTOMHqeCKHM npH3lfaK
Ha caMOM ,l{eJIe H3MepeH B HanpaBJIeHHOM IIlKaJIe HHTepBaJIOB.
TaKHM oopa30M, KaqeCTBeHHbIe ,l{HXOTOMHqeCKHe npH3Ha-
KH O,l{HOBpeMeHHO 51BJI5IIOTC5I KOJIHqeCTBeHHbIMH. n03TOMY B03-
MomHO KOJIHqeCTBeHHoe npe.L{CTaBJIeHHe KaqeCTBeHHbIX npH-
3HaKOB B apHcp1vleTHqeCKOM eBKJIH.L{OBOM npOCTpaHCTBe C no-
MOInblO CJIe.L{YIOIneM npone,nypbI aUxomOMU3al{UU.
Ka}K.L{bIM KaqeCTBeHHbIH npH3HaK X, HMelO HH m 3HaQeHHH,
npe,l{CTaBJI5IeTC5I npH 3TOM m .L{HXOTOMHqeCKHMH npH3HaKaMH Xs
(s = 1, ..., m), r,l{e Xs (i) = 1, eCJIH X (i) = s, H Xs (i) = 0,
eCJIH X (i) =1= S. 9TH m ,l{HXOTOMHQeCKHX npH3HaKOB Xs oopa-
3YIOT CTOJIOl(bI N X m MaTpHl(bI X C HYJIb-e,l{HHHqHbIMH 3JIe-
MeHTaMH. TaKHe MaTpHl(bI Ha3bIBaIOTC5I 6YlleBCKUMU. MaTpHu a
X 3a,l{aeT a,l{eKBaTHoe npe,l{CTaBJIeHHe npH3HaKa X, npH KOTO-
pOM Ka:>K,l{bIH 060eKT H306pa:>KaeTC5I CBoeM CTpOKOH KaK TOqKa
m-MepHoro npOCTpaHCTBa; caM :>Ke npH3HaK xapaKTepH3yeTcR
He O.L{HHM, a l(eJIOM cOBoKynHocTblO CTOJI6l(OB. TaKoe npe,l{CTaB-
JIeHHe B03MO)KHO H ,l{JI5I HeaJIbTepHaTMBHbIX npH3HaKOB, pea-
JIH3yeMhlx MHOr03HaQHhIMH oT06pameHH5IMH X: eCJIH x (i) =
== {Sl' ..., 5 p }, TO i-51 cTpoKa 1\1aTpH[(bI X 6Y.L{eT co,nep>KaTb
e.L{HHHl(bI B cToJI6l(ax 51' ..., 5 p .
PaCCMOTpHM CTaH,l{apTHbIe Onepal(HH eBKJIH.L{OBCKHX npo-
CTpaHCTB npHMeHHTeJIbHO K TaKHM 6YJIeBCKHM MaTpHl(aM. nYCTb
a = (ai' ..., am) - npOH3BOJIhHbIM BeKTOp C KOMnOHeHTaMH
as. KaKoB CMhICJI BeKTopa Xa? OQeBH,l{HO, Xa = asx s , r.n e
s
Xf? - 6YJIeBCKHH BeKTOp, COOTBeTCTBYIO HM S-MY 3HaqeHHIO
5
.. "-' :L._ ___ L _!! _.._
--. III
..,.
'i:
npH3HaKa x. BeKTOp asxs HMeeT KOMnOHeHTaMH JIHOO HYJIH npH
x (i) =1= S, JIHOO as npH x (i) = s, TaK liTO BeKTOp Xa CTaBHT B
COOTBeTCTBHe Ka:>K,!{OMY OO'beKTY i lIHCJlO as, r,!{e S - 3HalleHHe
npH3HaKa x Ha OO'beKTe i. npH HeaJIbTepHaTHBHOM xapaKTepe
npH3HaKa i-51 KOMnOHeHTa BeKTopa Xa paBHa cYMMe as! + ...
... + as BeJIHlIHH, OTBeqalO HX 3HalleHH5IM 51 ..., Sp npH3HaKa
p
H8 i-M OO'beKTe. TaKHM oopa30M, K03cpcpHnHeHTbI JIHHeHHOH
KOMOHHanHH HMelOT O,!{HOBpeMeHHO CMbICJI KOJIHtIeCTBeHHbIX
v
3HatIeHHH, MeTOK as, npHnHcaHHbIX 3HaqeHH5J.M S HOMHHaJIb-
Horo npH3HaKa npH OTOOpa:>KeHHH cp (s) = as. npH nonapHO-
HepaBHbIX as MHO)KeCTBO BeKTopOB Xa HCtIepnbIBaeT KJIaCC
B3aHMHO-0,!{H03HaQHbIX npeoopa30BaHHH, T. e. KJIaCC npeoopa-
v v
30BaHHH, ,!{onYCTHMbIX B HOMHHaJIhHOH III KaJIe.
CJIe,!{yeT OTMeTHTb, QTO MHO:>KeCTBO BceB03MO:>KHbIX JIHHeH-
HbIX KOMOHHanHH Xa = asxs C He 06513aTeJIbHO HepaBHbIMH
v
KOMnOHeHTaMH TaK:>Ke HMeeT HarJI5J.,!{HbIH CMbICJI: 3TO MHO:>KeCT-
BO npe06pa30BaHHH HOMHHaJIbHOrO npH3HaKa, npH KOTOP@X
,!{onYCTHMbI YKpynHeHH5J. (06'be,!{HHeHH5J.) 3HalleHHH. ,QeHCTBH-
TenbHO, eCJIH, HanpHMep, a 1 = a2' TO KOMnOHeHTbI BeKTopa
Xa, COOTBeTCTBYIOInHe 06'beKTaM C nepBbIM H BTOpbIM 3Halle-
HH5J.MH npH3HaKa, O,!{HHaKOBW. OnepanH5J. YKpynHeHH5I 3Ha-
tIeHHH npH3HaKOB, T. e. nepexo,!{ K MeHee J{pOOHbIM rpa-
J{anH5IM, IIlHpOKO HCnOJIb3yeTcH B aHaJIH3e J{aHHbIX. JII06a5J.
peaJIbHa51 KJIaCCH HKanH5I HaCQHTWBaeT MHoro YPOBHeH ,!{p06-
HOCTH paCCMaTpHBaeMbIX THnOB 06'beKTOB; YHHBepCaJIbHa51 J{e-
C5ITHtIHa5J. KJIaCCHcpHKanH5J. ,lI.8eT npOCTeHIllHH CTaH,!{apTHbIH npH-
Mep. npe,ll.CTaBJI5J.eTCH eCTeCTBeHHbIM paCCMaTpHBaTb UlKallbl,
H,OMUflallbflble 8 UlUpOKOM CMbLClle: nOM HMO B3aHMHO-O,!{H03HaQ-
HbIX OTOOpa:>KeHHH ,!{JI5J. HHX ,!{onYCTHMbIMH 5J.BJI5J.IOTC5J. H (He
06513aTeJIbHO Bce) 0,ll.H03HaQHble oT06pa:>KeHH5J.. OTo6pa:>KeHHe,
He 51BJI5J.IOIneeC5I B3aHMHO-0,ll.H03HallHbIM, 005J.3aTeJIbHO 06'be)(H-
H5J.eT HeKoTopbIe 3HalleHH5J. npH3HaKa. ABToMaTHlIeCK8H peaJIH-
3anHH TaKHX oT06pa)KeHHH B KJIaCCe JIHHeHHblx KOM6HHanHH
Xa cYIneCTBeHHo paCIllHp5J.eT B03MO:>KHOCTH cpopMaJIbHbIX MO-
v v
,!{eJIeH aHaJIH3a .n8HHbIX, OCHOBaHHbIX Ha JIHHeHHbIX onepa-
nH5J.X, a HMeHHO HM 6y,!{eT nocBSI eHa OOJIblIla5J. QaCTb ,lI.aJIb"
v
HeHlIlero H3JIO:>KeHHH.
B COOTBeTCTBHH CO CKa3aHHbIM ,lI.HXOTOMHtIeCKHH npH3HaK
x T8K:>Ke cne.nyeT 3a,!{aBaTb He O,!{HHM, a ,lI.BYMH 6YJIeBCKHMH
cToJI6naMH, COOTBeTCTBYIO HMH Ka:>K,lI.OMY H3 J{BYX 3HalleHHH
npH3HaKa. HMeHHO TaKoe npe,ll.CTaBJIeHHe rapaHTHpyeT B03-
MO:>KHOCTb JIHHeHHOro npe,!{CTaBJIeHH5I JII060ro ,lI.OnYCTHMoro
npeo6pa30BaHH5I. B TO :>Ke BpeM5J. ,!{HXOTOMHQeCKHH npJI3HaK
1\10)KeT 6hITb 3aJ{aH e,ll.HHCTBeHHbIM 6YJIeBCKHM CTOJI6UOM, B KO-
26
k
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,
i
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,
J
I
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II1I I ill' 1111 III J 11111 11
ropOM eAHHHl.(bI XapaKTepH3YlOT O)\HO, a HYJIH - )\Pyroe 3Ha-
qeHHe npH3HaKa. KaKOH CMbICJI HMeeT TaKoe npe,l{CTaBJIeHHe?
01JeB II,l{H 0 , BC5IKOe npeoopa30BaHHe BH,l(a ax, r,l{e a -
npOH3BOJIbHOe tlHCJIO, nepeBO,l{HT e,l{HHHl.(bI B a, a HYJIH OCTaB-
JISleT HYJIHMH. HHalle rOBopH, MHO>KeCTBO BeKTopOB ax xa-
paKTepH3yeT BceB03MO>KHble H3MeHeHHH MaCIIlTaoa npH CPHKCH-
pOBaHHOM HaqaJIe OTClIeTa, T. e. IllKaJIY OTHOIlleHHH. TaKHe
MHOJKeCTBa npeoopa30BaHHH B ,l{aHHOM CJIYtIae HMelOT Ba)KHhIH
CO,l{ep)l{aTeJIbHblH CMbICJI, cpHKCHPY5I BHHMaHHe Ha O,l{HOM, HeHY-
JIeBOM 3HatleHHH npH3HaKa. TaKHe npH3HaKIf, Bblpa)KalO He
HaJIHtIHe 1-IJIH OTCYTCTBHe Y OO'beKTa KaKOrO-JIHOO KalleCTBa,
qaCTO BCTpellalOTCSI B peaJIbHblX cHTyal.(H5IX. nO,l{tIepKHeM,
tITO B OTJIHtIHe OT TaKHX npH3HaKOB, KaK «nOJI», «HHTpaBepT-
HOCTb-3KCTpaBepTHOCTb», BTopoe 3HalleHHe OTHOCHTeJIbHblX
jlHXOTOMHQeCKHX npH3H8KOB Bblpa)KaeT He HaJIHlIHe KaKoro-
JIHOO KaqeCTBa, aJIbTepHaTHBHoro nepBOMY, HO JIHlIlb OTCYTCT-
BHe nepBoro. HanpHMep, MO}KHO YKa3blBaTb HaJIHtIHe HJIH OT-
CYTCTBHe MaTeMaTUtleCKHX cnOCOOHOCTeH Y YlIa HXCSI, npHtIeM
HH<popMa[(IIH 00 OTCYTCTBHH MaTeMaTHtIeCKHX cnOCOOHOCTeH
u
Y yqeHHKa He 3aMeH5IeT CBe,l{eHHH 0 HaJIHlIHH Y Hero KaKHX-
JIHOO ,l{pyrHx cnOCOOHOCTeH.
OTHOCHTeJIbHblH AHXOTOMHtIeCKHH npH3HaK cpaKTHlIeCKH Bbl-
,l{eJI5IeT no,ltMHO}KeCTBO OO'beKTOB, OOJIa,l{alO HX COOTBeTCTBYIO-
HM KaQeCTBOM, TOJIbKO B 3TOM CMbICJIe npOTHBonOCTaBJI5ISI
HX OCTaJIbHblM OO'beKTaM. B qaCTHOCTH, OT,l{eJIbHbIe CTOJIOl.(bl
MaTpHl.(bI X MorYT paCCMaTpHBaTbC5I KaK TaKHe Bbl,l{eJIeHHble
nO)\MHO)l{eCTBa.
TaKHM oopa30M, HMeIOTC5I ,l{Ba CO,l{ep>KaTeJIbHO H cpopMaJIb-
HO pa3JIHtlHbIX Tuna ,l{HXOTOMHtIeCKHX npH3HaKOB, KOTopble
HMelOT pa3JIHtlHbIe npe,l{CTaBJIeHH5I - O,l{HHM HJIH ,l{BYM5I OY-
JIeBCKHMH CTOJI6[(3MH, B 3aBHCHMOCTH OT Toro, B KaKOH IllKa-
JIe - OTHOllIeHHH HJIH HHTepBaJIOB - OHH H3MepeHbl, T. e.
B 3aBHCHMOCTH OT Toro, SlBJISlIOTC5I JIH OHM 06blllHblMH HOMHHaJIb-
HblMH npH3HaKaMM HJIH npOCTO BbI,l{eJISlIOT MHO)f{eCTBO OO'beK-
TOB.
OOCY,l{HM Tenepb B03MO}KHOCTb OYJIeBCKOro npe,l{CTaBJIeHH5I
rrpH3HaKOB, H3MepeHHbIX B ,l{PyrHx THllax IllKaJI. B COOTBeTCT-
BMH C npe,l{bI,l{ymUM paHrOBble npH3HaKH xapaKTepH3YIOTCSI
BblCKa3blBaHHSlMli BH,l{a x (i) x (j) (i, j Em). TaKHe Bbl-
CKa3bIBaHHH MO)KHO OOJIellb B cpOPMY npe,l{HKaTOB BH,l{a «x (i)
s» (s ==== 1, ..., m). TOtlHee rOBopSl, C Ka>K,l{bIM 3HalleHHeM S
npH3HaKa x accouHHpyeM npeOUKam, T. e. OYJIeBCKHH npH3HaK
hs Ha MHo2KeCTBe 06'beKTOB, KOTOpblH )\JISI BC5IKOrO i onpe,l{eJISI-
eT, npe IlIecTByeT JIH 3HaqeHHe x (i) ,l{aHHoMY S, nOJIaraSl
27
..
..A..
hs (i) = 1 ,IJ:JI5I X (i) S M hs (i) = 0 ,IJ:JI5I X (i) < s. TaKoe
npe,IJ:CTaBJIeHMe paHroBoro npH3HaKa Ha3hIBaeTC5I IllKaJIoii:rYT..
TMaHa, BnepBbIe npe,l{JIO}KHBIllerO cooMpaTb MHcpopMaUHIO 0 Ta-
KHX npH3HaKax B BH,l{e OTBeTOB Ha BonpocbI BH,IJ:a «x (i) s?».
AHaJIOrHQHbIe npe,l{HKaTbI, T. e. oYJIeBCKHe npH3HaKH, MO}K-
HO acCOnHHpoBaTb C KOJIHlIeCTBeHHbIMH npH3HaKaMH. Ha-
npHMep, npH3HaK, H3MepeHHbIH B IllKaJIe oTHollIeHHH, xapaKTe-
pIl3yeTc5I npeJJ.llKaTaMH :i:j = a npH BceB03MOLKHbIX a.
<PHKCHPY5I j H a, MbI nOJIyqMM ,IJ:HXOTOMHqeCKHH npH3HaK psa,
KOTOpbIH ,l{JI5I KaJK,IJ:OrO i onpe,l{eJI5IeT, BbIllOJIHeHO JIH YCJIOBHe
x(i) = a, T. e. npeBOCXOJJ:HT JIM x (i) 3Ha l leHMe S B ex pa3, H npH
s
YTBepJJ:HTeJIbHOM OTBeTe nOJIaraeT PS(J., (i) = 1 (s = x (j)).
TaKoro pO,l{a npe,lIJfKaTbI cO,l{epJKaT TeM oOJIee nOJIHYIO HH-
cpOpManHIO 0 npH3HaKe, lIeM oOJIee nJ10THO nOKpbIBalOT BbIOH-
paeMbIe a HHTepBaJI BeInecTBeHHoH OCH, H MorYT MCnOJlb30-
BaTbC5I B KalleCTBe HCXO,IJ:HbIX ,l{JI5I npHMeHeHH5I MeTO,l{OB aHa-
JIH3a ,l{aHHbIX. CYIneCTBeHHoe npeHMYIneCTBO 3TOH cpOpMbI
npe,l{CTaBJIeHH5I COCTOMT B TOM, tITO JIHHeHHbIe npeoopa30Ba-
HH5I OYJIeBCKHX npM3HaKOB COOTBeTCTBYIOT JIIOOhIM:, CKOJIb
yrO,l{HO CJIO}KHbIM, HeJIHHeHHhIM npeoopa30BaHH5IM HCXO,l{HbIX
npH3HaKOB. 3TO n03BOJI5IeT HCnOJIb30BaTb MO,l{eJIH H MeTO,IJ:hI
JIHHeHHOro aHaJIH3a OYJIeBCKHX npH3HaKOB ,l{JI5I HeJIHHeHHOro
aHaJIH3a HCXO,l{HOH HHcpopManHH, tITO HeB03MO}KHO ocy ecT-
BHTb nOKa HHKaKHM HH@M cnOCOOOM.
B npHHnHne MO}KHO CTpOHTb H3 ,l{aHHbIX npH3HaKOB JIIO-
ObIe npe,l{HKaTbI H HCnOJIb30BaTb HX B aHaJIH3e ,l{aHHbIX. J1o-
KaJIbHbIe MeTOJJ:bI n03BOJISlIOT paCCMaTpHBaTb npe,l{HKaTbI TOJIb-
KO npH HeOOXO,l{HMOCTH, Ha COOTBeTCTBYIO HX Illarax aJIro-
pHTMa. B 3a,l{aQaX onHCaHH5I 3TH H,l{eH ycneiliHo peaJIH30BaHbI
B [7, 20, 31, 55-571.
1.2. 3A,nAHHE CBOnCTB B TEPMHHAX
OTHOWEHHA H MATPHI\
1.2.1. a KBUBaJleflTHOCTU U cTpgKTypHble
OTflotueflUR
0003HallHM tIepe3 in x m MHO)KeCTBO Bcex yno..
p5l,l{OtIeHHbIX nap (i, j) 06'beKTOB i, j E m. nO,l{MHO}KeCTBa
p £ m X m Ha3hIBaIOTC5I 6UflapHblMU omflOtueflu.,flMU, TaK
KaK xapaKTepH3YIOT CBOHcTBa nap OO'beKTOB. nOCKOJIbKY HO-
MHHaJIbHbIH npH3HaK x xapaKTepH3yeTC5I reM, liTO ,l{JI5I ero 3Ha-
28
lIeHI,di OCMbICJIeH BOnpOC «paBHbI-He paBHbI», CB5DKel\f C HMM
6HHapHOe OTHOIlleHHe Hepa3JIHQHMOCTH OO'beKTOB no npH3HaKY:
p == {( i , j) I x (i) == x (j)}.
(1.1)
OTHOlIleHHe p 51BJISleTC5I pecplleKCuBflblM, T. e. (i, j) E P
,l(JI5I Bcex j E m, cUMMempU'lflblM, T. e. (i, j) E p-+ (j, i) E
E p .L{JISI Bcex i, j E 91, mpaH3umuBHblM, T. e, (i, j) E p H
U, k) E p (i, k) E p, nOCKOJIbKY TaKOBO OTHOIlleHHe pa-
BeHCTBa 3HaqeHHH npH3HaKa x. Pe JIeKCHBHoe cMMMeTpHQHOe
TpaH3liTHBHOe OTHOlIleHlie Ha3bIBaeTC5I BKBUBalleflmHocmblO, OHO
nOJIHOCTblO xapaKTepH3yeT nOpO)K.L{alOIII.HH ero HOMHHaJIbHbIH
npli3HaK B TOIYI CMhICJIe, qTO n03BOJI5IeT BOCCTaHOBHTb CHCTeMY
l\lHO)KeCTB X-I (s > 06'beKTOB, He pa3JIHQHMbIX no npH3HaKY,
KaK MHomeCTB BH.L{a p (i > == {j I (i, j) E p}. ,L{eHcTBHTeJIb-
HO, eCJIH p - npOH3BOJIbHOe OTHOIlleHHe 3KBHBaJIeHTHOCTIi,
TO HeCOBna.L{aIOInHe MHOjl{eCTBa p (i) 3KBHBaJIeHTHbIX OO'beK-
TOB oopa3YIOT pa30HeHHe MHO}KeCTBa 91. HanoMHHM, qTO co-
BOKynHocTb R == {R l' ... , R m} HenYCTbIX nO.L{MHO)KeCTB
Rs 91 oopa3yeT pa36ueHue m Ha KllaCCbl R I , ..., Rm, eCJI"
Kam,l(WH OO'beKT npHHa.L{JIe)KHT O,l(HOMY H TOJIbKO O.L{HOMY li3
MHO>KeCTB Rs. no pa30lieHMIO HOMHHaJIbHbIH npH3HaK BOCCTa-
HaBJIHBaeTC5I C TOqHOCTblO .L{O B3aHMHO-0.L{H03HaQHOrO npeoopa-
30BaHHSI: .L{OCTaTOQHO nOCTaBHTb B COOTBeTCTBHe OO'beKTaM
Kam.L{OrO KJIaCCa Rs KaKOe-JIHOO qHCJIO cP (s), TaK, qTOObI s *
=1= t -+ cp (s) =1= cp (t), T. e. cp (s) ObIJIO B3aHMHO-0.L{H03HaqHO.
TaKHl\1 06pa30M, OTHOlIleHHSI 3KBHBaJIeHTHOCTH qepe3 co-
OTBeTCTBYIO lie KJIaCCbI nepa3JIHqHMbIX OO'beKTOB 3 a.L{a lOT
H3M.epeHH5I B H01vIHHaJIbHOH IllKaJIe.
AHaJIOrHQHbIM OOpa301\1 C Kam.L{bIM paHrOBbIM npH3HaKOM
10)KHO CB5I3aTb OHHapHoe OTHOIlleHHe
p= {(i, j) I X (i) X (j)},
(1.2)
nOpOLK.L{aeMOe .L{OnycTHMhIMH cpaBHeHH5IMM (<<He xyme») ero
3HaQeHHH.
TaKoe oHHapHoe OTHOIlleHHe p SlBJI5IeTCSI pe JIeKCHBHbIM,
TpaH3HTHBHbIM H IlUHeUflblM, T. e. (i, j) E P HJIH (j, i) E p .L{JI5I
JIIOObIX i, j E p, nOCKOJIbKY TaKOBbIM SlBJI5IeTC5I qHCJIOBOe OT-
HorneHlie . Pe JIeKCHBHwe TpaH3HTHBHwe JIHHeHHWe OTHO-
rneHHSI npHH5ITO Ha3bIBaTb IlUfleUHblMu KBa3unopfliJKaMU.
JIHHeHHbIe KBa3HnOpSl.L{KH HecYT B ceoe nOJIHYIO HH opMa-
1.(lilO 0 paHroBoH IllKaJIe. ToqHee, BC5IKHH JIMHeHHbIH KBa3H-
nOp5l.L{OK nopO)K.L{aeT paHroBYIO IIlKaJIY. ,LLJI5I Toro qTOObI yoe-
29
t_'":_
,1IJITbCSI B .9TOM, COnOCTaBHM C npOH3BOJIbHbIM JIHHeMHbIM KBa.
,....,
3HDOpHAKOM P OTHOIlleHH6 3KBHB8JIeHTHOCTH p I{aK ero CHMMe-
TPHqHYIO qaCTb
p = {( i, j) , (i, j) E p H (j, i) E p}.
nYCTb R = {R 1 , ..., Rm} - pa30HeHHe MHO)KeCTBa m Ha
-
KJIaCCbI 3KBHBaJIeHTHbIX no p OO'beKToB. OKa3bIBaeTCR, 3TH
KJIaCCbI MO:>KHO nepeHYMepOBaTb TaKHM oopa30M, qTO p COB-
na,neT C omHOtu£HueM npeotueCmeOeaHUJl B nOJIYtIeHHOM TaKHM
06pa30M ynopRJ{OqeHHOM pa36HeHHH: (i, j) E p TOr){a If TOJIh-
KO TOr,na, KOr){a HOMep KJIaCCa, CO){epLKa ero i He npeBbIlllaeT
HOMepa KJIaCCa, Co){ep}Ka ero j. I1HbIMH CJIOBaMH, p =
= U Rs X Rt. 1JroObI YKa3aTh HaJIHtIHe ynop5l){OqeHHOCTH
s t
KJIaCCOB, COBna){alO eH C HYMepanHeii, oy){eM 3aKJIIOtIaTb KJIaC-
CbI ynop5l){OqeHHOrO pa30HeHH5I R He B <PHrYPHbIe, a B KpyrJIbIe
CKOOKH: R = (R 1 , ..., Rm). B JIHTepaType npHH5ITO TaKJKe Ha-
3bIBaTb ynop5l){OQeHHOe pa30HeHHe paH:J/Cupo8aflUe.M, a 3K-
BHBaJIeHTHhIe OO'beKTbI H3 O){HOrO H Toro }Ke KJIaCCa R s -
C8.,fl3aflliblMU. B TOM CJIyqae, KOr.L{a Hepa3JIHtIHMbIX OO'beKTOB
HeT, T. e. Bce KJIaCCbI Rs 0){H03JIeMeHTHbI, OTHOIlleHHe p Ha-
3bIBaeTC5I IlUfleUflblM nOpJlOKOM, a COOTBeTCTBYIO ee paHmHpo-
BaHHe - cmpoeuM.
nOcTaBHM B COOTBeTCTBHe Ka}K){OMY KJIaccy Rs KaKOe-JIHOO
qHCJIO cp (8) C coxpaHeHHeM ynop5l){OqeHHOCTH KJIaCCOB: 8 t
<p (8) cp (t). OtIeBH){HO, TaKoe H3MepeHHe onpe){eJIeHO C
TOqHOCTblO ){O npOH3BOJIhHOrO MOHOTOHHO B03paCTalO ero
npeoopa30BaHHH H, 3HaqHT, HMeeT paHrOBYIO IllKaJIY.
TaKHM oopa30M, HOMHHaJIbHbIe H paHrOBble IllKaJIbI xapaK-
TepH3YlOTCR OHHapHbIMH OTHOIlleHHSlMH 3KBHBaJIeHTHOCTH H
v
JIHHeHHbIMH KBa3HnOp5l){KaMH, KOTopbIe, B CBOIO Oqepe.L{h, nOJI-
HOCTblO onpe,neJI5IIOTC5I COOTBeTCTBYIO HMH pa30HeHH5IMH m
Ha KJIaCCbI 3KBHBaJIeHTHhIX OO'beKTOB. Cy ecTBeHHYIO pOJIb
B onpe,neJIeHHH III KaJIbI HrpalOT COOTHOIlleHH5I Me)K){y KJIaCCa-
MH 3THX pa30HeHHH. ,[{JIR JIHHeMHOro KBa3HnOp5l){Ka Ka)K.L{bIH
KJIaCC CTporo JIyqlIIe npe){bI){Y HX, a ,nJI5I OTHOIlleHH5I 3KBHBa-
JIeHTHOCTH KJIaCCbI CB5I3aHbI TOJIbKO C caMHMH COOOH.
HeTpY){HO npe,nCTaBHTb ceoe CBOHCTBa OO'beKTOB, KOTopble
xapaKTepH3YIOTC5I pa30HeHH5IMH C HHbIMH THnaMH OTHOIlleHHM
Me)K,ny KJIaCCaMH. npe)K,ne Bcero HanOMHHM 0 cy ecTBoBaHHH
ynopfl.L{OqHBalO Hx CBOHCTB, onpeAeJISlIO HX CB5I3H TOJIbKO Mem-
J!Y COCe){HHMH KJIaCCaMH, KaK, HanpHMep, ){JI5I rpynn oopa3a
LKH3HH Rl' ..., R m. ,[{JIH xapaKTepHcTHKH CB5I3eH Me)K,l{Y co-
ce,nHHMH KJIaCCaMH OTHOIlleHHe no J{aHHOMY npH3HaKY ){OJI)KHO
30
I
1
t
,
'"
,
,
I
BKJlIOQaTb TaKHe napbI (i, j), qTO 06'beKTbI i H j HMelOT JIHOO
OJlMHaKOBble, JIHOO coce,!{HHe THnbI o6pa3a >KH3HH:
p = Rl X (R 1 U R 2 ) lJ [R 2 X (R 1 U R 2 U Rs)] U [Rs x
x (R 2 U R3 U R 4 )] U... U [Rm-l X(Rm-1 U Rm)].
AHaJIOrHqHO CBOHCTBO «3Tan nepepaooTKH npO,!{YKTa, pea-
JIM3yeMbIM npe,!{npH5ITMeM» B TeXHOJIOrHtleCKOM UHKJIe nepe-
pa60TKH CbIpb51 B KOHeqHYIO npo.nYKUHIO TOqHee Bcero npe,!{-
CTaBJI5IeTC5I opHeHTHpoBaHHoH nenblO (pHC. 1.1), r,!{e BepI.IlHHbI
H300pa}Ka1OT rpynnbI npe,!{npHHTHM, peaJIH3YIO HX O,!{HH H
TOT }Ke 3Tan nepepaOOTKH npO,!{YKTa, a CTpeJIKH - nepe,!{aqy
npo.nYKTa C 3Tana Ha 3Tan. XOTH nepBbIM 3Tan npe,!{IIlecTByeT
o
>r0
:.0
:. . . .
PHC. 1.1
BceM nOCJIe,!{YIOInHM, OTHOIlleHHeM ,!{OMHHHpOB8HH5I OH CB5I3aH
TOJIbKO co BTOpbIM, TaK KaK oe3 BToporo 3Tana oopa60TKa npo-
,!{YKnHH HeB03MO}KHa. 3TO 03HaqaeT, 1ITO ,!{JI5I cooTBeTCTBYIO-
r.u.ero ynop5l,!{OqeHHOrO pa36HeHH5I npe.nnpH5ITHH no 3TanaM
R = (R 1 , ..., R m) OTHOI.IleHHe p COCTOHT H3 Tex H TOJIbKO Tex
nap (i, j), ,!{JI5I KOTOpbIX j co,!{ep}KHTC5I B KJIaCCe, Henocpe.nCTBeH-
m-l
HO CJIe,!{YIO e1vI 3a KJIaCCOM i: p = U Rs X RS+l.
s=1
KaK BH,!{HO, HCnOJIb30BaHHe 513bIKa OTHOIlleHHH n03BOJIHeT
oxapaKTepH30BaTb TOHKHe OTTeHKH B CTpYKType CB5I3eH Me}K-
v
,!{y KJIaCCaMH, OTJIHqalO He ,!{aHHbIH THn ynop5l,!{OqeHHOCTH OT
. U U u
CTaH,!{apTHOH CTPYKTYPW ,!{OMHHHpOBaHH5I, 3a,!{aBaeMOH JIHHeH-
HbIM KBa3HnOp5l,!{KOM.
B OOIneM CJIyqae B03MO)KHa npOH3BOJIbHaH CTpYKTypa CB5I-
3eM Me>K,!{Y KJIaCCaMH 06'beKTOB. TaK, no CBOHCTBY «CTpYKTypa
u u
B3aHMOCB5I3eH» rpYI?na JIIO.nen MO>KeT pa3,neJIHTbCH Ha TPH no)l.-
rpynnbI, nepBa51 113 KOTOpbIX B HeKOTopOM CMbICJIe onpe,neJISieT
nOBe,neHHe .nBYX .npyrHx, He CB5I3aHHbIX Me1K y C060H no,ll-
rpynn. TaKYIO CTPYKTYPY MO)KHO onHcaTb C nOMoInblO pa30He-
HH5I {R 1 , R2' R 3} ,naHHoM cOBoKynHocTH Ha no,!{rpynnbI, ,!{JI5I
KOToporo Ha MHOJKeCTBe HOMepoB KJIaCCOB {I, 2, 3} 3a,!{aHO
OTHOIlleHJle {(I, 1), (2, 2), (3, 3), (1, 2), (1, 3)}, nOKa3hIBalO-
mee, KaK CB5I3aHbI Me)K,!{y COOOM no,!{rpynnbI Rs: Ka}K,!{a51 Jl3
v
HHX COCTOHT H3 «B3aHMOBJIH5IIO HX» JIIO,neH, npHqeM nO,ll.rpyn-
na R 1 BJIH5IeT Ha R 2 H R 3. COOTBeTCTBYIO ee OTHOIlleHHe paB'"'
HO P=RIXRIUR2XR2UR3XR3UR1XR2URIXRs.
31
..
r \ ..t."..... L!.'S 'If. \ _'1..1.. .....
Har JI5I,l{HO ,l{aHHa51 CTpYKTypa MOLKeT ObITb npe,l{CTaBJIeHa B
BH,l{e rpacpa CB5I3eH 1\1em.L{Y KJIaCCaMH (pHC. 1.2).
PaccMoTpeHHble CTPYKTypHhIe CBOllCTBa MorYT ObITb oxa-
paKTepH30BaHbI B TepMHHax pa30HeHHH co CTPYKTYPOH. Pa3-
6uenueM co cmpYKmypor1 Ha30BeM napy (R, x), r,l{e R ===
== {R 1 , ..., Rm} - pa30HeHHe MHomeCTBa A C HYMepOBaHHhI-
MH KJIaCCaMH, a x {I, ..., m} X {I, ..., m} - OTHOlIleHHe
CB5I3H Mem):{y I\JIaCCaMH. npHHa.L{JIemHOCTb (s, t) E x 03Ha-
lIaeT, QTO HMeeTC5I (HanpaB-
JIeHHa5l) CB5I3b oooeKToB
KJIaCCa R 8 C 060eKTaMH
KJIaCCa R t. OTHOlIleHHe Px,
COOTBeTCTBYIO ee nape (R,
x), COCTOHT H3 Tex H TOJIb..
KO Tex nap (i, j), .L{JI5I KO-
TOphIX i H j CO.L{ep)KaTC5I B
KJIaCCaX R 8 H R t n pH (s, t)
E x. I1Halle rOBOp5l,
Px == U R s X R t' ( 1.3) ,
(S,t) Ex
P 1 2 ,LLJI5I HOMHHaJIbHbIX npH3.
HC. . {
HaKOB x == (s, t) I s == t},
,l{JI5I HanpaBJIeHHbIX paHroBbIx x = {(s, t) I s t}. OTHOlIleHH5I
COCe,l{CTBa C x = {(s, t) II s - t I I} BnepBble CHCTeMa-
THlIeCKH HCnOJIb30BaJIH B 3a,l{atIaX aHaJIH3a ,l{aHHbIX B. JI. Ky-
neplIlTox H B. A. TpOcpHMOB.
YKa}KeM B 3aKJIIOtIeHHe HeKoTopbIe THnbI CTPYKTypHbIX
u
OTHOlIleHHH, KOTophIe xapaKTepH3YIOT Ba}KHhIe qaCTHbIe THnbI
;naHHbIX. OTHOlIleHH5I C ,l{peBOBH,l{HOH: CTPYKTypoii xapaKTepH-
3YIOTC5I napaMH (R, x), ,l{JI5I KOTOphIX X - ,l{epeBo, T. e. CB5I3-
HbIH rpacp oe3 KOHTypOB (pHC. 1.3, a). TaKHe OTHOlIleHH5I MO-
rYT HCnOJIb30BaTbC5I, Han.pHMep, ,l{JI5I onHCaHH5I HepapXHtIe-
CKHX CHCTeM. OTHOlIleHH5I C nHKJIHtIeCKOH CTpYKTYPOH xapaK-
TepH3YIOTC5I napaMH (R, x), Y KOTOpbIX X - nHKJI (pHC. 1.3, 6).
TaKHe OTHOlIleHH5I MorYT xapaKTepH30BaTb nOCJIe.L{OBaTeJIb-
HOCTb nepexoAoB no yrJIOBoMY paCCT05lHHIO. OTHolIleHHe C KY-
cOllHo-nenHoH CTPYKTYPOH xapaKTepH3YIOTC5I napaMH (R, x),
Y KOTOpbIX X - cOBoKynHocTb neneH (pHC. 1.3, 8). TaKHe OT-
HOlIleHH5I onHCbIBalOT B03MO}KHOCTH pa3pbIBOB B ynop5lAOtIe..
HHH HJIH HaJIHlIHe p5l,l{a «napaJIJIeJIhHbIX» npOneCCOB. OTHO-
lIIeHH5I, BbI,l{eJI5IIOInHe nO,l{MHO}KeCTBa, xapaKTepH3YIOTC5I na-
paMH (R, x), Y KOTOphIX X COCTOHT H3 e HHCTBeHHoii neTJIH.
TaKoe oTHorueHHe HMeeT BHA x = {(s, s)}, r,l{e s - BepllIHH8
3
I#>
C neTJIeH (pHC. 1.3, 2), H BbI,l{eJISleT nO,l{MHO>KeCTBO R S. Onepa-
lI.H5I BbI,l{eJIeHH5I nO,l{MHO:>KeCTBa oocY>K,l{aJIaCb B n. 1.1.4.
PaccMoTpeHHbIe THnbI CTPYKTYP 0Y,l{YT aHaJIH3HpOBaTbC5I B 9 5.2.
B oo eM CJIYlIae BbIpa3HTeJIbHOCTb 5I3bIKa OTHOIlleHHH ,l{JlSI
onHCaHH5I KalleCTBeHHOH HH opManHH no cpaBHeHHID C 5I3W-
KOM TaOJIHn OO'beKT-npH3HaK onpe,l{eJISleTC5I TeM, liTO OTHorne-
HH5I B SlBHOH cpopMe 3a,l{aIDT H rpynnbI OO'beKTOB, H CBSl3H MeJK"
,l{Y HHMH B BH,l{e e,l{HHOro MaTeMaTHlIeCKoro OO'beKTa.
0 . . . 0
. . . . . . . . . .
. . . . 1ft
(1,) 6) 8) 2)
./
PHC. 1.3
1.2.2. OTHOUleHUJl U MaTpUl1bl CBR3U
BC5IKOe OTHOIlleHHe p m X m MO:>KHO npe,l{-
CTaBHTb C nOMO hID KBa,l{paTHOH OYJIeBCKOH (T. e. HYJIb-e,l{H-
HHlIHOif) MaTpHnbI pa3MepHOCTH N X N. nepeHYMepyeM OO'beK-
TbI H onpe,l{eJIHM OYJIeBCKYID MaTpHny r = (rij), nOJIara51
rij == 1 TOr,l{a H TOJIbKO TOr,l{a, Kor,l{a (i, j) E p. 5ICHO, QTO
H oopaTHO BCSlKa51 OYJIeBCKa51 N X N MaTpHna r nopO>K,l{aeT
OTHOIlleHHe {(i, j) I rij = I}, COCT05l ee H3 Tex nap (i, j),
,l{JISI KOTOpbIX 'ij == 1. YKa3aHHoe COOTBeTCTBHe Me:>K,l{Y OT-
HOllIeHH5IMH H MaTpHnaMH, OlleBH,l{HO, B3aHMHO-O,l{H03HallHO.
KaKHe MaTpHnbI r cooTBeTcTBYIDT OTHOlIleHH5IM 3KBHBaJIeHT-
HOCTH H nOpO>K,l{aID HM HX pa30HeHH5IM R? OlleBH,l{HO, 'ij = 1
TOf,l{a H TOJIbKO TOr,l{a, KOr,l{a i H j cO,l{ep:>KaTC5I B O,l{HOM H TOM
)Ke KJIaCCe R. OTCID,l{a CJIe,l{yeT, liTO O,l{HOBpeMeHHOH nepeCTa-
HOBKOH CTpOK H CTOJIOnOB MaTpHna r MO:>KeT ObITb npHBe,l{eHa
K OJIOllHO-,l{HarOHaJIbHOMY BH,l{Y pHC. 1.4. ,aJISI 3Toro nepecTaB-
JISIIOTC5I OO'beKTbl H OTBeqaID He HM CTpOKH H CTOJI0 nhI TaK,
lITOObI CHallaJIa IlIJIH Bce OO'beKTbI H3 KJIaCCa R l , 3a HHM-
06'beKThI KJIaCCa R 2 H T. ,l{. Ha pHc.l.4 3alIlTpHxoBaHHbIe KJIeTKH
Ha r JIaBHOH ,l{HarOHaJIH cooTBeTcTBYIDT OT,l{eJIbHbIM KJIaCCaM
Rs, TaK liTO rij == 1 ,l{JI5I i, j E Rs. OCTaJIbHble KJIeTKH MaTpH"
[(hI, OTpamaIDlI.(He CBSl3H Mem):(y pa3HbIMH KJIaCCaMH, 5IBJI5IIDTC5I
HYJIeBbIMH. Bo BHYTpeHHHx MaTpHlIHbIX TepMHHax 3TH MaTpH-
I.I.hI xapaKTepH3YIDTC5I CBOHCTBaMH rii = 1 (pecpJIeKCHBHoCTb),
rij = rji (cHMMeTpHqHOCTb) H rijrjk rih (TpaH3HTHBHOCTb)
):(JISI Bcex i, j, k E ffi.
2 3aK. 1 743
33
AH8JIOrHQHO MOmHO yoe HTbC5I, 1ITO P 51BJI5IerC5I JIMHeHHbIM
.KB83HnOpSl KOM TOr 8 H TOJIbKO TOr,!{8, KOr,!{8 COOTBeTCTBYIO-
851 MaTpH a r MOmeT ObITb npHBe,!{eHa K OJIOqHO-TpeyrOJIb-
HOMY BH,ny pHC. 1.5 C nOMO blO o,!{HoBpeMeHHoH nepecTaHoBKH
CTpOK H CTOJIO OB. npH 3TOM CTpOKH i ,!{J1S1 i E Rl ueJIHKOM
.cOCTOSlT H3 e,!{HHHU (<<He xY:>Ke» Bcex OCT8JIbHbIX), CTpOKH
i E R 2 co,!{ep)K8T HYJIH TOJIbKO B Tex CTOJIOuaX i, qTO i E Rl.
Booo e, ,!{JI5I CTpOK i E R s HY JIH HaXO,!{51TC5I B CTOJIO aX
i E Rt npH t < s. Bo BHYTpeHHHx M8TpHtlHhIX TepMHH8x
T8KHe MaTpH bI xapaKTepH3YIOTC5I YCJIOBH5IMH: fii = 1 (pe-
<pJIeKCHBHoCTb), fijfjh, fik (TpaH3HTHBHOCTh) H fij + fji 1
(JIHHeHHocTb) ,!{JI5I JIIOObIX i, i, k E ffi.
PHC. 1.4
PHC. 1.5
B oo eM CJIYQ8e ,!{JI5I OTHOllIeHH5I Px, cooTBeTcTBYIO ero
pa30HeHHIO co CTpYKTYPOH (R, x), MaTpHU8 r MOJKeT ObITb oxa-
paKTepH30BaHa CJIe,!{YIOInHM oopa30M. Pa36HeHHe R = {R l , ...,
.... , R m} nopom,!{8eT pa30HeHHe MaTpHUbI r Ha KJIeTKH
r (s, t) = (fij) (i E Rs, j E R i ) CB5I3eH Me:>K,!{Y KJIaCC8MIf Rs
H R t (s, t = 1, ..., m). 3TH KJIeTKM COCTOSIT JIM60 ueJIHKOM H3
e,!{HHMqHbIX, JIHOO eJIMKOM H3 HYJIeBbIX 3JIeMeHTOB. A HMeHHO
f (s, t) = 0 ,!{JI5I (s, t) $. x H r (s, t) = 1 ,!{JISI (s, t) E x.
K8)K,!{8R KJIeTK8 f (5, t) COOTBeTcTByeT, no CYTH ,!{eJI8, npOH3-
Be,!{eHHIO RsXRt II onpe,!{eJISleT, BKJIlOqeHO JIH RsXRt B Px
(npH f (s, t) = 1) HJIH HeT (npH f (5, t) = 0). T8KMM OOp830M,
M8TpHua CTPYKTYPbI pHC. 1.1 HMeeT BH,!{ pIlC. 1.6, 8 pHC. 1.2 -
-BH,!{ pHC. 1.7. npe,!{ocT8BJI5IeM QHT8TeJIIO C8MOCT05lTeJIbHO BOC-
CTaHOBHTb BM,!{ CTPYKTypbI M8TpH ,!{JI5I CTPY!{TYP pHC. 1.3.
5YJIeBcKYIO M8TPH[(Y ,n8HHoro OTHOllIeHH5I 1\10)KHO paCCMa-
TpHBaTb K8K MaTpHUY CBH3eH Me>K,l{Y OO'beKT8l\fH; e,l{JIHM[(8 MJIM
34
IIYJIb B nqeHKe (i, j) MO}KeT HHTepnpeTHpOBaTbCH KaK HaJIHqHe
lIJIH OTCYTCTBHe CBH311 no ,l{aHHOl\1Y CBOHCTBY Me*,l{Y 06'beKTa-
f\1H i H j. 3Ta MaTpH[(a KOJIHqeCTBeHHO npe,l{CTaBJIS!eT ,l{aHHOe
u u
KaqeCTBeHHoe CBOHCTBO - C nOMO hlO KOJIHqeCTBeHHOH xa-
paKTepHCTHKH CB5I3eH Mem,l{Y 06'beKTaMH.
Bo06 e rOBOp5l, HHOTKY,l{a He CJIe,l{yeT, qTO HHTeHCHBHOCTb
HaJIHqH5I HJIM OTCYTCTBH5I CB5I3H MeiK,l{Y 06'beKTaMH 3aMep5leT-
C5I 06S!3aTeJIbHO e,l{HHH eH H HYJIeM. L{JI5I 3a,l{aHH5I OTHOlIleHHH
MOfKHO paCCMaTpHBaTb 6YJIeBCKHe (a, )-MaTpHUbI, TaK qTO
rij == a )lJI5I (i, j) E p H rij == AJI5I (i, j) $. p. nOl\tHMO Ka-
qeCTBeHHOH HHcpOpMall.HH 0 CTpYKType oTHorueHH5I p (ex, )-
MaTpH a COAep*HT KOJUfqeCTBeHHYIO HHcpopMa[(HIO 06 HHTeH-
PHC. 1.6
PHC. 1.7
CHBHOCTH CB5I3H no OTHOIlleHHIO p; 3Ta HHTeHCHBHOCTb paBHa
ex npH (i, j) E p H npH (i, j) $. p. ECJIH YBeJIHQHTb 3TH QHC-
JTa, CKa>KeM, B,l{BOe, TO 3TO 6YAeT 03HaqaTb ,l{BOHHoe YBeJIHQeHHe
Beca ,l{aHHoro OTHOlIleHHH, TaK KaK HHTeHCHBHOCTH CBH3H no
OTHOlIleHHIO YBeJIHqHJIHCb BABoe. TOqHbIH CMbICJI TaKOH HH-
TepnpeTa HH BeJIHqHH a H 6Y,l{eT 06CYJK,l{aTbCH B 9 4.1 B paM-
Kax JIHHeHHbIX MOAeJIeH aHaJIH3a ,l{aHHbIX.
5YJIeBcKYIO (a, )-MaTpH[(Y h == (h ij ) MomHO BbIpa3HTb
qepe3 6YJIeBcKYIO (1, O)-MaTpH[(Y COOTBeTCTBYIO ero OTHOIlle-
HRS! KaK h = A,r + 11, r,l{e A, == a - , 11 = , npHqeM CJIO-
}ReHHe MaTpH[(bI A,r C qHCJIOM J..t nOHHMaeTC5I KaK npH6aBJIeHHe
f..t KO BceM 3JIeMeHTaM MaTpH[(bI. npH 3TOM MaTpH[(a r HeceT
B ce6e CTPYKTypHYIO HH<popMa[(HIO 06 OTHOlIleHHH p,OT,l{eJIeH-
HYIO OT HHTeHcHBHocTeH ex H , H 6YAeT Ha3bIBaTbC5I KapKacoM
MaTpU[(bI h. OqeBM,l{HO, KapKac He 3aBHCHT OT 3HaqeHHH a H p.
Bo MHorHX CJIYQa5lX Y 06HO B KaqeCTBe KapKaca HCnOJIb30BaTb
2* 35
....,--",. _.
_ i 'w "' ,
'Ii " . ..N....._( I!t-.:d."t ..... ,....J::'101.:.'''. - "" , ,
He (1, O)-MaTpMuy, a (1, -1 )-MaTpHuy r, B KOTOpOI1 npHHa)\-
JIe)KHOCTb HJIH HenpHHa,l{JIe>KHOCTb na pbl 06'beKTOB ,l{aHHOMY
OTHOlIleHHIO Bblpa)KeHa B CHMMeTpHqHOH cpopMe: ri j == 1 ,l{JI5I
(i, j) E p, T. e. ,l{JI5I h ij = ex, M ri j == - 1 ,l{JI5I (i, j) p, T. e.
,l{JI5I h ij == . TaKoH (1, - 1)-KapKac BhlpaLKaeTC5I Qepe3 Ma-
TpHU.Y h paBeHcTBoM h == 'Ar + J..t, r):(e A + f.t == ex, f.t-
- A == . B03MO)KHOCTb HCnOJIb30BaHH5I npOM3BOJIbHblX (ex, )-
npe,l{CTaBJIeHMH ,l{aHHoro oTHorueHH5I p MO)KeT paccMaTpHBaTb-
C5I KaK npM3HaHHe Toro aKTa, qTO MHTeHCHBHOCTb CB5I3M no
KaqeCTBeHHOMY npM3HaKY, 6Y,l{yqH Bblpa)KeHa B ,l{HXOTOMHqe-
CKOH <popMe, H3Mep5leTC5I B llIKaJIe HHTepBaJIOB. B HeKoTopbIX
cJIyqa5lx cO,l{epJKaTeJIbHblH CMhICJI 3a,l{aqH ,l{HKTyeT B03MO)KHOCTb
HCnOJIb30BaHH5I III KaJIbl oTHoIIleHHH, KOr,l{a OTCYTCTBHIO CB5I3H
Me)K,l{Y 06'beKTaMH no OTHOIlleHHIO p, (i, j) (f p, npHnHcbIBaeTC5I
HYJIeBOe 3HaqeHHe, TaK qTO HCnOJIb3YIOTC5I (cx, O)-MaTpHnbl CB5I3H.
Ha 513blKe MaTpHn CB5I3H MO)KHO Bblpa)KaTb H 60JIee TOH-
KHe BeCOBble xapaKTepHcTHKH CTPYKTYp. B qaCTHOCTH, XOpOIllO
H3BeCTHO, qTO pa3JIHqHble 3HaqeHH5I ,l{aHHOrO HOMHHaJIbHOrO
npH3HaKa MorYT OTJIHqaTbC5I no CTeneHH HCnOJIb30BaHH5I B 3a-
,l{aqax onHCaHH5I H KOHCTpYHpOBaHH5I, T. e. HMeTb pa3JIHqHYIO
HHcpopMaTHBHOCTb, HJIH 3HaqHMOCTb. HanpHl\fep, BO MHorHX
3a,l{aqaX qaCTO BCTpeqalO HeC5I 3HaqeHH5I MeHee HH opMaTHB-
Hbl, qeM Te, KOTopble BCTpeQaIOTC5I pe,l{KO (OCHOBHa51 H,l{e51
TeopHH HHcpopManHH llIeHHoHa). nYCTb npH3HaK HMeeT m
3HaqeHHH H xapaKTepH3yeTC5I pa36HeHHeM {R 1 , ..., R m} MHO-
)KeCTBa 91, npHqeM 3HaqeHM5I s xapaKTepH3YIOTC5I BeCOBbIMH KO-
3<P<PHnHeHTaMH As (8 == 1, ..., m). 3TO MO)KHO oTpa3MTb m X m
MaTpHneH CB5I3H r (A) == (aij), r,l{e aij == As ,l{JI5I i, j E Rs
(s = 1, ..., m) H aij == 0 ,l{JI5I (i, j) (f p. HHaqe rOBOp5l, HH-
TeHCHBHOCTb CB5I3H JII06blX 06'beKTOB s- ro KJIaCCa nOJIaraeTC5I
paBHoH As, a CB5I3b 06'beKTOB pa3Hblx KJIaCCOB - HYJIIO. 3Ta
MaTpHna CO,l{ep)KHT Y)Ke He ,l{Ba, a m + 1 pa3JIHQHbIX qHCJIOBbIX
3HaqeHHH HHTeHCHBHOCTH CB5I3H. BnpOqeM, ee MO)KHO npe,l{CTa-
BHTb KaK CYMMY 6YJIeBCKHX (As, O)-MaTpHn h s == (h j)' COOT-
BeTcTBYIOI.IJ.HX OT,l{eJIbHbIM KJIaCCaM Rs (s == 1, ..., m): hfj== As
,l{JI5I i, j E Rs H h j == O,l{JI5I (i, j) $. Rs X Rs.
Beca OT,l{eJIbHbIX 3HaqeHHH npM3HaKa MorYT 3a,l{aBaTbC5I
KaK H3 «BHeIllHHX», TaK H H3 «BHYTpeHHHx» co06pa)KeHHH,
onpe,l{eJI5IeMbIX TOJIbKO pa36HeHHeM R. B qaCTHOCTH, B COOT-
BeTCTBHH C H)J.eeH 06 YMeHbllIeHHH HH<POPM THBHOCTH C YBeo!lt
qeHHeM qaCTOTbl MO:>KHO nOJIO)KHTb As = N s HJIH As == log N s '
r,l{e N s == I Rs I - qHCJIeHHOCTb KJIaCCa Rs. 3TH Beca Bbl-
36
QHCJI5IIOTC5I HenOCpe.L{CTBeHHO no KaqeCTBeHHOH HH<popMa[(HH 0
npH3HaKe: no OTHOllleHIHO p BbI5IBJISUOTC5I KJIaCCbI R s If no,n-
CqMTbIBaIOTC5I HX qHCJIeHHOCTH. nO.L{OOHbIe MaTpHrU>I HrpalOT
BamHYIO pOJIb B .L{aJIbHeHI.I1eH TeopHH.
B oo eM CJIyqae pa30HeHMH co CTPYKTYPOH MaTpH[(a CBSI-
U u u u
3eM onpe.L{eJI5IeTCSI BeCOBOH' CTPYKTYPOH - MaTpH[(eH m X m
BeCOBhIX K03cp<pHIJJIeHTOB A == (Ast), TaK qTO rij (A) == Ast
.L{JI5I (i, j) E Rs x Rt. TIpH 3TOM Ast = 0 03HaqaeT, qTO (s, t) (f
(f x. TaKa51 MaTpM[(a r (A) B COOTBeTCTBMH C pa30HeHMeM
R nO pa3.L{eJiSleTCSI Ha KJIeTKM r (s, t) C O.L{HHaKOBbIMH 3JIeMeH-
TaMH Ast.
BecoBa51 CTpYKTypa A 51BJI5IeTC5I xapaKTepMcTHKOH CB5I-
3eM Me:>K.L{Y OT.L{eJIbHbIMH 3HaqeHH5IMH npH3HaKa H MOJKeT pac-
cMaTpHBaTbC5I He3aBMCHMO OT OO'beKToB. B 3TOM CMbICJIe MO:>KHO
U
paccMaTpHBaTb BeCOBbIe CTPYKTYPbI KaK TeOpeTHqeCKHH cno-
coo 3a,l{aHM5I CBOHCTB C nOMOInblO YKa3aHHSI CB5I3eM Me:>K,ny 3Ha-
qeHH5IMM [17, 66, 27]. OqeBM.L{Ho, nOH5ITMe BecoBoH CTPYKTYPbI
oooo aeT Bce npe,l{bI,l{Y He THnbI ,l{aHHbIX. n03TOMY BBH,l{Y He-
Tpa H[(HOHHOCTH 3Toro cnocooa 3a,l{aHH5I npH3HaKoB nOJIe3HO
OOCY,l{HTb B03MO:>KHOCTb ero HCnOJIb30BaHH5I ,l{JI5I 3a,l{aHH5I KO-
JIHqeCTBeHHbIX npH3HaKoB.
3Ta B03MO)KHOCTb CB5I3aHa npeJK,l{e Bcero C HCnOJIb30BaHHeM
pa3HOCTH Xi - Xj (HJIH ee MO,l{YJI5I) Me>K,l{Y 3HaqeHH5IMH Xi
H Xj ,l{aHHOrO npH3HaKa KaK xapaKTepHCTHKH OJIH30CTH 3THX
3HaqeHHH. IlpH 3TOM MaTpH[(a CBSl3eH (aij) Me}l{):{Y OO'beKTaMH
i, j E m no npH3HaKY X cpopMHpyeTcSI KaK MaTpH[(a BeJIHqHH
a (X (i) - X (j)), r,l{e a (x) - HeKOTopaSl, KaK npaBHJIO, MO-
HOTOHHa5l, CPYHK[(H5I Be eCTBeHHoro aprYMeHTa X, a X (i),
X (j) - 3HaqeHH5I x Ha OO'beKTaX i, j E m. B npOCTeHI.I1eM
CJIyqae aij == 1 - (x (i) - x (j))/D, r,l{e D - pa3Max BeJlH-
qMH x (i) ,l{JISI i E . B03MO:>KHOCTb HCnOJIb30BaHH5I TaKHX
MaTpH[( B TeopHH aHaJIH3a ,l{aHHWX BnepBwe oWJIa OTMeqeHa
r. L{3HHeJICOM [ 118], KOTOpbIH ,l{OKa3aJI, qTO K03qxpH[(HeHT
KOppeJISI[(HH MaTpH[( pa3HOCTeM (x (i) - x (j)) H (y (i) -
- Y (j)) ,l{JI5I npH3HaKoB x H y paBeH K03qxpM[(HeHTY KOppeJISI-
[(HH BeKTopoB (x (i)) H (y (i)), T. e. CTOJlO[(OB ,l{JISI 3THX )Ke npH-
3HaKoB l (CM. cp-JIY (4.110).
CJIe.L{yeT OTMeTHTb, qTO TaKHe KOJlIfqeCTBeHMble MaTpH[(bI
CB5I3eH, COXpaH5IIO He HHcpopMa[(HIO 0 CTpYKType paCCT05lHHH
1 HCnOJIb30BaHHe BeJIHtIHH Xij = X (i) - x (j) H aij = 1 -
- Xij/ D ,n;aeT o,n;HHaKoBbIe pe3YJlbTaTbl, TaK KaK K03cpcpHI1.HeHT KOp-
peJIHI1.HH He MeHSleTCSI npH JlHHeHHbIX npeoopa30BaHHSlX ,naHHblX
(CM. n. 4.2.4).
37
Me>K.L{Y 3HaqeHH5Il\1H npH3HaKa, HMelOT CneUHaJIbHYIO CTPYKTYPY.
B qaCTHOCTII, B KaJK OH CTpOKe MaTpHIJ.bI 3JIeMeHTbI CHaqaJIa
MOHOTOHHO YObIBalOT no .L{HarOHaJIH, a 3aTeM MOHOTOHHO B03-
paCTalOT, T. e. fvlaTpHlI.a CB5I3H 51BJI5IeTC5I JIHHeHHOH [49].
B 06 eM cJIyqae l'vlaTpH a CB5I3H MeLK.L{Y OO'beKTaMH no .L{aH-
HOMY CBOHCTBY f\10)KeT Hl'vleTb npOH3BOJIbHYlO CTPYKTYPY. 3TO
OTHOCHTCH npe)K.L{e Bcero K cJIyqalO, Kor.L{a CBOMCTBO onepa-
HOHaJIH3HpyeTC51 He O.L{HHM, a cpa3Y HeCKOJIbKHMH npH3Ha-
KaMH Xl, ..., Xn, KaK, HanpHMep, 3KOHOMHQeCKa51 OCBOeHHOCTb
reppHrOpHH HJIH nOrpeOHTeJIbCKOe KaqeCTBO npO.L{YKIIJiH
( CM . 4.2).
L{Pyra51 npHqHHa n05lBJIeHH5I npOH3BOJIbHOH CTPYKTYPbI
MaTpHlI.bI CB5I3H COCTOHT B TOM, qTO 3aqaCTYIO TaKHe MaTpHlI.bI
npe.L{CTaBJI5IIOT COOOR CaMOCT05lTeJIbHbIH THn 3MnHpHqeCKOM HH-
cpopMalI.HH, a He <POpMHPYIOTC5I Ha OCHOBe 3HaqeHHH npH3HaKOB.
HCXO.L{HbIe .L{aHHbIe C caMoro HaqaJIa MorYT HMeTb BH.L{ MaTpHUbI
CB5I3H. TaKOBbI CBOHCTBa «CTpYKTypa B3aHMOOTHOI.lIeHHH B KOJI-
JIeKTHBe» HJIH «B3aHMOCB5I3H MHorOOTpaCJIeBOrO KOMnJIeKca»,
u u u
BbIpa)KaeMbIe COlI.HOMeTpHqeCKOH MaTpHlI.eH H MaTpHlI.eH no-
TOKOB npO.L{YKlI.HH Me)K.L{Y OTpaCJI5IMH. TaK )Ke KaK H paHee pac-
CMOTpeHHbIe, 3TH MaTpHlI.bI CO.L{ep1KaT KOJIHqeCTBeHHbIe oueHKH
CB5I3H Me)K.L{Y OO'beKTaMH. T OJIbKO Ha.L{l> C.L{eJIaTb cyrneCTBeHHYIO
orOBopKY. B OTJIHqHe OT 06bIqHbIX npH3HaKOB TaKHe MaTpHlI.bI
xapaKTepH3YIOT He CTOJIbKO OT.L{eJIbHbIe OO'beKTbI, CKOJIbKO
CBH3H Me)K.L{Y HHMH, H B 3TOM CMbICJIe 3a.L{alOT CHCTeMHbIe CBOM-
CTBa, XapaKTepH3YIOrnHe Bce MHo:meCTBO OO'beKTOB.
TaKHM oopa30M, 513bIK MaTpHlI. CB5I3H BeCbMa YHHBepCaJIeH:
Ha HeM MO)KHO BbIpa>KaTb H OObIqHbIe OO'beKTHbIe, H 60JIee
u
CJIOJKHbIe CHCTeMHbIe CBOHCTBa.
MaTpHlI.bI CB5I3H Me)K.L{Y OO'beKTal\1H MorYT ObITb nOJIyqeHbI
H3 cooTBeTcTBYIOInHX TaOJIHlI. OO'beKT-npH3HaK. TaK, 6YJIeBCKa51
(1, O)-MaTpH a r .L{JI5I HOMHHaJIbHOrO npH3HaKa x MO>Ker ObITb
Bblpa)KeHa qepe3 6YJIeBcKYIO NXm MaTpHUY X, nOCTaBJIeHHYIO
eMY B COOTBeTCTBHe C .L{HXoTOMH3aUHeH, npocToH <POPMYJIOM
r = X XT. ,[{eMCTBHTeJIbHO, B Ka)K.L{oif cTpoKe MaTpHlI.bI X
HMeeTC5I pOBHO O.L{Ha e.L{HHHlI.a, TaK qTO CKaJI5IpHOe npoH3Be-
.L{eHHe .L{BYX CTpOK paBHO HYJIIO, eCJIH 3TH e,nHHHUbI Y HHX B
pa3HbIX CTOJI OlI.a X , If e,nHHHlI.e, eCJIH B O,l{HOM H TOM 1Ke, TaK qTO
(i, j)-if 3JIeMeHT h1aTpHlI.bI XXT paBeH e,l{HHHlI.e Tor,l{a H TOJIbKO
Tor,l{a, KOr,l{a x (i) = x (j), qTO H Tpe60BaJIOCb ,l{OKa3aTb.
B 06rneM CJIyqae, npH HaJIHqHH npOH3BOJIbHOM BeCOBOR
crpYKTypbI A = (A 8 t), MaTpHlI.a CB5I3eH Me>K,l{Y OO'beKTaMH HMe-
eT BH,l{ r (A) = XA XT, TaK KaK (i, j)-if 3JIeMeHT npOH3Be,l{e-
HH5I XAXT paBeH ')..,8t npH i E R 8 , j E Rt.
38
.
s
1.3. fPynnHPOBKH H PACnPEAEJlEHHSI
1.3.1. Paa6uefluR U ux pacnpeaeAeflUJl
B npe,l{bI,l{Y HX naparpaq,ax MbI yoe,l{HJIHCb,
QTO KaqeCTBeHHaH HHq,OpM8UHH 00 OO'beKTaX MO}KeT ObITb OX a-
paKTepH30BaHa pa30HeHHeM C TOM HJIH HHOM CTPYKTYPOH CB5I-
3eM Me1K,l{Y ero KJIaCCaMH. Ha nepBbIH: B3rJISI,l{ Ka:>KeTCH, qTO
nOHSlTHe pa30HeHHH - 3TO TeopeTHKO-MHO:>KeCTBeHHbIH CHHO-
HHM ,l{JI5I CTaTHCTHqeCKOrO TepMHHa «rpynnHpOBKa». O,l{HaKO
3TO He COBceM TaK. B nOH5ITHH CT8TI1CTHqeCKOM rpynnHpo KH
v v
CO,l{ep:>K8TCSI, no KpaHHeH Mepe, ,l{Ba MOMeHTa, OTJIltqalOIIIJfe
ee OT cOBoKynHocTH HenepeCeKalOInHXC5I HenYCTbIX KJIaCCOB-
p'a30HeHHSI. Bo-nepBbIX, KJIaCCbI rpynnHpOBKH. HH,l{eKCHpOBa-
HbI 3HaqeHHSlMH npH3HaK8, KOTopbIe HX onpe,l{eJI5IIOT, H, BO-
BTOpbIX, HeKOTopbIe KJIaCCbI rpynnHpoBKH MorYT OKa3aTbC5I
v
nYCThIMH, nOCKOJIbKY B HCCJIe,l{yeMoH cOBoKynHocTH MorYT OT-
CYTCTBOBaTb OO'beKTbI C TeMH HJIH HHbIMH 3HaqeHH5IMH npH3Ha-
Ka. TaKHM oopa30M, zpynnup08Ka - 3TO cOBoKynHocTb He-
nepeceKalOmHXCSI KJIaCCOB, HH,l{eKCHpOBaHHbIX 3JIeMeHTaMH He-
v
KOToporo MHO}KeCTBa 3HaqeHHH COOTBeTCTBYIOInerO npH3HaKa.
.aaHHoe pa3JIHQHe MO:>KeT nOKa3aTbCH BeCbMa CXOJlaCTHQHbIl\1,
O,l{HaKO OHO HMeeTCH, XOTH BO MHorHX cJlYQa5lX Hecy ecTBeH-
HO, H TaM, r,l{e 3TO He BbI30BeT nYTaHHUbI, MbI OY,l{eM paCCMaTpH-
BaTb TepMHHbI «rpynnHpOBKa» H «pa30HeHHe» KaK CHHOHHMbI.
KOJIHqeCTBO OO'beKTOB B KJIaCCe Rs 0003H8qHM Qepe3 N s ,
a HX OJIIO - Qepe3 Ps = Zs . nOCKOJIbKY BCSlKHH 06'beKT npH-
Ha,l{JIe:>KHT O,l{HOMY H TOJIbKO O,l{HOMY KJlaccy Rs, TO Ns = N,
Ps = 1. BeKTop (N I , ..., N m ) oY,l{eM Ha3bIBaTb pacnpeaeAe-
flueM, a BeKTOp (PI' ..., Pm)-pacnpeoeAeHueM ttacmom rpyn-
nHpOBKH R = {R l , ..., Rm}. nOCJle,l{Hee OY,l{eM 0003HaqaTb
(R) = (PI' ..., Pm). BeJIHQHHa Ps XapaKTepH3yeT Bep05lTHOCTb
(QaCTOTY) nqna,l{aHHSI OO'beKTa B KJIaCC 'Rs npH paBHOBep05lT-
HOM BbIOOpe OO'beKTOB MHO:>KeCTBa m.
Pacnpe,l{eJIeHHe qaCTOT HBJI5IeTC5I ynpOIlJ,eHHhIM KOJIHQeCT-
BeHHbIM OnHCaHHeM rpynnHpOBKH. TaKOll cnocoo KOJIHqeCTBeH-
HOrO npe,l{CTaBJIeHHH KaqeCTBeHHOM HH<pOpMaUHH, HeCMOTp51
Ha HenOJlHOTY, 113-3a KOTOpOH TepSlIOTC5I OT,l{eJIbHbIe OO'beKTbI,
SlBJI5IeTC5I OOI.IJ.enpHHSlTbIM B MaTeMaTHlJeCKOH CTaTHCTHKe. Ka-
u
qeCTBeHHbIe ,l{aHHbIe B03HHKaIOT, KOr,l{a BMeCTO H3MepeHHH npo-
H3BO,l{51TC5I nO,l{CQeTbI [154]. HCnOJIb30BaHHe pacnpe,l{eJIeHHH
v
BMeCTO HH,l{HBH,l{yaJIH3HpOBaHHbIX onHcaHHH MO:>KeT OKa3aTh-
CSI BeChMa nJlO,l{OTBOpHbIM, KaK CBH,l{eTeJlbcTByeT onbIT CTaTH-
CTHqeCKOH q,H3HKH.
39
....-
_ow
--
E'"
PaCCMOTpHM J{Ba pa36HeHHSI Ha MHO)KeCTBe m: R =
- {R 1 , ..., Rm} H R' = {Rl, ..., R:n,}. nYCTb, H8npHMep, m
COCTOHT H3 10 06'beKTOB, 3aHYMepoB8HHbIX nepBbIMH lIHCJI8MH
H8TypaJIbHOrO pSlJ{a, H R = {1-2-3, 4-5-6-7, 8-9-10},
R' = {1-4-8-9, 2-3-10, 5-6-7}. 3J{ecb J{JISI npocToTbI
0603HaQeHHH 3JIeMeHTbI OTJ{eJIbHbIX KJIaCCOB He 3aMKHYTbI B <pH-
rypHbIe CKo6KH, 8 38nHcaHbI tIepe3 THpe.
PaCCMOTpHM Tenepb nepecelleHHe Rs n RI. KaKOB ero
CMbICJI? B Rs n Rt cOJ{ep)KaTC5I 06'beKThI, npHHaJ{JIe}K8 He OJ{-
HOBpeMeHHO S-MY KJI8CCY pa36HeHH5I R H t-MY KJI8CCY p836He-
HHSI R'. ECJIH 3TH p830HeHHSI nOpO)KJ{eHbI HeKOTopbIMH npH-
3HaK8MH, T. e. SlBJI5IIOTCSI rpynnHpoBK8MH, TO Rs n R; eCTb
cOBoKynHocTb Bcex 06'beKTOB, HMelO HX OJ{HOBpeMeHHO s-e
3HaqeHHe OJ{Horo H t-e 3H8tIeHHe J{Pyroro npH3HaKa. MO)KeT
CJIytIHTbCSI, liTO npH J{8HHbIX s H t TaKHX 06'beKTOB He cy ecT-
ByeT BOBce, TorJ{a Rs n R t= 0. B H8111eM npHMepe Rl n Ri =
= {1,2,3}n {1,4,8,9} = {1},R1nRa = 0, R 2 nR2 = 0.
, OlleBHJ{Ho, K8)K.L{bIH 06'beKT npHH8.L{JIe)KHT OJ{HOMY H TOJIb-
KO OJ{HOMY H3 MHO)KeCTB Rs n Rt (s = 1, ..., m; t = 1, ..., m').
CJIeJ{oB8TeJIbHO, cOBoKynHocTb MHO)KeCTB Rs n Rt o6p83yeT
rpynnHpoBKY MHomeCTB8 m, KOTOpa51 Ha3bIBaeTCSI nepecetLe-
HueM rpynnHpoBoK R H R' H o603Hall8eTc5I RR'. T8KHM 06-
p830M,
RR' = {Rs n R; I s= 1, ..., m; t= 1, ..., m'}. (1.4)
KJIaCCbI p836HeHHSI RR' - 3TO rpynnbI 06'beKTOB, COOTBeTCT-
BYIOInHX pa3JIHlIHbIM KOM6HH8nH5IM (8, t) 3HalleHHH npH3HaKOB
R H R'.
COBoKynHocTb HenYCTbIX KJI8CCOB RR' X8p8KTepH3yeTcSI
HeKOTopbIM OTHOIlleHHeM 3KBHB8JIeHTHOCTH Ha MHOJKeCTBe A.
nYCTb p H p' - OTHorneHHe 9KBHB8JIeHTHOCTH, COOTBeTCTBYIO-
InHe p836HeHH5IM R H R'. K8K BbIpa3HTb lIepe3 HHX OTHorneHHe
3KBHB8JIeHTHOCTH J{JISI RR'? OKa3bIB8eTcSI, nepeCeqeHHIO RR'
COOTBeTcTByeT OTHOIlIeHHe p n p'.
,aeHcTBHTeJIbHO, KJI8CCbI rpynnHpoBKH RR' COCTOSIT H3
06'beKTOB, 3KBHBaJIeHTHbIX OJ{HOBpeMeHHO H no R H no R'.
3HallHT, JI106851 napa (i, j) 06'beKTOB TaKoro KJIaCCa COJ{ep)KHT-
C5I OJ{HOBpeMeHHO H B P H B p', T. e. B p n p'. 06p8THO, eCJIH
(i, j) $. p n p', TO (i, j) He COJ{epmHTC5I B p HJIH B p', TaK tITO
i H j He 3KBHBaJIeHTHbI JIH60 B R, JIH60 B R' H B cooTBeTcTBYIO-
eM p836HeHHH npHH8J{JIem8T pa3HbIM KJI8CC8M. CJIeJ{OBaTeJIb-
HO, H B RR' OHM non8J{YT B p83Hble KJI8CCbI, 1JTO H Tpe60BaJIOCb
J{OK83aTb.
40
1
1
If
f
LlJI5I pacnpe.L{eJIeHH5I nepeCelleHH5I rpynnHpoBOK R R' BBe-
J{eIVI CJIe YIO He 0003HaQeHH5I: N st - KOJIHqeCTBO OO'beKTOB
B MHO)KeCTBe Rs n Rt; N s . = N st - KOJIH1JeCTBO 001>eKTOB
t .
B MHO)KeCTBe Rs; Nt. = Nst-KOJIHtleCTBO 001>eKTOB B MHO-
s
}KeCTBe R;. 3.L{eCb H .L{aJIee TOllKa Ha MeCTe KaKOrO-JIHOO HH.L{eK-
ca 03HallaeT, liTO paCCMaTpHBaeTC5I pe3YJIbTaT CYMMHpOBaHH5I
no 3TOMY HH.L{eKCY. 0003HalleHH5I .L{JISI lIaCTOT aHaJIOrHlIHbI.
Pacnpe.L{eJIeHHe lIaCTOT rpynnHpoBKH RR' lIaCTO H300pa-
JKaeTC5I B BH.L{e CJIe.L{YIO eH MaTpHUbI, Ha3bIBaeMOH ma6/lul1eu
COnpflJlCeliHOCmu (KoMoHHaUHoHHoH TaOJIHueii):
Pl1 Pl2 . . . Plm'
Pml Pm2 . . . Pmm'
Pl.
Pm.
P.I
p. 2
P.rn'
1
CTpOKH 3TOH TaOJIHUbI COOTBeTCTBYIOT 3HalleHH5IM O.L{HOrO
npH3HaKa (HMeHaM KJIaCCOB pa30HeHH5I R), CTOJIOUbI - 3Ha-
qeHH5IM .L{pyroro (HMeHaM KJIaCCOB pa30HeHHSI R'), aHa nepece-
lIeHHH S-H CTpOKH H t-ro CTOJIOua HaXO.L{HTCR Pst- Q8CTOTa KOM-
OHH8UHH (5, t) (s = 1, ..., m; t = 1, ..., m'). B.C8MOH HH)KHeH
cTpoKe CO.L{ep)K8TC5I BeJIHlIHHbI p. t, B C8MOM npaBoM CTOJIOue-
BeJIHlIHHbI Ps.. 3TH CYMMapHble nOK838TeJIH H83bIBaIOTC5I HHor-
.L{a MapZUflallbflblMU t/,acmomaMU. CYMM8 HX (no CTpOKe MJIM
CTOJIOny) - e.L{HHMU8.
,aJI5I paCCMOTpeHHoro npHMepa MaTpHua COnpSl)KeHHOCTH
BhIr JISI.L{HT CJIe.L{YIO HM oopa30M:
0,1 0,2 0
0,1 0 0,3
0,2 0,1 0
,
0,3
0,4
0,3
0,4
0,3
0,3
1
41
-
"" M 'Ii:
_._ ,'IKh ,-.. g' " '¥ m !.t:.""_ ,!<il ""... . ,
AHaJIOrliQHO onpe.L{eJIHeTC5I nepeCeqeHHe OOJIblIlerO tIHCJIa
rpynnHpOBOK. ECJIIi R 1 , R 2 , ..., Rn - 3a.L{aHHbIe n rpynnH-
pOBOK, TO tIepe3 R 1 ... Rn 0003HallaeTC5I HX nepeCeqeHlie, T. e.
rpynnHpoBKa MHO}KeCTBa m Ha KJIaCCbI Rt n ... n R n
(r.L{e R: k - KJIaCC pa36HeHH5I Rk), OTBetIalOIII.He KOMoHHau.H-
5IM S = (SI' ..., Sn) 3HalleHHH npH3HaKOB Rl, ..., Rn. COOT-
BeTCTBYIOIII.He pacnpe.L{eJIeHH5I 0603Haq81OTCH 6YKBaMH N S1 ,...,s
n
Ii PSI".. ,sn. nepecelleHHe rpynoHpoBOK Rk C HH,l{eKCaMH k,
np06eralOlI.(liMH He Bce B03MO)KHbIe 3HatIeHH5I, a TOJIbKO HeKO-
Topoe nO.L{MHO)KeCTBO V, 6YAeT 0003HallaTbC5I RV = n Rk,
kEY
a 9JIeMeHTbI ero pacnpe,l{eJIeHH5I - qepe3 NV H p V . ECJIH
pa36HeHHSlM Rk COOTBeTCTBYIOT OTHOlIleHHH pk S;;; 91 X 91, TO,
o lIeBH,l{H 0 , rpynnHpoBKe RV COOTBeTcTByeT 9KBHBaJIeHTHOCTb
pV = n pk. MaprHHaJIbHbIe BeJIHlIHHbI N H P nOJIYll81OTCSI 3,l{eCb
kEY
npOCTO KaK 9JIeMeHTbI pacnpe,l{eJIeHHH rpynnHpoBOK RV.
BC5IKOe pacnpe,l{eJIeHHe (RV) 51BJISleTC5I no onpe,l{eJIeHHIO
.A-tap2UHallbflblM .L{JI5I (R 1 R 2 ... Rn). nepecelleHHe MHorHX rpyn-
llHpOBOK Ha3bIBaeTC5I TaK}Ke Mfl020MepHOU, HJIH KOM6ufla1{UOfl-
v II II
/iOU, HJIH KpamHOU 2pynnup08KOU, a pacnpe,l{eJIeHHe ero l1a-
CTOT - MHorOMepHOH Ta6JIHneH COnp5l}KeHHOCTH.
ECJIH pa36HeHHe Rk HMeeT mk KJIaCCOB (3HatleHHH), TO
o6III.ee tIHCJIO" KOM6HHanHH BH,l{a s = (SI, ..., sn) paBHo npoH3-
BeAeHHIO m 1 m 2 ... m n . B qaCTHOCTH, npH ,l{HXOTOMHqeCKHX npH-
3HaKax 3TO lIHCJIO paBHO 2 n H 6MCTpO pacTeT C YBeJIHlIeHHeM
ll. O,l{HaKO KOJIHlIeCTBO HenYCTbIX KJIaCCOB B nepecelleHHH
pa36HeHHH no nOH5ITHMM npHtlHHaM He MO}KeT npeBOCXO,l{HTb
KOJIHlIeCTBa 06'beKTOB N B paCCMaTpHBaeMOM MHO}KeCTBe.
npOaHaJIH3HPyeM Tenepb nOH5ITHe BKJIlOl1eHH5I pa36HeHHH.
rOBOp5lT, tlTO pa36HeHHe R 8KJllOtteflo B pa36HeHHe R', H nHIIlYT
R £;;; R', eCJIH BCSlKHH KJIaCC pa36HeHH5I R CO,l{ep}KHTC5I B Ka-
KOM-JIH60 KJIaCCe pa36HeHHH R'. 5ICHO, liTO Ka)K,l{bIH KJIaCC pa3-
6HeHH5I R' 51BJI5IeTC5I 06'be,l(HHeHHeM Tex KJIaCCOB Rs, KOTopbIe
CO.L{epmaTC5I B HeM. HHbIMH CJIOBaMH, R' nOJIYtIeHO YKpynHe-
HHeM R, TaK liTO npH3HaK R' MO}KHO paCCMaTpHBaTb KaK pe-
3YJIbTaT npHMeHeHH5I O,l{H03HallHoro (HO He 06paTHO O.L{H03HaQ-
Horo) npeo6pa30BaHH5I K npH3HaKY R (CM. c. 26).
OqeBH,l{HO, eCJIH R s;; R' H R' s;; R, TO R = R't T. e.
KJIaCCbI R s COBna,l{aIOT C nO,l{XO,l{51III.HMH KJIaCCaMH R t, B tlaCT-
HOCTH m = m'.
KaK CB5I3aHbI Me)K,l{Y C060H OTHOlIleHHSI 9KBHBaJIeHTHOCTH
p H p', eCJIH .L{JI5I COOTBeTCTBYIOIII.HX pa36HeHHH BbInOJIHeHO
42
f
J
.
(:
It
-
R R'? OKa3bIBaeTCSI, R R' TOr.L{a H TOJIbKO TOr.L{a, KOr,na
,
pS;;p.
,[{eHCTBHTeJIbHO, (i, j) E p 03HallaeT, liTO i H j CO,l{ep)KaTCSI
B O.L{HOM H TOM )Ke KJIaCCe Rs. Ho Rs c= Ri -.L{JISI HeKOTOpOrO t,
TaK qTO i, j npHHa,l{JIe)KaT R;, T. e. (i, j) E p'. TaKHM OOpa30M,
p p'. OOpaTHO, eCJIH p p', TO OO'beKTbI, KOTOpble 3KBHBa-
JIeHTHbI no p, ,l{OJI)KHbI ObITb 3KBHB8JIeHTHbI H no p'. Ho 3TO
B03MO)KHO TOJIbKO, eCJIH KJI8CCbI Rs neJIHKOM CO,l{ep)K8TC5I B
KJI8CC8X Rt, T. e. R R', tlTO H TpeOOBaJIOCb ,l{OKa3aTb.
OTMeTHM, B tlaCTHOCTH, liTO .L{JI5I JIIOObIX R H R' RR' R,
RR' R', p n p' c= p, p n p' £; p'. ,L{JI5I OTHOlIleHHH 3TH
cpaKTbI TpHBHaJIbHbI, nOCKOJIbKY OHHapHbIe OTHOlIleHH5I-
3TO MHO)KeCTBa, a OOIna51 lIaCTb pa3HbIX MHO)KeCTB BCer,l{a BKJIIO-
lIeHa B HCXO,l{HbIe MHO)KeCTBa.
OlleBH,l{HO TaKLKe, tlTO R c= R' -+ RR' = R, TaK KaK npH
p f; p' cnpaBe,l{JIHBO p n p' = p. B 3TOM CJIYlIae (RR') =
= (R); TaOJIHna COnpSl>KeHHOCTH B KaJK,l{OH CTpOKe CO,l{ep)KHT
u u u
e.L{HHCTBeHHbIH HeHYJIeBOH 3JIeMeHT H8 nepecelleHHH S-H CTpO- '"
KH H t-ro cToJI6na npH Rs Rt.
KaK Y)Ke OTMellaJIOCb, npH R c= R' Me)K,l{Y 3HalleHH5IMH npH-
3HaKOB R H R' cy ecTByeT <PYHKnHOHaJIbHa51 CB5I3b cp (s) == t
npH R s R t, n03BOJI5IlOma51 TOllHO npe,l{CKa3bIBaTb 3HalleHHe
t npH3HaKa R' KaK TOJIbKO 3a.L{aHO 3HalleHHe S npH3HaKa R.
OopaTHbIH nporH03 HeO,l{H03HalleH, BOOOIne rOBOpSl, TaK KaK
COOTBeTCTBHe cp He 005l3aTeJIbHO B3aHMHO-O.L{H03HallHO. npH
B3aHMHO-0,l{H03HallHOM cp pa30HeHHH COBna,l{aIOT, liTO COOT-
BeTcTByeT HaHOOJIblIleH CB5I3H npH3HaKOB (B03MO)KHOCTM O,l{-
H03HallHoro B3aHMHoro nporH03a). BKJIlOlIeHHe R c= R' xapaK-
TepH3yeT HaHOOJIblIlYIO HanpaBJIeHHYIO CB5I3b - OT R K R'.
nOKa3aTeJIH CBSl3H npH3HaKOB .L{OJI)KHbI npHHHMaTb B paCCMO-
TpeHHbIX CJIYlIaHX 3KCTpeMaJIbHOe 3HatJeHHe.
1.3.2. AKCUO.MaTut/,eCK,Oe nOCTpoeflue
paCCTOflflUfl .Me'JICay pa36uefluflMu
He3aBHcHMo OT paCCT05lHH5I Me)K,ny MaTpHnaMH
CB5I3H (CM. np HJIO>KeH He) , n03BOJISlIOInero B npHHnHne H3Me-
p5lTb paCCT05lHHe Me)K.L{Y pa30HeHH5IMH KaK paCCT05lHHe Me)K,l{Y
COOTBeTCTBYIOInHMH OTHOllIeHH5IMH 3KBHBaJIeHTHOCTH, nonbI-
TaeMC5I CKOHCTpYHpOBaTb MeTpHKY B MHomeCTBe pa30HeHHI1: BO
BHYTpeHHHX TepMHHax. B 3TOM pa3.L{eJIe 6y)(eM paCCMaTpnBaTb
TOJI bKO Heynop 51.L{OlleHHble pa30HeHH5I.
BBe.L{el'v1 HeKOTophIe BcnOMOraTeJIbHhIe nOH5ITH5I. I3Y.L{eM fO-
BOpHTb, liTO pa30HeHHe R" HaXO.L{HTC5I .Me'JICoy pa30HeHH5IMH R
43
-
. " ",,,,,'!!W'W_,"",,,,,,,..
H R' (0003HaqHM lR, R", R']), eCJIH npOH3BOJIbHbIe OO'beKTbl
i H j, HaXO)J.HCb B O)J.HOM H TOM }Ke KJIaCCe (B pa3HbiX KJIaCCaX)
B R H R', npHHa)J.JIe}KaT O)J.HOMY H TOMY }Ke KJIaccy (pa3HblM
KJIaCCaM) H B R". .1I.JIH COOTBeTCTBYIOIll.HX OTHOmeHHH p, p',
p" I KaK HeTpY)J.HO nOKa3aTb, lR, R", R'] TOr)J.a H TOJIbKO Tor-
,n.a, Kor,n.a
p n p' c.= p" e pUP' ·
MHO}KeCTBO S £; m oY)J.eM Ha3blBa Tb ceZ.MeHmOM pa30 e-
HHH R, ecJIH S HBJIHeTCH OO'be)J.HHeHHeM HeKOTOPbIX KJIaCCOB
R. 51cHo, ecJIH S - cerMeHT, TO H S = m - S - TO}Ke
cerMeHT. qepe3 Rs oo03HaQHM MHO)KeCTBO KJIaCCOB pa30He-
HHH R, oopa3YIOIll.HX B OO'be)J.HHeHH H cerMeHT S. CJIe)J.OBaTeJIb-
HO, Rs _ 3TO pa30HeHHe MHO}KeCTBa S, nOJIYQeHHOe Y)J.aJIe-
HHeM H3 R Bcex 06'beKToB, He BXO)J.HIll.HX B S.
PaCCMOTPHM CJIe)J.YIOIll.He TPe60BaHH H K npOH3BOJIbHOH CHM-
MeTpHQHOH Mepe 6JIH30CTH pa36HeHHH f (R, R'):
1. .1I.JIH JIlOObIX Tpex pa30HeHH H R, R', R" H3 lR, R",
R'] cJIe,nyeT paBeHcTBO
f (R, R') = f (R, R") + f (R", R') ·
2. ECJIH S £; m _ cerMeHT 060HX pa30HeHH H R H R',
npHQeM Ha S = m - S 3TH pa30HeHH H COBna)J.aIO T , TO
f (R, R') = f (Rs, Rs).
l'
3. PaccTOHHHe f (0, U) Me}K)J.Y YHu8epcaAbHbl.M pa36HeHH-
eM U, COCTOHIll.HM H3 e)J.HHCTBeHHoro KJIaCCa, BKJIIOQalOIll.ero
Bce N OO'beKTOB, H mpu8uaAbHblM pa36ueHUe.M 0 MHO}KeCTBa m
Ha N 0)J.H03JIeMeHTtlbIX KJIaCCOB 3aBHCHT OT N H onpe)J.eJIHeTCH
CPOPMYJIOM
r)J.e <p _ npOH3BOJIbHaH CTporo B03paCTalOIll.aH <PYHKlI. HH , Ta-
KaH, QTO qJ(X) > <p (1).
x
TpeooBaHHe 1 BbIpa}KaeT HHTYHTHBHO OQeBH)J.I-lOe paBeHcT-
BO paccToHHHH )J.JIH pa30HeHH H , Haxo)J.HIll.HXCH B OTHomeHH H
«Me}K)J.Y», npH 3TOM BOBce He TpeOYH cnpaBe)J.JIHBOCTH HepaBeH-
CTBa TpeyrOJIbHHKa. TpeooBaHHe 2 Bblpa}KaeT He3aBHCHMOCTb
paccToHHHH OT cOBna)J.alOIll.HX QaCTeH pa30HeHHi\, T. e. He3a-
BHCHMOCTb oO'beKTOB.Tpe6oBaHHe 3 <PHKcHPyeT MacmTa6 H3Me-
peHM5I paccTOSIHHSI.
OKa3blBaeTCH, 3TH TpeOOBaHHH n03BOJIHIOT 0)J.H03HaQHO Bbl-
pa3HTb f (R, U) Qepe3 <PYHKlI. HIO <p. A HMeHHO )J.JIH R =
44
f (0, U) = <p (N) - N <p (1),
= {Rt, ..., Rm}
m
f (R, U) = rp (N) - cp (N s ).
5=1
(1.5)
LLJI5I ,l{OKa3aTeJIbCTBa paCCMOTpHM pa30HeHHSI R 1 , R 2 , ...,
Rm, nOJIYlIaeMbIe B pe3YJIbTaTe nOCJIe,l{OBaTeJIbHOrO paCCbI-
naHH5I KJIaCCOB R Ha 0.L{H03JIeMeHTHbIe nO,l{KJIaCCbI:
RO = {Rt, R 2 , ..., Rm} = R;
R 1 = {iI' ..., iN 1 , R 2 , ..., Rm};
R 2 == {it, ..., iN 1 +N 2 , R 3 , ..., Rm};
Rm = {iI' ..., iN} === O.
OlIeBH,l{HO, 0 = Rm f;; Rm-I c= ... f;; R c= U. Tor,l{a cor-
JIaCHO TpeooBaHIHo 1
f (0, U) = f (0, R) + f (R, U),
. npHlIeM
m-l
f (0, R) = f (RS, RS+ 1).
s=O
CJIe,l{OBaTeJIbHO,
m-I
f(R, U)=rp(N)-Ncp(I)- f(RS,Rs+l).
s=O
Tenepb 3aMeTHM, lITO pa30HeHH5I. RS H Rs+ 1 COBna.L{aIOT
BCIO,l{Y, KpOMe MHO)KeCTBa Rs+ 1, SlBJI5I1O erOC5I cerMeHTOM liX
OOOHX, npHlIeM pa30HeHHe Rs+l Ha 3TOM cerMeHTe TpHBHaJIb-
HO, a pa30HeHHe Rs - YHHBepCaJIbHO. Tor,l{a no TpeooBaHHIO
3
f (RS, RS+ 1) = cp (N s + I) -N s + 1 cp (1)
Ii, CJIe.L{OBaTeJIbHO,
f(R, U)===rp(N)-Nrp(I)- (rp(NS+l)-Ns+1CP(1))=
s
In
=rp(N)- cp(N s ),
s= 1
qTO H TpeOOBaJIOCb .L{OKa3aTb.
Teneph Ha OCHOBaHHH (1.5) no TpeooBaHliIO 2 MO)KHO ,l{O-
Ka3aTb, lITO paCCT05lHHe f 0.L{H03HaQHO BbIpa)KaeTCSI lIepe3 <p
45
npM JIIOObIX R M R'. L{eHcTBHTeJIbHO, npM R c= R' HMeeM no
Tpe6oBaHHIO 1, TaK KaK R' U,
f (R, U) = f (R, R') + f (R', U).
OTCIO,n:a no cpopMYJIe (1.5)
m' m
f(R,R')=f(R, U)-f{R', U)= rp(N.t)- cp(N s .).
t = 1 s= 1
(1.6)
nOCKOJIbKY RR' HaXOJJ:MTC5I Me*,n:y R H R', npHQeM
RR' R H RR' R', TO no TpeooBaHHIO 1 H cpopMYJIe (1.6)
nOJIyqaeM
f (R, R') = f (R, RR') + f (R', RR') = cp (N s .) + rp (N. t )--
s t
-2 cp (N st ), (1.7)
s,t
qTO H Tpe60BaJIOCb ,n:OKa3aTb.
<1> OpMYJIa (1.7) npe,n:cTaBJI5IeT paCCT05lHHe, 0,AH03HaQHO
onpe,l{eJI5IeMOe TpeOOBaHH5IMH 1-3.
PaccMoTpHM ,l{Ba qaCTHbIX CJIyqa51 paCCT05lHHH (1.7), no-
JIyqaeMWX npH KOHKpeTHwx YHKnH5Ix cp.
nOJIO)KMM rp (x) = x 2 . TOf,l{a nOJIYQHM COfJIaCHO (1.5)
f(R. V) = N 2 - . t N: = N 2 (1- . t P:). (1.8)
m
llOJIyqeHHa51 BeJIHqHHa cS (R) = 1 - P HMeeT eCTeCT-
s= 1
BeHHoe HCTOJIKOBaHHe KaK Cpe,l{H5I5I OlliH6Ka nponopnHOHaJIb-
HOfO npOfH03a 3HaqeHHH npH3HaKa R.
.QJI5I YTOqHeHHSI CKa3aHHoro paCCMOTpHM CJIeJJ:YIO YIO 3a-
,l{aqy. OO'beKTbI H3 MHO)KeCTBa m n05lBJl5IIOTC5I paBHOBep05lT-
HO, H TpeoyeTcSI npe,l{CKa3bIBaTb 3HaqeHHSI npH3HaKa R Ha
3THX OO'beKTax. B03MO)KHbI pa3JIHqHble MeTOJJ:bI npOfH03a.
MHOfHe H3 HHX OCHOBaHbI Ha HH<popManHH 0 Bep05lTHOCT5IX Ps
nOSlBJIeHHSI 3HaQeHHH R s. DpOfH03HpOBaHHe 3HaqeHHH npH-
3HaKa R Ha3bIBaeTC5I nponopnHOHaJlhHbIM, eCJIH OHO ocy ecT-
tot
BJI5IeTCH C Bep05lTHOCT5IMH Ps JJ:JI5I 3HaqeHMH S, T. e. 3HaqeHHe s
npe,n:CKa3bIBaeTC5I C qaCTOTOH Ps. ECJIH npOfHo3HpyeMoe 3Ha-
qeHHe eCTb Rs, Bep05lTHOCTb qefO paBHa Ps, TO Bep05lTHOCTb
46
I
{
..
t
..-
OIJJH6KH paBHa 1 - Ps, TaK qTO Cpe H5I5I OJISI OIJJH60K npo-
nOpIJ.liOHaJIbHOrO npOrH03a paBHa:
Ps(l-ps)= I- P: = cS (R).
s s
TaKHM OOpa30M, cS (R) XapaKTepH3yeT TOqHOCTb npOrHO-
3lipOBaHH5I 3HaqeHHH R: qeM MeHblIle cS (R), TeM B Cpe,l{HeM
npOrH03 TOqHee. BeJIHqHHa cS (R) XapaKTepH3yeT CTeneHb He-
Onpe eJIeHHOCTH, pa30pOCa 3HaqeHHH npH3HaKa. OHa npHHH-
MaeT MaKCHMaJIbHOe 3HaQeHlie, eCJIH Bce N s paBHbI Me)K,l{Y COOOH
Ii B03MO)KHOCTH JI5I nporH03a HaHxY U1"e. MHHHMaJIbHOe
3HaQeHHe cS (R) paBHo HYJIlO, OHO ,l{OCTHraeTC5I, Kor,l{a Bce,
OO'beKTbI O,l{HHaKOBbI no R, T. e. Bce KJIaCCbI rpynnHpoBKH nyc-
TbI, KpoMe O,l{HOrO, cO,l{ep:>Ka ero Bce OO'beKTbI. YqHTbIBaH,
qTO BeJIHqHHa cS (R) = Ps (I - Ps) eCTb CYMMa HcnepcHH
Ps (1 - Ps) oliHOMHaJIbHbIX CJIyqaHHbIX BeJIHqHH, COOTBeTCT-
BYIO HX OT,l{eJIbHbIM 3HaqeHH5IM npH3HaKa, KOTopbIe npHHH-
MalOT e):{HHHQHOe 3HaqeHHe C Bep05lTHOCTblO Ps, a HYJIeBOe-
C -Bep05lTHOCTblO 1 - Ps, eCTeCTBeHHO Ha3BaTb 3TOT nOKa3aTeJIb
Kat/,ecm8eflflou oucnepcuet1 HO.M,UHallbfl020 npu3flaKa R..
BOJIblIlOe pacnpocTpaHeHHe nOJIyqHJIa ,l{Pyra51 Mepa He-
onpe,l{eJIeHHOCTH HOMHHaJIbHOrO npH3HaKa - 8flmpOnUJl:
. m m
h(R)=- Pslogps= Pslog . (1.9)
s= t s= 1 Ps
3Ta XapaKTepHCTHKa BO MHOrOM aHaJIOfliqHa cS (R), TaK
KaK <PYHKnHH log Ps H 1 - Ps Be,l{YT ceo51 CXO,l{HbIM oopa30M:
np" Ps = 1 npHHHMalOT HYJIeBOe 3HaqeHHe H C YMeHblIIeHHeM
Ps CTporo MOHOTOHHO B03paCTaIOT: 1 - Ps -,l{0 e,l{HHHnbI,
a log..!... - ,l{0 oeCKOHellHOCTH. BOJIhIllHH ,l{Hana30H H3MeHeHHH
Ps
.iIorapHcpMa OOeCneqHBaeT 3HTponHH OOJIbIllYIO qYBCTBHTeJlb-
HOCTb no cpaBHeHHIO C ,l{HcnepCHeA.
3HTponH5I xapaKTepH3yeT Cpe,l{Hee KOJIHqeCTBO HH<popMa-
nHH no llIeHHOHY, cO,l{ep}Ka eeCH B ,l{aHHOM npH3HaKe, H aHa-
JIOrHqHO ,l{HcnepCliH 51BJI5IeTC5I nOKa3aTeJIeM KaqeCTBa nporHO-
v
3a ero 3HaqeHHH.
OCTaeTC5I ,l{ooaBHTb, liTO npH cp (x) = log x
f (R. U) = N log N - N slog N s = N ( - Ps log p s) .
( 1.10)
1
TaR qTO N f (R, U) eCTb 3HTpOnH5I.
47
";t-.4<
W Ii::t.t. !.."", J....'!' ...
TaKHM OOpa30M, H 3HTpOnMH H HCnepCMH onpe enHIDTCH
TpeooBaHMHMM 1-3, TOJIbKO ,nJIH pa3JIHlIHbIX MaclliTaoHpYID-
III.HX <PYHK[(HH cpo Ha OCHOBe CKa3aHHoro MO)KHO YTBepm.L{aTb,
liTO H B oow.eM CJIYlIae f (R, U) no (1.5) xapaKTepH3yeT Heonpe-
eJIeHHOCTb 3HalleHHH npH3HaKa R H KalleCTBO ero nporHo3a,
OCHOB3HHoro H3 BeJIHlIMHaX PS. ¥ ,, rt "'*
MbI npHBeJIH npHMepbI 3a.L{aHHH KOHKpeTHoH M3cIlITaoHPYID-
eH cpYHKnHH ,l{JIH YTOllHeHHH p3CCTOHHHH (1.7). PaccMoTpHM
Tenepb .L{PyrYID B03MomHOCTb YTOllHeHHH paccToHHHH - C no-
MOW.bID .L{ononHMTeJIbHbIX pa3YMHbIX Tpe6oBaHHH. TaK, B pa-
oOTe [101] npe.L{JIaraeTCH BbI6HpaTb M3clllTao H3 OCHOBe cO.L{ep-
}K3TeJIbHbIX KOHlI.enUHH 0 COCe.L{CTBe pa30MeHHH. HanpHMep,
MO:>KHO ClIHTaTb COCe.L{HHMH TaKHe pa30HeHHH, O.L{HO H3 KOTOpbIX
nOJIYlIaeTcH pa3,l{eJIeHHeM KaKOrO-JIHOO KJIaCCa ,l{Pyroro pag-
oHeHHH Ha ,l{Be HenycTbIe lIaCTH HJIH nepeHOCOM O,l{HOro OO'beK-
Ta B ,l{pyroH KJIacc. ECTecTBeHHo TpeooBaTb, lITOObI paccToH-
lIHe Me)K,l{Y TaKHMH coce;L{HMMH pa30HeHHHMH paBHHJIOCb e,l{R-
HHl.(e.
BbISICHHM, KaKHe MaCI.I1TaoHpYIDIII.He <PYHKnHH ,l{JIH paccToH-
HHH (1.7) OneHHB3IDT pa3,l{eJIeHHe KaKOrO-JIHOO KJIaCCa pa30He-
HHH e,l{HHHlIHbIM paccToHHHeM. OlleBH,l{HO, eCJIH B pa30HeHHH
R KJIaCC C lIHCJIOM OO'beKTOB M pa30HTh Ha ,l{Be HenYCTbIe lIaCTH
lIHCJIeHHOCTblO M 1 H M2' TO B CHJIY (1.6) paccToHHHe Mem.L{y R
H HOBbIM pa30HeHHeM paBHO cp (M) - cp (M 1 ) - cp (M 2 ). Tor-
.L{a 3a,l{alla COCTOHT B TOM, lITOObI BbIHCHHTb, KaKHe CPYHKnHH (})
Y OBJIeTBOpHIDT COOTHOI.I1eHHID
cp (M 1 + M 2 ) = cp (M 1 ) + cp (M 2 ) + 1 (1.11)
,l{JIH JIIOObIX HaTypaJIbHbIX lIHCeJI M 1 H M 2 .
nOJIaraH M 1 = M 2 = 1, B CHJIY (1.11) nOJIYllHM cp (2) =
= 2cp (1) + 1. MeTo,l{OM HH,l{YKnHH I\fO)KHO ,l{OKa3aTb, liTO
<p (M) = Mcp (1) + M - 1 = M (cp (1) + 1) - 1. ,[{eHcTBH-
TeJIbHO, eCJIH 3TO TaK, TO cp (M + I) = cp (M) + cp (1) +
+ 1 = M (cp (1) + 1) - 1 + cp (1) + 1 = (1\1 + 1) (cp (1) +
+ I) - 1, tlTO H TpeOOB3JIOCb ,l{OKa3aTb.
0003HallaH cp (1) + 1 lIepe3 a, nOJIYlIHM BH,l{ HCKOMOH Mac-
I.I1TaoHpYIDIII.eii <PYHKnHH cp (x) = ax - 1, r,l{e a npOH3BOJIb-
HO. npH 3TOM
m
f(R,U)=(aN-I)- (aNs-I)=m-l, (1.12)
s= 1
T. e. B .L{3HHOM CJIYIIae paccToHHHe .L{O YHHBepCaJIbHOrO pa3-
oReHHH OneHHB3eTcH lIHCJIOM KJIaCCOB pa30HeHHH. MO)KHO no-
K83aTb, liTO TO }Ke paCCTOHHHe BbI,l{eJIHeTCH H3 ceMeHcTBa (1.7)
48
.1
I
I
I
I
I
t
j
,
'1"
f
-
YC OBHeM: pa30HeHHe, no yqaeMOe nepeHOCOM O HOrO OOneK-
Ta H3 K acca B K acc, orCTOHT OT HCXO Horo Ha 2. ECJIH lKe MbI
XOTHM, qTo6bI paCCT05lHHe Me)K,!{Y 3TH1\fH ,!{BYM5I pa30HeHH5I1fH
paBH5IJIOCb 1, TO CJIe,!{yeT B KaqeCTBe MaClIITaOHpYIO eH <PYHK-
[(HH B351Tb cp (X) = ax - 1/2.
1.3.3. PaccToflflue Jvte'JICay OTH:oUleflUflMU
B repJvtUflax zpynnupoBoK
PaCCT05lHHe Me)K,!{Y OTHOIlIeHH5IMH p, p' c= m X
X m onpe,!{eJI5IeTC5I KaK paCCT05lHHe (n. 2) 1e)K Y COOTBeTCT-
BYIO HMH (1, O)-MaTpHnaMH r == (rij) H r' == (rif), r,!{e rij ==
= 1 (i, j) E p, riJ == 1 (i, j) E p':
N N
d (p, p') = d (r, r') t::= (rij -rij)2 = I rij-rhI 2 . (1.13)
i,j=l l,j=l
nOc e,!{Hee paBeHcTBo OOn5lCH5IeTC5I TeM, qTO ,!{JI5I oYJIeB-
CKHX BeJIHqHH MO,!{Y b HX pa3HocTH COBna,!{aeT C KBa,!{paToM.
B TO }Ke BpeM5I OHO nOKa3bIBaeT, qTO paCCT05lHHe (1.13) paBHo
u ,
KOJlHqeCTBY n9pa3p5l,!{HbIX HeCOBna,!{eHHH MaTpHn r H r KaK
6YJIeBCKHX N2-MepHbIx BeKTopoB, T. e. X8MMUfl208Y paCCmOJl-
flUIO Me>K,!{Y HHMH. 3Ta BeJIHqHHa paBHa KOJIHqeCTBY nap (i, j),
npHHa,!{JIe}Ka HX p, HO He npHHa,!{JIe)Ka HX p', nJIIOC KOJIH-
qeCTBO nap (i, j), npHHa,!{JIe)KaWJlX p', HO He npHHa,!{JIe}Ka HX
p, qTO, KaK HeTpY,!{HO BH,!{eTb, paBHO 1 p I - I p n p' 1 nJIIOC
I p' I - 1 p n p' I,. TaKHM oopa30M,
d (p, p') = 1 p I + 1 p' 1- 2 , p n p' ,. ( 1. 14)
B TOM cJIyqae, Kor,!{a p H p' - OTHOIlleHHH 3KBHBaJIeHTHO-
CTH, cooTBeTcTBYIO He HOMHHaJIbHbIM npH3HaKaM, T. e. pa3-
6HeHHHM R == {R 1 , ..., R m} H R' == {R 1, ..., R:r,,}, paCCT05l-
HHe (1.14), KaK HeTpY,!{Ho BH,!{eTb, COBna,!{aeT C ,!{HcnepcHoH-
HhIM paCCT05lHHeM (1.7), nOJIyqaeMbIM npH BbIoope B KaqeCTBe
MacillTaoHPYIO eH CPYHKnHH cp (x) = x 2 .
,[{eHcTBHTeJIbHO, ,!{JI5I OTHOlIleHH5I 3KBHBaJIeHTHOCTH p =
m
= U Rs X Rs tlHCJIO npHHa,!{JIe}Ka HX eMY nap (i, j) paBHo
s= 1
CYMMe qHCJIeHHOCTeH MHO}KeCTB Rs X Rs (s == 1.. ..., m), paB-
.
HbIX, OqeBH,!{HO, N . CJIe,!{oBaTeJIbHO, 1 p I == N , TaK qTO
s= 1
d(R)=d(R, U)=lul-1 pl=N2_ N:,
s
49
1'::W' ".. "" ............, ,
TaR KaK B U = m X m CO,Aep)KHTCH pOBHO N2 nap, 3 3TO H
eCTb BbIpa)KeHHe (1.8).
B TepMHHax KallecTBeHHoM ,nHcnepcHH cpopMYJIa (1.14) npH-
MeT CJIe YIO HH BHA:
d(R, R')= N:. + N t -2 N:t,
s t 8,t
(1. 15)
HJIH
d (R, R') = 2d (RR') - d (R) - d (R').
( 1.16)
TIpH 3TOM, OlleBH,nHO,
d (R, RR') = d (RR') - d (R),
d (R', RR') = d (RR') - d (R').
(1 .17)
TaKHM oopa30M, paCCTOHHHe (1.13), BBe,neHHoe B TepMHHax
MaTpHn (rij) H (r!l) nOKa3aTeJIeH CBH3H Me)K,ny OO'beKTaMH H,
Ka3aJIOCb ObI, OT,neJIbHO YlIHTbIBalO ee Ka:>K,nylO napy OOneK-
TOB, ,nJIH HOMHHaJIbHbIX npH3HaKOB BbIpa}KaeTC5I B lIHCTO CTa-
THCTHlIeCKHX, HeHH,nHBH,nyaJIH3HpOBaHH@X TepMHHaX pacnpe-
,neJIeHHM COOTBeTCTBYIOInHX rpynnHpOBOK. npH 3TOM B CHJIY
(1.17) paCCTOHHHe BbIlIHCJIHeTCSI no <popMYJIe (1.15) KaK CYMMa
npHpa eHHii OIllHOOK nponopnHOHaJIbHOrO npe,nCKa3aHHH npH
tot
nepexo,ne OT pacnpe,neJIeHHH OT,neJIbHhIX npH3H3KOB K HX COB-
MeCTHOMY pacnpe,neJIeHHIO (CM. TaK}Ke n. 2.2.2).
TIpHMeH5I5I CPOPMYJIY (1.13) K OTHOIlleHH5IM, nopO)K,naeMbIM
npOH3BOJIbHbIMH pa30HeHHHMH co CTPYKTypLoH, MO>KHO nOJIY-
lIHTb oooo eHHe <POPMYJIbI (1.15). .QJIH 3Toro HeooxoJtHMO 3HaTb
KaKoe pa36HeHHe co CTPYKTypoii COOTBeTcTByeT nepeCeqeHHIO
Px n p ' OTHOlIIeHHH Px H P " COOTBeTCTBYIO HX napaM (R, x)
H (R', x'). Yoe,nHMcH, 1ITO 3TO - pa30HeHHe RR co CTPYKTY-
pOM x x', r,ne
«s, 5'), (t, t'» E x x' (5, t) E x H (5', t') E x'.
J
,
,lXeiicTBHTeJIbHO, (i, j) E P" n P I Tor,na H TOJlbKO Tor,n3,
Kor,na, C O,nHOH CTOpOHbI, i E Rs, j E Rt ,nJlSl HeKOTopOH napbI
(8, t) E x H, C ,nPyroH CTOpOHbI, i E R;,; j E Rt, ,nJISI HeKOTopbIX
s', i', (s', I') E X'. 3TO 3KBHBaJIeHTHO TOMY, qTO i E Rs n R;,
H j E Rt n R;" T. e. «s, s'), (i, I'» E x x', qTO H TpeOOBa-
JIOCb ,nOKa3aTb.
3aMeTHM Tenepb, liTO ,nJISI (R, x) qHCJIO nap B OTHOIlleHHH
p,,= U Rs X Rt, OlleBH,nHO, paBHO NsN t . 113 CKa3aHHoro
(s,t)E" (s. t)EH
Ha OCHOBe (1.13) nOJIYlIaeTCH cJIe,nYIOm.aH CPOPMY JIa ,nJIH pac-
50
-
CT05lHM5I Me)K.L{Y pa36HeHM5IMH co CTPYKTypoif:
d«R,x),(R',x'»=d(px,p ')= N s . N.t+
(s.t) Ex
+ N. s ,N.t,-2 NstNs,t,. (1.18)
(s'.t')EX' (s.t)EX
(s'.t')EX'
OTclO.L{a HeTpY.L{Ho' nOJIyqHTb CPOPMYJIbI paCCT05lHH5I l\1e)K-
JJ.Y pa30HeHH5IMM co CTpYKTypaMH KOHKpeTHoro BH.L{a: paHroBbIx
npH3HaKoB, Coce.L{CTBa, .L{peBoBM.L{HbIX If T. n.
npOaHaJIH3HPyeM oOJIee nO.L{pOOHO CJIyqaH paHroBbIx npM-
3HaKOB. B npocTpaHcTBe TOJIbKO paHroBbIx npH3HaKoB HCnOJIb-
30BaHHe oo eH <POPMYJIbI (1.13) He COBceM YMeCTHO, TaK KaK
nepeCeqeHHe p n p' OTHOIlleHHH JIHHeHHOro KBa3HnOp5l.L{Ka ca-
MO Y)Ke, Booo e rOBOp5l, He Y.L{OBJIeTBOp5leT YCJIOBMIO JIMHeii-
HOCTH (c. 29) H He COOTBeTcTByeT ynop5l.L{OqeHHOMY pa30HeHHIO.
. nYCTb, HanpHMep, R - npH3HaK, xapaKTepH3YIO HH CTe-
neHb 3.L{OpOBbSl qeJIOBeKa, a R' - CTeneHb ero ooraTCTBa.
Tor.L{a IIlYTOqHaH .L{HJIeMMa «qTO JIyqIIle - ObITb Oe.L{HbIM, HO
3.L{OpOBbIM HJIH ooraTbIM, HO OOJIbHbIM» XOp0I.I10 HJIJIIOCTpHpyeT
npMpo.L{Y HeJIHHeHHOCTH: OO'beKTbI HecpaBHHMhI no RR', eCJIH
O.L{HH H3 HHX JIyqIIle no R, a BTOpOH - no R'. ECJIH (s, t) -
KOMOHHanH5I 3HaqeHHH R H R', TO (s + a, t - b) HecpaBHHMa
C (s, t), eCJIH a, b > 0 HJIM a, b < O. O.L{HaKO MO>KHO .L{oonpe-
.L{eJIHTb nOp5l.L{OK Ha MHO)KeCTBe KOMoHHaUHH (s, t) TaK, qTOObI
OH CTaJI J1HHeHHblM. .QJI5I 3Toro MO)KHO HCnOJIh30BaTb JIeKCHKO-
rpa<pHqeCKOe ynop5l.L{OQeHHe: oY.L{eM CqHTaTb, qTO (s, t) JIyqIIle
(p, q), eCJIH S < p. ECJIH )Ke S = p, oY.L{eM cpaBHHBaTb no
BTOpOH KOMnOHeHTe: (s, t) JIyqIIle (s, q), eCJIH t< q. Y nop5l,nOqeH-
Hoe pa30HeHHe, KJIaCCaMH KOToporo 51BJI5IIOTC5I nepeCeqeHH5I
Rs n R t, ynop5l,nOqeHHbIe JIeKCMKorpa<pHqeCKH, 0003HaqHM
R * R'. AHaJlOrHqHO onpe,l{eJI5IeTC5I R' * R, eCJIH nepBoH
KOMnOHeHTOH CqHTaTb HOMep KJIaCCa R:, a BTOpOH - Rs.
OKa3bIBaeTC5I [66J, ,l{JI5I ynop5l,l{OqeHHbIX pa30HeHHH R H R'
d (R, R') = d (R, R * R') + d (R * R', R' * R) +
+ d (R' * R, R'). (1.19)
0003HaQHM COOTBeTCTBYIO He Heynop5l,l{OQeHHbIe pa30Me-
--- ,...,
HH5I BOJIHHCTOH qepTOH cBepxy: R, npHqeM p = p n p-.
MO>KHO nOKa3aTb [66], qTO
d (R, R*R') +d (R'*R, R')= 1.. d(R, R'), (1.20)
2
51
"-
TaK liTO (1.19) nepenHIlleTC5I CJIe,l{YlOrnHM 06pa30M:
d (R, R') = d (R, R') +d (R*R', R' *R). (1.21)
2
-'
BeJIHqHHa d (R, R') 3a,l{aeTC5I <POPMYJIOH (1.15). BbIpa3HM
,...., -'
d (R * R', R' * R) qepe3 pacnpe,l{eJIeHHe (RR'). ,[{JI5I 3TO-
ro nO,L{CqHTaeM KOJIHlIeCTBO nap (i, j), KOTopbIe npHHa,l{JIe)KaT
R * R', HO He npHHa,l{JIemaT R' * R. IlYCTb i E Rs nRt,
j E R p n R . Tor,n:a TOT <paKT, liTO (i, j) E R * R', 03HaQaeT,
liTO S < p HJIH S == p, HO t < q. B TO )Ke BpeM5I (i, j) $. R' *
* R, T. e. (t, s) JIeKCHKorpa<pHqeCKH xyme, lIeM (q, p), TaK liTO
t > q HJIH t == q, HO S > p. O,n:HOBpeMeHHoe BbInOJIHeHHe 3THX
YCJIOBHH B03MomHO TOJIbKO npH s<p H t>q. CJIe,n:OBaTeJIbHO,
NstN Pq nap (i, j) npH S < p H t > q npHHa,l{JIe)KaT R * R' -
- R' * R. 3TO 03HaqaeT, liTO
IR*R'-R'*RI= NstN pg . (1.22)
s<pt>q
AHaJIOrHqHO
IR'*R-R*R'I= NstN pq .
s>p t<q
( 1.23)
Ilepeo603HaqHM B (1.23) HH,l{eKCbI CYMMHpOBaHH5I: nOMe-
H5IeM MeCTaMH HH,l{eKCbI S H p, a TaK)Ke t H q. Tor,l{a
IR'*R-R*R'I= NpqNst=)R*R'-R'*RI.
p<'sq>t
OTclO,n:a CJIe,n:yeT, tlTO
d(R*R',R'*R)=2 NstN pq . (1.24)
s<p t>q
CPOPMYJIbI (1.21) BMeCTe C (1.15) H (1.24) ,l{alOT BbIpa)KeHHe
pacCTOSlHH5I d (R, R') Me)K,n:y paHroBbIMH npH3HaKaMH Qepe3
pacnpe,n:eJIeHHe nepeCeqeHH5I COOTBeTCTBYlOrnHX rpynnHpoBoK.
HeTPy,n:Ho y6e,n:HThc5I, qTO OHa 5IBJI5IeTC5I qaCTHbIM CJIyqaeM
<POPMYJIbI (1.18), o,n:HaKO OCHOBHaSl <popMYJIa (1.21) OCHOBaHa
TOJIhKO Ha HOMHHaJIbHbIX H paHroBbIx npH3HaKax. OHa nOKa-
3bIBaeT, qTO B cOBoKynHoM npOCTpaHCTBe HOMHHaJIbHbIX H paH-
rOBbIX npH3HaKOB paCCTOSlHHe Be,n:eT ce651 He COBceM «perYJI5Ip-
HbIM» OOpa30l\L <I>Op1\1Y JIa (1.21) ,n:aeT pa3JIOJKeHHe pacCTOSlHH5I F I
l'vlem,ny paHroBbIMH npH3HaKaMH Ha ,l{Be qaCTH. O,nHa QaCTb,
1 --' --' ,
"2 d (R, R), yqHTbIBaeT TOJIbKO pa3HH Y B «HanOJIHeHHH»
KJIaCCOB R H R', 6e30THOCHTeJIbHO K HX ynopSl,l{OqeHHIO, a ,n:PY-
52
I
.-..
ra5l, d (R * R', R' * R), Ha060pOT, Onpe.L{eJI5IeTC5I TOJIbKO
pa3HHu.eH B ynop5l.L{OqeHHH KJIaCCOB. IlpH 3TOM K03qxpHnHeHT
1
2' B nepBOM CJIaraeMOM nOKa3bIBaeT, qTO pa3 HH u a Me)K,l{Y He-
ynop5l,l{OqeHHbIMH pa36HeHH5IMH yqHTbIBaeTC5I C B,l{BOe MeHb-
IlIHM BeCOM, qeM pa3HHna B HX ynop5l.L{OQeHH5IX. 3TO MO)KeT
npHBO,l{HTb K CJIyqa5lM, KOr.L{a d (R, R') B.L{BOe MeHbIlle, qeM
,.....,
d (R, R'). ,aJI5I 3TOrO .L{OCTaTOQHO, lJTOObI R H R' nOpO)K.L{a-
JIHCb O.L{HHM H TeM )Ke CTpOrHM paHmHpOBaHHeM MHO)KeCTBa
C nOMO blO ero pa36HeHH5I Ha HHTepBaJIbI. nepeCeQeHH5I TaKMX
MHTepBaJIOB TaK)Ke 6Y.L{YT HHTepBaJIaMH HCXO,l{HOro CTpororo
paH)KHpOBaHH5I, TaK QTO R * R' H R' * R COBna.L{YT. IlYCTb,
HanpHMep, paccMaTpHBalOTC5I nepBbIe .L{eC5ITb HaTypaJIbHbIX
QHCeJI C HX eCTeCTBeHHbIM ynop5l.L{OQeHHeM, npHlJeM R nOJIyqe-
HO pa36HeHHeM 3Toro p5l,l{a nOcpe.L{HHe, a R' - B ,l{BYX Me-
CTax, TaK QTO
R = (1-2-3-4-5,6-7-8-9-10), R' = (1-2-3,
4-5-6-7, 8-9-10).
Tor,l{a Ta6JIHna cOnp5l)KeHHOCTH 3THX pa36HeHHH HMeeT BH,l{
0,3 0,2 0
0 0,2 0,3
0,5
0,5
0,3
0,4
0,3
1
PaCCT05lHHe d (R, K) = (52 + 52) + (3 2 + 4 2 + 3 2 ) -
- 2 (3 2 + 2 2 + 2 2 + 3 2 ) = 32, a d (R, R') = d (R. R') =
=== 16, TaK KaK d (R * R', R' * R) = O. COrJIaCHO OObIQ-
HhIl\1 npe.L{CTaBJIeHH5IM paCCT05lHHe He .L{OJI)KHO TaK pe3KO YMeHb-
llIaTbC5I TOJIbKO H3-3a Toro, lJTO CTaJIH YQHThIBaTb ynop5l.L{OQeH-
HOCTb KJIaCCOB rpynnHpOBOK. Q.L{HaKO, Ha Halli B3rJI5I,l{, pa3-
OMeHH5I R H R' ,l{eHCTBHTeJIbHO C6JIH3HJIHCb, KaK TOJIbKO MbI
YBM,l{eJIM H yqJIH TO, lJTO OHM O.L{HOHanpaBJIeHHbI.
TeM He MeHee cpopMYJIa (1.21) cepbe3Ho nO.L{pbIBaeT B03-
MO}I{HOCTb arperHpOBaHH5I KaQeCTBeHHbIX npH3HaKoB pa3HbIx
THnOB llIKaJI KaK paBHonpaBHbIx: eCJIH Bce npH3HaKH OTHOC5ITC5I
K O,l{HOMY ynop5l.L{OqHBalO eMY rJIyoHHHOMY cpaKTOPY, TO
HOMHHaJIhHble npH3HaKH 6Y.L{YT .L{pyr OT .L{pyra B,l{BOe .L{aJIbllie,
53
qeM paHfOBbIe, Y KOTOpbIX ynopH,nOQeHHa51 qaCTb (1.24) 6y-
,neT BeCbMa MaJIa H3-3a HX o,nHOHanpaBJIeHHOCTM.
Pa3HHna B CTpYKType HOMHHaJIbHbIX H paHfOBbIX npH3Ha-
KOB 06HapY)KHBaeTC5I H npH aHaJIH3e MX ,nHcnepCHH. ,[{JI5I CBOM-
CTBa, 3a,naHHoro OTHOlIleHHeM p, no aHaJIOrMH c ,nHcnepCMeM HO-
1
MMHaJIbHbIX npM3HaKOB cS (R) = N2 d (U, R) onpe,neJIHM Ka-
1
tleCmeeflflYIO iJucnepculO KaK cS (p) = N2d (u, p), r,ne u =
= m X m. OqeBH,nHO, ,nJI5I pa36HeHHH co CTPYKTYPOH (R, x)
KaqeCTBeHHa51 ,nMcnepcHH
cS (R, x) = 1 -- Ps Pt. (1.25)
(s. t) E x
,lXJI5I paHfoBofO npM3HaKa R B CMJIY (1.21) nOJIyqaeM
d (R) = d (R, V) = ...!... d (R, U) = ..!.. d (R), (1.26)
2 2
{
TaK KaK R * U = U * R. OTcIO,na CJIe,nyeT, qTO ,nJI5I paH-
rOBoro npH3HaKa R efO KaqeCTBeHHa51 ,nHcnepCH5I B,nBoe MeHb-
,....J
lIle ,nHcnepCHH HOMHHaJIbHOrO npH3HaKa R, nOJIyqaeMOrO H3
R oT6paCbIBaHHeM HH<popMaQHH 06 ynopH,nOqeHHOCTM ero 3Ha-
qeHHH. 3TO He napa,nOKC. B CHJIY (1.25)
6(R)= s t Ps (1- t /t),
T. e. BeJIHqHH8 cS (R) paBHa cpe,nHeM OlIlH6Ke nporH03a, COC-
T05l erO B YTBep)K,neHMH, qTO 3HaqeHHe R Ha OO'beKTe He JIyq-
lIle JIID6oro ,naHHoro 3HaqeHH5I. I1Haqe rOBOp5l, () (R) xapaKTe-
pH3yeT cpe,nHIOIO OIllHOKY paHrOBbIX YTBep)K eHMM BH,na
«x (i) He xYJKe, qeM S» (CM. c. 28), npH KOTOpbIX 3aBbIIlIeHHe HC-
THHHOM OneHKH S He CqHTaeTC5I OIliHOKOH, TOf,na KaK () (R)
xapaKTepM3yeT OIIlHOKY HOMHHaJIbHbIX YTBeplK,neHMM «x (i)
paBHo S». B 06IneM c.TIyqae pa30MeHHM co CTPYKTYPOH (R, x)
KaQeCTBeHHa51 ,nHcnepCM5I paBHa cpe,nHeM olliHoKe YTBep)K,ne-
HHH BM,na «x (i) CBSl3aHO C x (j) no CTpYKType x».
OCTaeTC5I 3aMeTHTb, qTO paCCT05lHMe Mem,ny HeynopH,l{OqeH-
HbIM pa36neHHeM if H ynop5l)lOlleHHbI1\1 pa30HeHHeM R' BbIpa-
1KaeTC5I CJIe,nYIOIneM cpopMYJIoii:
d (R, R') =+ ( N t'-2 N t +N 2 ) ,
t s, t
(1.27)
54
I
--
J3blTeKaIOmeH 113 (1.18). KaK BH HO, paCCT05IHHe (1.27) BO-
06 e He 3aBHCH'f OT ynOp5l OlleHHOCTH 3HaqeHHH npM3HaKa R' .
PaCCMOTpHM Tenepb CTaTHCTHlIeCKYIO HHTepnpeTaQHIO pac-
CTOSlHH5I Me)K,l{Y OTHOlIleHH5IMH HenOCpe,l{CTBeHHO B TepMHHax
OTHOlIleHHH. BMecTo (1.13) 6Y,l{eM paccMaTpHBaTb Bep05lTHOCT-
HbIR aHaJIOr paccToSIHlHI () (R. R') = 2 d (R. R') - BepoSIT-
HOCTb TOfO, qTO npoH3BOnbHa51 ynOp5l,l{OlleHHa51 napa 06'beK-
u
TOB npHHa,l{ne:>KHT O,lJ.HOMY H TOJIbKO O,l{HOMY H3 OTHOlIIeHHH p
,
H p.
Cq;OPMYJIHPyeM 31'0 B 60nee HafJI5I,l{HbIX TepMHHax. BYAeM
rOBopHTb, liTO npH3HaKH R H R' cozIlaco8aHbl Ha 06'beKTaX i H j,
eCJIH rij == rEi H 'ji == 'Ii. B npOTHBHOM CJIYQ8e R H R' Ha-
3bIB8lOTC5I He COrJIaCOBaHHblMH H8 OO'beKT8X i H j. HOMH-
- -
HanbHbIe npH3HaKH R H R' He cornaCOBaHbI Tor,l{a H TOJIbKO
TOf,l{a, KOf,l{a O,l{HH H3 HHX HMeeT O,l{HHaKOBbIe 3HatIeHH5I Ha
060HX 06'beKTaX, a BTOpOH - pa3HbIe. 3HatIHT, cS (R, R')
eCTb Bep05lTHOCTb HecornaCOBaHHOCTH npH3HaKoB R H R'.
STY Bep05lTHOCTb HeTpY,l{HO BbIpa3HTb B TepMHHax Ta6JIH[(hI CO-
np5l:>KeHHOCTH C nOMO blO cpopMynbI, OTnHlIHOH OT BbIlIlepac-
CMOTpeHHbIX.
,lXn51 HOMHHaJIbHbIX R H R', ecnH i E Rs n Rt (c Bep05lT-
HOCTblO Pst), TO j ,l{On}l{eH nonaCTb B Rs-Rt (c Bep05lTHOCTblO
Ps. - Pst) nH60 B Ri - Rs (c Bep05lTHOCTblO p.t - Pst).
3HallMT, Bep05lTHOCTb HecornaCOBaHHOCTH R H R' paBHa
6 (R, R') = P 8t (Ps. + p. t -2pst),
s. t
( 1.28)
TaK liTO () (R, R') MomHO paCCMaTpHBaTb KaK cpe,nHee 3H8qe-
HHe CJIYlIaHHOH BenHI:{HHbI, npHHHMalOIneH 3HaqeHHe Pa. +
+ p.t - 2p8t C Bep05lTHOCTblO Pst.
ECJIH npH3HaKH R H R' paHfOBbIe, TO OHH He COrJIaCOBaHbI
Ha i H j Tor,l{a H TonbKO TOf,l{a, Kor.na i xy:me, qeM j, no O,lJ.HOMY
H3 HHX, a no BTOPOMY - He xyme. EcnH i E Rs n R; (c Be-
p05lTHOCTblO Pst), TO npH HeCOfJIaCOBaHHOCTH 06'beKT j ,non:>KeH
npHHa,l{ne>KaTb KaKOMY-JIH60 H3 R p n R , r,ne P S H q > t
MJIH P > S H q t. Cne,l{OBaTenbHO,
(R, R') = Pst ( Ppq + ppq ) . (1.29)
s. t . p s q>t p>s q t
.l{JIH paHroBhlx npH3HaKOB YMeCTHO rOBopHTb T8K)Ke 0
C021lQC08aHflOcmu npU3HaK08 8 cmp020M CMbICJle: OHa HMeeT Me-
CTO, eCJIH O,l{HH 06'beKT xyme ,l{pyrofo no 060HM npH3H8KaM
O,l{HOBpeMeHHO. 5Y,l{eM rOBopHTb TaK:>Ke, liTO npH3HaKH cmpo20
55
tir ' '" .. ,
:::1.7 ...... ,, j......,.......r...,.i -.. '. "". _"
He C021laCOeaHbl, eCJIH i no O HOMY npH3H3KY CTporo JIyqrne, qe1\1
j, 3 no BTOPOMY - CTporo xy*e. HeTPYJ(HO BHJ(eTb, liTO Be-
pOSlTHOCTb CTporOH HeCOrJI3COB3HHOCTH p3HrOBbIX npH3H3KOB
COBn3,l{3eT C () (R * R' , R' * R), KOTOp3H p3BH3 B CHJIY
( 1 .24)
6 (R*R', R'*R) = 2 Pst ( ppq ) . (1.30)
s. t p>s q<t
<I>OPMYJI3 (1.21) nOK33bIB3eT TOr,l{3, liTO BepOHTHOCTb Heco-
rJI3COB3HHOCTH p3HrOBbIX npH3H3KOB p3BH3 CYMMe BepOHT-
HOCTeH HX CTporOH HeCOrJI3COB3HHOCTH H nOJIOBHHbI BepOHT-
HOCTH HeCOrJI3COB3HHOCTH COOTBeTCTBYIO HX p330HeHHiL
OlleBH,l{HO, npH3H3KH CTporo COrJI3COB3HbI TOr,l{3 H TOJIbKO
TOr,l{3, KOr,l{3 CTporo He COrJI3COB3HbI npH3H3KH R- H R', r,l{e
R- - 3TO npH3H3K R C OOp3THbIM nopH,l{KOM 3H311eHHH. CJIe-
,l{OB3TeJIbHO, BepOHTHOCTb cTporoH COrJI3COB3HHOCTH p3HrOBbIX
npH3H3KOB
fJ (R-*R',R'*R-)=2 P8t ( PP9 ) . (1.31)
s, t 1><.s q<t
<I>OPMYJIbI (1.28)-(1.31) ,l{31OT Cpe,l{HHe 3H311eHHH HeKO-
u u
TOpbIX CJIYtI3HHbIX BeJIH1JHH; CTaTHCTHtIeCKHH 3H3JIH3 HX npo-
Be,l{eH JI. H. nOJIH YKOM [79], KOTOpbIH nOK333JI, B 1I3CTHOCTH,
liTO p3CcToSlHHe (1.18) ,l{3eT HeCMe eHHYIO . COCTOHTeJIbHYIO
u
3CHMnTOTHlIeCKH-HOPM3JIbHYIO OneHKY ,l{JIH cooTBeTcTBYIo eH
MO,l{eJIbHOH BeJIHtIHHbI (CM. T3K)Ke [66]).
P3CcToHHHe B MHO:>KeCTBe p3H}KHpOB3HHH p3CCM3TpHB3JIOCb
B [44, 41, 118, 163], 3 B MHO:>KeCTBe p330HeHHH - B [159,72,
158] (He33BHcHMO ,l{Pyr OT ,l{Pyr3). AH3JIH3 cOBoKynHoro npo-
CTp3HCTB3 OTHOllleHHH npOBO,l{HJICH 3BTOpOM COBMeCTHO C
JI. D. 1JepHbIM (CM., B 1I3CTHOCTH, [66]).
.
r lla8a 2
AHAnH3PAcnPEAEnEHHR
KAIJECTBEHHbIX nPH3HAKOB
2.1. JIOfJIHHEAHbIE MOAEJIH
2.1.1. 061JRCHRIOU{ue t/JaKTopbl
B .L{aHHOH rJIaBe 0Y.L{YT paccMoTpeHbI HeKoTopbIe
MeTO.L{bI KOHCTpYHpoBaHH5I H onHCaHH5I B TepMHHax TaOJIHu.
cOnp5l)KeHHOCTH npH3HaKoB, T. e. MHoroMepHbIx .L{HcKpeTHbIx
pacnpe,l{eJIeHHiL HecMoTp51 Ha TO qTO 3Ta TepMHHOJIOrH5I HaH-
oOJIee CTaTHCTHqeCKa5l, aHaJIH3 TaOJIHn cOnp5lJKeHHOCTH, Ha-
qaTbIH K. nHpCOHOM [156] H .a*. IOJIOM [144], ,l{OJIroe BpeM5I
OCTaBaJIC5I BHe OCHOBHbIX HanpaBJIeHHH pa3BHTH5I MaTeMaTHKO-
CTaTHCTHqeCKHX MeTO.L{OB, KaK, BnpOqeM, M aHaJIH3 K qeCT-
BeHHbIX npH3HaKoB BOOoIne.
B ,l{OKJIa,l{e M. KeH,l{aJIJIa 0 oY.L{y eM CTaTHCTHKH [146],
npOqHTaHHOM B KOpOJIeBCKOM CTaTHCTHqeCKOM oo ecTBe AH-
r JIMH B 1942 r., npOOJIeMaTHKa aHaJIH3a Ta6JIHn COnpSl)KeHHO-
CTH .L{a)Ke He ynoMHHaJIaCb. nOJIOmeHHe H3MeHHJIOCb B 60-x
rO,l{ax; cnYCT5I 25 JIeT B aHaJIOrHqHOM BbIcTynJIeHHH [147]
M. KeH,l{aJIJI Ha3BaJI aHaJIH3 Ta6JIHn cOnp5l)KeHHOCTH O,l{HOH H3
rJIaBHbIX TOqeK pocTa CTaTHCTHqeCKOH HaYKM. B nOCJIe,l{Hee Bpe-
M5I ony6JIHKOBaH p5l,l{ MOHorpacpHH, paccMaTpHBalO Hx MO-
.L{eJIH aHaJIH3a MHoroMepHbIx ,l{HCKpeTHbIx pacnpe,l{eJIeHHH
(CM., HanpHMep, [134, 154]).
Pa3pa6oTaHHbIe MeTO.L{bI B TOM HJIH HHOM CTeneHH CBSl3a-
HbI C TaK Ha3bIBaeMhIMH JIOrJIHHeHHbIMH MO.L{eJI5IMH MHorOMep-
HbIX pacnpe,l{eJIeHHH, KOTopbIe OCHOBaHbI Ha npe.L{CTaBJIeHHH
pacnpe,l{eJIeHHH CHCTeMaMH ({BKJIa,l{OB», ,l{aBaeMbIx TeMH HJIH
HHbIMH MHO)KeCTBaMH npH3HaKoB. I1cnoJIb30BaHHe 3THX MO,l{e-
JIeH .L{JISI nOCTaHOBKH H perneHH5I 3a.L{aQ KOHCTpYHpoBaHH5I <paK-
TOpOB Ii onHCaHH5I npH3HaKoB MbI npOBO.L{HM B paMKax KOH-
u.ennHH .lI. rY.L{M3H8 [128], KOTopbIe npe.L{CTaBJI5IIOTC5I HaM HaH-
oOJIee eCTeCTBeHHbIMH .L{JISI 3THX neJIeH.
npe)K,l{e Bcero paccMoTpHM npOOJIeMY KOHCTpYHpoBaHH5I
HOMHHaJIbHOrO cpaKTopa, KOTOpbIH ObI HaHJIY1JIllHM oopa30M
COOTBeTCTBOBaJI HCXO,l{HbIM .L{aHHbIM, npHqeM KOHCTpYHpOBa-
1
57
_'t""'" ''''' ,."",,:I:!... b.',_ '
HHe ,l{OJL>KHO npOH3BO,l{HThC5I B TepMHHaX Bep05lTHOCTeH KJIaC-
COB OO'beKTOB, KaK 3TO ,l{HKTyeTc5I BbIOpaHHbIM 513bIKOM.
CJIe yeT OTMeTHTb, qTO nOH5ITHe 0 HaHJIyqllIeM COOTBeTCT-
BHH <t>aKTOpa ,l{8HHbIM ,l{8JIeKO He 0.L{H03HaqHO. Ha nepBbIH B3rJI5I.L{,
MO>KeT nOKa3aTbC5I, qTO e,l{HHCTBeHHbIM KpHTepHeM KaqeCTBa
51BJI5IeTCR CTeneHb COOTBeTCTBH5I 3HaqeHHH CKOHCTpYHpOBaH-
Horo <t>aKTopa 3HaqeHH5IM HCXO,l{HbIX npH3HaKOB, T. e. OIllHOKa
npe,l{CTaBJIeHH5I npH3HaKOB C nOMO bID CKOHCTpYHpOBaHHoro
aKTopa. qeM TOqHee annpOKCHMaUH5I 3HaqeHHH HCXO.L{HWX
npH3HaKOB, TeM JIyqIIle KaqeCTBO cpaKTopHoro npe,l{CTaBJIeHH5I,
TaK qTO npOOJIeMa KOHCTpYHpOBaHH5I nOJIHOCTbID YTOqH5IeTC5I
BbIOOpOM nOKa3aTeJI5I KaqeCTBa annpOKCHManHH. I1MeHHo TaK
B HaCT05lrnee BpeM5I H OOCTOHT ,l{eJIO, KOr,l{a aHaJIH3 ,l{aHHbIX Be-
,l{eTCR B TepMHHax TaOJIHU 00'beKT-npH3HaK HJIH OO'beKT-OO'beKT
(CM. rJIaBbI 3, 4). B OTJIHqHe OT 3TOR KOHnennHH npeOCma8JlJlIO-
U{ux fjJaxmopo8 HCnOJIb30BaHHe 513bIKa pacnpe,l{eJIeHHR eCTe-
CTBeHHbIM oopa30M npHBo,l{HT K KOHnenUHH OO'b5lCH5IID erO aK-
Topa. A HMeHHO <t>aKTop Ha3bIBaeTC5I 067JRCHJlIOU{UM, eCJIH npH
Ka:>K,l{OM ero KOHKpeTHOM 3HaqeHHH HaOJIID,l{eHHbIe HCXO,l{Hhle
npH3HaKH 51BJI5IIDTC5I CTaTHCTHqeCKH He3aBHCHMbIMH. BoooIne
rOBopH, 3TH npH3HaKH, KaK npaBHJIO, B3aHMOCB5I3aHW, KaK,
HanpHMep, BonpOCbI nCHXOJlOrHqeCKOrO TeCTa. Q,l{HaKO C TOq-
KH 3peHH5I OT,l{eJIbHbIX 3HaqeHHH cpaKTopa OHH CTaHOB5ITC5I He-
3aBHCHMbIMH, T. e. yqeT 3HaqeHHH aKTopa ,l{eHCTBHTeJIbHO
00'b5lCH5IeT CB5I3H HaOJIID.L{eHHbIX npH3HaKOB.
nYCTb R = {R 1 , ..., Rm} - pa30HeHHe MHO:>KeCTBa OO'beK-
TOB no OO'b5lCH5IID eMY HOMHHaJIbHOMY <PaKTOPY. L{JI5I TOqHOH
OPMYJIHpOBKH OY,l{eM 0003HaqaTb HH,l{eKCOM a cBepxy Te xa-
paKTepHCTHKH, KOTophle OTHOC5ITC5I K KJIaccy Ra. HanpHMep,
u ( ) I n
Bep05lTHOCTb 3HaqeHHM SI, ..., Sn npH3HaKOB x , ..., x npH
YCJIOBHH, qTO 3HaqeHHe 00'b5lCH5IIDInerO cpaKTOpa eCTb a, 000-
3H8qHM p .. .5n. Tor,l{a no onpe eJIeHHID 067J.flCH.,fllOU{ezo ifJaKmopa
.L{J151 JIIDObIX a, S = (S1, ..., sn)
,
I
,
f
i
P a _ p a _ p a P a P a
5 - 51. . . Sn - 51 52. .. Sn'
(2.1 )
r,l{e p - Bep05lTHOCTb 3HalJeHM5I Sit npH3HaKa Xk ,l{JI5I OO'beK-
TOB 113 KJIaCCa R a.
Y CJIOBBe (2.1) BbIpaLKaeT <paKT cTaTHCTUlJeCKOIl He3aBHCH-
MOCTH npH3HaKOB Xl,..., x n npM KaLK,l{OM 3HaqeHHH <paKTOpa R.
BnepBble OHO CHCTeMaTHqeCKM MCnOJIb30BaJIOCb n. JIa3ap-
C eJIb,l{OM, KOTOpbIH Ha3BaJI ero aKCHOMOH /lOKa/lbflOt'j, He3a-
8UCU.M.OCmu, a nOCTpoeHHe 00'b5lCH5IIDInerO <paKTopa - /la-
meflmHo-cmpYKmypflblM aHa/lU30M [151J.
58
-
PaCCMOTpHM CJIe.L{YIOIlJ,HH npHMep (151). rpynny li3 TbICR-
qH qeJIOBeK onpotHJIH, KaKHe *ypHaJIbI OHM qHTaIOT. OTBeTbI
o qHTaeMOCTH *ypHaJIOB A H B CBe.L{eHbI B CJIe.L{YIOlI.\YIO Ta6-
JlHny conp5DKeHHOCTH:
A
A
B
260
240
500
B
140
360
500
400
600
1000
3.L{eCb oYKBaMH A H B 0003HaqeH nOJlO)KHTeJlbHbIH, a 6YK-
BaMH A Ii B -oTpHnaTeJlbHbIH OTBeTbI Ha Bonpoc 0 tITeHHH
COOTBeTCTBYIO ero )KypHaJla. IlpH3HaKH A H B He 51BJl5IIOTC5I
CTaTHCTHqeCKH He3aBHCHMbIMH, 0,26 = PAB =1= PAPB = 0,2.
3HaqHT, B )KypHaJlaX A H B eCTb qTO-TO oo ee, XOT5I OHH H He
51BJl5IIOTC5I O.L{HHaKOBbIMH C TOqKH 3peHH5I qHTaTeJleH. OO'b5lC-
HHTb HX CB5I3b MO)KeT yqeT TpeTbero npH3HaKa - «ypOBeHb
00pa30BaHH5I» - CO 3HaqeHH5IMH C - «BbICIllHH» H C - «HH3-
llIHH». Pacnpe.L{eJleHHe 3THX Tpex npH3H3KOB H300pa>KeHO
B CJIe.L{YIO eH TaOJlHne:
c
c
B
A A A A
240 160 400 20 80
60 40 100 80 320
100
B
400
300
200
500
100
400
500
t
TIpH <pHKCHpOB3HHbIX 3H3QeHH5IX npH3HaK3 «ypOBeHb
oopa30BaHH5I» npH3HaKH A H B CT3THCTHQeCKH He3aBHCHMbI:
B ooeHX nO.L{T30JlHnaX CTOJIOnbI nponopnHOHaJIbHbI, XOT5I H no-
pa3HoMY. 3HaQHT, npH3HaK C 06'b5lCH5IeT CB5I3b Me>K.L{Y }KYP-
HaJIaMH A H B. OHH ooa paCCQHTaHbI B OCHOBHOM H3 BbICOKO-
o6pa30BaHHbIx QHTaTeJIeH, 80% KOTOpbIX QHTaeT mypHaJl A H
59
(
,
60 % - mYPHaJI B. B TO )Ke BpeM5I Cpe.rUI HM3KOOOpa30BaHHbIX
qMTaTeJIeH TOJIbKO 20% lIHTalOT *ypHaJI A M 20% - B. TIpM
3TOM ,l{JI5I KaJK,l{Oll rpynnbI lIHTaTeJIeH A M B CTaTHCTMlIeCKH
He3aBHCHMbI. 3HallMT, nepBYIO rpynny, COCTaBJI5IIOInYIO 50%
OilporneHHbIX, MomHO oxapaKTepH30BaTb BeKTopoM (80 %,
60 %): Bep05lTHOCTb Toro, 111'0 CJIYllaHHo BblopaHHbIli npe):(cTa-
BHTeJIb qHTaeT mypHaJI A, paBHa 0,8, He3aBMCHMO 01' Toro, lIM-
TaeT JIM OH :tKypHaJI B, Bep05lTHOCTb lITeHMSI KOTOpOrO paBHa 0,6.
AHaJIOrHlIHO BTOpa51 nOJIOBMHa onporneHHblx xapaKTepH3yeT-
CH BeKTopoM (20 %, 20 % ).
B oOIneM cJIyqae cOBeprneHHo HeBep05lTHO, QT06bI B Ka-
lIeCTBe 06'b5lCH5IIOInerO cpaKTopa HCnOJIb30BaJIaCh CTOJIb npOCTa51
M HenOCpe,l{CTBeHHo HaOJIIO.L{aeMa51 xapaKTepHcTMKa, KaK «06-
pa30BaHMe» B npMBe,l{eHHOM npHMepe. TI03TOMY He06xo,l{HMO
OilHcaTb TeXHHKY BbIQHCJIeHM5I 06'b5lCH5IIOInerO cpaKTopa no
3a.L{aHHbIM npH3HaKaM. C STOll neJIblO 0603HallMM lIepe3 Va Be-
p05lTHOCTb TOrO, 111'0 CJIYllallHO BbI6paHHbIll 06'beKT npMMeT
3HalleHMe ex no 06'b5lCH5IIO eMY cpaKTOpy. Tor.L{a, 011 eBM,l{H 0 ,
Bep05lTHOCTb PS h 3HalleHH5I 5h npH3HaKa Xk eCTb cpe,l{HeB3Be-
rneHHa51 BeJIHlIHH P h: PS h == VaP . AHaJIOrHlIHO npe,l{CTaB-
a
JI5IeTC5I Bep05lTHOCTb PSI". Sn Ha60pa (51' ..., 5n) 3HalleHMH npM-
3HaKOB Xl, . . ., X n :
PS""Sn = Va P ",sn' (2.2)
I
06'be,l{HH5I5I (2. I) M (2.2), nOJIYllHM
PSI" .Sn = Va p ... p ·
a 1 n
(2.3)
j,
f
"
I
3,l{eCb CJIeBa CT05lT Ha6JIIO,l{aeMble, a cnpaBa - HeHa6JIIO.L{ae-
MbIe BeJIHlIHHbI, TaK 111'0 (2.3) MO:>KHO paccMaTpMBaTb KaK CHC-
TeMY ypaBHeHHll ,l{JI5I onpe,l{eJIeHH5I HeM3BeCTHbIX va,P ,. ;., P n
no H3BeCTHbIM BeJIMlIHHaM Ps = P SI...Sn.
5Y.L{eM ,l{aJIee paccMaTpHBaTb TOJIbKO ,l{HXOTOMHlIeCKHe
nepeMeHHbIe H <jJaKTopbI. 31'0 ynpOInaeT 0003HalleHH5I. Qepe3
Ph 0603HallaeTC5I Bep05lTHOCTb O,l{HOrO, nOJIO:>KMTeJIbHOrO, OTBe-
Ta Ha Xk H lIepe3 Pi = 1 - Ph - Bep05lTHOCTb ,l{pyroro, 01'-
pMnaTeJIbHOrO, OTBeTa. AHaJIOrHlIHO onpe,l{eJISlIOTC5I BeJIHlIHHbI
PhZ, Pill' Pili M T. ,l{. YpaBHeHH5I (2.3) MO>KHO TpaKToBaTb TOr,l{a
KaK ypaBHeHH5I ,l{JI5I Bep05lTHOCTell nOJIOmHTeJIbHbIX OTBeTOB.
Ho Bep05lTHOCTH CMellIaHHbIX Ha60poB (SI' S2 ,53, ..., sn) MO:>K-
60
J
"
I
'* 1
I
"1
II
"
,
_r_
HO BblqMCJIHTb Ha OCROBe TOJIbKO 3THX BepOSlTHOCTeH. HanpM-
Mep,
PkZ=Ph-Pkh Pklm=Phm-Pklm, Pkl;;"=PkY--Pklm=
= Ph - Phl-- Phm + Pklm.
n03TOMY KOJIH1JeCTBO He3aBMCHMbIX ypaBHeHHH (2.3) paB-
HO qHCJIY BceB03MomHblX Ha60pOB (51' ..., 8m) HOMepOB npM3Ha-
KOB H3 {I, 2, ..., n}. KaK M3BeCTHO, npM <pHKcHpoBaHHoM In
u e m n!
9TO qHCJIO COqeTaHMH fl == m!(n-m)l' a npH npOH3BOJIb-
n
HOM m - cZZ = 2 k (paBeHcTBY m == 0 cooTBeTcTByeT YCJIO-
m=O
BHe va = 1). TaKMM oopa30M, HMeeTC5I Bcero 2 n ypaBHeHHit
a
HeH3BeCTHbIMH SlBJI5IIOTCSI 3HaqeHH5I Va, P , ..., P - Bcero
(n + 1) ,l{JISI Ka)K,l{OrO cx. llpM ,l{HXOTOMHqeCKOM <paKTope nOJIY-
qaeTC5I 2 (n + 1) HeM3BeCTHblX. qMCJIO HeM3BeCTHblX COBna,l{aeT
C qMCJIOM M3BeCTHblX TOJIbKO npH n = 3. npH 3TOM He rapaHTH-
pOBaHO, qTO HeH3BeCTHble BeJIHqHHbI 0Y,l{YT 3aKJIIOqeHbI l\1em y
HYJIeM H e,l{MHHneH, KaK 3TO ,l{HKTyeTcSI HX BepOSlTHOCTHbIM CMbIC-
JIOM.
npH m-3HaqHOM <paKTope qHCJIO nepeMeHHbIX paBHO
m (n + 1). ECJIH qHCJIO nepeMeHHbIX m (n + 1) OOJIbille qMC-
JIa ypaBHeHMH 2 n , TO perneHHe onpe,l{eJISleTC5I HeO.L{H03HaqHO,
OT.L{eJIbHbIe KOMnOHeHTbI MorYT ObITb OTpMnaTeJIbHbIMH MJIH
OqeHb 60JIhIllHMH H, CJIe.L{OBaTeJIbHO, He nO.L{XO,l{SlT KaK BepOSlT-
. HOCTHbIe xapaKTepHCTMKH OO'b5lCHSlIO ero <paKTopa. ,[{JISI CJIY-
qa51 )Ke 2 n m (n + 1) n. .lIa3apc<peJIb.L{OM Hero COTPY,l{HM-
KaMH npe,l{JIO)KeH p5l.L{ OCTPOYMHbIX cnOCOOOB peIlleHHSI, HC-
nOJIb3YIO HX TOJIbKO qaCTh ypaBHeHMH (2.3), .L{OCTaTOqHYIO .L{JISI
3Toro (CM. [151], a TaK)Ke Ha PYCCKOM 513bIKe [66]). 11M npoaHa-
JIH3HpOBaHbI TaK)Ke OOJIee CJIO)KHbIe MO.L{eJIH, HanpMMep paH-
rOBoro OO'b5lCHSlIO ero <paKTopa (MO,l{eJIb JIaTeHTHbIX paCCTOSl-
HHH [151 ]). Ho H 3.L{eCb peIlleHHe, Booo e rOBOp5l, MomeT ObITb
He,AOnycTHMbIM KaK BepOSlTHOCTHoe pacnpe,l{eJIeHHe, npHqeM
pe3YJIbTaT 3aBHCMT OT Toro, KaKMe H3 2 n ypaBHeHMH HCnOJIb-
30BaHbI ,l{JISI ero nOJIyqeHHSI.
.lI. fY,l{M3H npe.L{JIOmHJI .L{PyroH cnocoo nOCTpoeHMSI OO'bSlC-
HSlIO ero <paKTopa, HCXO.L{SI M3 Toro nOJIOmeHM5I, tJTO BOBce
He 005l3aTeJIbHO CTpeMHTbC5I K TOqHOMY COOJIIO,l{eHHIO paBeHCT-
Ba (2.3) 3MnHpHqeCKOrO H TeOpeTHqeCKOrO pacnpe,l{eJIeHHH
[ 129]. 0003HaqHM TeOpeTHqeCKHe Bep05lTHOCTH
Ps = PS1...Sn =}J Va P ... P ; (2.4)
a 1 n
61
HX OTKJIOHeHHe OT 3MnMpHlJeCKHX OObIlJHO H3Mep5leTC5I KBa,npa-
THlJeCKHM nOKa3aTeJIeM
2 ( - " ) 2 '"
X ==N PS- Ps Ips,
s
(2.5)
KOTOphIH aCHMllTOTHqeCKH npH N -+ 00 HMeeT pacnpe,l{eJIeHHe
X 2 npH rHnOTe3e, liTO reHepaJIbHOe pacnpe,l{eJIeHHe P: (BbIOOp-
KOM H3 KOToporo nOJIYlJeHbI HaOJIIO,l{eHHble 3HalJeHH5I Ps) COB-
na,l{aeT C ;;s; qMCJIO CTeneHeH CBOOO,l{bI 3Toro pacnpe,l{eJIeHH5I
paBHO oo eMY KOJIHqeCTBY BceB03MOLKHbIX KOMOHHanHH S =
= (S1' ..., Sn) (B ,l{MXOTOMHqeCKOM CJIyqae 2 n ) MHHYC KOJIHlJeCT-
BO OneHHBaeMbIX napaMeTpOB.
BMeCTO KBa,l{paTHqeCKOrO nOKa3aTeJI5I (2.5) HCnOJIb3yeTC5I
TaK:>Ke nOKa3aTeJIb, oCHoBaHHbIH Ha OTHOIlIeHHH OneHOK MaK-
cHMYMa npaB,l{OllO,l{OOH5I [129]:
h 2 =2N Pslog s .
s Ps
(2.6)
A
npH BbIllOJIHeHHH rHnOTe3bI paBeHCTBa P = Ps BeJIHqHHa
h 2 pacnpe eJIeHa TaK :>Ke, KaK H X 2 . 9TO MOJKHO OO'b5lCHHTb
CJIe,l{YIO HM oopa30M. .QJI5I JIIOOoro X > 0 BbInOJIHeHO log X
1 - X, TaK qTO npH X, y > 0
I
x-y log x-y ,
x y y
i
npHqeM paBeHCTBO ,l{OCTHraeTC5I npH x = y. 3HaqHT, npH
x y
x 1 ( x-y x-y ) x2_y2 _ _ (x-y)2+2y (x-y)
log- - --+ =
y 2 x Y 2xy 2xy
Ilo,l{cTaBJI5I5I 3TO B (2.6), nOJIyqaeM
h 2 2 Ps (Ps-Ps)2+2 s(Ps-Ps) (ps:ps)2 =X 2 ,
s 2ps Ps s Ps
(2.7)
QTO ,l{eMOHcTpHpyeT CB5I3b (2.5) H (2.6). npH HCnOJIb30BaHHH
MeTO,l{a MaKCHMaJIhHOrO npaB,l{Ollo,nOOH5I ,l{JI5I nyaccoHoBCKoro
pacnpe.L{eJIeHH5I nOKa3aTeJIb (2.6) 5IBJI5IeTC5I ,l{a)Ke oOJIee 3,l{eK-
BaTHbIM [129, 130].
62
I
---
B 3THX TepMHHaX 3a.L{aqa CTaBHTC5I CJIe,l{YIO HM OOpa30M:
u a ct
HaHTH TaKHe HeOTpHnaTeJIbHbIe BeJIHqHHbI Vat PS1' ..., Ps ,
n
Y.L{OBJIeTBOp5lIOInHe YCJIOBH5IM:
va=l, p =I,..., p:=I, (2.8)
a 51 1 Sn n
KOTopbIe MHHHMH3HPYIOT KpHTepHH (2.6) (HJIH (2.5)), BbIQHC-
JIeHHbIH B COOTBeTCTBHH C (2.4).
0003HaqHM qepes PS Bep05lTHOCTb O.L{HOBpeMeHHoro Ocy.
ll(eCTBJIeHH5I KOMOHHaUHH 3HaqeHHH 5 = (51' ..., Sn) npH3HaKOB
H a <paKTopa. B CHJIY MO.L{eJlH Ps = VaPs ... p . BepOHT-
HOCTb p (a/s) 3HaqeHH5I <paKTopa a npH YCJIOBHH, qTO npH3Ha-
KH HMelOT 3HaqeHH5I S = (51' ..., Sn), no onpe,l{eJIeHHIO paBHa:
p (a/s) = P a _
Ps
a a
ON Ps ... Ps
\N 1 n
a a
Va PSI .. .psn
a
(2.9)
HCHO, qTO
Va= Psa;
s
(2.10)
VaP = Psa,
s/s1l.
(2.11)
. r.L{e qepe3 S/Sh, 0003HaqeH TOT <paKT, qTO CYMMHpOBaHHe npo-
H3BOAHTCH no BceM KOMOHHaUH5IM s C <pHKCHpOBaHHbIM 3Haqe-
HHeM Sh,.
B CHJIY (2.9) 3TH <POPMYJIbI MO>KHO nepenHcaTb B TepMHHax
u
YCJIOBHOH Bep05lTHOCTH:
va= PsP(a,/s); (2.12)
a
p = Ps P (a/s)/v a ,
h, s/8k
(2.13)
rJ(e slsh, 03HaqaeT KOMOHHanHIO (51' ..., Sn) C cpHKCHpOBaHHbIM
3HaqeHHeM Sh,.
COOTHOIlleHH5I (2.12), (2.13) npHBO)lHT K CJIe,l{YIO el\lY HTe-
paTHBHOMY MeTO.L{Y OneHKH BeJIHqHH Va' p Ha OCHOBe Ha6JIIO-
,l{eHHbIX 3HaqeHHH PS. BbIoepeM HaqaJIbHbIe HeOTpHnaTeJIbHbIe
3HaqeHH5I Va (0), P h, (0) BeJIHqHH Va; H P h, TaK, QT06bI BbIITOJI-
H5IJIHCb YCJIOBH5I (2.8). Tenepb B COOTBeTCTBHH C (2.9) MO>KHO
63
I
-<II
BblqMCJIMTb 3HaqeHM5I P (als); nO,l{CTaBJI5I5I HX B (2.12), (2.13),
nOJIyqHM HOBbIe 3HaqeHH5I Va (1), P k (1). Ha oo eM, (t + 1)-M
mare nponecca BbIqHCJIeHM5I npOHCXO,l{51T no CJIe,l{YIO HM <pop-
MYJIaM:
Va (t + 1) = Ps Pt (als);
s
(2.14 )
P (t -t- 1) = Ps Pt (al s)/v (t),
k s/sk
(2.15)
r,l{e YCJIOBHa51 Bep05lTHOCTb Pt (als) onpe,l{eJI5IeTC5I <POPMYJIOH
Va (t) p (t) . . . p (t)
Pt (als) = 1 n .
Va (t) p (t) ... p (t)
a
( 2.16)
TIpH t 00 3HaqeHM5I P (t), Va (t + 1) CXO,l{51TC5I K pemeHMIO
CMCTeMbI (2.12), (2.13) npH ,l{OCTaTOqHO mMpOKMX npe,l{nOJIO-
LReHH5IX 1 . ECJIM npH KaKOM-JIHOO a Va (t) 0, TO 3TO 3HaqHT,
qTO CJIe,l{yeT Y,l{aJIHTb JIaTeHTHbUi KJIaCC Ra H npOBO,l{MTb BbI-
qMCJIeHM5I C OCTaBlIlHMMC5I KJIaCCaMH. B TOM CJIyqae, KOr,l{a
BeJIHqMHbI Va H p MOrYT ObITb OneHeHbI no MeTO,l{Y MaKCMMY-
Ma npaB,l{OnO,l{OOM5I, OHM Y,l{OBJIeTBOp5lIOT ypaBHeHH5IM (2.12)
M (2.13) [1301. OneHKH (2.14), (2.15) He OOSl3aTeJIbHO CXO,l{SlTCSI
K OneHKaM MaKCHMaJIbHOrO npaB,l{OnO,l{OOH5I, eCJIH BblOOp Ha-
lJaJIbHbIX 3HaqeHHH HeY,l{aqeH. Tor,l{a, KaK OObIqHO npH HTepa-
TMBHbIX npOne,l{ypax, HymHo npoBecTH BbIqHCJIeHH5I HeCKOJIbKO
pa3, HaqHHaSl C pa3JIHqHbIX HaqaJIbHbIX 3HaqeHHH. BbIoHpaeT- ' I
CSI TO perneHMe, KOTopoe ,l{aeT MeHbrnee 3HaqeHHe KpHTepMIO
(2.6) MJIM (2.5). .
,aaJIee MbI nOKameM, KaK 3TH M,l{eH OOOO aIOTC5I ,l{JISI 3a,l{aQ
onHCaHH5I, a TaKme onHCaHH5I H KOHCTpYHpOBaHM5I O HOBpe-
MeHHO Ha OCHOBe npMMeHeHMSI oo eH MO,l{eJIH rY,l{M3Ha, K M3-
u u u
JIO)KeHMIO KOTOpOH MbI CeHQaC M nepeH,l{eM.
1 MeTo,n (2.15), (2.16) HBJIHeTCH peaJIH3au.HeH 06bIQHOrO MeTo,na
peIlIeHHH (BeKTopHoro) ypaBHeHHH F (X) = O. STOT MeTO COCTOHT
B TOM, QTO ypaBHeHHe F (X) = 0 npHBo,nHTCH K BH,ny X = f (X),
nOCJIe tJero paCCMaTpHBaeTCH nOCJle,nOBaTeJlbHOCTb X (t), onpe,neJleH-
HbIX YCJlOBHeM X (t + 1) = f (X (t)). npH IlIHpOKHX npe,nnOJlO)KeHHHX
X (t) X (t 00), rJ.(e F (X) = o.
64
2.1.2. MYAbTunAuKaTu8floe npeacra8AeHue
pacnpeaeAeHuR
BC5IKOMY HenycToMY MHO}KeCTBY npH3HaKoB
V c= {I, 2, ..., n} nOCTaBHM B COOTBeTCTBHe BeJIHlIHHY TV > 0,
xapaKTepH3YIO YIO BKJIa,l{ MHO)KeCTBa V B pacnpe,l{eJIeHHe
Ps === PS1.. .sn (Sit = 1, 0; k = 1, ..., n). 5Y,l{eM CqHTaTb, qTO
,l{JI5I JIlOOOrO S = (8}, ..., Sn)
p 8 = 11 n T ' (2.17)
V v
r,l{e npOH3Be,l{eHHe oepeTcSI no BceM nO,l{MHO>KeCTB8M V, a
Try B 3aBHCHMOCTH OT KOHKpeTHOrO Haoopa 3HaqeHHH Sy =
1
= (Sit), k E V paBHO JIHOO TV, JIHOO v. .QJISI onpe,lteJIeHHO-
't'
CTH npHMeM T V = TV, eCJIH KOJIHqeCTBO nOJlO}KHTeJIbHbIX
3HaqeHHH (e,l{HHHn) B Haoope Sv HMeeT TY }Ke qeTHOCTb, qTO
H oo ee KOJlHlIeCTBO 3HaqeHHH, T. e. I v I - MO HOCTb V.
ECJIH }Ke pa3H H na I V I II KOJIHqeCTBa e,l{HHHn B Sv HeqeTH8,
v 1 TX V V
OYAeM rrOJIaraTb Tsv = V. .YIHbIMH CJlOBaMH, Tsv = 't ,
eCJlH KOJIHlIeCTBO OTpHnaTeJIbHbIX 3HaqeHHH (HYJIeH) B Sv qeT-
HO, H T = I/T v , eCJIH STO KOJIHlIeCTBO HeqeTHO. TaKHM 00-
pa30M, B MO,l{eJIH (2.17) OTpHnaTeJIbHbIe 3HaqeHHH KaK ObI no-
rarualOT ,l{pyr ,l{Pyra cor JIaCHO npaBHJIY «MHHYC H8 MHHYC
. ,l{aeT nJIlOc». B COOTBeTCTBHH C STHM ,lJ.JI5I V = {I, 2, 3, 4}
'I" V -- 'I" V -- 'I" V - 'I" V 'I" V - 'I" V - 1 1 '1" V
1100 -- 0101 -- oooo -- , lOOO -- 1101 -- .
Bcero HMeeTC5I 2 n COOTHOIlleHHH BH,lJ.a (2.17) - no qHCJIY
oHHapHbIx HaoopoB S = (s}, ..., sn). MO)KHO JIH Onpe,l{eJIHTb
3HaqeHHH 1') H TV no 3a,l{aHHbIM Bep05lTHOCT5IM Ps? 1JHCJIO B03-
MO:>KHbIX HenYCTbIX nO,l{MHOmeCTB V {I, ..., n}, KaK H3-
BeCTHO, paBHO 2 n - 1, TaR qTO BMeCTe C 1') nOJIyqaeTCSI pOBHO
2 n HeH3BeCTHbIX. PeIlleHHe cy ecTByeT H onpe,lJ.eJI5IeTCH O,lJ.HO-
3HallHO, eCJIH BCe BeJIHqHHbI Ps nOJIOJKHTeJIbHbI (npH Ps = 0
B (2.17) XOTH ObI O,lJ.HH H3 COMHO}KHTeJIeH 005l3aH paBH5ITbCSI
HYJlIO, qTO npOTHBOpeqHT CMbICJIY BeJIHqHH 'tv H 1')).
.QOKa3aTeJIbCTBO SToro YTBep)K,l{eHHH npOBe,lteM npSlMbIM pe-
rneHHeM ypaBHeHHH (2.17), T. e. BbIp83HB TV qepe3 PSt ,[{JIH
,l].aHHOrO MHO:>KeCTBa npH3HaKOB V paCCMOTpHM Bce B03MO>KHbIe
HaoopbI HX 3HaqeHHH Sv = (Sit), k E V H 0003H8qHM qepe3
V+ MHOjJ{eCTBO Tex sv, B KOTOpbIX <PHrYPHpyeT qeTHOe qHCJIO
OTpHu.aTeJIhHbIX 3HalleHHH Sit, a qepe3 V- - MHO>KeCTBO Tex
3 3SK. 1143
65
r- " I' o""IL'iI'1J. U\.'J'"'' r"lt:.
u
Sv, qHCJIO OTpHnaTeJIbHblX 3HaqeHHH Sit B KOTOpbIX HeqeTHO.
3TO 03HalIaeT, lITO V+ = {sV l'tr = 'tV} H V- = {SV I 't =-{;}.
I V V 'C
,[(JI5I Toro qTOObI 51BHO BbI,l{eJIHTb HH,l{eKCbI npH3HaKOB nO,l{MHO-
)KeCTBa Y B npOH3BOJIbHOM Haoope s = (S1' ..., Sn), OY,l{eM npe,l{-
CTaBJI5ITb ero B BH,l{e s = (SV, S v ), r,l{e sv = (Sk)' k $. Y.
B 3THX 0003HaqeHH5IX CnpaBe,l{JIHBO CJIe,l{YIO ee paBeHCTBO:
n P8 V S V
n SVEV+ 2
T Y =
n P8 8_
s_ S EV- V V
V Y
(2.18)
,[{eHCTBHTeJIbHO, COrJIaCHO (2.17) n Ps s- CO,l{epmHT npo-
svfV+ V Y
H3Be,l{eHHe I Y+ I qJIeHOB TV, a n Ps s- - npOH3Be,l{eHHe
svEV- V V
I V- I qJIeHOB 2...., TaK qTO qaCTHOe ,l{aHHbIX BbIpa)KeHHH CO-
'tv
,l{ep}KHT npOH3Be,l{eHHe I Y+ I + I V- I = 2 1V , qJIeHOB TV,
TaK KaK oo ee tIHCJIO HaoopOB 5y paBHO 2I V I. BHeillHee
npOH3Be,l{eHHe 3TH X qaCTHbIX no BceB03MO)KHbIM S y YMHomaeT
KOJIHqeCTBO tlJIeHOB 'tV eIne Ha 2 n -I Y I no qHCJIY Bcex Sy =
= (Sit), k $. V. Bcero B KBa,l{paTHOH CKOOKe CO,l{ep)KHTCSI
2 n -I V I . 2 1Y I = 2 n qJIeHOB TV, tlTO nOCJIe H3BJIeqeHHH KOpH5I
CTeneHH 2 n KaK pa3 ,l{aeT TV.
nO Ka1KeM, qTO Bce OCTaJIbHbIe COMHO:>KHTeJIH B KBa,l{paTHOH
CKOOKe B (2.18) B3aHMHO cOKpa aIOTcH. PaccMoTpHM npoH3-
BOJIbHOe MHO)KeCTBO W c= {I, ..., n}, W =1= V, H COOTBeTCT-
BYIO He eMY qJIeHbI 'ts . OqeBH,l{HO, BH,l{ 'ts B BbIpa)KeHHH
B KBa,l{paTHbIx cKooKax CPOPMYJIbI (2.18) npH ,l{aHHOM s=
= (51' ..., sn) onpe,l{eJISleTC5I COOTHOllIeHHeM KOJIHqeCTB OTPH-
naTeJIbHbIX 3HaqeHHH npH3HaKoB B Sy H Sw. ECJIH 3TH KOJIH-
qeCTBa HMelOT o,nHHaKOBYIO qeTHOCTb (qeTHbI MJIH HeqeTHbI O,l{HO-
BpeMeHHO), TO B Ps BXO,l{HT COMHO)KMTeJIb 't w . ,aeHCTBHTeJIbHO,
u
eCJIH B Sw KOJIMqeCTBO OTpHnaTeJIbHbIX 3HaqeHHH qeTHO, TO
B P8 COMHO)KHTeJIeM 51BJI5IeTCSI 't W , npHqeM Ps HaXO,l{HTC5I B qHC-
JIHTeJIe, TaK KaK"'"svFE V+. ECJIH }Ke KOJIH1JeCTBO OTpHnaTeJIb-
. .
HbIX 3HaqeHMH B Sw HeqeTHO, TO B P8 BXO,l{HT: COMHO)KHTeJIeM
I/TW, npHqeM P8 HaXO,l{MTC5I B: 3HaMeHaTeJIe, TaK KaK Sy E V-,
TaK qTO B (2.18) OllSlTh BXO,l{MT 't w .
AHaJIOrM1JHO B BbIpameHHe B KBa,l{paTHhIx cKooKax <POPMY-
JIbI (2.18) BXO,l{HT l/'t w , eCJIM KOJIHtleCTBO OTpHnaTeJIbHbIX 3Ha-
66
qeHHH B O,l{HOM H3 Ha60pOB Sy, Sw npH .L{aHHOM S = (51' ..., Sn)
qeTHO, a B .L{pyroM - HeqeTHO.
BC5IKHH Haoop S = (SI' ..., Sn) MO:>KHO npe.L{CTaBHTb B BH-
.L{e S = (Svnw, Swn v ' S w n y , S W n y ). 113 .L{OKa3aHHoro CJIe-
,nyeT, qTO BH,l{ 'ts B COCTaBe Ps B (2.18) COBna,l{aeT C BH,l{OM
1: 5 : B COCTaBe Ps', eCJIH S' OTJIHqaeTC5I OT S 3HaqeHH5IMH npH-
3HaKOB H3 W n V HJIH -W n V. .QeHCTBHTeJIbHO, COBnaAe-
HHe HJIH pa3HHna B qeTHOCTH KOJIHqeCTB OTpHnaTeJIbHbIX 3Ha-
qeHHH B Sv H Sw onpe,l{eJI5IeTC5I TOJIbKO nO,l{HaoopaMH swn v
H S w nv.
TIpH 3TOM H3MeHeHHe 3HaqeHH5I O,l{Horo npH3HaKa C HH.L{eK-
caMH H3 V n W HJIH V n W «nepeBOpaqHBaeT» 'TS; B (2.18):
nepeBo,l{HT COMHO)KHTeJIb 't W B 1/'t w H oopaTHo. 3acpHKCHPY-
eM KaKOH-JIHOO H3 npH3HaKOB k E (W n \7) u ( W n V) (3TO
MHomeCTBO HenYCTO, TaK KaK V=/= W) H nOCTaBHM B COOTBeTCT-
BHe BC5IKOMY S = (SI' ..., Sit, ..., Sn) Haoop S' = (S1' ..., Sk, ...,
..., Sn), OTJIHqaIOInHHC5I OT S TOJIbKO 3HaqeHHeM Sit. OqeBH,l{HO,
3TO COOTBeTCTBHe B3aHMHO-0.L{H03HaqHO, npHqeM B (2.18) npH
S H S' BXO.L{51T B3aHMHO COKpaInaeMble 't W H 1/'t w . nOCKOJIbKY
3TO BepHo .L{JI5I JIlOooro S, cpopMYJIa (2.18) ,l{OKa3aHa.
<PopMYJIa (2.18) nOKa3bIBaeT, KaK xapaKTepH3yeTC5I BKJIa,l{
'tv MHO:>KeCTBa V s;; {I, ..., n} B pacnpe,l{eJIeHHe Bep05lTHOCTeH
Ps (s E S). OTHoIlleHHe
(8 V ) =
n P8 V Sv
Sy EY+
n Psv Sv
syEV
1
2t V I
(2.19)
Bhlpa:>KaeT B3aHMOCB5I3b npH3HaKOB MHO)KeCTBa V npH <pHKCH-
pOBaHHbIX Ha ypOBHe 5 V 3HaqeHH5IX OCTaJIbHbIX npH3HaKOB,
TaK qTO (2.18) npe.L{CTaBJI5IeT C060H reOMeTpHqeCKOe Cpe,l{Hee
3THX BeJIHqHH:
'tV = [ (sv) r !2 n -/V1 ·
TaK, npH n == 3
(2.20)
1
) - [ P18 2 S 3 J -;- _
':, (8 2 , 8 3 --
P082 83
(2.21 )
3*
67
- .......
_ : {. <: ij.:.. .... »J. .u .. l
."'..r _ ""''''- _.=- ---w- ..... ..t1...'..x..{ ';;,.t-1: itL .."""'Ir,.AJ.'h ,.. .... JlIAJIIIIIII, ,> J
u u
KOpeHb KBa.L{paTHbIH OTHOCHTeJIbHOH Bep05lTHOCTH nepBOrO 3Ha-
qeHHH npH3HaKa Xl no cpaBHeHHID C BepOHTHOCTbID BToporo
3HaqeHHH, a
I
(Sa) = [ PllS 3 P008S ] 4 - (2.22)
PlOSS P01Ss
u
KopeHb qeTBepTOH CTeneHH TaK Ha3bIBaeMOrO nepeKpecmH020
omHOtueHUJl B Ta6JIHne COnpSl)KeHHOCTH nepBoro H BToporo npH-
3HaKOB npH <pHKCHpOBaHHOM 3HalleHHH TpeTbero npH3HaKa,
t
= [ PIll PIOO POlO POOl ] 8 - (2.23)
Pooo POll PIOI PliO
u
KopeHb BOCbMOH CTeneHH nepeKpeCTHoro OTHorneHHH B Tpex-
MepHOM pacnpe.L{eJIeHHH BceX Tpex npH3HaKOB.
TaKHM 06pa30M, 'tV - 3TO OneHKa CBH3H npH3HaKOB MHO-
}KeCTBa V. PaBeHCTBO 'tV = 1, 03HallaID ee, liTO V He .L{aeT
BKJIa,na B pacnpe,l{eJIeHHe P8' CBH,l{eTeJIbCTByeT 06 OTCYTCTBHH
CBH3H Me}K,l{Y npH3HaKaMH MHO}KeCTBa VB neJIOM, XOT5I CBH3H
Me}K)l.Y npH3HaKaMH nO,l{MHO}KeCTB W s;; V B03MO}KHbI Ha
CKOJIb yrO,l{HO CHJIbHOM ypOBHe.
nYCTb V - KaKOe-JIH60 HenYCToe nO,l{MHO>KeCTBO MHO}Ke-
CTBa {I, 2, ..., n} II k E V. BbIpa>KeHHe (2.19) MO}KHO nepenH-
caTb B TepMHHax k II W = V - {k} CJIe,l{YID HM 06pa30M:
t
(sv) = ( (S W l)/ (s w o) 2 ,
f,l{e I, 0 - 3HaqeHHH She ,[(eHCTBHTeJIbHO,
n PSWl8 W
S EW+
( SW I):::; W
n PsW 18 W
s eW-
W
(2.24)
(2.25)
.
n P8 W 08 W
S EW+
(S W ' 0) = w (2.26)
n P8 W 0SW
'w EW -
MHO}KeCTBa Sw H Sv HMelOT B (2.25) O,l{HHaKOBYID lIeTHOCTb
KOJIllqeCTB OTpHnaTeJIbHbIX 3HalleHHH npH3HaKoB, a B (2.26)-
pa3JIHlIHYID, TaK KaK B (2.25) Sh = 1, a B (2.26) Sh = 0, npH-
lIeM Sw H Sv OTJIHlIaIDTC5I KaK pa3 Ha Sh. CJIe,l{OBaTeJIbHO,
68
J
'.
,I
f
)
.,,111 II III IUUWU .
- I JlIIIIII 11111111 IIIIIIJ 11111111111
- II I
(sw, 1 )/ (sw, 0) 5IBJI5IeTCH CTeneHbK> 21 W I npoH3Be,neHHH
PSVs y no BCeB03i\10)KHbIM S v E V+, ,neJIeHHbIX Ha npoH3Be,ne-
HHSI Ps 8- no BceB03MO)KHbIM S v E W-, liTO H ,nOKa3bIBaeT
y y
(2.24) .
113 (2.24) CJIe,l{yeT, liTO paBeHCTBO (sV) == 1 03HaQaeT
paBeHcTBo (S y , 1) == (sV' 0), T. e. npH (s v ) == 1 OneH-
Ka CBSl3H ( Sy ' Sk) B MHO)KeCTBe V - {k} He 3aBHCHT OT 3Ha-
qeHH5I Sh (npH JII060M k E V). HHalle rOBOp5l, CBSl3b B MHO-
)l{eCTBe V paBHa e,l{HHHne TOr,l{a H TOJIbKO TOr,l{a, KOr,l{a paBHbI
CB5I3H B MHO)KeCTBe W == V - {k} npH JII060M 3HalleHHH Sh.
TaK, B npHMepe C TpeM5I npH3HaKaMH COOTHOIlleHH5I (2.21)-
(2.23) 3KBHBaJIeHTHbI CJIe,l{YIO HM:
1
-
=( (sJ==1)/ (s3=0) 2 ;
1
(S3) = ( (I, s3)/ (0, S3) 2 ,
TaK liTO
1
.,
= [ (1, 1) (0, 0) ] 4 _
(1, 0) (0, 1)
u
KopeHb lIeTBepToH CTeneHH nepeKpecTHoro OTHOllIeHHSI CB5I-
3eM (S2' S3).
C ,l{pyroii CTOpOHbI, MbI HMeeM
1
,&12 = ( (S3 = 1) (S3 = 0) 2 ,
TaK liTO, YMHO)Ka51 ,&12 Ha == ,&123, nOJIYllHM
,&12,&123 = (S3 = 1),
a ,l{eJISI ,&12 Ha = ,&123, npHXO,l{HM K
t ,;12
(S3 = 0) = .
,;123
BOT KaK CB5I3aHbI Me}K,l{Y C060H nOKa3aTeJIH H T. AHa-
JIOrHlIHbIe COOTHOllleHH5I MomHO nOJIYlIHTb H ,l{JISI o6 ero
CJIYlIaSl.
BY,l{eM r6BopHTb, tlTO npH3HaKH MHO)KeCTBa V He 8saUMO-
CBJlSaHbl 8 l{e.llo.Jt, eCJIH npH JII060M S y (S y ) = 1. B 3TOM
CJIyqae 'ty == 1 (B COOTBeTCTBHH C (2.18), II BKJIa,l{ MHomeCTBa
V B Ps OTCYTcTByeT.
69
I
l
K CO}KaJIeHHIO, oopaTHoe He BepHO: 83 TV = 1 He CJI, e.L{yeT
liTO (s v ) = 1 npH JIIOOOM S v . PaCCMOTpHM, HanpHMep, CJIe-
,l{YlOrnee pacnpe,l{eJIeHHe Tpex npH3HaKOB:
81=1
8 2 = 1 8 2 =0
8 3 = 1 8 3 =0 83=1 8 3 == 0
0,05 0,15 0,15 I 0,05
0,15 0,15 0,15 0,15
.,.
J
;:
:t-
81 == 0
i
B COOTBeTCTBHH C npe,l{bI.L{YInHM
1
1
't 12 = [ (S3 = 1) (S3 = 0)]"2 = [ PIll POOl . PlIO POOO ] 2 = 1.
PlOI POll PIOO POlO
TOr,l{8 KaK nepeKpeCTHbIe OTHOIIleH8S1 He p8BHbI:
1
(s3=1)=-, (s3=O)=3,
3
TaK liTO, XOT5I ,&12 = I, npH3H8KH I, 2 B3aHMOCBSl3aHbI_
PaBeHcTBo 3THX nepeKpecTHblx oTHoIlleHHM rapaHTHpyeT-
C5I, KaK rOBOpHJIOCb BbIIIle, YCJIOBHeM = ,&123 = I, TaK KaK
1
= ( (S3 = I)/ (S3 = 0»2. CJIe,l{OB8TeJIbHO, eCJIH ,&123 = I,
TO ,&12 = 1 paBHOCHJIbHO TOMY, liTO (S3 = I) = (S3 =
= 0) == I, T. e.
PIts P008 = I.
PIOS POlS
(2.27)
PaBeHCTBO nepeKpecTHoro OTHOIIleHHSI ,l{BYX npH3HaKoB
e,l{HHHne CBH,l{eTeJIbCTByeT 0 TOM, liTO 3TH npH3HaKH CTaTHCTH-
qeCKH He3aBHCHMbI (npH BC5IKOM cpHKCHpOBaHHOM 3HalleHHH Tpe-
Tbero npH3HaKa). ,[{eHCTBHTeJIbHO, .L{JISI 3HalleHHH SI = i H
S2 = j npH S3 = k HMeeM no onpe.L{eJIeHHIO
Pi-A P-jh = (Pilh + PiOh) (Pl}h + POih).
70
nYCTb ,l{JISI Onpe,l{eJIeHHOCTH, HanpHMep, i = 1, j = 0,
TOf,l{a, pacKp BaH CKOOKH, nOJIyqHM
Pt.'k p.}", = P'J'k (PI1h. + Ptok + POlk) + Pl1h POOh-
nO,l{CTaBJISlH BMeCTO PllkPOok paBHoe eMY PI01tP61h, nOJIY-
qHM
Pi.k P.jk = Pijh (PII1t. + PIOk + POlk + POOh) = Pijk Polk,
T. e.
Pl.k P .jk _ Pijk
- --
P..h P..k P"h
Ka:>K,l{bIH H3 qJleHOB STOfO paBeHCTBa eCTh YCJlOBHaSl Be-
pOHTHOCTb COOTBeTCTBYIO HX 3HalleHHH 81 H 8 2 npH YCJlOBHH
83 = k. CJle,l{OBaTeJlbHO, ,l{OKa3aHO CJle,l{YIO ee YTBep>K,l{eHHe:
YCJlOBHbIe BepOSlTHOCTH P (81 = i, 8 2 = j I 83 == k) Y)(OBJleT-
BOpRIOT' YCJIOBHIO CTaTHCTHlIeCKOH He3aBHCHMOCTH npH J11000M
3HalleHHH 83 = k. HeTpy,nHO ,l{OKa3aTb H oopaTHoe: He3aBHCH-
MOCTb P (81' 8 2 I 83) = P (81 I 83) P (82 I 83) BJIeqeT paBeHcTBo
(2.27) .
CJIe,l{OBaTeJlbHO, paBeHcTBo 't 12 == 't 123 = 1 CnpaBe,l{JIHBO
TOfAa H TOJlbKO TOf,l{a, KOfAa pacnpeAeJleHHe npH3HaKOB Xl
H x 2 npH nr060M <pHKCHpOBaHHOM 3HaqeHHH x 3 Y,l{OBJIeTBOp5leT
v
YCJlOBHIO CTaTHCTHqeCKOH He3aBHCHMOCTH.
,[{ooaBJl5IR YCJIOBHe paBeHCTBa 't 13 = 1, HeTpYAHo yoe,l{HTb-
C5I, liTO He TOJIbKO ,l{JISI YCJIOBHbIX, HO H ,l{JIH MaprHHaJIbHOfO
pacnpe,l{eJIeHHR npH3HaKOB Xl H x 2 CTaHOBHTCR cnpaBeAJIHBbIM
CBOHCTBO CTaTHCTHqeCKOH He3aBHCHMOCTH. ,[{eHCTBHTenbHO,
't 123 = 't 13 == 1 aHaJIOfHlIHO npeAbIAy eMY BJIelleT paBeHcTBo
Pij. P.jk = Pijk p.;..
Pa3AeJlHB STO paBeHCTBO Ha npeAbIAy ee COOTHOIlleHHe
Pi.k P.jk = Pijk P..k'
nOJIyqHM
Pil. P"h = Pi.k p.}. . (2.28)
npOCYMMHpyeM nOCJIe,l{Hee COOTHOIIIeHHe no k. 9TO ,naCT
Pij. = Pi" p.}.,
TaK KaK P..k = 1, !Pl.k = Pl... nOJIyqeHHOe paBeHCTBO
k k
BbIpa:maeT <t>aKT He3aBHCHMOCTH npH3HaKOB Xl H X 2 . CYMMHPO-
BaHHe (2.28) no j ,l{aCT He3aBHCHMOCTb npH3HaKOB Xl H X3:
Pi" P..k = Pi.k.
71
...."....,.... <;. t< -
TaKHM OOpa30M, MbI ,!{OKa3aJIH, qTO paBeHCTBO ,&123 = ,&12 =
= 1'13 = 1 03HaqaeT, qTO KaK YCJIOBHbIe, TaK H MaprHHaJIbHbIe
pacnpe,!{ JIeHH5I nap -npH3HaKOB (Xl, x 2 ) H (Xl, x 3 ) Y,!{OBJIeTBO-
p5lIOT YC:nOBHIO CTaTHCTHqeCKOH He3aBHCHMOCTH.
PaCCMOTpHM Tenepb cJIyqaii, Kor,!{a e,!{HHHqHbIMH 51BJI5IIOTC5I
Bce B3aHMO.L{eHCTBHH Tpex npH3HaKOB: ,&123 = ,&12 = ,&13 =
= ,&23 = 1. 113 npe,!{bI,!{y ero CJIe,!{yeT, tlTO npH STOM JIIOObIe
,!{BYMepHble pacnpe,!{eJIeHHH (X 1 X 2 ) , (X1X3), (x 2 x3) Y,!{OBJleTBO-
v
p5llOT YCJIOBHIO CTaTHCTHqeCKOH He3aBHCHMOCTH.
Ha caMOM ,!{eJIe MomHO YTBepm,!{aTb OOJIbllIee: npH3HaKH
CTaTHCTHqeCKH He3a5HCHMbI B cOBoKynHocTH, T. e.
PiJh = Pi.. p.J. P..h-
(2.29)
l!eHcTBHTeJIbHO, npM ,&123 = ,&12 = ,&13 y*e BbIBe,!{eHO COOT-
HOllIeHHe
Pi}k p.}. = PiJ. P.Jh.
B CHJIY nOnapHOH CTaTHCTHqeCKOH He3aBHCHMOCTH npH3Ha- i
KOB OTCIO,!{a HMeeM Ii
PiJh p.}. = Pi" p.}. p.}. P..h,
qTO H TpeOOBaJIOCb ,!{OKa3aTb.
OqeBH,!{HO, H3 (2.29) B CHJIY (2.21-2.23) CJIe,!{yeT ,&123 =
= ,&12 = ,&13 = ,&23 = 1. MbI ,lJ.OKa3aJIH, qTO npH3HaKH CTaTH-
CTHqeCKH He3aBHCHMbI B cOBoKynHocTH, T. e. Y,!{OBJIeTBOp5lIOT
paBeHCTBY (2.29) Tor,!{a H TOJIbKO Tor,!{a, KOr,!{a ,&123 = ,&12 =
= ,&13 = ,&23 = 1.
I1HbIMH CJIOBaMH, COOTHOlIleHH5I (2.17) ,!{JI5I He3aBHCHMbIX
npH3HaKOB HMelOT BH.L{
PiJk = rrri '&; '&1 (2.30)
H He BKJIlOqalOT nOKa3aTeJIeH B3aHMOCB5I3H nap H OOJIblIlHX co.
BOKynHocTeH npH3HaKOB. AHaJIOrHqHOe BepHO H ,!{JI5I OOInero
v
CJIYtla51 n-1\1epHbIX pacnpe.L{eJIeHHH.
B 3aKJIIOqeHHe o6Inero onHCaHH5I Mo,!{eJIH (2.17) 3aMeTHM,
qTO B03MOjJ{Ha ee 3anHCb B aMHTHBHOM BH.L{e C nOMOInblO JIora- ,
pH<pMHpoBaHH5I. 0603HaqHM 1t s = log Ps, e = log 1'], As =
= log '&s , r,!{e qepe3 l?g 0603HaqeHa OnepalIJf51 B351TH5I (Ha-
TypaJIbHoro) JIOrapH Ma ,!{3HHOH BeJIHqHHbI. B STHX 0603Ha-
QeHH5IX COOTHOlIleHH5I (2.17) nepenHIlIYTcH cJIe,!{YIO HM SKBH-
BaJIeHTHbIM. 06pa301\1:
1t8=e+ AV ,
v Sv
(2.31 )
72
npHlJeM
A = 0, (k E V),
sk k. V
(2.32)
r,l{e CYMMHpOBaHHe npOH3BO,l{MTCR no 3HalJeHHSlM Sk ,l{JISI npo-
H3BOJIbHOrO k E V.
B qaCTHOCTH, ,l{JI5I TpeXMepHoro pacnpe,l{eJIeHH5I (2.31) HMe-
eT CJIe,l{YIOlI.(HH BH,l{:
:rttJh = e+ At +"" +Al + Aljl +'Afk3 +All + "'hI 3 , (2.33)
npHlJeM
'A + A = 0, A + A = 'A;5 + A:: = 0, (k =1= 1 = 1, 2, 3);
'1 123 + '1 123 _ '1 t 23 + '1 123 _ '1 123 + 'l t 23 - 0
"'Ojk "'ljk -"'iOk "'Ilk - "'liO "'ijl - ·
HeTpy,nHO nepenHcaTb BbIpa}KeHH5I (2.18) H (2.19) ;l(JIH
xapaKTepHcTHK CB5I3H nO,l{MHO)KeCTB npH3HaKoB B TepMHHax A,
npHqeM cooTBeTcTBYIO He BeJIHqHHbI H3 Cpe,l{HHX reOMeTpHlJe-
CKHX npeBpaT5ITCR B cpe,l{HHe apHcpMeTHqeCKHe. BeJIHqHHa A V
xapaKTepH3yeT a,lJJ{HTHBHbIH BKJIa,l{ MHO}KeCTBa V B pacnpe-
,l{eJIeHHe; paBeHcTBY 'tv = 1 cooTBeTcTByeT paBeHcTBo ",v = 06
HanpHMep, npH3HaKH CTaTHCTHqeCKH He3aBHCHMbI B cOBoKyn-
HOCTH TOr,l{a H TOJIbKO Tor,l{a, Kor,l{a
1tijk = log Pijh = e+ Al +Al + AI,
T. e. 1tijk eCTb CYMMa BKJIa,l{OB, ,l{aBaeMbIX Ka}K,l{bIM npH3Ha-
KOM B OT,l{eJIbHOCTH. MO,l{eJIb BH,l{a (2.31), (2.33) ,l{JIR JIorapH<p-
u v v
MOB pacnpe,l{eJIeHHH npHHSlTO Ha3bIBaTb Jl02JlUneunou..
JIorJIHHeHHbIe MO,l{eJIH oe3 YCJIOBHH (2.32) Ha napaMeTpbI
paccMaTpHBaJI P. nJI3KeTT [154]. OHH Y,l{OOHee,l{JI5I xapaKTepH-
CTHKH nOJIHTOMHqeCKHX npH3HaKoB, O,l{HaKO He BbIpa)KalOTCR
HenOCpe,l{CTBeHHO B TepMHHax BKJIa,nOB 'tV MHO}KeCTB npH3Ha-
KOB B pacnpe,l{eJIeHHe. BnpOqeM, YCJIOBH5I (2.32) HMelOT CMbICJI
BbIoopa HaqaJIa OTCtleTa ,nJIH napaMeTpoB 'A, TaK qTO JIHHeHHbIM
npeoopa30BaHHeM COOTHOlIleHHH (2.31) MO)KHO Bcer,l{a ,nOOHTb-
C5I BbInOJIHeHHH (2.32).
2; 1.3. J( epapxut/,ecKue MOaeAu U ol{eHKa
. ux napaMerpo8
MO,l{eJIb (2.17) (HJIH (2.31» nOJIHOCTblO xapaKTe-
pH3yeT pacnpe,l{eJIeHHe (npH Ps > 0) - B TOM CMbICJIe, tlTO ee
napaMeTpbI 'tV (HJIH A V ) O,l{H03HaqHO onpe,l{eJI5IIOTCH no pac-
npe)I.eJIeHHIO. llepeBO,l{ pacnpe,l{eJIeHHSI Ha Sl3bIK BeJIHqHH 'tv
Y,l{ooeH B TOM OTHOlIleHHH, qTO n03BOJI5IeT B e,l{HHooopa3HbIX
I
I
I
L
, ......' ;">l,,,,,,"
73'
-..
TepMHHaX paBeHCTB 'tV = 1 (HJIH A,v = 0) Bblpa>KaTb pa3JIHQ-
HbI CBOHCTBa pacnpe.L{eJIeHH5I. TaK, B npe bI.L{y eM pa3.lleJIe
MbI yoe.L{HJIHCb, liTO He3aBHCHMOCTb TpeX npH3HaKOB B cOBoKyn-
HOCTH 3KBHBaJIeHTHa YCJIOBHIO paBeHCTBa 'tV = 1 .L{JI5I Bcex V,
CO.L{ep1K8II.(HX oOJIee O.L{HOrO 3JIeMeHTa. Y CJIOBHe 't 123 = 'tl = 1
BbIpa:>K8eT cpaKT He3aBHCHMOCTH npH3H8KOB Xl H X 2 npH Ka>K-
.L{OM <pHKCHpOB8HHOM 3HaqeHHH X 3 , T. e. X3 SlBJI5IeTCH OO'bHCHH-
IO HM <paKTOpOM .L{JI5I Xl H X2. Booo e, Ha 513bIKe 'tV Y)(OOHO
CPOpMYJIHpOB8Tb rHnOTe3bI 0 TOM, liTO HeKOTOpbIe MHO}KeCTBa
npH3H8KOB V He .L{alOT BKJIa)(a B pacnpe)(eJIeHHe.
PaCCMOTpHM CJIe.L{YIOInHH npHMep. AMepHKaHCKHe COnHOJIO-
rH H3YQ8JIH <pOpMHpOBaHHe B3aHMOOTHorneHHH Me:>K.L{Y pa3HbI-
MH 3THHQeCKHMH rpynnaMH B YCJIOBHHX OOJIblIlOrO ropO.L{a.
OHH npe.L{nOJIO:>KHJIH, qTO CTeneHb .L{oopO:>KeJIaTeJIbHOCTH Me)K.L{Y
rpynnaMH onpe)(eJISleTC5I CTeneHblO (qaCTOToii) B3aHMHbIX KOH-
TaKTOB, 8 TaK:>Ke CO HaJIbHbIMH HopMaMH B rpynnax. 3TH .L{Ba
<paKTOp8 He 51BJISUOTC5I He3aBHCHMbIMH: OJI8rOnpH5ITHbIe HOpMbI
YBeJIHIIHBalOT tlaCTOTY KOHTaKTOB (H CTeneHb )(OOpO:>KeJI8TeJIb-
HOCTH), a qaCTbIe KOHTaKTbI cnoco6CTBYlOT H3MeHeHHIO BHYTpeH-
HHX HOpM nOBe.L{eHH5I B rpynnax. B CBOIO Ollepe.L{b, OOJIee qaCThIX
KOHTaKTOB H OJIarOnpH5ITHbIX HOpM MO:>KHO .L{OOHTbC5I, YBeJIH-
tlHB85I CTeneHb HHTerpHpOB8HHOCTH npO:>KHBaHH5I HH)(HBH.L{YY-
MOB p83HbIX rpynn. T8KHM OOp830M, O>KH.L{aJIOCb nOnO:>KHTeJIb-
Hoe BJIH5IHHe OJIH30CTH npO}KHB8HHH H8 B38HMOOTHOrneHH5I
Me1K Y rpynnaMH tIepes YBeJIHlleHHe qaCTOTbI KOHTaKTOB If H3- I
MeHeHHe rpynnoBMx HOpM B CTOPOHY 6naronpHHTCTBOBaHHH
[1691. ECJIH 0603HaqHTb COOTBeTCTBYIOInHe npH3HaKH: B
(OJIH30CTb npO}KHBaHH5I C ceMbeH .npyroii 3THHqeCKOH rpyn-
nbI); K (qaCTOTa KOHTaKTOB C tlJIeH8MH .L{pyroii 3THHllecKoH
rpynnbI); H (OJI8rOnpHHTcTBoBaHHe rpynnoBbIx HOpM); lJ (cTe-
neHb IIYBCTB8 ,noopO:>KenaTeJIbHOCTH K npe.L{CTaBHTeJIHM ApyroH
3THHllecKoH rpynnbI), TO rHnOTeTHtleCKHe CB5I3H Me)K.L{Y HHMH
MO:>KHO H300pa3HTb B BHJ\e rp a <p 8 (CM. pHC. 2.1). B Tep- i:
u
MHH8X p8Cnpe.L{eJIeHHH STY cxeMY MO:>KHO BbIp8SHTb CJle,ll.YIO-
rnHM oopa30M: tleTbIpeXMepHc>e pacnpe)(eJIeHHe (B, K, H, lJ)
MO:>KeT ObITb OO'b5lCHeHO .TOJIbKO napHbIMH pacnpeJ{eJIeHH5IMH
npH3H8KOB, COOTBeTCTBYIO HMH pHC. 2.1, T. e. oe3 pacnpe)(e-
neHH5I (B, lJ). Ha 513bIKe BeJIHqH{I 'tv 3TO 03Ha1l8eT, tlTO B <pop-
MYJIe (2.17) BeJIHqHHbl 'tv C V, cOJ{ep:>Ka HMH oonee .L{BYX
HHJ{eKCOB, 8 TaK)Ke C V = {B, lJ} paBHbI e,ll.HHHne: p8cnpe-
eJIeHHe Ps npe,ll.CTaBHMO BKJIaJ{aMH TOJIbKO OTJ{eJIbHbIX npH-
SH8KOB H HX nap (HCKJ1lOlIaSl napy {B, If}):
PiJIH = rl'rf 't 't 'tr1: K T H 'tfkH 1: Y 1:Z q . (2.34)
74
PHC. 2.1
TaKYIO yceqeHHYIO MO,l{eJIb 6y,neM H83bIBaTb q,acmHou B QT-
JIHqHe OT nO/lflOU MO,l{eJIH (2.17).
LlaHHbIe COl.(HOJIOrHQeCKOrO 06CJIeAOB aHHH [169]. 0 pacnpe,ne..
JIeHHH ,l{HXOTOMH3HpOBaHHblX nepeMeHHbIX B, K, H, l{ HMelOT
BH,l{, H306pa}KeHHbIH B Ta6JI. 2.1 (nHTHpyeTcH no [130]).
TaoJIHua 2.1
q
B
K
H
+
+ + + 77 32
+ + 30 36
+ + 14 19
+ 15 27
+ + 43 20
+ 36 37
+ 27 36
41 118
B03HHKaeT np06JIeMa: KaK npoBepHTb CnpaBe,l{JIHBOCTb
qaCTHOH MOAeJIH?
B03MO}KHbI ABa nYTH perneHHSI 3TOH np06JIeMbI. nepBbIH
nYTb COCTOHT B npHHSlTHH nOJIHOM MO,l{eJIH (2.17), OneHKe napa-
MeTpOB TV no HHcpopManHM Ta6JI. 2.1 C nOMOInhlO <POPMYJI
(2.18) M BbIHCHeHHH Taro, paBHHIOTC5I JIM e,l{HHHu.e Bce TV
75
r,..., ......
C V = {B, lJ}, a TaK)Ke C V, cO,l{ep)Ka MMH oOJIee ,l{BYX 3JIe-
MeHTOB.
Ha caMOM ,l{eJIe TV, BbIqHCJIeHHbIe no peaJIbHOM HH<popMa-
nHH, MorYT npHH5ITb 3HaqeHHe e,!{HHHna pa3Be qTO CJIyqaHHO,
H 3HaqeHH5I TV, OJIH3KHe K e,!{HHHne, TaK)Ke MOLKHO CqHTaTb
npaKTHqeCKH Hecy ecTBeHHbIMH ,l{JI5I Ps. J1. rY,!{M3H [129]
u u U
npe,l{JIO)KHJI CJIe,!{YID HH CTaTHCTHqeCKHH KpHTepHH ,l{JI5I OneH-
KH cy ecTBeHHocTH OTKJIOHeHHH 3HaqeHH5I TV OT e,l{HHHUbI.
0003HaqHM pacnpe,l{eJIeHHe reHepaJIbHOH COBoKynHocTH qepe3
p:, a Ps paCCMOTpHM KaK CJIyqaMHYID He3aBHCHMYID BbIOOp-
KY BeKTopOB S = (S1' ..., Sn) C Bep05lTHOCT5IMH p:. Tor,l{a Ha-
OJIID,!{eHHbIe qaCTOThI Ps 0Y,l{YT OneHKaMH MaKCHMaJIbHOrO npaB-
,l{OnO,l{OOH5I ,l{JI5I P:. B npe,l{nOJIO>KeHHH, qTO TV = 1 HJIH, qTO
TO )Ke CaMOe, IvV = log TV = 0, - paCnpe,l{eJIeHHe BeJlHqHHbI
IvV/a CTpeMHTC5I npH N 00 K HOpMaJIbHOMY pacnpe,l{eJIeHHID
C HYJIeBbIM MaTeMaTHqeCKHM O)KH,!{aHHeM H e,l{HHHqHOH ,l{HCnep-
CHelL BeJIHqHHa C1 - 3TO cTaH,l{apTHOe OTKJIOHeHHe pacnpe-
,l{eJIeHH5I Iv V , onpe,!{eJI5IeMOe CPOPMYJIOH
0 2 = (f. *) j22n,
(2.35)
r,l{e n - qHCJlO npH3HaKOB, a N s - KOJIHtJeCTBO OO'beKTOB,
HMeID HX KOMoHHanHID 3HalJeHHH npH3HaKoB S = (S1' ..., sn).
B COOTBeTCTBHH C 3THM ,l{JI5I npHH5ITH5I rHnOTe3hI 0 paBeHcTBe
TeOpeTHQeCKOrO 3HaqeHH5I Iv V HYJIID MO)KHO npHoerHYTb K
CTaH,l{apTHoH CTaTHCTHqeCKOH npoue,!{ype [46]. HanoMHHM
Bep05lTHOCTb Toro, qTO HOpMaJIbHO pacnpe,l{eJIeHHa51 BeJIHqHHa
npHHa,l{JIe)KHT HHTepBaJIY (a, b), paBHa <D (b) - <D (a), r,l{e
<D - <PYHKnH5I HOpMaJIbHOrO pacnpe,!{eJIeHH5I (B ,l{aHHOM CJIY-
Qaec napaMeTpaMH (0,1)). B qaCTHOCTH, ,l{JI5I HHTepBaJIa (-1,96;
1, 96) 3Ta Bep05lTHOCTb paBHa 0,95. npH Iv V =1= 0, MbI paCCMa-
TpHBaeM TOT HHrepBaJl (- a, a) (MHHHMaJlbHhIH no a >"0),
KOTOpbIH CO,l{ep)KHT 3HaqeHHe Iv V . E,!{HHH[(a MHHYC Bep05lT-
HOCTb 3Toro HHTepBaJIa ,l{aeT OneHKY Bep05lTHOCTH Toro, qTO
MbI HenpaBHJlbHO npHMeM rHnOTe3Y 0 paBeHCTBe IvV = 0.
PaCCMOTpHM 3HaqeHH5I napaMeTpOB nOJlHOM Mo,!{eJlH (2.17),
OneHeHHbIe Ha OCHOBe TaOJI. 2.2 no <popMYJIaM (2.18).
nOCJIe,!{HMH CTOJIOen TaOJIHnhI paCCqHT8H no npe,!{noc.ne,!{He-
MY npH (] = 0,04385.
¥l3 TaOJIHUbl BH,!{HO, qTO ocooeHHO OJIH3KH K e,!{HHHne
TBKH TBKlJ T EHlJ TKHlJ TEKHl[ TEl[ ( TT T O COOTBeTcTB y eT MO-
, , , , , '"1
,!{eJIH (2.34)), a TaKme T lJ - BKJIa,!{ npH3HaKa, Bblpa)KalOrnero
qYBCTBO ,!{oopO}KeJIaTeJIbHOCTH K ,l{pyrHM 3THHlJeCKHM rpyn-
76
1
I
1
....'
T a 0 JI H U a 2.2
v
'tV
'iv V
'ivV/a
B 0,84 -0,18 -4, 10
K 1,12 0,11 2,51
H 0,90 -0 ) 10 -2,28
q 0,95 -0,05 -1 14
,
EK 1,33 0,28 6,38
EB 1,16 0,15 3,42
Eq 1,03 0,03 0,68
KH 1,16 0,15 3,42
Kq 1,25 0,22 5,24
HY 1,20 0,18 4,10
BKH 1,00 0,00 0,00
BKY 0,97 -0,03 -0,68
EHY 0,99 -0,01 -0,23
KHq 1,05 0,05 1,14
BKHY 1,05 0,05 1,14
naMe TaKHM oopa30M, ,l{aHHbIe TaOJI. 2.2 He npOTHBOpetIaT rH"
nOTe3e (2.34).
PaccMoTpHM Tenepb BTOpOH nYTb BWHCHeHHH cnpaBeAJIH-
BOCTH tlaCTHOH MO,l{eJIH. 9TOT nYTb OCHOBaH Ha HenOCpe,l{cTBeH-
HOM OneHKe napaMeTpoB tlaCTHOH MO,l{eJIH C nOCJIeAYIOw.eH npo..
A
BepKOH COrJIaCHH HaOJIIOAeHHbIX tlaCTOT Ps H tlaCTOT Ps, BbItlHC-
u
JIeHHbIX Ha OCHOBe ,naHHOH MOAeJIH C nOMO blO nOKa3aTeJIH
X 2 BH,l{a (2.5) HJIB (2.6). -
KaK OneHHTb napaMeTpbI tlaCTHOH MOAeJIH? Be,l{b Tenepb
HeJIb351 HCnOJIb30BaTb <POPMYJIbI (2.18), OCHOBaHHwe Ha Ha-
OJIIOAeHHbIx tlaCTOTax - oueHKax MaKCHMaJIbHOrO npaB,lI.OnO-
,l{OOHH Bep05lTHOCTeH reHepaJIbHOH cOBoKynHocTH. TpeooBa-
HHe 'tv = 1 ,l{JI5I V, He BKJIlOlIeHHbIX B MOAeJIb, anpHopHO B He
MOJKeT 3aBHceTb OT KOHKpeTHbIX HaOJIIO,l{eHHbIX tlaCTOT. OueH-
KH MaKCHMaJIbHOrO npaB,l{onO,l{OOHH ,l{JI5I napaMeTpOB tlaCTHOH
MO,l{eJIH -
A n v
p, .= 'r s ,
VeF V
, r,l{e F - MHO>KeCTBO TeX V, tleH BKJIa,l{ yqTeH B tlaCTHOH MO-
,l{eJIH, H eTCH npH YCJlOBHH, liTO TeopeTHlIeCKOe paCnpe,l{eJIe-
HIfe Ps, XOT5I CaMO If He COBna,l{aeT C HaOJIIO,l{eHHbIM Ps, HO ,nJIH
KaJK,l{Oro V E F nopO)K,naeT TaKoe MaprHHaJIbHOe pacnpe,l{eJIe-
""V
HIfe Ps , KOTopoe COBna,naeT C COOTBeTCTBYIO Hl\'{ HaOJIIO,l{eH-
V
HbIM MaprHHaJIbHbIM pacnpe,ne.rreHHeM pry:
(2.36)
77
---
a,..,.... . , ".' \ .. .,.... ... ,
r.......... .i....;f.t ...-, '(I '!I,-'i; "'u i:'\I-" ;,'"", . ......... ...... .....y{ .....1.....!o'r.. 'L L
.....V_ A _ V
Psv- Psvsv-Ps ·
S_ V
V
(2.3i)
CJIe yeT 3aMeTHTb, qTO eCJIH YCJIOBHe (2.37) BbInOJIHeHO
JI5I MHO)KeCTBa npH3HaKoB V, TO OHO TeM oOJIee BbInOJIHeHO
.L{JI5I JIlOooro HenYCToro nO.L{MHO)KeCTBa W £ V, TaK KaK
pW = pv , a CYMMHpoBaHHe paBHbIx BeJIHqHH K HepaB-
sw S v
sv_w
HbIM qHCJIaM npHBeCTH He MomeT. B qaCTHOCTH, ,l{JI5I nOJIHOH
MO,l{eJIH (2.17) YCJIOBH5I (2.37) BblnOJIHeHbI ,l{JI5I JIlOooro V.
CKa3aHHoe npHBO.L{HT K CJIe,l{YIOIneMY onpe,l{eJIeHHIO. Mo-
,l{eJIb (2.36) oY,l{eM Ha3bIBaTb uepapXU'leCKOU, eCJIH BMeCTe co
BC5IKHM V E F cOBoKynHocTb F CO,l{ep)KHT H JIlOooe ero Henyc-
Toe nO,l{MHO)KeCTBO. HepapXHqeCKa51 MO,l{eJIb nOJIHOCTblO xapaK-
TepH3yeTC5I MaKCHMaJIbHbIMH (T. e. He BKJIlOqeHHbIMH HH B Ka-
Koe ,npyroe) MHO)KeCTBaMH cOBoKynHocTH F: BC5IKOe ,l{pyroe
MHO:>KeCTBO H3 P nOJIyqaeTC5I KaK nO,l{MHO)KeCTBO KaKOrO-JIHOO
H3 3THX MaKCHMaJIbHbIX MHO)KeCTB. DOJIee Toro, YCJIOBH5I
(2.37) .L{OCTaTOqHO paccMaTpHBaTb TOJIbKO ,l{JI5I MaKCHMaJIbHbIX
MHO:>KeCTB V E P, TaK KaK ,l{JI5I OCTaJIbHbIX W E F HX BbInOJIHe-
HHe oY,l{eT rapaHTHpoBaHo aBTOMaTHqeCKH. B paccMoTpeHHoM
npHMepe (MO,l{eJIb (2.34» MaKCHMaJIbHbIMH SlBJI5IIOTC5I MHO:>KeCT-
Ba {B, K}, {B, H}, {K, H}, {K, If}, {H, If}.
H3JIO:>KHM Tenepb cJIe,l{YIOInHH HTepaTHBHbIH aJIrOpHTM Bbl-
QHCJIeHH5I OneHOK HaHOOJIbmerO npaB,l{OnO,l{OOH5I ,nJI5I MO.L{eJIH
(2.36) C yqeTOM OrpaHHqeHHH (2.37) [1291. nOCKOJIbKY cnpa-
Be,l{JIHBOCTb (2.37) ,l{JI5I MHOJKeCTBa V BJIelleT cnpaBe,l{JIHBOCTb
(2.37) t ,l{JI5I JIlOooro ero nO,l{MHO)KeCTBa, 3TOT aJIrOpHTM npHMe-
u
HHM TOJIbKO ,l{JI5I HepapXHqeCKHX MO,l{eJIeH.
B aJIrOpHTMe HenoCpe.L{CTBeHHo BbIQHCJI5IIOTC5I 3HaqeHH5I
QHCJIeHHOCTeH N s , TaK qTO Ps = N s / N. Ha HaqaJIbHOM mare
nOJIaraeM Bce KOMoHHanHH S = (S1' ..., sn) paBHOBep05lTHbIMH:
N s (0) = 1.
Ha OOIneM, (t + 1)-M mare aJIrOpHTMa, npOHCXO,l{HT nepe-
ctJ.eT paHee nOJIyqeHHbIX BeJIHqHH N s (t) B HOBbIe BeJlHqHHbI
N s (t + 1) C yqeTOM HeOOXO,l{HMOCTH BbInOJIHeHH5I OrpaHHqe-
HHH (2.37) ,nJI5I MaKCHMaJIbHbIX V E F. 0003HaqHM qepe3 P'
MHO:>KeCTBO MaKCHMaJIbHbIX V E F. Tor,l{a
N V
N,(t+ 1) =N,(t) n Sv (t=O, 1, ...), (2.38)
N V (t )
VeF' Sv
18
I
t
i
I
I
r e, KaK OOblqHO,
N = N 8 8-.
V S_ V V
V
,[{JI5I Mo,neJIH (2.34) 3Ta cpopMYJIa npHMeT CJIe,nYIO HH BH,n:
Niilll(i+l)=Niilll(t) ANij.. ANi 011. 'i' ill. 'i,}ol 'i, okl .
Nij.. (t) Ni.k. (t) N.jk. N.j.z N..kl
(2.39)
r,ne, KaK OObIqHO, TOqKa B HH,neKce 03HaqaeT, qTO paccMaTpH-
BaeTC51 cYMMa BeJIHqHH no ,u.aHHoMY HH,neKcy.
Ilpe,neJIbHbIe 3HaqeHHSI N 8 = lim N 8 (t) paccMaTpHBalOTcSI
t-+oo
B KaqeCTBe TeOpeTHqeCKHX OneHoK. 3HaqeHH5I napaMeTpoB
'tv MOJKHO paCCqHTaTb no HHM C nOMo blO <POPMYJI (2.18).
BnpOqeM, MO)KHO cpOPMYJIY (2.38) nepenHcaTb B TepMHHax 'tv.
BOCnOJIb30BaBlIlHCb COOTHOlIleHH5IMH qaCTHOH Mo,neJIH (2.36), HO
nOJIyqeHHOe TaKHM cnocoooM BbIpaJKeHHe oy,neT CJIHIIlKOM rpo-
M03,nKHM.
npO,neJIaeM COOTBeTCTBYIO He BbIqHCJIeHHH ,Z(JIH paccMa-
TpHBaeMoro npHMepa. PaccMoTpHM MaprHHaJIbHbIe pacnpe-
,neJIeHH5I ,nJI5I MaKCHMaJIbHbIX MHOJKeCTB npH3HaKoB {B, K},
{B, H}, {K, H}, {K, lJ}, {H, lJ}:
H
+
142 108
126 232
q
+
186 125
97 200
K
+
175 75
136 222
H
+
172 139
96 201
+
+
B
B
+
+
K
K
q
+
+ 161 107
H
122 218
79
1 . i!' "t :'" . .....,... 'I / '% clo.l.li. \ . K - . t ..".....
3HalleHH5I N s (t), BbIlIHCJI5IeMbIe no <popMYJIaM (2.39),
CTaOHJIH3HPYIOTCSI K 25-MY llIary Ha ypoBHe, npHBe,l{eHHOM
B TaOJI. 2.3.
T aOJIHna 2.3
q
E
H
K
+
+
-1-
+
+
+
+
+
+
74,80
32,28
14.75
11,30
42,94
35,99
28,51
42,44
34,48
33,45
17,98
30,97
19,79
37,29
34,76
116 , 29
+
+
+
+
KaK H CJIe,l{oBaJIO O}KH,l{aTb, OneHKH napaMeTpoB MO)(eJIH,
npHBe,l{eHHbIe B TaOJI. 2.4, MaJIO OTJIHQaIOTC5I OT OneHOK napa-
MeTpoB nOJIHoH MO,l{eJIH (TaK KaK B nOJIHoH.. MO,l].eJIH 't Il ,l{JISI
.V $. F BeCbMa OJIH3KH K e,nHHHne):
TaoJIHua 2.4
v
"v
A V
B
K
H
g
BK
EH
KH
Kg
BY
0,83
1,13
0,91
0,94
1,35
1 ,18
1,15
1 ,28
1,22
-0,19
0,12
-0,9
-0,6
0,30
0,17
0,14
0,24
0,20
-HenOCpe)(CTBeHHbIH aHaJIH3 TaOJI. 2.3 nOKa3bIBaeT xopornee
A
COOTBeTCTBHe SHalleHHH p, If PS. KOJIHlleCTBeHHO CTeneHb CO-
OTBeTCTBHH BbIpa}KaeTC5I 3HalleHHeM nOKaSaTeJIH (2.6):
1. 2 = P810g 8 ,
S p,
paBHbIM B ,lJ.aHHOM CJIYllae 2,53. npH npHHSlTHH CTaTHCTHtIeCKOH
MOJl.eJIH (BeKTopbl S BbIoHpalOTcH He38BHCHMO ,l{pyr OT ,npyra C
BepoBTHoCTHMH Ps) 3Ta BeJIHIIHHa HMeeT pacnpe,lJ.eJIeHHe X 2
C IIHCJIOM creneHeH CBOOO,l{bI, paBHhIM oOlI1eMY KOJIHlleCTBY CJIa-
80
!
I
t
r
I
I
1
I
II
I
l
t
,
t
4
J
raeMbIX (rneCTHa,l{uaTb) MHHYC KOJIHqeCTBO OrpaHHqeHHH BH.L{a
(2.37), T. e. KOJIMqeCTBO I10,lJ.MHO}KeCTB B F (.L{eC5ITb, BMeCTe C
nycTbIM nOAMH01KeCTBOM t OTBeqalO HM napaMeTpy 1'). CJIe-
,l(OB8TenbHo, B Jl.aHHOM cnyq8e X 2 HMeeT 16 - 10 = 6 CTene-
HeR CBOOO,nbI, H TaOJIHUhI pacnpe,l{eJI.eHH5I X2 nOKa3bIB8IOT, qTO
3HaqeHHe X2 = 2,53 (npH IlleCTH I CTeneHHX CBOOO.L{bI) OTJIHqa-
eTCH OT HYJIH CTaTHCTHqeCKH He3HaQHMO: Bep05ITHOCTb 60Rb-
mero SHaqeHHH paBH8 0,85.
0003HaQHM MO,l].eJIb (2.34) qepe3 HI H pacCMoTpHM TPH
,l{pyrHe qaCTHbIe Mo,neJIH: H2' yqHTbIBalOmylO Bce napHbIe CB5I-
3M (T. e. F 2 = FI U {{B, lJ}}, r,l{e F i - MHO}KeCTBO nO.L{MHO-
}KeCTB npH3HaKoB V, BKJIIOqeHHbIX B Hi); H 3, yqHTbIBalO YIO
TOJIbKO BKJI8,l{b1 OT,lJ.eJIbHbIX npM3HaKOB (F 3 = {{E}, {K},
{H}, {lJ} }), H H4' ,nJI5I KOTOpOH F, co,nep}KHT TOJIbKO nYCTOe
nOJ{MHO)KeCTBO, T. e. Bce TV = 1, a 1'), OqeBH,lJ.HO, paBHo
I/2 n = :6 . KaK CJle.u.yeT H3 paccy*.u. eHHH npe.ZJ.bI.u.ym. ero
nYHK'Ia, B MO,lJ.eJIH H 3 Bce npH3H8KII paCCMaTpHBaIOTC5I KaK
CTaTHC'IHqeCKH He38BHCHMbIe B cOBoKynHoCTH. MO)KHO nO,l{CqH-
T8Tb, qTO nOKa3aTeJIb (2.6) ,nJI5I STHX Mo,neJIeR npHHHMaeT 3Ha-
tleHH5I 'X, (H 2 ) = 2,25, X 2 (H g ) = 178,37, X2 (H 4 ) = 209,43,
npHqeM pa01pe,neJIeHH5I X2 (H 2 ), X2 (H 3)' X2 (H 4 ) HMelOT 16 -11 =
= 5, 16 -I':) = 11 II 16 - 1 = 15 CTeneHeii CBOOO,l{bI COOT-
BeTCTBeHHO.
B COOTBeTC'U3HH C STHM MO,l{eJIH H 3 II H4 npHXO,l{HTC5I OT-
BeprHYTb, KaK Hl corJIaCYIO HeCH C ,l{aHHhIMH, TOr,l{a KaK H 2
eme nyqme corJIaU!eTCf( C pacnpe,l{eJIeHlleM PSt qeM HI' qTO,
BnpOtJ:eM, SICHO H Des OaCtleTOB, TaK KaK F 2 BKJIlOqaeT Fl. Pa3-
HHna X2 (HI) - X2 (H , - 3TO TO KOJIHqeCTBO HeCOOTBeTCTBH5I
MO,neJIII II ,naHHbIX, KO'l')poe yqHTbIBaeTCH BKJIlOtleHHeM 't 5 '1
B MO,neJIb. B CTaTHCTHtll KOH MOJl.eJIH pa3HOCTb X2 (HI) -
- X2 (H 2 ) HMeeT pacnpe,ne, HHe X2 C 6 - 5 = 1 CTeneHblO
CBOOO):{bI, T8K tlTO MO)KHO onehtTb cTaTHCTHqecKYIO 3H8t1HMOCTb
BKJIlOtleHHSI 'fEY B MO,l{eJIb 7151 y,,'IYllIlleHHH OO'b5lCHeHHH
,naHHbIX. BeJIHqHHa 0,28 cTaTHcTb.{ecKH He3HaqHMa npH ypOB-
He nopH,lJ.Ka 60 %, TaK qTO BKJItqeHHe "EY B Mo,neJIb HI
He onp8B,n8HO.
OTMeTHM B 3aKJIIOqeHHe t tJ:TO K JIOrJIh'eHHbIM Mo,neJI5IM OqeHb
OJIH3Ka MO,l{eJIb aHaJIH3a CTPYKTypbI MHt..OMepHoro pacnpe,l{e-
JIeHHH, pa3paOaTbIBaeMaH 10. H. raBpHJl1- 0M Hero COTPY,n-
HHKaMH [19]. B STOR MO,l{eJIH oo ee pacnpeAT{eHHe npe,l{CTaB-
JISleTCH B BH,ne npoH3Be,neHH5I Henocpe,l{CTBeHHo 4cnpe,l{eJIeHHH
MeHbIllero qHCJIa nepeMeHHblx, a He HX BKJIa,l(Oh KaK B JIor-
JIHHeHHOR Mo,neJIH. O,nHaKO pacnpeJl.eJIeHHe pv' 1" nO,l{MHO-
81
LKeCTBY npH3HaKOB V nOJIHOCTblO OnHCblBaeTCSI CHCTeMOH BKJIa-
,ZI-OB 't W HJIH AW no BceM W <= V (CM. n. 2.2.1). llO-BH,lI.HMOMY,
. u
HepapXHQeCKaH JIOfJIHHeHHaSl MO,l{eJIb C MaKCHMaJIbHbIMH MHO-
LKeCTBaMH V E F MOLKeT 6bITb CBe,lI.eHa K MYJIbTHnJIHKaTHBHoH
Mo,neJIH 10. H. raBpHJIbna, Bblpa>KalOII.(eH Ps lIepe3 npOH3Be-
,lI.eHHe Ps (V E F). O)lHaKO TOqHbIM H3yqeHHeM CBH3H 3THX
v
u
Mo,neJIeH flORa HHKTO He 3aHHMaJICSI.
2.1.4. OnucaHue npu3HaKo8 U KaY3allbHble
Mo8eAu
nOCKOJIbKY JIOrJIHHeHHaSl MO,lI.eJIb CO)lep}KHT BCIO
cYl.I.\eCTBeHHYIO HH<popManHIO 0 pacnpe)leJIeHHH (cTeneHb 3TOH
cYl.I.\eCTBeHHocTH OQeHHBaeTCSI nOKa3aTeJIeM (2.6)), ee MO)KHO
HCnOJIb30BaTb )lJIH pemeHHH pa3JIHQH@X 3a aq aHaJIH3a ,lI.aH-
HbIX.
B ,naHHOM pa3,lI.eJIe paccMoTpHM np06JIeMY onHcaHHH HeKO-
Toporo, HanpHMep n-ro, npH3HaKa npH3HaKaMH MHO}KeCTBa
V = {I, ..., n - I}.
0603HaQHM ,nJISI Y)106CTBa S = (SI' ..., sn) lIepe3 S = (v, sn),
r,lI.e v = (Sl' ..., Sn-1)' Tor,na No. = Nul + N vo . Qepe3
Pu 0003HaQHM BepoSlTHoCTb nepBoro 3HalIeHHSI npH3HaKa x n
npH YCJIOBHH, QTO 3HaQeHHSI OCTaJIbHbIX COCTaBJIHIOT KOMoHHa-
QHIO v:
N
P = vt
v N ·
v.
B 3THX TepMHHax onpe)leJIHM Be HQHHY
Pv N V1 _ Pvt
q,,= =---,
I-pv N oo Pvo
(2.40)
KOTopaSl xapaKTepH3yeT, HaCKoJIbKO qa e BCTpeqaeTCSI nepBoe
3HaQeHHe xn, lIeM BTopoe, npH YCJIOBHH, lITO 3HaQeHH5i OCTaJIb-
HbIX npH3HaKOB COCTaBJI51.lOT v. BeJIHQHHY qv Ha30BeM om-
&I &I
HocumellbHou 'laCmomou.
BBe,neM napaMeTpbl yW ,nJISI W c= V KaK KBa)lpaTbI COOT-
BeTCTBYIO HX - napaMeTpOB Mo,neJIH (2.17):
YW=('t Wn )2, (2.41),J
.;
r,ne 't Wn = 't'w' ,nJISI W' = W U {n}, B QaCTHOCTH Y =
= (t n )2. TaKHM oopa30M, yW npe,nCTaBJIHeT COOOH KBa,npaT
BeJIHQHHbI BKJIa,na MHO}KeCTBa W U {n}.
82
113 (2.17) CJIe,nyeT, tITO ,nJIH OTHOCHTeJIbHbIX tIaCTOT CnpaBe,n-
JIHBbI CJIe,nYlOrnHe paBeHCTBa:
qv = n 1' , (2.42)
WcV W
r,ne v = (51' ..., 5n-1)' 5W = (Sk)' k E W - KOMOHHalI.HH
3HatleHHH npH3HaKOB MHO)KeCTB V H W. npH 9TOM 1' w npH-
HHMaeT 3HatIeHHe .,w HJIH l/yW, B 3aBHCHMOCTH OT Toro, HB-
JIHeTCH JIH tIeTHbIM KOJIHtIeCTBO OTpHlI.aTeJIbHbIX 3HatIeHHH
npH3HaKOB B KOMOHHalI.HH W'. ,[{eHCTBHTeJIbHO, B CHJIY (2.17)
Pv1 = 11 n T W , Pvo =: 11 n T W ,
W Sw W Sw
npHtIeM 3HatIeHHH ,nJIH W, He CO,lJ.ep1KarnerO n, B ooeHX
BeJIHtIHHaX o,nHHaKOBbI, a ,nJIH W, Co,nep1KarnerO n - B3aHM-
HO OOpaTHbI. n09TOMY, pa3,neJIHB nepBYIO H3 HHX Ha BTOpylO,
nOJIYtlHM (2.42) , tITO H TpeOOBaJIOCb ,nOKa3aTb.
CooTHollIeHHH (2.42) BbIpa)KalOT OTHOCHTeJIbHbIe tIaCTOTbI
n-ro npH3HaKa Qepe3 napaMeTpbI .,W MYJIbTHnJIHKaTHBHo.
COOTBeTCTBYlOrnaH aMHTHBHaH Mo,neJIb HMeeT BH,n
Xv = ' (2.43)
wcv W
r,lJ.e ,nJIH BCHKoro k E W
= o.
s W
k
(2.44)
3,neCb
"
Xv = log qv, ;V = log y .
,W w
BeJIHtlHHbI w OlI.eHHBaIOT B3aHMOCB5I3b n-ro npH3HaKa
C npH3HaKaMH MH01KeCTBa W H B 3aBHCHMOCTH OT 3HatIeHHH
5w ,nooaBJI5IIOTC5I HJIH BbItlHTaIOTC5I H3 Xv. BeJIHtlHHa BH,na
X = log p
I-p
Ha3bIBaeTC5I /102Um, a HCnOJIb30BaHHe Mo,neJIefi BH,na (2.43) -
/lOZUm-aflallU3. npH H3MeHeHHH P OT HYJIH ,no eAHHHlI.bI JIOrHT
X npHHHMaeT 3HatleHH5I OT MHHYC oeCKOHetIHOCTH .no nJIIOC
oeCKOHetIHOCTH H n09ToMY ocooeHHo y,nooeH ,nJI5I onMcaHHH,
TaK KaK JIlOowe 3HatIeHHH napaMeTpoB He BWBO,lJ.HT orHT
u
3a npe,neJIbI ,nonYCTHMbIX H3MeHeHHH.
83
B CHJIY (2.41) II (2.31)
w = 2 log 't Wn = 2 ",Wn ,
TaK liTO a,ZJJ(HTHBHbIe COOTHOIlleHHH (2.43) .l{JIH JIOfHTOB Xv
U U
JIHIIlb nOCTOHHHbIM MHO}KHTeJIeM OTJIHllalOTCH OT JIOrJIHHeHHOIi
MO.l{eJIH (2.31).
AHaJIOrHtIHO Onpe,l{eJIHIOTCH qaCTHbIe MO,neJIH J{JIH OTHOCH-
TeJIbHbIX tIaCTOT H HX JIOrapH<pMoB. DapaMeTpbI .9THX Mo,neJIeH
MO}l{HO ou.eHHBaTb Ha OCHOBe paBeHcTB (2.41) tIepe3 napaMeTpbI
COOTBeTCTBYlOlllJIX MO){eJIeH BH){a (2.34) ,l{JIH HaOJIIO,neHHbIx tIa-
CTOT. HeTpY){HO TaK}Ke npHcnocooHTb HTepaTHBHbIM aJIrOpHTM,
U U
H3JIOLKeHHbIH B npe,l{bI,nymeM pa3){eJIe, K HenOCpe,l{cTBeHHoH
OI.(eHKe OTHOCHTeJIbHbIX tIaCTOT.
MO){eJIb BH,na (2.43) onHCbIBaeT «nOBe,neHHe» n-ro npH3HaKa
tIepe3 nOBe,l{eHHe OCT JIbHbIX. OHa BnOJIHe aHaJIOrHtIHa Tpa,nH-
I.(HOHHbIM MO,l{eJISIM perpeCCHH n-ro npH3HaKa Qepe3 npH3HaKH
MHO}KeCTBa V. 9Ta aHaJIOrHH CTaHeT COBceM nOJIHOM, eCJIH ne-
peHTH K TepMHHOJIOrHH TaOJIHI.( OO'beKT-npH3HaK.
DYCTb Xk (i) = 1, eCJIH 3HatIeHHe k-ro npH3HaKa ,nJIH
i E m nOJIOLKHTeJIbHO, H x k(i) = - 1, eCJIH OHO oTpHI.(aTeJIb-
HO. Tor,l{a COOTHOllIeHHe (2.43) MO}KHO nepenHcaTb c/Ie,nYIOID-HM
oopa30M:
= + w D x k .
We: V kEW
(2.45)
npH KaLK,nOM BeKTope v = (SI' ..., Sn-l) , eCJIH Sw co-
,nep}l{HT tIeTHOe KOJIHtIeCTBO oTpHuaTeJIbHbIX KOMnOHeHT, TO
II Xk nOJIOLKHTeJIbHO H 3HatIeHHe W ,n06aBJIHeTcH B ; eCJIH
w
}Ke Sw co,nepLKHT HetIeTHOe KOJIHtIeCTBO OTpHuaTeJIbHbIX KOM-
nOHeHT, TO w BbItIHTaeTCSI H3 )G, IlTO H ,l{OKa3bIBaeT .9KBHBa-
JIeHTHOCTb (2.45) H (2.43).
KaK BH,l{HM, perpeccHoHHasr 3aBHCHMOCTb (2.45) B OOllI-eM
CJIYtIae HeJIHHeHHa. OHa CTaHOBHTCH JIHHeMHOH TOJIbKO eCJIH
B HeM YtIHTbIBalOTCH He 60JIee tIeM napHbIe CBH3H n-ro H OCTaJIb-
HbIX npH3HaKoB.
PaccMoTpHM npHMep C HCCJIe,l{OBaHHeM <pOpMHpoBaHHH qYB-
CTBa ,noopO)KeJIaTeJ.I»HocTH K .l{pyrHM .9THHtIeCKHM rpynnaM.
B KatIeCTBe aHaJIH3HpyeMoH 3aBHCHMOM nepeMeHHoH eCTeCT-
BeHHO BbIoHpaeTcH lJ. Mo,neJIb (2.34) npHBo,nHT K cJIe,nYlOllI-eMY
onHcaHHIO OTHOCHTeJIbHbIX tIaCTOT:
q . . k - I\,I\,K I\,H
l} - f f l f k .
(2.46)
OCraJIbHble lJJleHbI COKpamalOTCSI npH ,AeJleHHH Pijkl Ha
PijhO. CooTHoIlleHHe (2.45) .l{JIH JIOrHTa HMee! JIHHeHHbul BH){,
84
TaK KaK B (2.46) cpHrYPHPYIOT TOJIbKO napHble CBSl3H (K, 4)
H (H , l/) :
)( = + K xK + H xH.
B COOTBeTCTBHH C TaOJI. 2.2 B (2.46) "( = (T CJ )2 = 0,95 2 =
= 0,90, "(K = (TKlj)2 = 1,56, "(H = (T HlJ )2 = 1,44. JIo-
rapHcpMHPYSI, nOJIYlIHM = - 0,10, pK = 0,40, H =
= 0,36. 3HalleHH5I K H H BbIpamalOT BKJIa,l{ KaLK,l{OrO
H3 npH3HaKoB K, H B JIOrHT HCCJIe.L{yeMoro nOKa3aTeJISI q
H aHaJIOrHlIHO Ko3$PH[(HeHTaM OObIlIHbIX perpeccHoHHhIX ypaB-
u
HeHHH MorYT HHTepnpeTHpOBaTbC5I KaK nOI<a3aTeJIH CB5I3H
BXO,l{HbIX npH3HaKoB K, H C BbIXO,l{HbIM npH3HaKoM 1] (lIacT-
HbIe Ko3$PH[(HeHTbI KOppeJI5I[(HH).
XOTSI (2.46) nOJIYlIeHO H3 (2.34), oopaTHbIM nepexo,l{, BO-
oow.e rOBopSl, HeB03MO)KeH: B MO,l{eJIH (2.46) HHKaK He OTpa)Ke-
Ha HHcpopMa[(H5I 0 TOM, qTO 't EKH = 1, nOCKOJIbKY 't EKH
COKpaIII.aeTCSI npH nepeXO,l{e -K OTHOCHTeJIbHbIM lIaCTOTaM. HHbI-
MH CJIOBaMH, (2.46) 3KBHBaJIeHTHa nOJIHOH MO.L{eJIH lIeTbl-
pex nepeMeHHblx C e,l{HHHlIHbIMH 3HalleHHSlMH 't Eg , 't EKlJ ,
't EHlJ , 't KHlJ, 't EKHlJ , H B HeM HHKaK He OTpaLKeHO TO
OOCTOHTeJIbCTBO, qTO B (2.34) 't EKH = 1. 1JTOObI YlIeCTb BCIO
HHcpopMa[(HIO MO,l{eJIH (2.34),. HeOoXo.L{HMO paccMoTpeTb Bblpa-
u
meHHSI nepeKpecTHbIx OTHOIlIeHHH ,nJI5I ,l{PyrHx npH3H8KOB, pac-
cMaTpHBaeMbIx B KalleCTBe BbIXO,nHbIX.
Oo03HallHM
q.jhZ = PljkZ/ POjkZ, qi.hl Pilhl/ PiOkZ,
qij.Z = P ijlZ/ P ijOZ,
TOr,l{a B CHJIY (2.34)
q .. = ",H ",pH ",KH ",lj H .
rl .. rt rj r1 ,
q . J.. - ",K ",BK ",HK ",lJK ·
. nl -.. .. irk r 1 ,
q .J.. - ",E ",KE ",H K
.}N.l -.. "j rk ·
(2.47)
( 2.48)
(2.49)
t
3,neCb BCIO,l{Y B l' cpHrYPHpyeT ,l{onOJIHHTeJIbHbIH HH,neKC Bbl-
XO.L{HOro npH3HaKa, lITOObI MOLKHO OblJIO pa3JIHlIHTb COOTBeTCT-
u
BYIOIII.He BeJIHlIHHbI pa3HblX ypaBHeHHH.
B Ka)K,nOM H3 ypaBHeHHH (2.47)-(2.49) OTpa)KeHa HH<pOP-
Ma[(HSI 0 TOM, liTO TEKH = 1, TaK liTO Ka:>K,l{Oe H3 HHX BMeCTe
C (2.46) HeceT nOJIHYIO HHcpopMaUHIO 0 lIaCTHOH MO.L{eJIH (2.34).
HeTpY.L{HO yoe,l{HTbC5I TaKjJ{e, liTO JIIOOOe nO.L{MHOJKeCTBO ypaB-
HeHHH: (2.46)-(2.49), KOTOPOMY npHHa.L{JIemHT He MeHbllle
.QBYX ypaBHeHHH, CO,l{ep)KHT nOJIHYIO HHcpopMa[(HIO 0 e.L{HHHlI-
85
HbIX napaMeTpaX MO,l{eJlH (2.34), B TO BpeMH KaK JUOOOe H3
OT,l{eJIbHbIX ypaBHeHHH (2.46)-(2.49) BKJIIOqael' JIHIllb qaCTHq.
HYIO HHcpopMal.(HIO.
BepHeMc5I K pHC. 2.1, H306pa)KalO eMY rpa<t> B3aHMOBJlH5I.
HHH nepeMeHHbIx B, K, H, l/. B BepIllHHY l/ H,l{YT .nyrH H3
K H H, B BeplIlHHY K - H3 B, H, l/, B BepllIHHY H - H3
E, K, l/, a B B HeT BXO,l{51 HX ,l{yr. CooTBeTcTBeHHo 3TOMY ypaB-
HeHHe (2.46) xapaKTepH3yeT BXO,l{bI B BepIllHHY If, (2.47)-
B BepIllHHY H, (2.48) - B BeplIlHHY K. 3TH TPH ypaBHeHHH ,l{a-
lOT KOJIHqeCTBeHHoe BbIpa:>KeHHe CTPYKTYPbI BJIH5IHHH pHC. 2.1,
npHqeM 3HaqeHH5I w = log r W = 2 log ,&W 51BJI5IIOTC5I Ol.(eH-
KaMH COOTBeTCTBYIO HX CB5I3eH. 3aMeTHM, qTO, XOTSI CB5I3H
a)
oj
6)
PHC. 2.2
MorYT HMeTb O,l{HOCTopOHHee HanpaBJleHHe, 3HaqeHH5I COOT-
BeTCTBYIO HX W CHMMeTpHqHbI B CHJIY CHMMeTpHqHOCTH T W .
TaK, EK = KE = 2A5K = 0,56, HK == KH = 0,30,
pEH = pHE = 0,30.
CTPYKTYPbI B3aHMOBJIHSlHHH BH,l{a pHC. 2.1 H COOTBeTCT-
BYIO He ypaBHeHH5I BH,l{a (2.46)-(2.48) Ha3bIBaIOTC5I B JIHTe-
paType KayaallbflblMU (npU'lUflHblMU) MOOe/lflMU, a Ol.(eHKa OT-
u u u
HOCHTeJIbHOH CTeneHH Ka)K,l{OH H3 ,l{yr KaY3aJIbHOH MO,l{eJIH -
nyme8blM aflallU3OM.
CJIe,l{yeT OTMeTHTb, qTO MO,l{eJIb pHC. 2.1 51BJI5IeTC5I He e,l{HH-
u u u
CTBeHHOH KaY3aJIbHOH MO,l{eJI, blO COOTBeTCTBYIOIneH MO,l{eJIH
(2.34). B 3aBHCHMOCTH OT Toro, KaKoe H3 nO,l{MHOJKeCTB ypaB-
HeHHH (2.46)-(2.49), 3KBHBaJIeHTHbIX MO,l{eJIH (2.34), MbI Bbl-
oepeM, TaKYIO KaY3aJlbHYIO MO,l{eJIb MbI H nOJIyqHM. Ha pHC. 2.2
H300pa)KeHbI rpacpbI npHqHHHOCTH ,l{JI5I CHCTeMbI (2.46)-(2.47)
(pHC. 2.2, a), CHCTeMbl (2.46-2.49) (pHC. 2.2, 6) H CHCTeMbI
(2.46), (2.49) (pHC. 2.2, 8). Ka)K,l{a51 H3 KaY3aJIbHbIX MO,l{eJIeii
pHC. 2.2 (a TaK)Ke MO,l{eJleH, cooTBeTcTBYIOInHX OCTaJIbHbIM no,l{-
MHO:>KeCTBaM ypaBHeHHH (2.46)-(2.49)) 3KBHBaJIeHTHa MO,l{eJIH
6
(2.34), T. e. npHBo,l{HT K TeM me TeopeTHlIecKMM tIaCTOTaM P8'
QTO H (2.34), eCJIM o eHKY napaMeTpoB npoH3Bo.L{HTb Heno-
Cpe.L{CTBeHHo ,l{.lISI STHX MO,l{eJIeH. TaKoe MHoroo6pa3He MO,l{e-
u
JIeH, O,l{HHaKOBO XOpOI.I1O onHCbIBalO HX .L{aHHbIe, MO)KeT BbI3BaTb
pa3,ltpa1KeHHe Y HCCJIe,l{OBaTeJI5I, pa3pa6aTbIBalO ero TeopHIO
KOHKpeTHoro H3YlIaeMoro SlB.lIeHH5I. G,l{HaKO cpe,ltH CTaTHCTH-
KOB cy ecTByeT MHeHHe, liTO BHHOBaT 3,l{eCb HCC.lIe,l{OBaTe.lIb,
a He MeTO,l{ 06paOOTKH. aTO MHeHHe OCHOBaHO Ha KOHuen HH,
COr.lIaCHO KOTOpOH npHlIHHHa51 CTpYKTypa SlB.lIeHHH He M01KeT
6bITb HenOCpe,l{CTBeHHbIM pe3Y.lIbTaTOM oopaOOTKH ,l{aHHbIX, a
.L{O.lIJKHa OpMHpOBaTbC5I Hccne,l{OBaTeJIeM KaK OCHOBa TeopHH
,l{aHHOro 51B.lIeHH5I. CTaTHCTHKa 3aHHMaeTC5I He BbIOOpOM MO,l{e-
u u
.lIe", HO .lIHlllb o eHKOH napaMeTpOB - BOT KBHHT.9CCeH MH
STHX npe,l{CTaBJIeHHH, KOTopbIe nO,l{TBep:>K,l{aIOTCSI onHcaHHbIMH
3,l{eCb pe3YJIbTaTaMH.
B oo eM CJIyqae rpa npHqHHHOH CTPYKTYPbI MO}KeT ObITb
pa30HT Ha C.lIOH VI' V 2 , ..., V p , TaKHe, llTO ,l{yrH Be,l{YT H3 CJIOH
V Z B CJIOH C OO.lIbIlIHMH HOMepaMH, HO He C MeHbI.I1HMH. nYTeBOH
aHaJIH3 npH STOM BbInOJlH5IeTCSI nOC.lIe,l{OBaTeJIbHO: CHallaJIa .L{.lISI
BHYTpeHHHX ,l{yr MH01KeCTBa V l Ha OCHOBe COBMeCTHoro pac-
npe,l{eJIeHHSI npH3HaKOB Vl' 3aTeM - ,l{yr, Be,l{YInHX B V 2 (H3
VI H V 2), Ha OCHOBe COBMeCTHoro pacnpe,ne.lIeHH5I npH3HaKOB
MHO}KeCTBa V l U V 2 H T. ,l{.
B JIOrHKY KaY3aJIbHbIX MO,l{eJIeH MorYT ObITb BKJIlOqeHbI H
MO.L{e.lIH nOCTpOeHH5I OO'bSlCHHIOIII.HX cpaKTopOB (n. 2.1.1).
PaccMoTpHM .L{.lI5I npOCTOTbI 3a)(aqy nocTpOeHH5I OO'bSlCHSlIO-
ero cpaKTopa ,l{.lI5I Tpex npH3HaKOB A, Bile co 3HatIeHHSlMH
i, j H k COOTBeTCTBeHHO. OO'b5lCHSlIO HH cpaKTOp, KaK onpe,l{eJIe-
HO B n. 2.1.1, STO TaKOH npH3HaK X co 3HaqeHHSlMH i, tITO
COBMeCTHoe pacnpe,l{eJIeHHe (ABCX) y.L{OBJIeTBOp5leT YCJIOBHIO
Pijhl = pz p pi pt,
r,l(e pz - Bep05lTHOCTb i-ro 3HaqeHIISI x, a pi (pj, p )-
Bep05lTHOCTb 3HalleHHH npH3HaKa A (H.lIH B, C) npH yc..
JIOBHH, llTO 3HalleHHe X paBHO i.
rpacp cooTBeTcTBYIO eH KaY3aJIbHOH CTPYKTYPbI H30opa-
}KeH Ha pHC. 2.3. MYJIbTHnJIHKaTHBHaSl 3anHCb .lIOr.lIHHeRHOH
MO,l{eJIH ,l{JISI STOR CTPYKTypbI HMeeT C.lIe,l{YIO HH BH,l{:
P t .J.. = 'n1':4 1' B '(c 't X 'tAX '( x T CX
1 If, l · I t i k l II J l kl.
MbI Y}Ke ,l{OKa3bIBaJIH, liTO eC.lIH 't 123 = ,&12 = 1, TO npH3HaK
x3 HBJIHeTC5I OO'bSlCHSlIO HM cpaKTopOM .L{.lISI Xl H x 2 : OHH He3a-
BHCHMbI npH BCSlKOM cpHKCHpOBaHHOM 3HalleHHH x 3 . AHa.lIOrHlI-
(2.50 )
87
f
110 1\10}l{HO nOKa3aTb, tITO Mo,neJlb (2.50) B TOqHOCTH xapaKTe-
pH3yeT cnyqaw, Kor a npH3HaKH A, B, C CTaTHCTHqeCKH He-
3aBHCHMbI npM BC5IKOM <pHKcHpoBaHHoM 3HaqeHHH X; 3TO ra-
P HC. 2.3
paHTHpyeTC5I paBeHcTBoM e,l{HHHne Bcex 3HaqeHHH 'tV, He
<PHrypHPYIOInHX B (2.50).
,[{J1R OTHOCHTeJlbHbIX qaCTOT
q. jnl = PlikZ/ POjhZ, qi.kl = Pilhl/ PiOhZ, qij.l = Piill/ PijOl
MO,l{eJlb (2.50) ,l{aeT CJle,l{YIOInHe BbIpa}KeHH5I:
q _ I\,A I\,AX q - I\,B I\,BX q . - I\,C I\,CX
· jhl - r r l ' i. hl - r r 1 , ir z- r r 1 ,
JIOrapHcpMHpoBaHHe KOTOpbIX ,l{aeT 3HaqeHHH OneHOK cBH3eH
KaY3aJlbHOH CTPYKTYPbI pHC. 2.3.
M3 onpe,l{eJleHH5I 06'b5lCH5IIO ero cpaKTopa CJIe,l{yeT, qTO
q.jhl = P / P .
0003HaqHM 3TY BeJlHQHHY, BbIqHCJl5IeMYIO no pacnpe,l{eJle-
HHIO TOJlbKO napbI A, X, qepe3 ql. Tor,l{a, OqeBH,l{HO,
ql = VA rfx,
TaK qTO eCJIH X HMeeT ,l{Ba 3HaqeHH5I, TO
1 1
yA=(qlqO) 2 , yAX=(ql/qO) 2 ,
npHQeM yfX = yAX = l/y X. 113 onpe,l{eJIeHH5I ql H qO He--
TPY,l{HO nOJlyqHTb, qTO yAX B ,naHHOM CJIyqae - He qTO
88
.,
HHoe, KaK KopeHb KBa,l{paTHbIH H3 nepeKpeCTHoro OTHOllIeHHH
pacnpe,l{eJIeHH5I (AX).
MeTo,l{ OneHHBaHH5I napaMeTpOB MO,l{eJIH OO'b5lCH5IIO erO
<paKTopa, onHcaHHbIH B n. 2.1.1, HBJI5IeTC5I eCTeCTBeHHOH MO,l{H-
<pHKau.HeH oo ero MeTO,l{a n. 2.1.3 ,l{JI5I MO,l{eJIH (2.50).
TaKHM oopa30M, JIOrJIHHeHHbIe MO,l{eJIH MorYT ObITb HCnOJIb-
30BaHbl He TOJIbKO ,nJIH onHCaHH5I, HO H ,Z(JIH KOHCTpYHpOBaHH5I.
B paooTe [127) npHBo,l{51TCH npHMepbI HCnOJIb30BaHH5I 3THX MO-
,neneii ,l{JI5I O,l{HOBpeMeHHoro perneHH5I 3a,l{atl KOHCTpYHpoBaHH5I
H onHcaHHH.
2.2. nOKA3ATEnH CBjl3H HOMHHAJlbHbIX
nPH3HAKOB
2.2.1. CTanaapTHblu noaxoa 1(, ot{eHKe C8Jl8U
BbItlHCJIeHHe K03cpcpHnHeHToB CB5I3H Me:m,ny npH-
3HaKaMH - O,l{HH H3 HaHOOJIee pacnpocTpaHeHHblx npHeMOB
npe,l{BapHTeJIbHOrO aHaJIH3a ,l{aHHhIX. B COOTBeTCTBHH C oo eH
., u.,
MeTO,l{OJlOrHeH aHaJIH3a ,l{aHHbIX, pa3BHBaeMOH B ,l{aHHOH KHH-
re, Mbl CIIHTaeM, lITO pOOJIeMa H3MepeHH5I CB5I3H Me}K,l{Y npH-
u
3HaKaMH He 51BJI5IeTCH caMOCT05lTeJIbHOH: B KalIeCTBe Mep CB5I-
3H CJle,l{yeT opaTb Te nOKa3aTeJIH, KOTopbIe B03HHKalOT B MO,l{e-
JI5IX KOHCTpYHpOBaHH5I H onHCaHH5I npH3HaKOB. Q,l{HaKO npoo-
JIeM H3MepeHH5I CB5I3H KalIeCTBeHHWX npH3HaKOB KaK caMO-
CT05lTeJIbHbIe npOOJIeMbI aHaJIH3a ,l{aHHbIX BMeCTe C npe,l{JIO}KeH-
HbIMH pellleHH5IMH 3aHHMalOT HaCTOJIbKO OOJIhrnOe MeCTO B cy-
meCTBYIO eii JIHTepaType, lITO Harne H3JIO)KeHHe ObIJIO ObI cy-
meCTBeHHO HenOJIHO oe3 ynoMHHaHH5I XOT5I ObI HeKOTopbIX CTaH-
,l{apTHhIX nOKa3aTeJIeH H3 orpOMHoro tJHCJIa K03<p<PHQHeHTOB,
npe,l{JIaraBIlIHXC5I B JIHTepaType H3 Tex HJIH HHbIX cooopa:me-
HHH [22, 29, 90, 131, 133, 144, 156].
PaccMoTpHM CHalIaJIa CHCTeMY ,l{BYX ,l{HXOTOMHlIeCKHX npH-
3HaKOB, TaOJIHna cOnp5I)KeHHOCTH KOTOpbIX HMeeT cJIe,l{YIO HH
BH,A:
A A
B a b a+b
B c d c+d
a+c b+d
89
..
nYCTb, HanpHMep, pellb H.L{eT 00 H3YlIeHHH CBSl3H ypOBHH
00pa30BaHH5I (A - BbICOKHH, A - HH3KHH) H ypOBHH 3apa-
OOTHOH nJIaTbI (B - BbICOKHH, B - HH3KHH). qeM OTJIHlIaIOT-
C5I cOBoKynHocTH, HMelO He pacnpe,l{eJIeHHSI BH,na
B B B E
A 220 180 400 A 20 380 400
A 80 520 600 A 280 320 600
300 700 1000 300 700 1000
COOTBeTCTBeHHO? 5ICHO, B nepBoH H3 HHX ypoBeHb 3apaooTHoH
nJIaTbI nOJIO>KHTeJIbHO CB5I3aH C ypoBHeM OOpa30B8HHH, 8 BO
BTOpOH - OTpHnaTeJIbHO. '
.l{eHcTBHTeJIbHO, B nepBOH cOBoKynHocTH H3 40 % JIHn C
BbICOKHM ypoBHeM 00pa30BaHH5I 22 % JIHn nOJIyqalOT BbICOKYIO
3apaooTHYlO nJIaTY, TaR liTO HX ,l{OJI5I (22/40 = 55%) 3HaIlH-
TeJIbHO BblIlle Cpe.L{HeH ,l{OJIH JIHn, nOJIYlIalO HX BbICOKYIO 3a-
paooTHYIO nJIaTY (30%), Tor,l{a KaK .L{OJI5I HH3KOOnJIaqHB8e-
MbIX JIHn C BbICOKHM ypoBHeM 00pa30BaHH5I (45?4) 3H8I1HTeJIb-
HO MeHbIlIe, lIeM cpe,l{HSISI .L{OJISI HH3KOOnJIallHBaeMbIX JIHn (70 %).
HanpoTHB, .L{OJIH JIHn C BbICOKHM ypoBHeM oopa30BaHHSI, nOJIY-
lIalO HX BbICOKYIO 3apaooTHYIO nJIaTY BO BTOpOH cOBoKyn-
HOCTH (20/400 = 5%), 3HallHTeJIbHO HH>Ke Cpe.L{HeH .L{OJIH BbI-
COKOOnJIallHBaeMbIX JIHn, Tor,l{a KaK .L{OJIH HH3KOOnJIallHBaeMbIX
Cpe,l{H JIHn C BbICOKHM ypOBHeM oopa30BaHHSI (95 %) 3Ha1lHTeJIb-
HO BbIrne Cpe.L{HeH.
B oo eM CJIyqae rOBOp5lT 0 nOJIO)KHTeJIbHOH CB5I3H npH3Ha-
KOB, eCJIH ,l{OJI5I A Cpe,l{H B OOJIbIlle, qeM BO BceR cOBoKynHo-
CTH, T. e.
a > a+b (2.51)
a+c N'
H 00 oTpHnaTeJIhHOH CB5I3H B npoTHBHOM CJIYllae. npeoopa3YH
(2.51), nOJIYlIHM 3KBHBaJIeHTHOe COOTHorneHHe
a> (a- b)(a+c) = a(N-d)+bc = a- +
'N N N N'
TaK KaK a + b + c = N - d. CJIe,l{OBaTeJIbHO, (2.51) 3KBH-
BaJIeHTHO TO MY, liTO ad > be. TaKHM oopa30M, no BeJIHIIHHe
T = ad = ..!:.. j .!!.- (2.52)
be cd'
90
.
u
Ha3bIBaeMOH nepeKpeCmHblM OmHOLUenUeM, MO)KHO CY,ZIJITb 0
CBSl3M Mem.L{Y npH3HaKaMH A ME. npM 't> 1 CB5I3b nOJIO)KH-
TeJIbHa, npM 't < 1 CBH3b OTpMnaTeJIbHa, a npH 't = 1 npM3Ha-
KM CTaTHCTMlIeCKH He3aBMCMMbI, TaK KaK ale = bid.
JII060nbITHo, liTO OO'be.L{MHeHMe .L{BYX paCCMOTpeHHbIX co-
BOKynHocTeH .L{aeT TaOJIHny COnpH)KeHHOCTM
B B
A 240 560 800
A 360 840 1200
600 1400 2000
.L{JISI KOTOpOH 't = 1, TaK KaK 240/360 = 560/840, T. e. CTpOKM
nponopnMOHaJIbHbI APyr Apyry. STOT npMMep HarJISlAHO no-
Ka3bIBaeT, K8K npM OO'beAMHeHMM pa3HopoAHbIX rpynn MorYT
noraCMTb APyr Apyra npOTMBOpeqMBbIe 3aKOHOMepHOCTM. llpM-
Mep oopaTHoro SlBJIeHM5I - JIOLKHOH KOppeJI5InMM, B03HMKalO-
meii npM OO'beAMHeHMM ABYX cOBoKynHocTeH, Ha KOTOpbIX npM-
3HaKM He38BMCMMbI,-npMBOAMJIC5I npM oocymAeHMM KOHnen-
nMO npeACT8BJIHIOmerO cpaKTopa (c. 59).
A. 9,nBapAc pacCMOTpeJI CJIeAYlOmee TpeooBaHMe K Mepe
CB5I3M AMXOTOMMlJeCKMX npM3HaKOB [122]: Mepa CB5I3M AOJI)KHa
u
MOHOTOHHO M OAMHaKOBO 38BMceTb OT AOJIeM nOJIOLKMTeJIbHOrO
3HalleHMSI npM3HaKa B B KJIaCCaX A AOJIeii nOJIOLKMTeJIbHOrO
3HalleHMH npM3HaKa A B KJIaCCaX B. HHbIMM CJIOBaMM, Mepa
CBSl3M cp AOJI)KHa y,lJ.OBJIeTBOp5lTb YCJIOBHIO
cp ( a:b · C:d ) = cp ( a:c ' b:d )'
OTKY.L{a CJIeAyeT, liTO cP 3aBMCMT OT nepeKpeCTHoro OTHOIlleHM5I T.
Erne paHbIlle aHaJIOrMtIHbIH BbIBOA ObIJI nOJIYlleH ,[{)K. IOJIOM
[144], KOTOpbIH TpeOOBaJI, lITOObI BeJIMqMHa MepbI CB5I3M He Me-
H5IJIaCb npM YMHO)KeHHM CTpOK MJIM CTOJIOnOB TaOJIMnbI CO-
npSl)KeHHOCTM Ha npOM3BOJIbHbIe lIMCJIa: cp (a, b, c, d) =
= cp (aa, ab, c, Pd) = cp ('Va, {)b, 'VC, ()d). HeTpYAHo AOKa3aTb
3KBHBaJIeHTHOCTb TpeooBaHMH IOJIa 11 3.L{BapAca.
llIMpoKo M3BeCTHbI CJIeAYlOmMe nOKa3aTeJIM CB5I3M, YAOB-
JIeTBOp5llOInMe YCJIOBHHM IOJIa M 3ABapAca: Koacpl/Jutfuenm
FOil a
Q== '(-1 _
't+ 1
ad- be
ad+be
(2.53)
91
'"1 ;ll'. .. ...
H KOacpcpUl1Ueflm KO/l/lU2al1uu
y= V-r-l = l,tJQd-V& .
V't +1 V ad + V bc
9TH MepbI npHHHMalOT HYJIeBOe 3HalleHHe TOr,!{a H TOJIbKO
TOf,!{a, KOr,!{a npH3HaKH CTaTHCTHqeCKH He3aBHcHMw, npHqeM
HX 3HaK COOTBeTcTByeT 3HaKY CB5I3H npH3HaKOB. 3KCTpeMaJIb-
Hoe nOJIO:>KHTeJIbHOe HJIH OTpHnaTeJIbHOe 3HaqeHHe + 1 ,!{o-
CTHraeTC5I, eCJIH O,!{Ha H3 BeJIHlIHH a, b, C, d paBHa HYJIIO. 3T0
03HaQaeT, liTO MepbI Q H Y MorYT ,!{aBaTb 3KCTpeMaJIbHOe 3Ha-
u u
lIeHHe H Tor,!{a, Kor,!{a 0,!{H03HallHbllI nporH03 3HaqeHHII
O,!{Horo npH3HaKa no 3HalleHH5IM ,!{Pyroro HeB03MO:>KeH.
.QJI5I He,!{HXOTOMHlIeCKHX HOMHHa bHbIX npH3HaKOB 3a-
KJIlOlIeHH5I 0 3HaKe CB5I3H HeYMeCTHbI. OTHOCHTeJIbHO oeccnop-
HbIM 3,!{eCb OCTaeTC5I JIHIllb BbIBO,!{ 00 OTCYTCTBHH CB5I3H npH3Ha-
KOB npH HX CTaTHCTHQeCKOM He3aBHCHMOCTH. 3TO n03BOJI5IeT
HCnOJIb30BaTb B KaqeCTBe H3MepHTeJI5I CB5I3H npH3HaKOB CTaH-
J{apTHYIO Mepy OTKJIOHeHH5I HX COBMeCTHoro pacnpeJ{eJIeHH5I OT
u
CTaTHCTHlIeCKOH He3aBHCHMOCTH:
(2.54 )
X 2 = N I (Nst-N s . N.tI N )2j .
S. t N s. N. t
9TOT CTaHJ{apTHbIM nOKa3aTeJIb, Ha3bIBaeMhIH qaCTO KO-
afjJt/JUl,fueHmOM K8aapamUtteCKOU COnp.fl3ICeflIiOCmu, TeM Donee
YJ{ODeH, liTO npH CTaH,!{apTHOH CTaTHCTHqeCKOH MOJ{eJIH BbIOOp-
KH B npe,!{nOJIO:>KeHHH He3aBHCHMOCTH npH3HaKOB aCHMnTOTH-
qeCKH no N nOJ{lIHH5IeTC5I pacnpe,!{eJIeHHIO X 2 C (m 1 - 1) X
X (m2 - 1) CTeneH5IMH CBOOOJ{bI. 3TO HCnOJIb3yeTC5I ,!{JI5I npo-
BepKH rHnOTe3bI 0 He3aBHCHMOCTH npH3HaKoB.
PacKpbIBa51 CKODKH, nOJIYlIHM oOJIee npocTYIO, XOT5I II Me-
Hee HarJIH,!{HYIO <POPMYJIY
( N2 ,
X 2 =N st -1 ) .
,5, t N s. N. t
(2.55)
(2.56)
113 (2.55) CJIe,!{yeT, liTO BeJIHqHHa X 2 HeOTpHnaTeJIbHa H
npHHHMaeT MHHHMaJIbHOe HYJIeBOe 3HaqeHIIe TorJ{a H TOJIbKO
TOr,ll.a, KOr,ll.a npH3HaKH CTaTIICTHqeCKH He3aBHCHMbI, T. e.
N st = Ns.N.t/N. nOCKOJIbKY MaKCHMaJIbHOe 3HaqeHHe X2
YBeJIHlIHBaeTC5I C pOCTOM N, qaCTO HCnOJIb3yeTC5I HOpMHpOBaH-
HhIM nOKa3aTeJIb
X 2 N 2
<1>2 == _ === st -1 ,
N s, t N s. N. t
(2.57)
92
MaKCHManbHoe 3HaqeHHe KOTOpOrO, KaK HeTpY HO nOKa3aTb,
paBHO min (m 1 - 1, m2 - 1). n03TOMY r. KpaMep npe JIo-
jKHn paccMaTpHBaTb KosqxpHUHeHT
K 2 =: a>2 (2.58)
min(ml-l, m2-1)'
KOTOpbIH npHHHMaeT 3HaqeHH5I B MHTepBane [0, 1 J. AHanOrMq-
Ha51 MO,lJ.H<p HKa UH5I obIna npe,lJ.nO)KeHa A. lJ ynpoBbIM:
T2=
V(ml-J) (m2- 1 )
.OlleBHAHo, npH m 1 = m 2 K09qxpHUHeHTbI 1JynpoBa
Mepa paBHbI, HO B oow.eM cJI Yllae T K, TaK KaK
K2=T2 V(ml-l)(ml1- 1 )
min (ml-l, m 2 -1) ,
a>2
(2.59)
H Kpa-
r e qHCnHTeJIb 3aBeAOMO npeBocxo MT 3HaMeHaTenb npH
m 1 =1= m2.
.QnH ,nHXOTOMHlleCKHX npH3HaKoB nYTeM HeCnO:>KHbIX, XOT5I
H rpoM03AKHX, npeoopa30BaHHH, MO>KHO nOKa3aTb, liTO Be-
nMlIHHa <D KOpH5I KBaApaTHoro H3 cp2 paBHa:
ad-be
Vfa+c) (b+d) (a+b) (c+d)
cp=
(2.60)
3Ta BeJIHlIHHa 3aMeqaTenbHa TeM, liTO paBHa Ko3<fxPHUHeHTY KOp-
PeJI5IUHH COOTBeTCTBYIO MX ,lJ.HXOTOMHlIeCKHX npH3HaKOB, pac-
CMaTpHBaeMbIX KaK KOJIHqeCTBeHHbIe.TIpH STOM COOTBeTCTBYIO-
He OYJIeBCKHe CTOJIOUbI HMelOT a + b eAHHHU H C + d HY-
neii JIH nepBoro npH3HaKa H a-t- c eAMHHU H b +d HYJIeR AJIH
a+b
BToporo, TaK liTO cpe HHe STHX BeKTopOB paBHbI H
a+c
7r- COOTBeTCTBeHHO. nOCJIe BbIlIHTaHHH cpeAHHX CKaJI5IpHOe
npOH3Be)(eHHe CTonOUOB paBHO (ad - be)/ N, HX CKaJI5IpHbIe
KBa paTbI - (a + b) (e + d)/N H (a + c) (b + d)/N, tlTO
H aeT (2.60) B COOTBeTCTBHH C onpe eJIeHHeM K09qxpHUHeHTa
KOppen5lUHH (c. 298).
TaKHM oopa30M, n51 AHXOTOMHlIeCKHX npH3HaKoB <D 2 eCTb
o HoBpeMeHHo KosqxpHUHeHT eTepMHHaUHH HX KonHlIeCTBeH-
Horo npe CTaBJIeHH5I CTOJIOuaMH TaOJIHUbI OO'beKT-npH3HaK
(c. 135). LlaJlee MbI YBHAMM, KaK STOT nOKa3aTeJIb B03HHKaeT
B MO eJI5IX onHCaHH5I H npM npOH3BonbHbIX HOMHHanbHbIX npM-
3HaKax ( 3.2, 4.2).
93
-- -. ........ ,
2.2.2. Mepbl CBfl3U KaK .Mepbl KatleCTBa
npozHo3a
llOTpeOOBaJIOCb HeCKOJIbKO ,l{eC5ITHJIeTHH ,l{JI5I <pop-
u
MHpOBaHH5I npe,l{CTaBJIeHHH 0 TOM, liTO Mepa CB5I3H npH3HaKOB
,l{OJI)KHa He CTOJIbKO o eHHBaTb CTeneHb OTKJIOHeHH5I HX COB-
MeCTHoro pacnpe,l{eJIeHH5I OT He3aBHCHMOCTH, CKOJIbKO xapaKTe-
v
pH30BaTb B03MO)KHOCTb nporH03a 3HalleHHH O,l{HorO H3 HHX no
3HalleHH5IM BToporo.
JI. rY,l{M3H H ,[{)K. KpYCKaJI B cepHH CTaTeH [131] ,l{aJIH 00-
30p 3HallHTeJIbHOrO KOJIHqeCTBa K03<p<pH HeHToB CB5I3H Kalle-
u
CTBeHHWX npH3HaKoB C TOqKH 3peHH5I onepanHOHaJIbHOH HH-
u u
TepnpeTa HH HX KaK OneHOK CTeneHH ,l{OCTH:>KeHH5I TOH HJIH HHOH
eJIH. B 3TOM CMbICJIe Mepbl CB5I3H KaK MepbI KalleCTBa npor-
H03a HJIH onHCaHH5I MorYT paccMaTpHBaTbC5I KaK OneHKH CTe-
neHH ,l{OCTH}KeHH5I perneHH5I OCHOBHbIX 3a,l{all aHaJIH3a ,l{aH-
HbIX.
CaMbIH TOllHbIH nporHo3 ,l{OCTHraeTC5I npH O,l{H03HallHOM HJIH
B3aHMHO-O,l{H03HallHOM COOTBeTCTBHH 3HaqeHHH npH3HaKoB. Ta-
Ka51 cHTyanH5I o eHHBaeTC5I 3KCTpeMaJIbHbIM 3HalleHHeM MepbI
CB5I3H.
KpHTepHeM OTCYTCTBH5I CB5I3H OObIlIHO ClIHTaeTC5I CTaTHCTH-
QeCK851 He3aBHCHMOCTb npH3HaKoB.O,l{HaKO nOCJIe,l{OBaTeJIbHa51
peaJIH3a H5I H,l{eH 0 Mepax CB5I3H KaK o eHKax B03MO:>KHOCTH
npe,l{cKa3aHH5I MO>KeT npHBO,l{HTb, KaK MbI YBH,l{HM ,l{aJIee, H K
HHbIM npe,ACTaBJIeHH5IM 0 M8KCHMaJIbHO HecB5I3aHHbIX npH3-
HaK8X.
,[{JI5I YTOllHeHH5I M'O,l{eJIH nporHo3a HeOOXO,l{HMO sa,naTb, BO-
nepBbIX, YCJIOBH5I npOBe,l{eHH5I nporH03a H, BQ-BTOpbIX, pe-
I.I1alO ee npaB'HJIO nporH03a.
CHallaJIa OOCY,l{HM YCJIOBH5I npoBe,l{eHH5I nporH03a. nYCTb
HMeIOTC5I ,nBa npH3HaKa-pa30HeHH5I R = {R1' ..., Rm 1 } H
S = {S1' ..., S m 2 } ,l{aHHOrO MHO)KeCTBa OO'beKToB. PeI.I1alO ee
YCTPOHCTBO Ha OCHOBe HH<popMa HH 0 lIaCTOTax Pst = N st/ N,
Ps. = Pstt p. t = Pst COBMeCTHoro pacnpe,l{eJIeHH5I ,l{OJI}K-
t s
HO ,l{JI5I Kam,l{OrO OO'beKTa i no HH<popMa HH 0 TOM, KaKOMY
KJIaccy Rs OH npHHa,l{JIemHT, ,l{aBaTb nporH03 Toro KJIaCCa St
pa30HeHH5I S, KOTOPOMY npHHa,l{JIemHT i. llo,l{o6Ha51 nOCTa-
HOBKa xapaKTepHa, B lIaCTHOCTH, ,l{JI5I 3a,nallH pacn03HaBaHH5I
oopa30B: KaJK,l{bIH I{JIaCC St 1\10mHO OTOm,l{eCTBHTb C pacn03Ha-
BaeMbIM oopa30M, a pa30HeHHe R MO)KeT paCCMaTpHBaTbC5I
KaK pe3YJIbTaT arperHpOBaHHoro npe,l{CTaBJIeHH5I CHCTeMW HC-
XOJ(HbIX npH3HaKOB, TaK liTO KJIaCCbI Rs COCT05lT H3 OO'beKTOB,
94
...i
t
t
t
SKBHBaJIeHTHbIX C TOqKH 3peHH5I HCXO,l{HbIX npH3HaKOB (CM.
n. 3.2.8).
B STOR cHTyanHH YCJIOBH5I npoBe,l{eHHH nporHo3a nOJIHOCTblO
onpe,l{eJIHIOTC5I 3a,l{aHHeM MexaHH3Ma nocTynJIeHH5I OO'beK-
TOB B peI.I1alO ee YCTPOHCTBO. STOT MexaHH3M ,l{OJI:>KeH ,l{JI5I BC5I-
KHX S H t HKCHpOBaTb Bep05lTHOCTb n05lBJIeHHH OO'beKTOB H3
R s n St. CJIe,l{YIO He TPH MO,l{eJIM ,l{OCTaTOqHO xoporno HJIJIIO-
CTPHPYIOT pa3Hooopa3Me B03MO)KHOCTeH cpOpMMpOBaHMH TaKHX
MexaHH3MOB.
M 0 ,n e JI b 1. OO'beKTbI BbloHpalOTC5I M3 m paBHOBep05lT-
HO (c B03Bpa eHHeM), TaK qTO OO'beKTbI MHO:>KeCTBa Rs n St
nOpO>K,l{aIOTC5I C Bep05lTHOCTblO p (s, t) = Pst = Nstl N.
M 0 .n e JI b 2. CHaqaJIa C paBHoH Bep05lTHOCTblO BbI,l{e-
JI5IeTC5I oopa3 St (t = 1, ..., m 2 ), a Y:>Ke H3 Hero OO'beKTbI Bbl-
1
OHpaIOTC5I paBHOBep05lTHO. Bep05lTHOCTb BbIoopa St paBHa -,
m 2
a Bep05lTHOCTb Bbl60pa i E Rs npH YCJIOBHH i E St paBHa YC-
JIOBHOH BepOHTHOCTH p (sit) = 8t . CJIe)J.OBaTeJIbHO, 06'beKTbI
· t '}
MHO}KeCTBa Rs n St nOpO:>K,l{aIOTcH C -Bep05lTHOCTblO N N st = I
m2 .t
=P (s, t). I
M 0 ,l{ e JI b 3. OO'beKTbI n05lBJI5IIOTC5I paBHOBep05lTHO,
u .
HO B peI.I1alO ee YCTPOHCTBO nocTynalOT He cpa3Y, a nOCJIe
H,l{eHTM MKanHH cO,l{ep:>Ka HX MX KJIaCCOB R s pa30MeHH5I R.
I1,l{eHTMcpHKanH5I KJIaCCa Rs ,l{JI5I OO'beKTa i npOH3BO,l{HTC5I C no-
tI
MOInblO Bep05lTHOCTHoro MexaHH3Ma, KOTOpbIH Ha3blBaeT 3Ha-
qeHHe s C Bep05lTHOCTblO Ps.. ECJIH KJIaCC Rs Ha3BaH npaBHJIb-
HO, OO'beKT i paCCMaTpMBaeTC5I peI.I1alO HM YCTPOHCTBOM, eCJIH
HenpaBHJIbHO - OO'beKT He paCCMaTpHBaeTC5I. OqeBM,l{HO, OO'beK-
Tbl MHO:>KeCTBa R s n S t nocTynalOT B pernalO ee YCTPOHCTBO
C Bep05lTHOCTblO p (s, t) = PstPs., nOJIyqeHHOH YMHo:meHHeM
u
BepOHTHOCTH HX nOpO)K,l{eHH5I Pst Ha Bep05lTHOCTb npaBMJIbHOH
M,l{eHTH HKanHH Rs.
PaccMoTpMM Tenepb npaBMJIa npHH5ITM5I perneHH5I 0 TOM,
KaKOMY o6pa3Y St npHHa,l{JIe}KHT OO'beKT M3 ,l{aHHOrO KJIaCCa R s.
HCHO, QTO Bep05lTHOCTb npaBHJIbHOCTH npe,l{CKa3aHM5I KJIaCCa
St ,l{JI5I KaKOrO-JIHOO 06'beKTa H3 Rs xapaKTepM3yerC5I ,l{OJIeH
001>eKTOB H3 Sf, cO,l{ep}Ka HXC5I B Rs, T. e. YCJIOBHOH Bep05lT-
HOCTblO p (tis) =" :: . B qaCTHOCTH, npH p (tis) = 1 Bce 06'beK-
ThI Rs CO,l{ep}KaTCH B St, npH p (tis) = 0 HHKaKOH OOneKT H3
Rs He MO)KeT npHHa,l{JIe:>KaTb St.
PaCCMOTpHM ,l{Ba peI.I1alO HX npaBHJIa:
95
. '"
1. l{eTepMHHHcTcKoe (onTHMaJlhHbIH npofH03): B KatIeCT-
Be 00pa3a, COOTBeTCTBYIO efO OO'beKTaM KJlaCCa Rs, YKa3bIBa-
eTCH TOT KJlaCC Sl o ' J{JIH KOTOpOfO ,nOCTHfaeT MaKCHMYMa
BeJlHtIHHa p (tis): p (to/s) = max p (tis).
t
2. CTOXaCTHtIeCKOe (nponopnHOHaJlbHbIH npofH03): J{JIH
OO'beKTOB H3 R s MO)KeT ObITb YKa3aH JIIOOOM 00pa3 8 t C 1JaCTO-
TOM, paBHOH p (tis) = N Nst .
s.
KOMOHHHPYH KOHKpeTHYIO MO.L{eJlb nOpO>KJ{eHHSI OO'beKTOB
C KOHKpeTHbIM npaBHJIOM npHHHTHH perneHHH, MO)KHO BbI1JHC-
JlHTb cpeJ{Hee 3HatIeHHe .L{OJIH npaBHJIbHbIX npofH030B Q (R, S)
H J{OJIH OIllHOOK 1 - Q (R, 8), KOTopbIe II SlBSITCH xapaKTepH-
CTHKaMH KalleCTBa nporH03a.
OJ{HaKo caMa no ce6e TaKaSl xapaKTepHcTHKa CJlHrnKOM
rpyoo OneHHBaeT CBH3b npH3HaKOB R H 8, nOCKOJIbKY ee Be-
JIHtIHHa 3aBHCHT He TOJIbKO OT CBH3H R H S, HO H OT TOfO, Ha-
CKOJIbKO XOpOllIO nporH03HpYIOTCSI 3Ha1JeHHSI 8 npH OTCYTCTBHH
HH<popMaQHH 00 R. n03TOMY HeooxoJ{HMO BbIlIeCTb BeJIHlIHHY,
xapaKTepH3YIO YIO KalleCTBO npOfH03a 3HalleHHH S npH OT-
CYTCTBHH HH<popManHH 0 pa30HeHHH R. BeJIH1JHHa pa3HOCTH
Q (R, S) - Q (S) (2.61)
u
xapaKTepH3yeT npHpa eHHe J{OJIH npaBHJIbHbIX npeJ{CKa3aHHH
H YMeHbrneHHe J{OJIH OIllHOOK, nOJIYtIeHHOe 3a C1JeT HCnOJIb-
30BaHHSI HHcpopManHH 0 3HatIeHHH npH3HaKa R. qacTo pac-
CMaTpHBaIOT TaKme OTHOCHTeJIbHYIO BeJIH1JHHY THna (Q (R, 8)-
- Q (S)/Q (8) HJIH (Q (R, S) - Q (8»/ (1 - Q (8» - OT-
u
HOCHTeJIbHbIH npHpOCT npaBHJIhHbIX npOfH030B HJIH OTHOCH-
TeJIbHOe YMeHhllIeHHe OIliHOOK npH YtIeTe HH<popManHH 00 R.
TaKHM 00pa30M, BeJlHtlHHa Qij (R, S) J{JIH Ka)KJ{oH H3
meCTH KOMoHHaUHH MO.L{eJIeM nOpOm.L{eHHSI OO'beKTOB i = 1, 2, 3
C npaBHJIaMH npHHHTH5I perneHHSI j = 1, 2 paBHa:
Qil= P(S, t)maxp(tls)-
s, t t
,lJ.JIH .L{eTepMHHHpoBaHHofo npofH03a H
Qi2= P(S, t)p(tls)-
s, t
JI.JI5I npOnOplI.HOHaJIbHOfO nporH03a. TaKHl\1 00pa30M, nOJIYlIaeM:
Qll = }: Psi max Pst = max Pst; Q12 = Nit ; (2.62)
s, t t Ps. s t s, tNNs.
1 Nst Q _ 1 N: t
Q21 = - max N st; 22 - - ; (2.63)
m2 s. t N 8. N. t t m" s. tN.. N. t
96
f
-
Qal = Pa. max Pst;
s t
Q32 = plt.
s. t
(2.64)
PaCCMOTpHM Tenepb Mepbl CBSl3H npH3HaKOB, OCHOB8HHhle
Ha BbInHcaHHbIX nOKa3aTeJIHX KalleCTBa nporH03a. CHall8JIa
npoaHaJIH3HpyeM COOTBeTCTBYIO He xapaKTepHCTHKH, nOJIY-
lJaeMble npM HCnOJIb30BaHHH MO,!{eJIH 1 nOpO}K,!{eHHR OO'beKTOB.
BeJIHlIHHa OTHOCHTeJIbHOrO YMeHblIleHHH OIIlHOKH OnTHMaJIb-
Horo nporH03a B STOM CJIYlJae paBH8 B CHJIY (2.62):
A=
max Pst -max p. t
s t
I-max p. t I
(2.65)
r,!{e Q (S) BhIIIHCJIeHO KaK Ql1 (U, S) = max p.t; U - YHH-
BepCaJIbHOe pa36HeHHe, COCTOHIl\ee H3 e,!{HHCTBeHHoro KJIaCCa,
co,!{ep>Karnero Bce 06'beKTbl.
KaK Mepa CBH3H npH3HaKoB STa BeJIHlIHH8 OhlJIa npe,!{JIo}Ke-
H3 B pa60Te [133] II Ha3bIBaeTCR KoafjJfjJUl{UeHmOM rymmMaHa.
OlleBH,!{Ho, A, - aCHMMeTpHtIHaH Mepa CBSl3H, npHHHM81O-
maR 3HalleHHH B HHTepBaJIe OT HYJIH ,!{O e,!{HHHnbI. 113 onpe,!{eJIe-
HMH HCHO, tITO A, = 1 Tor,!{a H TOJIbKO Tor,!{a, Kor,!{8 HMeeTCH
O,!{H03HaIlHoe COOTBeTCTBHe Me>K,!{Y S H t. ECJIH R II S CTaTHCTH-
lJeCKH He3aBHCHMbI, TO A, = 0, HO paBeHCTBO A, = 0 MO>KeT ,!{O-
CTHraTbCH H B ,!{PyrHx CJIYtI8HX. ,[lJIH SToro ,!{OCTaTOIIHO, llTo6b1
max Pst ,!{ocTHraJICH npH O,!{HO¥ H TOM >Ke t = to ,nJIH Bcex 3Ha-
t
qeHHM S = 1, ..., mi. Tor,na, OlleB H,!{H 0 , p.t o = Psto MaKCH-
s
MaJIbHO cpe,nH Bcex p. t, TaK liTO lIHCJIHTeJIb A, oopa aeTCH B
HYJIb.
YKa3aHHoe CBOHCTBO CYIneCTBeHHO CHH>KaeT SBpHCTHtle-
CKYIO neHHOCTb A, KaK MepbI CBH3H, TaK KaK A, He n03BOJIHeT OT"
JIMtIaTb ,npyr OT ,nPyra TaKHe npH3HaKH R, KJIa CCbI KOToporo
HMelOT O,nHO H TO me 3HatIeHHe S B KatIeCTBe MO.L{aJIbHOrO, T. e.
HaHOOJlee BepoHTHoro. A TaKHe CJIYlIaH B03HHKalOT ,!{OBOJIbHO
lJaCTO npH aHaJlH3e .L{aHHbIX, B KOTOpbIX O,!{HO H3 3HalleHHH St
BCTpetIaeTCH tIaIne OCTaJIbHbIX. C STOM TOlJKH 3peHHH nOKa38-
TeJIbHO pacnpe.L{eJIeHHe, nOJIYlIeHHoe ,[l}K. IOJIOM H 11. rpHH-
BY.L{OM B....1915 r. no pe3YJlbTaTaM HCCJIe.L{OBaHH5I npHBHBoK H8
3a6oJleBaeMOcTb XOJlepOM 1 [42]. _
I
j
II
f
J
I
1 nO,n06Hble pacnpe,neJleHHH tJaCTO BCTpetJalOTCSI npH aHaJIH3e ,naH-
HblX COil.HaJlbHO-3KOHOMHtJeCKHX HCCJleAOBaHHH.
4 3aK. 1743
97
T a 0 JI H U. a 2.5
He sa 60n eJJH 3a 60JI eJJH Bcero
npHBHTbie 276(98,9) 3(1,1) 279( 100)
HenpHBHTbIe 473(87,8) 66( 12,2) 539(100)
Bcero 749(91,6) 69(8,4) 818(100)
B CKOOKax YKa3aHbI npOneHTbI, KOTopbIe XapaKTepH3YIOT
YCJIOBHbIe pacnpe.L{eJIeHH5I p (tis).
OlleBH.L{HO, Iv = 0, TaK KaK nO.L{aBJI5IIO a51 lIaCTb Bcex 00-
CJIe.L{OBaHHbIX He 3aOOJIeJIa, KaK B KJIaCCaX R (C.L{eJIaHa HJIH He
C.L{eJIaHa npHBHBKa), TaK H BO BceH cOBoKynHocTH, T. e. Iv He
pearHpyeT Ha OlleBH.L{HYIO CB5I3b npHBHBOK C 3aOOJIeBaeMOCTblO.
PaCCMOTpHM Tenepb OiliHOKY nponopnHOHaJIbHOrO nporHO-
3a Q12' a TaK:>Ke ee YMeHbIIleHHe If OTHOCHTeJIbHOe YMeHbllIeHHe
N: t
cx= 1- ·
s, t N N 8. '
(2.66)
=] Nfft -...l.-"2:.N t = ] (Pst-Pso p.t)2
s, t t N N 8. N" s, t p. s
(2.67)
w=
1- P t
t
(2.68)
PaBeHCTBO (2.67) HeTpY.L{HO .L{OKa3aTb, pacKpbIBa51 CKOOKH
H npoH3BO.L{H HeOOXO.L{HMbIe cOKpa eHHH; OHO nOKa3bIBaeT,
liTO YMeHhIIleHHe OIllHOKH nponopnHOHaJIbHOrO nporH038 npH
MO.L{eJIH 1 MomeT paccMaTpHBaTbcH KaK aCHMMeTpHtlHa51 Bep-
CH5I nOKa3aTeJI5I KBa.L{paTHlIecKoH cOnp5l:>KeHHocTH. npH 3TOM,
OlleB H.L{H 0 , = w = 0 TOr.L{a H TOJIbKO Tor.L{a, KOr.L{a npH3HaKH
CTaTHCTHlIeCKH He3aBHCHMbI. MaKCHMaJIbHOrO 3HalleHH5I 3TH
u
nOKa3aTeJIH J{OCTHraIOT, KOr.L{a B03MO:>KeH 0.L{H03HallHbIH npor-
H03 St no Rs (HO He HaooopOT). npH 3TOM = 1 - P t=
t
= () (8) H W = 1. HanOMHHM, liTO () (8) eCTb HOMHHaJIbH8H .L{HC-
nepCH5I npH3HaKa 8.
JII060nbITHO, liTO nOKa3aTeJIb W, Ha3bIBaeMbIH K09<W>HnHeH-
TOM BaJIJIHCa [29, 131], MOjJ{eT ObITb HHTepnpeTHpoBaH B TepMH-
Ha x KalleCTBeHHOH J{HcnepCHH KaK aHaJlOr KOJIHlIeCTBeHHoro
93
KOppeJI5III.HOHHOro OTHOIlleHHSI (CM. c. 136), KOTopoe MO>KeT
6WTb onpe eJIeHO OPMYJIOH
11 2 = Var y-M (Var (y/x»
V ar y ,
r,ne Var H M - onepanHH HaXOJK eHHSI HcnepCHH H MaTeMa-
TIItleCKOro O}KH aHH5I COOTBeTCTBeHHO. ECJIH nOJIO)KHTb
Var S=(, (S) = 1 - '2.P t, a Var (S/s) = 1 - ( PS/ ) 2, TaK
8,t PS.
qTO Var (8/8) eCTb KalleCTBeHHa51 HcnepcHSI npH3HaKa 8,
paccMaTpHBaeMoro TOJIbKO JI5I OO'beKTOB Rs, TO cpe Hee 3Ha-
qeHHe YCJIOBHOH HcnepCHH Var (8/5) no BceM 8 paBHO:
2
Ps. Var (8/8) == 1 - .EE., lITO B CHJIY (2.66) eCTb Cpe H5ISI
S 8,t Ps.
olliHoKa nponopnHOHaJIbHOrO nporH03a 8 no R. B 3THX
u
TepMHHax, eHCTBHTeJIbHO,
VarS-M (Var (S/R»
w= ·
VarS
3aMeTHM, lITO 3Ta cpopMYJIa HMeeT CMbICJI He TOJIbKO JISI
HOMHHaJIbHbIX, HO H JI5I paHrOBbIX H Booo e CTPYKTypHbIX
npH3HaKOB B COOTBeTCTBHH C onpe eJIeHHeM KalleCTBeHHOH HC-
nepCHH. npH 3TOM, OlleBH HO, BeJIHqHHa w JI5I paHrOBbIX npH-
3HaKOB He H3MeHHTC5I, eCJIH paCCMaTpHBaTb HX KaK HOMHHaJIb-
Hwe, nOCKORbKY paHrOBaSl ,nHcnepCHSI npH3HaKa COCTaBJISleT
pOBHO nOJIOBHHY ero HOMHHaJIbHOH ,Z(HcnepCHH, a K03<p<pHnHeH-
TbI 1/2 B IIHCJIHTeJIe H 3HaMeHaTeJIe w cOKpa aIOTC5I.
PaccMoTpHM Tenepb K03qxpHnHeHTbI CB5I3H, B03HHKalO He
npH HCnOJIb30BaHHH Mo,neJIH 2 ,nJISI nOpO}K eHHSI ,naHHbIX.
npH ,neTepMHHHpoBaHHoM npaBHJIe npHH5ITHSI peIlleHHH nOKa-
3aTeJIb OTHOCHTeJIbHOrO YMeHbIIleHH5I OIllHOKH B 3TOM CJIYlIae
B CHJIY (2.63) paBeH:
Pst
m2 8, t P s. p. t
1
max Pst--
t m 2
1
1--
m2
N st
.... maxNst-l
s, t N 8. N. t t
m2- 1
B CHJIY MO,neJIH reHepHpOBaHHH OO'beKTOB 3,neCb BepOSlT-
HOCTh Q (8) npaBHJIbHOrO nporH03a npH OTCYTCTBHH HH<popMa-
I
HH 00 R paBHa Bep05lTHOCTH BbIoopa Sf, T. e. -.
m2
4*
(2.69 )
99
nOKa3aTeJIb (2.69) npHHHMaeT MaKCHMaJIbHOe 3HaqeHHe 1
TOf.L{a H TOJIbKO Tor,na, KOr,l{a S R, T. e. E03MomeH 0,l{H03Haq-
HbIM npofH03. ECJIH npH3HaKH R H S CTaTHCTHqeCKH He3aBHCH-
MbI, TO BeJIHqHHa (2.69) paBHa (m 2 max p.t - 1)/(m 2 - I),
qTO ,l{aeT 0 TOJIbKO npH paBHOMepHOM paCnpe,l{eJIeHHH npH3Ha-
1
Ka S, KOf,l{a p. t = m 2 ,l{JI5I Bcex t.. TaKHM OOpa30M, HaJIHqHe
KJIaCCa St C OOJIblIlHM 3HaqeHHeM p.t ,l{ame npH OTHOCHTeJIbHOM
He3aBHCHMOCTH npH3HaKOB npHBe,l{eT K OOJIbllIHM 3HaqeHH5IM
nOKa3aTeJI5I (2.69). B qaCTHOCTH, .L{JI5I .L{aHHbIX TaOJI. 2.5 ero
BeJIHqHHa paBHa 0,802.
bi\ npH CTOXaCTHqeCKOM npaBHJIe npHH5ITH5I pemeHHH MO.L{eJIb
2 ,l{aeT CJIe,l{YIO ee BbIpameHHe ,l{JI5I OTHOCHTeJIbHOrO YMeHbllIe-
HH5I OlIlHOKH nporH03a:
2
Pst
m2 s, t P s. p. t
1
1--
m2
1
m2
1 <D 2
,
m2- 1
(2.70)
r,ne <1>2 = X 2J N - CTaH.L{apTHbIH K03c}xPHUHeHT COnp5l:>KeHHO-
CTH (2.57). K03<P<PHnHeHT 1/(m 2 - 1) OTpa)KaeT HepaBHonpa-
BHe npH3HaKoB R H S B MO,l{eJIH npofH03a. OqeBH,l{HO,
<D 2 /(m 2 -1) paBHo 1 TOr,l{a H TOJIbKO TOf.L{a, KOf,na S;2 R, T. e.
KJlaCCbI St 0,nH03HaqHO onpe,l{eJI5IIOTC5I KJIaCCaMH, Ra. Cpe,A-
Hee reOMeTpHqeCKOe aCHMMeTpHqHbIX nOKa3aTeJIeH BH,l{a (2.70),
nOJIyqaeMbIX npH nporH03e S no R, a TaK)Ke R no S, paBHo,
01leBH,l{HO, KO.9Q?<PHnHeHTY 1JynpoBa (2.59).
51cHo, qTO nOKa3aTeJIb (2.69) Bcef,l{a npeBOCXO,l{HT (2.70).
,lXJIH .naHHbIX TaOJI. 2.5 <1>2 = 0,036.
3aMeTHM Ta K}Ke , llTO nOKa3aTeJIb <D 2 nOJIyqeH 3)J.eCb
KaK OTHOCHTeJIbHOe npHpa eHHe ,nOJIH npaBHJIbHbIX npofHo-
u
30B B ,naHHOH MO,neJIH.
TaKHM oopa30M, cTaH,napTHbIH nOKa3aTeJIb X 2 xapaKTepH-
3yeT KaqecTBo nporH038 npH reHepHpoBaHHH OO'beKTOB B co-
OTBeTCTBHH C pacnpe,l{eJIeHHeM R BHYTPH oopa30B St. Bonpe-
KH pacnpocTpaHeHHoMY MHeHHIO 0 TOM, liTO x2 Bcero JIHllJb
Mepa OTKJIOHeHH5I pacnpe,l{eJIeHH5I OT He3aBHCHMOCTH, 3TO
,l{aeT onepanHOHaJIbHYIO HHTepnpeTanHIO X2 KaK MepbI CB5I3H
npH3HaKOB.
PaCCMOTpHM Tenepb MepbI CB5I3H, B03HHKaIOInHe, KOr,l{a
OO'beKThI nocTynalOT Ha BXO,l{ peI.I1alO efO YCTpOHCTBa B COOT-
100
j
J
BeTCTBHH C MO,l{eJIblO 3. OpH OnTHMaJIbHOM nporHo3e OTHOCH-
TeJIbHOe npHpa eHHe OIllIfOKH paBHO
max p. t - p . s max Pst
s t
l-rnax p. t
MbI rOBopHM 3,l{eCb 0 npHpa eHHH, a He YMeHbIlleHHH OIllHO-
,
KIf, TaK KaK ,l{OOaBJIeHHe HHcpopManHH 00 R B ,l{aHHOM CJIYtIae
MO)l{eT TOJIbKO YXY,l{IllHTb nporH03 3a CtIeT OIllHOOK YCTPOHCT-
Ba, H,neHTHcpHnHPYIO ero 3HatIeHH5I R. OlleBH,l{HO, BeJIHlIHHa
(2.71) HeoTpHnaTeJIbHa, TaK KaK Pst p.t, H, CJIe,l{OBaTeJIb-
HO, ,l{JI5I BC5IKOrO s max Pst max p. t, a cpe,l{Hee MeHbIllHX
t t
BeJIIftIHH He MO)KeT ObITb OOJIbIIle, lIeM max p.t.
t
AHaJIOrHlIHbIM 00pa30M BbIlIHCJI5IeTC5I npHpa eHHe OIllHO-
KH npH nponopnHOHaJIbHOM nporH03e. B CHJIY (2.64) OHO
paBHo:
.
(2. 71 )
P t- P:t = B (RS)-B (S) = B (8, RS). (2.72)
t s. t
nOCJIe,l{Hee paBeHcTBo BbITeKa.eT H3 (1.17) H ,l{aeT, TaKHM 00-
pa30M, onepanHOHaJIbHYIO HHTepnpeTanHIO pacCTORHH5I Me}K-
):(Y pa30HeHH5IMH KaK MepbI KalleCTBa nporH03a. B CHJIY (1.16)
paCCT05lHHe Me)K):(Y pa30HeHH5IMH B (R, S) = d (R, S)/ N2 paB"
HO CYMMe aCHMMeTpHtIHbIX Mep (2.72) B (S, R S) H B (R, R S),
npe,l{cTaBJI5IR COOOH, TaKHM oopa30M, cHMMeTpH30BaHHbIH Ba-
pHaHT (2.72).
OtIeBH,l{Ho, B (S, RS) = 0 Tor,na H TOJIbKO TOr,l{a, Kor,l{a
S = RS, T. e. S R H KJIaCCbI R. 0,l{H03HaQHO nporH03HpYIOT-
C5I BKJIlOlIeHHbIMH B HHX KJIaCCaMH St.
npH cTaTHcTHlIecKoH He3aBHCHMOCTH R H S paCCT05lHHe
B (S, RS) = P t (1- p:.) = (l-B (S» B (R),
liTO ,l{OCTHraeT TeM OOJIbIllerO 3HalleHH5I, lIeM MeHbIlle B (8) H
60JIbIlle B (R), T. e. lIeM OJIH}Ke R K TpHBHaJIbHOMY, a 8 K YHH-
BepCaJIbHoMY pa30HeHH5IM, ,l{JI5I KOTOpbIX B paBHO e,nHHHne.
B oo eM CJIYlIae CTaTHCTHlIeCKaR He3aBHCHMOCTb Booo e He
CB5I3aHa C KaKHMH-JIHOO 3KCTpeMaJIbHbIMH CBOHCTBaMH pac-
CT05lHH5I. .aeJIO B TOM, QTO npH CTaTHCTHlIeCKOH He3aBHCHMO-
CTH KatIeCTBO nponopnHOHaJIbHOrO nporH03a SHatIeHHH S O,l{H-
HaKOBO ,l{JI5I JIIOOoro KJIaCCa Rs. PaCCTORHHe 1Ke YtIHTbIBaeT
Ka)K,l{bIH KJIaCC Rs He O,l{HHaKOBO, a nponopUHOHaJIbHO ero Be-
P05lTHOCTH. n03TOMY MomHO YXY,l{IllHTb KaqeCTBO nponopnHO-
101
I
I
j
I
...
HaJIbHOrO npOrH03a B MO.L{eJIH 3, eCJIH .L{OnOJIHHTeJIbHO YXY.L{-
illHTb B03MO)KHOCTH nporH03a St H3 HanOJIHeHHhIX KJIaCCOB
R s, ,l{JISI KOMneHcaUHH YJIYlIIllHB B03MO}KHOCTH nporH03a H3
MaJIbIX KJIaCCOB Rs (no cpaBHeHHIO co CJIYlIaeM CTaTHCTHlIec-
KOH He3aBHcHMoCTH). npOHJIJIIOCTpHPyeM CKa3aHHoe Ha npH-
Mepe .L{HXOTOMHlIeCKHX npH3HaKoB. .QJI5I HHX paCCT05lHHe
d (R, S) == 2 (a + d) (b + c) == 2 (a + d) (N .L- a -d),
(2.73)
liTO HMeeT BH,l{ x (N - x) npH x N. 5ICHO, liTO MaKCHMYMa
N u
3Ta BeJIHlIHHa ,l{OCTHraeT npH x = 2' TaK liTO HaHXY.L{IlIHH npor-
H03 on pe,l{eJI 51 eTCH paBeHCTBOM a + d =:: b + c, a He CTaTHCTH-
lIecKoH He3aBHCHMOCTblO.
BnpolleM, eCJIH MaprHHaJIbHble qaCTOTbI cpHKcHpoBaHbI 3a-
paHee, 3TO paBeHcTBo Mo:meT H He ,nOCTHraTbC5I. .QeHCTBHTeJIbHO,
np H 3a,l{aHHbIX P s. H p. t MaKCHMH3anH5I paCCT05lHH5I (2.73)
3KBHBaJIeHTHa MHHHMH3anHH P:t. 9TOT MHHHMYM, KaK He-
TPY.L{HO nOKa3aTb, .L{OCTHraeTC5I npH
Pst= Ps.+p.t _ (8,t=I,2),
2 4
eCJIH Bce 3TH BeJIHlIHHbI HeOTpHnaTeJIbHbI, T. e. eCJIH Ps. +p.t
0,5. ECJIH }Ke Ps. + p. t < 0,5 XOT5I ObI ,l{JI5I O,l{HOH napbI
s, t (a 3TO B03MO}KHO He OOJIee lIeM ,l{JI5I O.L{HOH napbI), TO nOJIa-
raeM Pst = 0, nOCJIe lIero OCTaJIbHbIe BeJIHlIHHbI onpe,l{eJI5IIOTC5I
no 3a.L{aHHbIM aBTOMaTHlIeCKH.
nYCTb, HanpHMep, (R) == (0,1; 0,9), (S) == (0,2; 08).
Tor.L{a npH CTaTHCTHlIeCKOH He3aBHCHMOCTH (RS) == (0,02;
0,08; 0,18; 0,72) H cS (S, R) == 0,44. ECJIH }Ke onpe,l{eJIHTb
(RS) == (0; 0,1; 0,2; 0,7), TO nOJIYlIHTC5I OOJIbIllee 3HalleHHe
cS (R, S) == 0,46.
CJIe,l{yeT OTMeTHTb TaK}Ke, liTO npH MaJIbIX ml H m 2 pac-
CT05lHHe cS ,l{aJIeKO He ,l{OCTHraeT CBoero MaKCHMaJIbHOrO 3Halle-
HH5I 1, 0Y,l{YlIH orpaHHlIeHo BeJIHlIHHOH (m - 1 )/m, r,l{e m =
== min (m 1 , m 2 ). B lIaCTHOCTH, npH ,l{HXOTOMHlIeCKHX npH3Ha-
Kax cS 1/2, TaK liTO 3HalleHHSI cS COcpe,l{OTOlleHbI B ,l{OCTaTOll-
HO Y3KOM HHTepBaJIe H npH aHaJIH3e npHXO,l{HTC5I PYKOBO,l{CT-
BOBaTbC5I H MaJIbIMH OT JIHlIHSlMH cS.
npH CTaH,l{apTHbIX CTaTHCTHlIeCKHX npe,l{nOJIOJKeHH5IX pac-
npe.L{eJIeHHe BeJIHlIHHbI V N (cS - cS*), r,l{e cS* - paCCTOSlHHe
B reHepaJIhHOH cOBoKynHocTH, CTpeMHTC5I K HOpMaJIbHOMY C
HYJIeBbIM Cpe,l{HHM H KOHellHOH ,l{HcnepCHeii. STOT <paKT, cnpa-
u u
Be.L{JIHBbIH H ,nJI5I OCTaJIbHbIX onHcaHHhIX nOKa3aTeJIeH, MO)KeT
102
ObITb OObIlIHbIM 00pa30M HCnOJIb30BaH ,nJIH npOBepKH rHnOTe3
u u
o paBeHCTBe paCCTOHHHM ,npyr ,npyry HJIH 3a,naHHOM BeJIHlIHHe
(CM. [66]).
C ,npyroH CTOpOHbI, paCCT05lHHe MO)KeT BbIlIHCJIHThCH no
MaTpHll.aM OO'beKT-OO'beKT H, CJIe,nOBaTeJIbHO, YHHBepCaJIbHO
npHMeHHMo K KalleCTBeHHbIM npH3HaKaM JIIOOOH CTPYKTYPbI.
3Ta B03MO}KHOCTb, BnpolleM, xapaKTepHa ,nJISI Bcex nOKa3a-
TeJIeH, OCHOBaHHbIX Ha MaTpHll.aX OO'beKT-OO'beKT (CM. n. 4.2.4),
H B HaCTOH:r.nee BpeMH HCCJIe,nOBaHa BeCbMa MaJIO. BnpolleM,
OcooeHHocTH cOBoKynHoro npocTpaHcTBa pa30HeHHH H paH}KH-
pOBaHIrli, OTMelleHHbIe Ha c. 53, He ,nalOT OCOObIX OCHOBaHHH ,nJIH
u
onTHMH3Ma no nOBO,ny B03MO}KHOCTeM o,nHOBpeMeHHoro aHa-
.nH3a pa3HOTHnHbIX npH3HaKOB oe3 CBe,neHH5I HX K O,nHOMY
THny llIKa.n.
OTMeTHM Tenepb, liTO Mo,neJIH nporH03a, paCCMOTpeHHbIe Ha-
MH, MorYT ObITb oooo:r.neHbI B HanpaBJIeHHH M3MeHeHHSI KpHTe-
pHeB npHHHTH5I pellIeHHH.
TaK, CToxaCTHlIeCKoe npaBHJIO npHHHTHH pellIeHHH MO}KeT
ObITb OCHOBaHO He TOJIbKO Ha P (tIs), HO H Ha JIIOOOH MOHOTOHHOH
<PYHKIl.HH cp (p (tIs)). HanpHMep, lIaCTOTa npHnHCbIBaHHH 00-
pa3a St OO'b€KTaM KJIaCCa Rs MO}KeT onpe,neJIHTbCH npOnOpll.HO-
HaJIbHO - log p (tIs) = log Ps. -log Pst. B 3TOM CJIYllae npH
Mo,neJIH 1 nOpO}K,neHH5I OO'beKTOB KalleCTBO nporH03a onpe,ne-
'" '"
JI5IeTCH YCIl08flOU 8flmponueu
H(SIR)= - logPstlogp(tls)=H(RS)-H(R), (2.74)
s, t
r,ne H (R) - 3HTponHH (1.9) pacnpe,neJIeHHH (R).
I1pH OTCYTCTBHH HH<popManHH 00 R KalleCTBO nporH03a
H (SI U) = H (S), TaK liTO npHparqeHHe nOKa3aTeJIH KalleCTBa
OT nOJIYlleHHSI HH<popManHH 00 R B .cpe,nHeM paBHo:
I (R, S) = H (SIR)-H (S) = H (RS)-H (R)-H (S).
(2.75)
3Ta BeJIHlIHHa Ha3WBaeTCH KOJIHlIeCTBOM HH<popManHH B R
00 S I1JIH, YllHTbIBaH CHMMeTpHlIHOCTb npaBoH lIaCTH paBeHcT-
Ba (2.75), 8saUMflOU uflcpopMal{ueU R H S. OTHOCHTeJIbHOe npH-
pa:r.neHHe nOKa3aTeJIH (2.74)
I (R, S)
H (S)
,naeT HanpaBJIeHHYIO HH<popMaIl.HOHHYIO Mepy KalleCTBa npo-
rH03a. Ooa nOKa3aTeJIH (2.75) H (2.76) lliHpOKO HCnOJIb3YIOT-
CH B HCCJIe,nOBaHH5IX [47, 102]. B paMKax cTaH,napTHoH CTaTHC-
u
THQeCKolf Mo,neJIH npH rHnOTe3e He3aBHCHMOCTH BeJIHlIHHa
(2. 76)
103
21 aCHMnTOTHtIeCKH pacnpe,neJleHa TaK :H<e, KaK H 'X". H3BecT-
HO, tITO 1 = 0 Tor,na H TOJlbKO Tor,na, Kor,na R H S CTaTHCTHtIeC-
KH He33BHCHMbI.'
AHaJIOrHtIHbIM oopasoM MO:H<HO MO,nH<pHII.HpOBaTb ,neTepMH-
HHCTCKoe npaBHJIO npHHSlTHSI pellIeHHSI, npHqeM 3,neCb ,llOCTa-
TOtIHO ,naJKe, tITOObI He Bce 3HatIeHHSI OOoOIneHHOro KpHTepHH
OblJIH nOJIO>KHTeJIbHbI, a XOTSI ObI O,nHO, MaKCHMaJIbHoe.
o,nH<pHKaII.HSI KpHTepHSI npH ,neTepMHHHCTCKOM nporH03e
HMeeT caMOCTOSlTMbHoe 3HatIeHHe C TOtIKH 3peHHSI pellIeHHSI npo-
6JIeMbI onHcaHHSI. IleHcTBHTeJlbHO, pe3YJIbTaT ,neTepMHHHCT-
CKoro nporH03a M HO p3CCM3TpHBaTb KaK onHcaHHe 3Haqe-
HHM St npH3HaKa S TeMH 3HatIeHHSlMH Rst KOTOpbIM OH nOCTaB-
JIeH B COOTBeTCTBHe npH nporH03e. HCnOJIb30BaHHe p (tIs)
,llJISI npHHHTHH pemeHHH MO:H<eT npHBo,nHTb K TpHBHaJIbHMM
onHcaHHHM npH HaJlHtIHH KJlaCCa St, CO,llep:H<a ero OOJIbllIHH-
CTBO OO'beKTOB. OO'beKTbI SToro KJlaCCa MorYT COCTaBJIHTb
60JlbllIHHCTBO cpe,nH OO'beKTOB KJIaCCOB R 8. nOJIyqeHHOe B pe-
SYJlbTaTe onHcaHHe «KJlaCC Sf xapaKTepeH ,nJIH Bcex Rs»
oy,neT Onpe,lleJIHTbCH <paKTHtIeCKH pacnpe,neJIeHHeM S, oe3
KaKOrO-JlHOO YtIeTa COBMeCTHoro pacnpe,neJleHHH. TaK, ,ne-
TepMHHHCTCKoe npaBHJlO npHHHTHH pemeHHH npHMeHHTeJIbHO
K ,naHHbIM rpHHBy,na -IOJl3 (TaOJl. 2.5) npHBO,llHT B JlIOOOM
CJIyqae K nporH03Y «OTCYTCTBHe 3aOOJIeBaHHH», H CJIe,llOBaTeJIb-
.,
HO, K BMBO,llY 0 TOM, tITO JlIO,nH npeHMYIneCTBeHHO XOJIepOH He
OOJleIOT, He3aBHCHMO OT cpaKTa npHBHBKH. HCHO, tITO .9TOT Bbl-
BO,n Ha caMOM ,neJle nOpOJK,neH CTPYKTYPOH pa30HeHHH S:
B paccMaTpHBaeMoH cOBoKynHocTH C caMoro Haq3JIa 92 % He-
OOJIeBllIHX JIIO,neft. nO.9TOMY npH pacn03HaBaHHH oopa30B KJlaC-
Cbl Sf OObItIHO oepYTCH paBHOtIHCJIeHHbIMH, HO npH 3HaJlH3e
.9KCnepHMeHTaJlbH@X MHorOMepH X aHHMX pacnpe,neJIeHHe
pa30HeHHH S He KOHTpoRHpyeTcH H ccne,nOBaTeJI eM , a onpe,lle-
JIHeTCH rrpHpo,nOH H3yqaeMOH CHCTeMbI. B qaCTHOCTH, ,nJIH COnH-
aJIbHbIX HCCJIe,llOBaHHH no,nOOHble CJIYtIaH BeCbMa xapaKTepHbI.
npH OneHKe CBSl3H npH3HaKOB MbI YtIJIH OTMeqeHHOe OOCTOH-
TeJIbCTBO 3a CqeT HCnOJlb30B3HHH He aOCOJIIOTHbIX SH3qeHHH
BepOHTHOCTeH npaBHJIbHbIX nporH030B, 3 HX npHpaIneHHH. no- ;
BH,nHMOMY, npHpaIneHHHMH LKe eCTeCTBeHHO BOCnOJIb30BaTbCH
H B sa,naqe onHcaHHH. By,neM OCYIneCTBJlHTb npe,nCK3saHHe
Ha OCHOBe BeJIHtIHH 8t = P (tIs) - p.t, Bblpa1KalOInHX
H3MeHeHHe ,nOJIH OO'beKTOB oopa3a St B KJIaCCe Rs no cpaBHe-
HHIO C 9TOH ,nOJIeH BO BceH cOBoKynHocTH m. B tIaCTHOCTH,
.neTepMHHHCTCKHH nporH03 no MaKCHMYMY f1st Ha ,llaHHbIX
rpHHBy,na - IOJIa ,naeT BnOJlHe npHeMJIeMOe C co,nepJKaTeJIb-
u
HOH TOtIKH speHHH onHcaHHe «HenpHBHTbIe npeHMYIneCTBeH-
104
j
.........
HO 3a60JIeBalOT (TaK KaK L\21 = 87,8% - 91,6% < L\22 =
= 12,2% - 8,4%), a npHBHTbIe - HeT (L\11 > L\12)>>.
OqeBH,l{HO, B CHJIY JIHHeHHOCTH BbIpa:meHHH Q npH CToxac-
THQeCKoM nporH03e BeJIHllHHa npHpa eHHH Q (R t S) - Q (S)
MOLKeT 6b1Tb npe,l{CTaBJIeHa KaK SHalleHHe Q OT npHpaw.eHHH
st, TaK llTO Q (R, S) - Q (8) eCTb cpe,nHee 3HalleHHe Q, no-
JIyqaeMOe npH HCnOJIb30BaHHH A.t. npH ,lJ.eTepMHHHcTCKOM
u
LKe nporH03e B CHJIY HeJIHHeHHOCTH onepaUHH BSHTHH MaKCH-
MYMa BeJIH1JHHa npHparneHHH Q He COBna.naeT C SHalleHHeM Q
OT npHparneHHH L\st. B lIaCTHOCTH, npH MO eJIH 1 nOpO>K,l{eHHH
06'beKTOB KalleCTBO onHcaHHSI no MaKcHMYMY L\st xapaKTepH-
u
3yeTC5I BeJIHlIHHOH
11 = Psl max L\8t = max (Ps£-Ps. p.t).
s. t t s t
(2.77)
9Ta xapaKTepHcTHKa KalleCTBa onHcaHHH npe,lJ.JIO>KeHa
TI. C. PocToBueBbIM [91] B CBH3H C aBToMaTH3aUHeR npouec-
u
ca nOJIyqeHHH cO,l{ep:maTeJIbHO npHeMJleMbIX onHcaHHH ,lJ.JIH MHO-
rOMepHoH COUHOJIOrHlIeCKoH HHcpOpMaUHH (CM. n. 3.2.8). Hs
BbIpa)KeHH5I (2.77) CJIe,l{yeT, qTO f.t = 0 Tor,l{a H TOJlbKO Tor.na,
KOr,l{a npH3HaKH CTaTHCTHlleCKH He3aBHCHMbI.
2.2.3. 8KcnepuMeflTaAbfloe cpaBfleflue Mep
CBJl3U
KaK BH,nHO H3 BbIllieCKa3aHHoro, KJIaCC Mep CB5I-
3H, HHTepnpeTHpyeMbIx KaK MepbI KalleCTBa nporH03a, BeCbMa
IlIHpOK. KaKYIO >Ke Mepy BbI6paTb HCCJI e,nOBaTeJI 10 , aHaJIH3H-
PYlOrneMY KOHKpeTHbIe ,naHHbIe, eCJIH HeT HHKaKHX cooopa}Ke-
HHH B nOJIb3Y TOR HJIH HHOH MO,neJIH nporH03a?
KaK Y}Ke rOBOpHJIOCb, B STOM CJIYllae Mepa CB5ISH .nOJI>KHa
,l{HKTOBaTbC5I BbI6paHHbIM MeTO,l{OM peweHHH 3a,lJ.all KOHCTPYH-
pOBaHH5I HJIH onHCaHH5I. C STOM TOllKH 3peHH5I npe,l{CTaBJI5IIOT
OC06bIH HHTepec MepbI a, (0, <D 2 , cS, B03HHKalOrnHe np" CToxac-
THlIeCKOM npaBHJIe npHH5ITHH pelIIeHHH, nOCKOJIbKY HMeHHO OHH
nOHBJI5IIOTC5I B paMKax MO,l{eJIeH KOHCTpYHpoBaHH5I H onHca-
HH5I B TepMHHax Ta6JIHU 06'beKT-npH3HaK H 06'beKT-06'beKT
(CM. 9 3.2, 4.2)
B TO )Ke BpeMH ,l{JI5I ueJIeH HenOCpe,l{CTBeHHoro HCnOJIb30Ba..
HHH B aHaJIH3e KOHKpeTHbIX )(aHHbIX npe,nCTaBJIHeT HHTepec
cpaBHeHHe pa3JIHIIHbIX Mep CB5I3H He no 3aJlO:>KeHHbIM B HHX
, .
npe,l{nOCbIJIKaM, a no HX nOBe,l{eHHIO Ha:BceM MHO>KeCTBe nap
npH3HaKOB HJIH, TOllHee, Ta6JIHn' COnpH}KeHHOCTH. Kor,l{a
MOJKHO rOBopHTb 06 O,l{HHaKOBOM nOBe,l{eHHH Mep CB5I3H? Kor,l{a
105
"'no "' _"
- -----------oi
OHn COB n a.L{aIOT , HJIH nponopUHOHaJIhHbI, HJIM JIHHeHHO npe.L{-
CTaBMMhI .L{Pyr qepe3 .L{pyra? nO-BM.L{Hl\10MY, HaH60JIee eCTeCT-
BeHHO CqHTaTb O.L{HHaKOBbIM nOBe.L{eHHe Tex Mep CBH3H, KOTO-
pbIe O.L{HHaKOBO ynOp5I.L{OqHBaIOT napbI npH3HaKOB no CTene-
HH CBH3H. YTOqHHM CKa3aHHoe. nYCTb cP (Pst) H 'lI' (Pst) -
,l{Be MepbI CBH3H, onpe,l{eJIeHHbIe Ha BceB03MO)KHbIX Ta6JIHuaX
COnpH)KeHHOCTH {Pst I Pst 0, Pst = I}. BY.L{eM rOBO-
S ,t
pHTb, qTO cp H 'I' KOMOflomOHHble, eCJIH cy ecTByeT TaKaH MO-
HOTOHHa51 <PYHKUHH f (x), qTO cP = f '1'. HHbIMH CJIOBaMH,
.L{JI5I JII06bIX npH3HaKOB x, y, x', y' HepaBeHCTBO <p (x, y) >
> cp (x', y') HMeeT MeCTO TOr,l{a H TOJIbKO Tor,na, Kor.L{a'l' (x,y) >
> 'I' (x', y') (HJIH 'I' (x, y) < cP (x', y'».
OKa3bIBaeTC5I, KOMOHOTOHHbIX Mep CBH3H OqeHb MaJIO. Toq-
Hee, eCJIH rJIa,l{KHe, KaK CPYHKUMM OT Pst, MepbI CBH3H cp H 'I'
KOMOHOTOHHbIe, TO rpa)J.HeHT (BeKTOp npOH3BO)J.HbIX : t }
u u
O,l{HOH H3 HHX .L{OJIJKeH JIHHeHHO BbIpa)KaTbCH qepe3 BeKTOp
npOH3BO,l{HbIX a a'i' ,l{pyroiL ECJIH )Ke rpa,l{HeHTbI He CB5I3aHbI
Pst a
COOTHOlIIeHHeM a acp = ex a '1'. + , a 3TO TaK ,l{JI5I Bcex paCCMOT-
Pst Pst
peHHbIX KOHKpeTHbIX Mep, TO cP H 'P He KOMOHOTOHHbI [90].
n03TOMY ,l{JI5I 60JIbllIHHCTBa Mep CBH3H MO}KHO rOBopHTb
He 0 KOMOHOTOHHOCTH, HO JIHIllb 06 onpe,l{eJIeHHOH CTeneHH
u
CXO,l{CTBa, H3MepHH ero ,l{OJIeH MHomeCTBa TaKHX qeTBepOK npH-
3HaKOB x, x', y, y', T. e. nap Ta6JIHU (Pst), (p;t) , Ha KOTOpbIX
MepbI <p H 'I' Be,l{YT ce651 O,l{HHaKOBO. ECJIH O,l{HHaKOBoe nOBe,l{e-
HHe Mep cp H 'P HMeeT MeCTO pOBHO ,l{JIH nOJIOBHHbI Bcex B03- 1
MO)KHbIX CJIyqaeB, TO MOJKHO rOBopHTb 0 TOM, qTO cp H 'I' CTa-
THCTHqeCKH He CBH3aHbI. BJIH30CTb MepbI MHO}KeCTBa O,l{HHaKO-
Boro nOBe,l{eHH5I K e,l{HHHue rOBopHT 0 60JIbIllOM CXO,l{CTBe
cp H 'P, a 6JIH30CTb K HYJIIO - 0 CXO,l{CTBe cp H - 'P.
OnHllleM pe3YJIbTaTbI 3KcnepHMeHTaJIbHOrO aHaJIH3a CXO,l{-
CTBa Mep CBH3H, npOBe,l{eHHOro n. C. PocToBueBbIM [90],
KOTOpbIH HCnOJIb30BaJI CJIe,l{YIO HH BapHaHT MeTO,l{a MOHTe-
KapJIo.
CHaqaJIa CJIyqaHHO BbI6HpaIOTC5I pa3MepbI Ta6JIHU (Pst) H
(p;t) (paBHoBepoHTHo H3 qHCeJI 2, 3, 4). 3aTeM npH BbI6paHHbIx
pa3MepHOCT5IX CJIyqaHHO BbI6HpaIOTC5I Ta6JIH bI cOnp5l}KeHHO- :
CTH (Pst) H (p t). Ta6JIHua cOnp5l}KeHHOCTH {Pst I Pst 0,
Pst = I} reHepHpyeTcH B npe,l{nOJIO}KeHHH paBHoMepHocTH
pacnpe,l{eJIeHH5I ee 3JIeMeHTOB. 3TO npOH3BO,l{HTC5I nOCJIe,l{OBa-
TeJIbHbIM reHepHpoBaHHeM BeJIHqHH Pst Ha OCHOBe CJIe,l{YIO ero
CBOHCTBa. ECJIH BeKTop = ( 1' ..., n) paBHoMepHo pacnpe-
106
):{eJIeH B 06JIaCTH {6 I 6i O, 6i == I}, TO npH <pHKCHpOBaH-
HbIX 61 = al'...' 68 == a 8 KOMnOHeHTa 68+1 pacnpe.L{eJIeHa TaK
)Ke, KaK (1 - i l ai}(1 - 'YJII (n-s-I»), r)J.e 'YJ - paBHoMepHo
pacnpe.L{eJIeHHaH BeJIHlIHHa.
KaK nOKa3aJIH paClIeTbI, MaTpHQbI COnp5l)KeHHOCTH, creHe-
plipoBaHHbIe 3THM cnoc060M, cooTBeTcTBYIOT OTHOCHTeJIbHO
CHJIbHO CBH3aHHbIM napaM npH3HaKOB. ,LLJI5I o6eCneQeHH5I B03-
MO)l{HOCTH BapbHpOBaHHH CTeneHblO CBH3H B aHaJIH3e HCnOJIb-
30BaJIHCb MaTpH[(bI Pe == EP + (1 - E) 8, r.L{e b 8t = P8. X
X p. h TaK liTO MaprHHaJIbHble pacnpe.L{eJIeHHH .L{JIH P e H P
cOBna.L{aIOT. npH E == 0 nOJIYlIaeM Pe = B, T. e. npH3HaKH
CTaTHCTHlIeCKH He3aBHCHMbI. YBeJIHlIeHHIO E OTBellaeT YBeJIH-
lIeHHe CBH3H npH3HaKOB.
PaCCMaTpHBaJIHCb CJIe.L{YIOIII.He K03<P<P H UHeHTbI CBH3H: A(2. 65),
w (2.68), (2.77), I (2.75), R (2.76), <D 2 (2.57), K 2 (2.58),
T2(2.59), a TaK)Ke l-<S, r.L{e -paccToHHHe Me)K,l{Y HOMHHaJIb-
HbIMH npH3HaKaMH (1.15), 1 - , r,l{e & - paCCTOHHHe Me}K,l{Y
paHroBbIMH npH3HaKaMH, nOJIYlIaeMbIMH <PHKCa[(HeH nOp5l,l{Ka
3HalleHHH, 3a,l{aBaeMOro HX HYMepa[(Heii. KpoMe Toro, HCnOJIb-
30BaJICH K03qxpH[(HeHT paHroBoH KOppeJIH[(HH KeH,l{aJIJIa, KOTO-
pbIH MO)KHO onpe,l{eJIHTb lIepe3 paCCTOHHHe CPOPMYJIOH (CM.
c. 222)
6(R-, S)-6(R, S)
17 b ==::
Vi) (R) 6 (8) ,
r.L{e R- nOJIYlIaeTC5I H3 R 06paTHbIM ynopH,l{OlleHHeM KJIaCCOB.
MHoroKpaTHoe reHepHpoBaHHe Ta6JIH[( cOnp5l)KeHHOCTH
P H P' ,l{aJIO B03MO)KHOCTb nO.L{ClIHTaTb ,l{OJIIO e CJIYlIaeB O)(H-
HaKOBoro ynopH,l{OlleHHH nap npH3HaKoB ,l{JI5I Kam.L{WX BYX
li3 3THX nOKa3aTeJIeH. 3Ta BeJIHqHHa 51BJIHeTCH O[(eHKOH Bepo-
51THOCTH O.L{HHaKOBoro nOBe,l{eHHH Mep CBH3H, cTaH,l{apT Ha51
ollIH6Ka KOTOpOH MOJKeT 6bITb OneHeHa KaK Ve (1 - e)/ L,
r.L{e L - lIHCJIO HcnbITaHHH. B qaCTHOCTH, npH L == 1000 MaK-
CHMaJIbHaH 0lIlH6Ka paBHa 0,0158. no npaBHJIY 30' npaKTHlIec-
KH HeBepoHTHo OTKJIOHeHlie e OT HCTHHHoro 3HalleHH5I 60JIb-
we qeM Ha 0,05.
HIi)Ke npHBe,l{eHbI nOJIYlIeHHbIe 3HalleHli51 e ,l{JI5I Kam,l{oH
napbI npH3HaKOB npH E == 1 (Ta6JI. 2.6) H g == 0,25 (Ta6JI. 2.7).
Ha nepeCeqeHHH CTpOKH H cToJI6[(a CTOHT BeJIHlIHHa 1000 e
u
,nJI5I cooTBeTcTBYIOI..I.(eH napbI Mep CB5I3H, TaK liTO CXO.L{CTBO Mep
CB5I3H BbIpa)KeHO B Ta6JI. 2.6 H 2.7 B npOMHJIJIe.
113 3THX Ta6JIH[( BH,l{HO, liTO H3MepliTeJIH paHrOBOH CBH3 H
(2.78)
101
!!F - ........-;.:;v-
" -
Ta6JIHIJ.a 2.6
1 1-0 "
A w p. R 0 1 T K 1-0 't
A w p. R 1 cJ>1 T K 1-0 1- S "C
A '10001 838 I 823 1 812 I 747 , 743 I 780 I 736 I 705 , 546 I 494
w I 110001 869 1 928 , 775 , 769 I 837 1 820 I 690 1 549 1 495
f.t J I /10001 867 1 843 I 833 I 835 f 766 I 769 571 I 515
R J I I I 10001 825 I 814 I 845 , 812 I 716 I 547 r 489
I I I , , 1 1000/ 963 I 809 I 727 I 825 I 559 / 509
<1>2 I I I I I 1 1 000 I 81 7 , 740 1 819 1 561 , 509
T I I I I I , I 10001 875 r 697 1 571 I 473
K I I I I 1 I 1 I 1000r 605 I 533 I 485
I-B I , 1 I I I , I 110001 586 , 522
1-6 I I I I / I I I I / 1000/ 851 ,
1
TaoJIHua 2.7
A J 10001 610 1 614 I 604 J 603 I 605 , 567 I 532 r 568 I 454 I 456
w I 110001 886 I 908 I 774 I 768 1 836 , 801 , 505 , 478 I 484
1 ill 0001 882 I 814 , 814 I 842 I 763 I 569 1 502 I 498
R I I , , 10001 862 1 854 , 880 I 817' 553 , 490 1 504
I I , , r 110001 986 I 826 I 751 , 649 I 506 I 512
a>2 1 I I 1 r , 10001 822 I 749 , 655 I 504 I 504
T J , I , 1 1 J 10001 885 I 495 I 482 I 498
K 1 I I 1 I 1 , I 1000' 404 I 439 1 485
1 - i) I I 1 I I 1 1 I , 1 000 I 581 I 501
1- 6' , 1 I I I , I I I 1 1 000 I 790
,!{alOT pe3YJI, bTaTbI npaKTHQeCKH He3aBHCHMbIe OT H3MepHTe-
.lIeB HOMMHaJIbHOH CB5I3H. KpOMe Toro, BH,!{HO 3HallHTe.lIbHOe
CXO,!{CTBO Me:H<,!{Y HH<popManHOHHbIMH nOKa3aTeJI5IMH I H R II
cooTBeTcTBYIO HMH ,!{HcnepcHoHHbIMH xapaKTepHCTHKaMH <1>2 H
w. JIlOoonbITHO, liTO nOKa3aTe.lIH <D, K, T pa3JIHlIalOTCH ,!{O-
BO.lIbHO CH.lIbHO, liTO OO'bSlCH5IeTCSI nO-BH,!{HMOMY, B.lIHSlHHe f
BapbHpOBaHH5I pa3MepOB TaOJIHU cOnp5lJKeHHocTH.
108
Heo6xo,nMMO OTMeTMTb TaK>Ke, qTO SKCnepHMeHTaJlbHbIH
aHaJIH3 Mep CBH3H Ha OCHOBe ,npyrHx C006pa)KeHHH npOBo,nHJlCH
paHee B [139, 161]. I1cCJle,noBaHHe [161) BocnpoH3Be,neHo U
npO OJI1KeHO B [29].
2.2.4. Mepbl CBR3U, B03flUKalOU{ue
8 AOZAUfleUHblX Mo8eARX
nOcJle npe,nhI.L{y ero 0630pa Mep CBH3H, KOHCT-
PYHpyeMbI B :epMHHax Ta6JIHUbI COnpH}KeHHOCTH, 06paTHMCH
K JIorJIHHeHHOH Mo,neJlH KaK cpe,nCTBY nOpO>K,neHH5I Mep CBH3H.
MbI Y1Ke Heo,nHOKpaTHO OTMellaJlH B03MO}KHOCTb HCnOJlb30Ba-
HH5I nOKa3aTeJIeH TV KaK xapaKTepHCTHK CBH3H npH3HaKOB
MH01KeCTBa V (CM., B lIaCTHOCTH, aHaJIH3 Ha c. 69-71).
,aJl51 CHCTeMbI ,nBYX ,nMXOTOMHlIeCKHX npH3HaKOB A H E
nOJlHa51 MY JIbTHnJIHKaTHBHa51 MO,neJIb (2.17) co,nep}KHT e,nHH-
cTBeHHbIM napaMeTp TAB, OTHOCHInHHCH 'K B38HMOCB5I3H STHX
npH3HaKOB. B COOTBeTCTBHH C (2.17) -
TAB = ad .
be
aTO - He qTO HHoe, °KaK no,npOOHO paCCMaTpHB8BlIIeeCH
B n.2.2.1 nepeKpeCTHoe OTHOIlleHHe T.
AMHTHBHa51 <popMa JiorJlHHeHHOH MO,neJIH ,naeT Mepy CB5I311
A, = log TAB = log ad -log be. (2.79)
CPOPMYJIbI (2.17) H (2.18) n03BOJlHIOT paccMaTpHBaTb TaK}Ke
nOKa3aTeJIb TAB CBH3H A H B B CHCTeMe MHorHX npH3HaKOB
KaK cpe,nHee reOMeTpHlIeCKoe nepeKpeCTHbIX OTHOllieHIIH ( s AB )
pacnpe,neJIeHHH (A, B), nOJIyqaeMbIX npH BceB03MO}KHbIX KOM-
I 611Hau.HSIX S AS 3HaqeHHH: OCTaJIbHbIX npIl3HaKOB. .
JI. ry,nM3H npe,nJIOmHJI TaK)Ke HCnOJIb30BaTb ,nJIH oueHKH
CB5I3H npH3HaKOB TO YMeHbllIeHHe OTKJIOHeHHH TeopeTHlIeCKoro
pacnpe.L{eJIeHH5I OT Ha6JIIO.L{eHHOrO, KOTopoe ,nOCTHraeTCH YlIe-
TOM STOM CB5I3M:B JIOrJIMHeHHOH l Mo,neJIH. B o6 eM BH,lte STY
H,l(elO MOJKHO C<POPMY JIHpOBaTb · CJIe,nYIOInHM 06pa30M.
nYCTb F 1 H F 2 - ,nBe cOBoKynHocTH 'no,nMHOJKeCTB npH3Ha-
KOB, t onpeJ{eJI5IIOInHe :HepapXHlIeCKHe MO,neJIH HI' °u H 2" 'npH-
I j qeM"P 1"S;; 'F 2.' Tor,na pa3HocTb 3HaqeHHit' nOKa3aTeJI5I t (2.17)
I 'X.2 (Hl) - 1. 2 (H 2 ) _ xapaKTepH3yeT TO :.YMeHbllIeHlle «Heo6'bsrc-
j
-
1 HMeHHO T3KOB CMbICJI K09cpcpHI.(HeHTOB MHO:>KeCTBeHHOH H tJa-
CTHQHOH eTepMHHau.HH B perpeCCHOHHOH MO eJlH KOJIHtJeCTBeHHoro
aHaJIH3a ( M. [42], a TaK2Ke 3.2).
103
,., { -" ""r"'-J
HeHHOH BapHanHH» pacnpeAe eHHH (H3MepHeMOH nOKa3aTe eM
(2.67», KOTopoe AocTHrHYTo YlIeToM B MOAe H Be HlIHH v
.n H V F 2 - Fl. ECJIH npHHHTa CTaHAapTHaH CTaTHCTHlIe-
CKaH MOAe b H 'X,2 (HI)' 'X,2 (H 2 ) HMelOT paCnpe,l{eJIeHHH 'X,2
c n i H n2 CTeneHHMH CBo6oAbI COOTBeTCTBeHHO, TO pa3HOCTb
X 2 (HI) - 'X,2 (H 2) HMeeT pacnpe,ne eHHe 'X,2 C n i - n 2 CTene-
HSlMH CBo6oAbI (no lIHCJIY MHO}KeCTB B F 2 - F I) [128]. OTclOAa
C e,l{yeT, liTO pa3HOCTb 'X,2 (HI) - 'X,2 (H 2 ) MO}KeT npHB eKaTbCH
u
.n H OneHKH CTaTHCTHlIeCKOH 3HallHMOCTH BK lOlIeHHSI MHO-
1KeCTB V E F 2 - FIB MOJl.e b.
B COOTBeTCTBHH C (2.6)
"'2
X 2 (Hl}-X 2 (H2}= P8log :. , (2.80)
s ps
rAe p , P: - TeOpeTHQeCKHe OneHKH pacnpe,l(e eHHSI P: Ha
OCHOBe HI H H 2 cooTBeTCTBeHHo, TaK liTO
" 2
log : = log n 'r v = A V.
ps V E F 2 -F I V.E F 2 -F I
CJ1eAOBaTeJIbHO,
'X,2 (HI) - X 2 (H 2 ) = Ps A V-
S VEF 2 -F I
cpeAHee 3HalleHHe Be HQHH + A V .
B qaCTHOCTH, ,l{ H ,nBYMepHoro pacnpe,ne eHHH npH F I =
= {{A}, {B}}, F 2 = {{A}, {B}, {A, B}} B KalleCTBeMepbI CBH-
3H A H B nO YllaeTcH
A = 'X,2 (Hl)-'X,2 (H 2 ) = Pij A B = AAB (a + d -b-c).
i, j
OTHOcHTe bHaH BeJIHlIHHa YMeHbllleHHH HeOO'bHCHeHHoH
BapHan HH
d 2 .= x;2 (Hi) _X;2 (H 2 ) (2.81)
x 2 (H j )
npHHHMaeTcH B KalleCTBe MepbI CBSl3H Me}K,ny pacnpeAe eHHeM
Ps H cOBoKynHocTblO F 2 - Fl npH YlIeTe cOBoKynHocTH Fl.
.-.. 1
B lIaCTHOM CJIYlIae, KorAa FI = 0, TaK liTO P =1') = 2 n '
nOKa3aTeJIb (2.81) no aHaJIOrHH C KOJIHlIeCTBeHHbIM aHaJIH30M
MO>KeT 6bITb Ha3BaH K03<pqn1nHeHToM MHO}KeCTBeHHoH AeTepMH-
HMHH pacnpe,neJIeHHH Ps H Mo,neJIH H 2 . B APyroM lIaCTHOM c y-
lIae, KorAa F 2 OTJIMlIaeTCSI OT Fl Bcero OAHHM MHO}KeCTBOM W,
nOKa3aTeJIb (2.81) MO}KeT 6bITb Ha3BaH K09<p<pHnHeHToM lIaCTHOH
110
JlerepMHHaUHH pacnpe,l{eJIeHH5I P8 H MHO)l{:eCTBa npH3HaKOB
W npH yqeTe HH<pOpMaII.HH, ,l{aBaeMOH MO.L{eJIblO HI. n03TOMY
B o6meM cJIyqae nOKa3aTeJIb (2.81) MO)KeT ObITb Ha3BaH Koat/J-
t/JutiUeflmoM 'l,acmU'I,flO-MflOJ/CecmBeflHOU aemepMuflatfuU pac-
npeAeJIeHH5I H CHCTeMbI F 2 - FI npH <pHKCHpOBaHHOH MO,l{eJIH
HI.
AHaJIOrH5I C KOJIHqeCTBeHHbIMH nOKa3aTeJI5IMH COCTOHT B
TOM, qTO H B TOM H B ,l{pyroM CJIyqae K03<P<PHII.HeHT onpe,l{eJIH-
eTC5I KaK qaCTHOe OOo5lCHeHHOH H HeOOn5lCHeHHOH BapHaUHH,
TOJIbKO B KOJIHqeCTBeHHOM CJIyqae BapHaQH5I H3MepHeTC5I ,l{HC-
nepCHeH, a 3,l{eCb- nOKa3aTeJIeM X 2 . 3HaqHTeJIbHO OOJIee cy e-
CTBeHHOH OTJIHqalO eH ocooeHHOCTblO npe,l{CTaBJI5IeTC5I TO, qTO
B KOJIHqeCTBeHHOM CJIyqae 3TH nOKa3aTeJIH H3Mep5llOT COOT-
BeTCTBHe onHCaHH5I 3HaqeHHH: O.L{HHX npH3HaKOB qepe3 3Haqe-
HH5I ,l{pyrHx, TOr,l{a KaK 3,l{eCb peQb H,l{eT 0 COOTBeTCTBHH pac-
npe,l{eJIeHH5I no BceM paCCMaTpHBaeMbIM npH3HaKaM ero onH-
caHHIO. B03MO)KHO, HMeHHO 3TO OOCT05lTeJIbCTBO OQ'b5lCH5IeT
cy ecTBeHHbIH He,l{OCTaTOK K03<PCPHII.HeHTOB CB5I3H (2.53)-
(2.54), OCHOBaHHbIX Ha 't: OHH nJIOXO «CHrHaJIH3HpYIOT» 0 B03-
u
MO)KHOCTH 0,l{H03HaqHOrO nporH03a 3HaqeHHH O,l{HOrO npH3Ha-
Ka no 3HaqeHH5IM ,l{pyroro ,l{a)Ke KOr,l{a TaKa51 0,l{H03HaqHOCTb
HMeeTC5I. MO,l{eJIeH onHCaHH5I H KOHCTpYHpOBaHH5I B TepMH-
Hax pacnpe,l{eJIeHHH, KOTopbIe ObI npHBO,l{HJIH K TaKHM npor-
H03HbIM nOKa3aTeJl5IM, nO-BH,l{HMOMY, nOKa He cy ecTByeT.
. .... .
-- "
r .IlaBa 3
AHAJ1H3 TA5J1HU. 05'bEKT-nPH3HAK
3.1. J1HHEnHA AnnpOKCHMAI.{HSI
6ynEBCKHX TA6nHQ 1
3.1.1. 3aaatlu KOH,CTpyupOBaflUR
KOAUtf,eCTBeHflblX ljJaKTopoB
PaccMoTpHM TaOJIHUY oooeKT-npH3HaK pa3Mep-
HOCTH N X M, KOTopaH COCTOHT H3 n nO,l{TaOJIHU X It pa3MepHo-
TH Nxmh, (k = 1, ..., n), r,l].e mk =M. Ka}K,l{aSl MaTpHu a
X k cooTBeTcTByeT 3a,lJ.aHHOMY npH3HaKY x k . ,L{JIH npOCTOTbI i
OrpaHHQHMCSI CJIyqaeM, KOr,l{a Xh, - 6YJIeBCKHe MaTpHUbI HO- f
MHHaJIbHbIX npH3HaKOB. B npHHUHne TaKHe MaTpHUhI MorYT
HMeTb <POPMY O,l{HorO H3 CJIe,l{YIO HX THnOB:
A) Xh, - oYJIeBCKaSl MaTpHua N X mh, C mlt opTOrOHaJIb-
HbIMH CTOJIOuaMH, npHtJ:eM CYMMa STHX CTOJIOUOB ,!{aeT CTOJIOeu,
Bce KOMnOHeHTbI kOToporo paBHbI e,l{HHHue; Ka}K,l{bIH CTOJIOeu
OTBeqaeT OT,l{eJIbHOMY 3H8QeHHIO npH3HaKa, TaK QTO SJIeMeHT
(i, s) MaTpHUbI X It paBeH e,!{HHHue B TOM H TOJIbKO B TOM CJIyqae,
Kor,l].a Xk (i) = s; B Ka}K,l{oH cTpoKe X It COAep}KHTCH O,l{Ha H
TOJIbKO O,l{Ha e,l{HHHua, T. e. npH3HaK 3a,l{aeT pa30HeHHe Ha MHO- I ' .
1KeCTBe oooeKToB;
B) .X k - oYJIeBCKHH N-MepHbIH CTOJIOe , BbI,l{eJISlIO HH He-
KOTopoe MHO}KeCTBO OO'beKTOB; STOT CTOJIOeu xapaKTepH3yeT j
,lJ.HXOTOMHtJ:eCKHH npH3HaK C cpHKCHpOBaHHbIM HYJIeM, T. e. f
u u
H3MepeHHbIH B lIIK8JIe OTHoweHHH;
B) Xh, - npOH3BOJIbHaH oYJIeBCK8H M8TpHua N X mh"
TaK tJ:TO ee CTOJIOUbI He OOH38TeJIbHO opTOrOHaJIbHbI ,!{pyr ,lJ.py-
ry, npHQeM HeKOTopbIe CTpOKH MorYT ObITb HYJIeBbIMH. nep-
Boe 03HatJ:aeT, tJ:TO 3HaqeHHH npH3HaK8 HeaJIbTepHaTHBHbI:
npH3H8K HeO,l{H03HaQeH, BTopoe - tJ:TO ,!{JI5I HeKOTophIX 000-
eKTOB 3H8qeHHH npH3HaKa He YKa3aHbI. TaKYIO nO,l{MaTpHUY J
tJ:8CTO ObIBaeT YMeCTHbIM paCCMaTpHBaTb KaK cOBoKynHocTb
1 OnHcaHHble pe3YJI'bTaTbJ nOJlytIeHbI COBMeCTHO C H. B. MytIHH-
ROM (67J.
112
CTOJI61l. 0B , COOTBeTCTBYIO HX OT,!{eJIbHbIM 3HaqeHH5IM npH3H8-
Ka, T. e. KaK cOBoKynHocTb npH3H8KOB, 38,!{8HHbIX B cpopMe B.
06bIqHO MbI oy,!{eM CqHT8Tb, qTO M8TpHnbI X k 38,lJ.aHbI B <pop-
Me A, XOT5I ,!{aJIbHeHlIIee C COOTBeTCTBYIO HMH MO,lJ.H<p HKa UH5IMH
ciipaBe,lJ.JIHBO H AJISI ,!{pyrHx <jJOpM. ..po"'; _is
npOOJIeMa, KOTOpa51 6Y,lJ.eT paCCMOTpeHa B ,lJ.aHHOM pa3,!{eJIe,
COCTOHT B TOM, qTOObI CKOHCTpYHpOBaTb TaKOH KOJIHqeCTBeH-
HbIM npH3HaK z, KOTOpbIH HaHOOJIee OJIH30K K MaTpHnaM X k ,
H YKa3aTb npaBHJIa ero HHTepnpeTaUHH B TepMHHax HCXO,lJ.HbIX
npH3HaKOB x k .
K8K OneHHTb OJIH30CTb Z K HCXO,lJ.HbIM npH3H8KaM?
npe,lJ.CTaBJI5IeTC5I eCTeCTBeHHbIM HCnOJIb30BaTb ,!{JI5I 3TOM
1l.enH paCCT05lHH5I Me*,lJ.Y npH3HaKaMH H HX npOeKnH5IMH Ha
COOTBeTCTBYlOmHe nO,lJ.npOCTpaHCTBa. TIpH 3TOM B03HHKalOT ,!{Ba
B HeKOTopOM CMbICJIe CHMMeTpHqHhIX nOKa3aTeJISi.
TIepBbIH Ji3 HHX oueHHB8eT OJIH30CTb Me>KJ(Y HCXO,!{HbIMH MaT-
pHnaMH H ox npoeKUHSlMH Ha OCb KOHCTpyopyeMoro nOKa3aTe-
JI5I :
n n
IXh-PzXkI2= d(Xk,PzXk). (3.1)
k=l k=1
BTOpOH OCHOB8H H8 oueHKe OJIH30CTH MeLK,!{Y KOHCTPYHPy-
eMbIM npH3HaKoM Hero npOeKnH5IMH Ha no,!{npOCTpaHCTBa
L (X k ), nopO*,lJ.aeMbIe HCXO}J.HbIMH npH3HaK8MH:
n n
d (z, Ph Z) = I Z-P k Z 1 2 , (3.2)
k=l k=l
r,lJ.e qepe3 Ph 0003HaQeH npOeKTOp Ha L (X k ): Ph =
= Xk(XIXk)-lXI.
3a,lJ.aqa KOHCTpYHpOBaHH5I Z CPOPMYJIHpyeTC5I TOrJ(a K8K
v
3a,!{aqa MHHHMH3aUHH O,lJ.HOrO H3 3THX nOKa3aTeneH no BceM
N-MepHbIM Z, Y,lJ.OBJIeTBOp5l1O HM YCJIOBHIO
ZTZ = 1 (3.3)
Toro, qTO Z HMeeT e,lJ.HHHQHYIO JtJIHHY.
PerneHHe 3a,lJ.aQH MHHHMH38UHH (3.1) npH Orp8HHqeHHH
(3.3) oY,lJ.eM Ha3bIBaTb tjJaKmopoM-npoeKmopoM, a peweHHe
38,!{aQH MHHHMH38UHH (3.2) npH OrpaHHtleHHH (3.3) - tjJaK-
mopoM-npoeKt{ueu.
PaCCMOTpHM 3TH 3a,!{aQH no OT,lJ.eJIbHOCTH.
n 0 C T P 0 e H H e H a H a JI H 3 <p 8 K TOp 0 B-
n p 0 e K TOp 0 B. npe>K,lJ.e Bcero 3aMeTHM qTO
n
d(X h , PzXh)=d(X, PzX)=IX_pzXJ2, (3.4)
k= 1
113
r,l{e X = (Xl' ..., X n ) - BC5I TaOJIH[(a HCXOJ(HbIX ,l{aH-
HhIX N X M. 3TO CJIe,l{yeT H3 TOrO, qTO paCCTOSlHHe d (X It ,
P z X k ) paBHO CYMMe paCCTOSlHHH d (X kS , P zX ks ) ,l{JI5I OT,l{eJIb-
HbIX CTOJIO[(OB X ks MaTpH[(bI X k, TaK qTO oo aSl CYMMa no k
H no S ,l{aCT KaK pa3 CYMMapHoe paCCT05lHHe Me*,l{Y BceMH
CTOJIO[(aMH H HX npOeK[(H5IMH Ha {az}.
B COOTBeTCTBHH C pa3):(eJIOM n.8 npHJIO:>KeHHSI 3a,l{aqa MM-
HHMH3a[(HH (3.4) npH YCJIOBHH (3.3) eCTb He qTO HHoe, KaK 3a-
,l{aqa 0 rJIaBHbIX KOMnOHeHTax MaTpH[(bI X, T. e. 00 OTbICKaHHH
COOCTBeHHbIX BeKTopOB MaTpH[( XTX H XXT, COOTBeTCTBYIO HX
HaHOOJIbIlleMY COOCTBeHHOMY qHCJIY, peIlleHHe KOTOpOH onHca-
HO B pa3,l{eJIe n.4 npHJIOmeHH5I.
KaKOB CMbICJI MaTpH[( XTX H XXT npHMeHHTeJIbHO K ,l{aH-
HOMY CJIyqalO?
3JIeMeHTaMH MaTpH[(bI X TX SlBJI5IIOTC5I CKaJI5IpHbIe npOH3-
Be,l{eHH5I OYJIeBCKHX CTOJIO[(OB x ks H XU, COOTBeTCTBYIO HX S-MY
3HaqeHHIO k-ro H t-MY 3HaqeHHIO I-ro npH3HaKOB. OqeBH,l{HO, !
(x ks , xlt) = N:f:, (x ks , x ks ) == N: (XU, XU) = N:, (3.5)
r,l{e N:, N:, N= - KOJIHqeCTBa OO'beKTOB, HMelO HX s-e 3Ha-
qeHHe npH3HaKa Xk, t-e 3HaqeHHe npH3HaKa Xl, s-e 3HaqeHHe
Xk H l-e 3HaqeHHe x t O,l{HOBpeMeHHO. CJIe,l{OBaTeJIbHO, MaTpH-
U3 XTX pa3MepHOCTH M X M COCTOHT H3 OJIOKOB SkZ =
= xIx z = (N ) pa3MepHOCTH mk X mZ, SlBJISlIO HXCH
TaOJIHuaMH COnpSl:>KeHHOCTH npH3HaKOB Xk, Xl (k, 1 = 1, ..., n),
npHqeM ,Z(HarOHaJIbHbIe OJIOKH Sk = xIx k 51BJISIIOTCSI ,l{HarO-
HaJIbHbIMH MaTpH[(aMH pacnpe,l{eJIeHHH npH3HaKOB Xk: Sk =
= Diag (N , ..., N ,).
nYCTb A - MaKCHMaJIbHOe coocTBeHHoe qHCJIO MaTpH[(bI
XTX, a a = (aks) - cooTBeTcTBYIO HH coocTBeHHbIH BeKTOp. J
KOMnoHeHTbI aks BeKTopa a xapaKTepH3YlOT MacIllTaobI H3Me-
peHHSI ,l{JISI OT,l{eJIbHbIX 3HaqeHHH S npH3HaKoB x k . ,[{eHcTBH-
TeJIbHO, rJIaBHaSl KOMnOHeHTa Z MO:>KeT ObITb npe,l{cT3BJIeH3
COrJIaCHO cpopMYJIe (0.14) B BH,l{e JIHHeHHOH KOMoHHaUHH 6y- I
neBCKHX CTOJIO[(OB x ks MaTpH[(bI X:
1 1 n mk
Z = _ Xa = _ ahB x ks ,
VA VA k=-l 5= 1
TaK qTO B Z CYMMHPYIOTC5I 3HaqeHHSI npH3HaKOB, O[(eHHBae-
MbIe BeJIHIIHHaMH aks, BbIpa:>KaIOInHMH BKJIa,l{ COOTBeTCTBYIO HX
u
3HaqeHHH B Z.
114
OlI.eHKY 3HaqHMOCTH npH3HaKOB H HX 3HaqeHHH Y,l{06HO npo-
H3BO HTb no HX BKJIa.L{Y B BeJIHqIfHY Iv == ZTXXTZ, nOCKOJIbKY
MaKCHMYM Iv COOTBeTcTByeTMHHHMYMY I X-PzX 12=Tr (XTX)-
_ ZTXXTZ = nN - A. npH 3TOM (CM. 0.4)
Iv =ZT XXT Z = aT XT Xa = N ahs alt. (3.6)
k. I s. t
OTCIO,l{a BH,l{HO, 1ITO Iv eCTb CYMMa BeJIHqHH
st N kl
akl = 5t ahs au,
(3.7)
KOTopbIe xapaKTepH3YIOT BKJIa,l{ Ka)K,noH KOM6HHaUHH (s, t)
3HaqeHHH Xk, Xl B onTHMH3HPyeMbIH nOKa3aTeJIb. BeJIHqHHa
BKJIa,l{a B A, OT,l{eJIbHOrO 3HaqeHH5I S npH3HaKa Xk paBHa CYMMe
n m l
k N k1 ksT X
as=aks stalt=aks X a.
1=1 t=1
(3.8)
AHaJIOrHQHO BeJIHqHHa BKJIa,na B Iv npH3HaKa Xk paBHa:
k m k n ml kl T T
ah = as = ahs N st alh =ak Xk Xa, (3.9)
5 5=1 1=1 t=1
r.L{e ak = (akt, ..., ahmk)-BeKTOp MaCllITa6HpYIO HX MHO)KH-
TeJIeH npH cToJI6uax MaTpHUbI X k .
BBe,l{eHHbIe BeJIHqHHbI (3.7) - (3.9) xapaKTepH3YIOT 3Ha-
u u
qHMOCTb npH3HaKoB, OT,l{eJIbHbIX 3HaQeHHH H nap 3HaqeHHH
npH3HaKoB C TOqKH 3peHH5I HX BKJIa,l{a B onTHMH3HPyeMbIH no-
Ka3aTeJIb H MorYT HCnOJIb30BaTbCH ,l{JIH HHTepnpeTaUHH nOJIY-
qaeMOrO cpaKTopa. qeM 60JIblIle BeJIHqHHa 3Ha1lHMOCTH, TeM
60JIbllie cooTBeTcTBYIO ee 3HaqeHHe (HJIH npH3HaK, HJIH KOM-
6HHaUHH 3HaqeHHH) np05lBJI5IeTC5I B z, TaK qTO Z MO)KeT pac-
CMaTpHBaTbCH KaK cpopMaJIbHOe npe,l{CTaBJIeHHe Toro o606 alO-
ero nOH5ITHH, KOTopoe npOHBJIHeTCH B HaH60JIee 3HaqHMWX
npH3HaK8X H 3HaqeHHHX. TaK, cpaKTOp Z, ,l{JIH KOToporo HaH6o-
JIee 3HaqHMbIMH OKa3aJIHCb Te npH3HaKH nCHXOJIOrH1JeCKOrO
TeCTa, KOTopbIe xapaKTepH3YIOT HHTeJIJIHreHTHOCTb, MO)KeT 6bITb
Ha3BaH «<t>aKTopOM HHTeJIJIHreHTHOCTH». nO.L{06HbIH cnoco6 HH-
u
TepnpeTaUHH no BKJIa,l{Y B onTHMH3HpyeMbIH nOKa3aTeJIb pe-
KOMeHAyeTC5I H B ,l{aJIbHeHllleM TeKCTe.
n
PaCCMOTpHM Tenepb MaTpHQY XXT == XkXr, co6CTBeH-
k=l
HhIe BeKTopbI KOTOpOH H HBJIHIOTCH rJIaBHbIMH KOMnOHeHTaMH
(CM. pa3,l{eJI n.4). OqeBH,l{HO, (i, j)-H 3JIeMeHT MaTpHUbI xkxI
115
.f
!
paBeH e,l{HHM[(e TOr,l{a H TOJIbKO TOr,l{a, KOr,l{a Xk (i) = Xk (j)
(CQHTaeTC5I, qTO Xk 3a,l{aHO B cpopMe A, T. e. xapaKTepH3yeT pa3-
6HeHHe) , a B OCTaJIbHbIX CJIYlIa5lX paBeH HYJIIO. 3TO 03HaqaeT,
liTO xkxI - 6YJIeBCKa51 N X N MaTpHna pa36HeHH5I no <
k-MY npH3HaKY. 06 HH 3JIeMeHT CYMMapHOH MaTpHnbI XXT
npe,l{CTaBJI5IeT C060H KOJIHqeCTBO npH3HaKOB, HMelO HX OAH-
HaKOBbIe 3HaqeHH5I Ha COOTBeTCTBYIO HX 06'beKTax. 3Ta BeJIH-
lIHHa qaCTO HCnOJIb3yeTC5I B aHaJIH3e ,l{aHHbIX KaK 3BpHCTHlIec-
Ka51 OneHKa CB5I3H 06'beKTOB. B ,l{aHHOH KHHre OHa B03HHKaeT B
u u u
paMKax MO,l{eJIeH JIHHeHHOH annpOKCHManHH, paCCMaTpHBaeMbIX
u u
B 3TOH H CJIe,l{YIO eH rJIaBax.
rJIaBHa51 KOMnOHeHTa KaK c06CTBeHH H BeKTOp MaTpHnw
CB5I3H XX T MOJKeT ObITb HaH,l{eHa C nOMO hlO npOne,l{ypbI HTe- '
paTHBHoro nepeClIeTa no cpopMYJIe (n. 15). npH BbIoope BeK-
Topa u = (1,1, ..., 1) B KalleCTBe HallaJIbHOrO nOJIYlIHM BeK-
TOp z (1) = XXT U , i-51 KOMnOHeHTa KOToporo paBHa cYMMe
CB5I3eH i C OCTaJIbHbIMH OO'beKTaMH. ,[{aJIbHeihIlHe HTepanHH MO-
rYT npHBecTH K H3MeHeHHIO CTPYKTYPbI KOMnOHeHT BeKTopa
z (t), HO OObIqHO He OlleHb cy ecTBeHHOMY. n03TOMY ,l{JI5I nep-
BOHaqaJIbHOH npHKH,l{KH 3HalleHHH cpaKTopa MO)KHO HCnOJIb30-
BaTb BeJIHlIHHbI CYMMapHbIX CBSl3eH 06'beKTOB C OCTaJIbHbIMH.
CJIe,l{yeT TaK:>Ke OTMeTHTb, liTO HTepaTHBHa51 npone,l{ypa
nOCTpOeHH5I BeCOBbIX oueHOK 06'beKTOB KaK KOMnOHeHT C06CT-
BeHHoro BeKTopa, COOTBeTCTBYIO ero MaKCHMaJIbHOMY COOCT-
BeHHOMY lIHCJIY, npHMeH5IeTC5I ,l{OBOJIbHO ,l{aBHO Ha OCHOBe 3B-
pHCTHlIeCKHX C006pa:>KeHHH. B qaCTHOCTH, I.I1HpOKO H3BeCTHO
nO,l{OOHOe perneHHe «3aAaqH 0 JIH,l{epe» [6] ,l{JI5I MaTpHn ,l{OMH-
HHpOBaHH5I.
B TOM CJIYlIae, KOr,l{a MHHHMaJIbHOe 3HalleHHe (3.1) He CJIHI.I1-
KOM MaJIO, 3TO MOJKeT paCCMaTpHBaThC5I, KaK YKa3aHHe Ha TO,
liTO HCXO,l{HbIe ,l{aHHble He MorYT 6bITb oxapaKTepH30BaHbI e,l{HH-
CTBeHHbIM cpaKTopOM. B 3TOM CJIYlIae B COOTBeTCTBHH C OO HMH
nOJIOJKeHH5IMH MeTO.L{a rJIaBHbIX KOMnOHeHT KaK MeTO,l{a an-
npOKCMManHH ,l{aHHbIX CJIe.L{yeT nepeHTH K paCCMOTpeHHIO MaT-
pH OCTaTOqHbIX CB5I3eH 8 1 = XTX - 'Aaa T , XXT - AZZ T H
HCKaTb HX MaKCHMaJIbHOe coocTBeHHoe lIHCJIO "'2 H COOTBeTCT-
BYIOInHe HOpMHpOBaHHbIe COOCTBeHHbIe BeKTopbI a 2 H Z2.
rJIaBHbIe KOMnOHeHTbI Z H Z2 opTOrOHaJIbHbI H nOpO:>K.L{aIOT Ta-
KYIO nnOCKOCTb, KOTOpa51 HaHJIyqIIlHM oopa30M annpOKCHMH- ]
PyeT TaOJIHny .L{aHHbIX B CMbICJIe KpHTepH5I d (X, PX),' r,ne
P - onepaTOp npoeKTHpOBaHH5I Ha 3TY n:rrOCKOCTb. AHaJIOrHlI-
HO nepBbIe p r JIaBHbIX KOMnOHeHT Z1, Z2, ..., zP, KOHCTpYHpye-
Mble KaK COOCTBeHHhIe BeKTopbI, COOTBeTCTBYIOInHe MaKCHMaJIb-
HbIM COOCTBeHHbIM lIHCJIaM OCTaTOllHbIX MaTpHU XXT-
116
p
_ AtZtztT, MHHHMH3HPYlOT IX-PXI 2 , r,ll.e P-MaTpH",a npOeK-
t= 1
TOpa Ha L (Zp). KpHTepHeM OCTaHOBKH npo",ecca KOHCTPYHPO-
BaH HH rJIaBHbIX KOMnOHeHT B COOTBeTCTBHH c n.B 51BJI5IeTC51
p
6JIH30CTb K e,l{HHH[(e BeJIHqHHbI }2Asl N n' nOCKOJIbKY
5=1
N n m k
A.s=Tr(XXT)=Tr{XTX)= N =nN.
5=1 k=15=1
OObIqHO xoporno HHTepnpeTHpyeMble KOMnOHeHTbI BbI,l{e-
JI5IlOTC5I npH 3HaqeHH5IX STOM BeJIHqHHbI nOp5l,l{Ka 0,7-0,8.
n 0 C T P 0 e H H e H H H T e p n peT a "' H 51 <p a K-
TOP a-n p 0 e K "' H H. KpHTepHH (3.2) MO)KeT ObITb npeoopa-
30BaH CJIe,l{YID HM oopa30M:
d (z, Ph z) = (z-P h Z)T (z-P h z) =
k k
n
= ZTZ- ZTPhZ,
k=1 k
r,l{e nOCJIe,l{Hee BbIpa)KeHHe nOJIyqeHO B CHJIY PkP k = Ph. TaK
KaK no YCJIOBHID (3.3) ZTZ = 1, TO MHHHMH3a",H5I (3.2) CBO,ll.HTCH
u
K 3KBHBaJIeHTHOH 3a,l{aqe MaKCHMH3a",HH
ZTPZ, (3.10)
n
r,l{e P = Pk' npH YCJIOBHH (3.3).
k=1
9Ta 3a,l{aQa ObIJIa paccMoTpeHa r. CanopTa [160]; KaK H
npe bI,l{YIna5l, OHa CBO,l{HTC5I K 3a,naqe 0 COOCTBeHHbIX BeKTopax
H 3HaQeHH5IX, HO TOJIbKO He ,l{JI5I MaTpH",bI XXT, a ,nJI5I MaTpH-
[(hI P === Pk. PerneHHeM 3a.L{aQH MaKCHMH3a",HH (3.10) npH
k
YCJIOBHH (3.2) 51BJI5IeTC5I coocTBeHHbIH BeKTop Zl MaTpH",bI P,
cooTBeTcTBYIDlI.(HH ee HaHOOJIbrneMY coocTBeHHoMY qHCJIY A 1 =
r. ZlTPZl (CM. 0.4).
r.. O,l{HaKO B OTJIHqHe OT 3a,l{aqH 0 cpaKTope-npoeKTope 3,l{eCb
HMeIOTC5I TpHBHaJIbHble «napa3HTHqeCKHe» perneHH5I, KaK B 3a-
n
,Qaqe KaHOHHqeCKOrO aHaJIH3a. JIIDOOH BeKTop zEn L (X k )
k==1
51BJI5IeTC5I COOCTBeHHbIM BeKTopOM MaTpH",bI P, COOTBeTCTBYID-
HM MaKCHMaJIbHOMY coocTBeHHoMY 3HaqeHHID n. ,[{eHcTBHTeJIb-
HO, PkZ = Z ,l{JI5I JIIDOOrO k, eCJIH zEn L (X k ). DOSTOMY
117
.,. ....-'V4 'II"
Pz= PhZ = nz. To, 'ITO n L (X k ) He CBOAHTC5I K HYJIIO, ra-
k
paHTHpyeTC5I TeM, 'ITO BeKTOp U = (1,1,..., 1) C eAHHHlIHbIMH
KOMnOHeHTaMH 3aBeAOMO npHHaAJIeLKHT L (X k ) AJI5I JIlOooro k
KaK CYMMa CTOJIOUOB MaTpHUbI X k (npH3HaKH 3aAaHbI B <popMe
A). 9TO TpHBHaJIbHOe pemeHHe HaB5I3aHO cpOpMOH npeACTaBJIe-
HH5I AaHHbIX H He OTpaLKaeT HX COAepLKaHH5I. n03TOMY npa-
uecc nOHCKa COOCTBeHHbIX BeKTopOB CJIeAyeT HallHHaTb C MaT-
PHUbI OCTaTOtIHbIX CB5I3eH P - n uu T , rAe u = ulV N -
,...,J
pe3YJIbTaT HOpl\fHpOBaHH5I BeKTopa u, TaK liTO I u I = 1.
aTO COOTBeTcTByeT MeTOAY, onHcaHHOMY B pa3AeJIe ll.4. MaTpH-
,...,J,...,J n
ua nuu T COCTOHT H3 OAHHaKOBbIX 3JIeMeHTOB N ' TaK liTO Ha ca-
MOM ,neJIe pellb HAeT 0 TOM, 11 TOObI B HallaJIe H3 Ka*Aoro 3JIe-
MeHTa MaTpHUbI P BbItIeCTb noporOBoe 3HalleHHe nl N, 3JIHMH-
HHpYIOInee TpHBHaJIbHOe pemeHHe u = (1, ..., 1). npH 3TOM
Bce cJIeAYlOmHe coocTBeHHbIe BeKTopbI ZS OYAYT opTOrOHaJIbHbI
U, TaK 'ITO (ZS, u) = z = 0, T. e. OYAYT HMeTb HYJIeBbIe
l
cpeAHMe.
PaccMoTpHM Tenepb CMbICJI MaTpHUbI, P = Pk. MaTpHua
k
P k paBHa:
P k = X h (XI Xh)-l xI = (p ),
(3. 11 )
rAe
p. _ { II N=, eCJIH Xk (i) = Xk (j) = s,
lJ - 0, eCJIH Xk (i) i= Xk (j),
TaK KaK XIXk=Diag(N ,..., N k).
TaKHM oopa30M, npoeKTop Ha L (X k ) HMeeT BHA MaTpHUbI
CB5I3eH cooTBeTcTBYlOmero pa30HeHH5I C eCTeCTBeHHbIMH Beca-
MH KJIaCCOB: CB5I3H Me*AY OO'beKTaMH pa3HbIx KJIaCCOB HYJIe-
Bble, a MeLKAY OO'beKTaMH OAHoro H Toro *e KJIaCCa paBHbI Be-
JIHlIHHe, oopaTHoH ero lIHCJIeHHOCTH.
CYMMapHa51 MaTpHua P = (Pij) eCTb MaTpHua CB5I3eH MeLKAY
OO'beKTaMH, rAe Pij - 3TO cYMMa BeCOB II N Tex 3HalleHHH S
npH3HaKoB Xk, KOTopbIe COBnaAalOT AJI5I OO'beKTOB i H j. CYMMa
mh
3JIeMeHTOB MaTpHUbI Ph COCTaBJI5IeT (I/N ) N N= = N,
s= 1
TaK liTO ee cpeAHee 3HalleHHe paBHo II N, a cpeAHee 3HalleHHe
MaTpHUbI P, CJIeAOBaTeJIbHO, eCTb nl N. I1MeHHo 3Ta BeJIHlIHHa
BbIlIHTaeTC5I H3 3JIeMeHTOB MaTpHIl.bI P AJI5I 3JIHMHHHpOBaHH5I
118
TpHBHaRbH ro aKTOpa, TaK qTO Cpe.L{Hee 3HaqeHHe OCTaTOq-
HbIX CB5I3eH paBHO HYJIIO.
MaTpHUa P MOLKeT ObITb npe.L{CTaBJIeHa qepe3 X CJIe.L{YIOIIIJIM
06pa30M. 0003HaqHM qepe3 S .L{HarOHaJIbHYIO M X M MaTpH[(Y,
COCTaBJIeHHYIO H3 .L{HarOHaJIbHbIX KJIeTOK Sk = xlx k :
8 1 0 . . . 0
S == 0 S2 ... 0
o o. .. Sn
TOf.L{a HeTpY.L{HO BH.L{eTb, qTO P == XS-IXT, npHqeM P k =
=XkSi l xI.
DepeH.L{ervl K oocy:m.L{eHHIO BonpOCOB HHTepnpeTa[(HH nOJIY-
qaeMbIX cpaKTopOB.
KaK BbIpa3HTb Z B BH.L{e JIHHeHHOH KOMOHHa[(HH CTOJIO[(OB
MaTpH[(bI HCXO.L{HbIX .L{aHHbIX X?
0003HaQHM qepe3 Xk = Pkz npOeK[(HIO cpaKTopa Z Ha no.L{-
npOCTpaHCTBO L (X h ). OqeBH.L{HO, i-51 KOMnOHeHTa 3Toro BeK-
Topa npH i E R: paBHa Cpe.L{HeMY 3HaQeHHIO Z Ha OO'beKTaX
KJIaCCa R::
1
Zl$ = N k
s
] Zi.
iER k
s
npH 3TOM Xk = Xhah, r,ne ak = ( , ..., Zm h) - BeKTOp BHYT-
pHKJIaCCOBbIX Cpe.L{HHX .L{JI5I pa30HeHHSl Rk. DOCKOJIbKY BeKTop
Z coocTBeHHbIH .L{JI5I MaTpH[(bI P, TO pz = AZ, TaK 1ITO
1 1 1 1
z=-Pz= - Xk= - Xkak= --Xa, (3.12)
A Iv k= 1 Iv k Iv
r.L{e a = (a l , ..., an) - BeKTOp, COCTaBJIeHHbIH H3 ak KaK no.L{-
BeKTopOB. 3TO .L{aeT npe.L{CTaBJIeHHe Z B BHAe JIHHeHHOH KOMOH-
Ha[(HH CTOJIO[(OB MaTpH[(bI X, npHQeM COOTBeTCTBYIO He K03cp-
cpH[(HeHTbI (MaCI.IlTaOHPYIO He MHO)KHTeJIH) npOnOp[(HOHaJIb-
HbI Cpe.L{HHM 3HaQeHH5IM z no BbI.L{eJI5IeMbIM cToJiO[(aMH MHo:me-
CTBaM OO'beKTOB.
BeKTOp a Ha caMOM .L{eJIe 51BJI5IeTCH COOCTBeHHb1M .L{JISI MaT-
pH bI S-lX T X pa3MepHOCTH M X M H COOTBeTcTByeT TOMY
:iKe COOCTBeHHOMY QHCJIY A, QTO H z. B 51QeHKe ((k, s), (I, i»
MaTpH[(bI S-lXTX HaXO.L{HTC5I BeJIHQHHa N:fIN: - YCJIOBHaH
BepoSlTHOCTb i-ro 3HaQeHH5I Xl, npH YCJIOBHH, QTO Xk npHMeT 3Ha-
qeHHe s.
119
.... "..)... . "
...;
....
i
1JTo6bI y6e.L{HTbC5I B CKa3aHHOM, YMHO:mHM paBeHCTBO
'Az = Xa CJIeBa Ha (X X k )-IXI:
A, (xi X k )-l x z = (xi X h )-l xl Xa,
HO (xIx h )-1 Xkz = ak, TaK tlTO Aak = Ski XIXa. B TepMHHax
BeKTopa a = (aI' ..., an) H MaTpHnbI X = (Xl' ..., X n ) nOCJIe,l].-
HHe paBeHCTBa 03HatlalOT, tlTO 'Aa = S-IX1Xa, tlTO H Tpe6oBa-
JIOCb ,nOKa38Tb. ..
KaK OneHHTb 3HaqHMOCTb npH3HaKOB H HX 3HaqeHHH? TaK
:me, KaK H B npe.L{bIAy eM CJIyqae, - HX BKJIaAOM B BeJIHqHHY
A = ZTPZ. nOCKOJIbKY A, =, ZT ( Pk)Z = ZTPkZ, TO BeJIHqHHa
k k
etA = ZTPk Z M01KeT CJIY}KHTb OneHKOH 3HaqHMOCTH npH3HaK8
Xk ,l].JISI KOHCTpYHpyeMoro <t>aKTopa z. OqeBHAHO,
(X,k = ZT X k ak = (z, aks x ks ) = aks (z, x ks ),
s s
QTO ,naeT pa3JIOJKeHHe etk Ha COCTaBJISlIOIII.He aks = aks X
X (z, x ks ) , BbIpa*alOIII.He 3HaqHMOCTb OT,lJ.eJIbHbIX 3HaqeHHH
npH3HaKOB.
HCnOJIb30B8HHe OneHOK 3HaqHMOCTH etk H etka npH HHTep-
npeTanHH <t>aKTopa nOJIHOCTblO aH8JIOrHQHO paCCMOTpeHHOMY
B 38,lJ.8Qe 0 cpaKTope- npOeKTope.
CJIeAyeT OTMeTHTb, tlTO ooe p8ccMoTpeHHhIe 3aA8QH KOHCTPYH-
pOBaHHSI CXOAHbI no MaTeMaTHtleCKOH CTpYKType. Hx MO}KHO
paccMaTpHBaTb KaK QaCTHbIe CJIYQ8H OOJIee oo eH 3a,naQH npo-
u
eKTHpOB8HH5I, OCHOB8HHOH H8 paCCMOTpeHHH «KOCbIX» CKaJISlp-
HhIX npOH3BeAeHHH BH,na <x, y>=x Ay, 8 He (x, y) = xTy,
H «KOCbIX» npoeKTopOB XB-IXT. npH A = S-I, B=(XT X)
nOJIYQHM KpHTepHH (3.1),_.8 npH A ;:::; E, B = E - KpHTe- I
PHH (3.2). 6 iI ...v
LPaCCMOTpHM CJIe.L{YIOInHH -npHMep. nYCTb::HMeeTc5I 6 OOD-
eKTOB H 5 npH3HaKoB, 3HatleHHfI KOTOPbIX. COAep}KaTCSI B
TaOJl. 3 .1. t ij)--
nepenHlIIeM STY TaOJIHny B - BH,l].e M8TpHl(bI H3 OYJIeBCKHX
CTOJIOnOB, COOTBeTCTBYIO HX OT.L{eJIbHhIM 3HaQeHHflM npH3Ha-
KOB. npH STOM B03HHK8eT Bonpoc 0 TOM, CKOJIbKHMH CTOJIOua-
MH 3aMeH5ITb HMelO HeCH TPH ,l(HXOTOMHQeCKHX npH3HaKa.
B COOTBeTCTBHH C n. 1.1.4 3TO 38BHCHT OT Toro, pacCMaTpHBalOT-
CfI JIH HX 3HaQeHHSI KaK paBHonp8BHbIe HJIH HeT. B .L{aHHOM CJIY-
Qae Bce 3HatleHH5I oYAeM CtlHTaTb cYIII.ecTBeHHhIMH. B oYJIeB-
CKOH <popMe TaOJI. 3.1 HMeeT BH T80Jl. 3.2.
120
Ta 6JIHua 3.1
-
- Xl Xl Xl x" Xi
- I
1 + + a + 6
2 + - 6 + B
3 + - B + a
4 - - a + a
5 - + B - B
6 - + B - a
Ta6JIHl1a3.2
Xt x. x. x, XI
1 1 0 1 0 1 0 0 1 0 0 1 0
2 1 0 0 1 0 1 0 1 0 0 0 1
3 1 0 0 1 0 0 1 1 0 1 0 0
4 0 1 0 1 1 0 0 1 0 1 0 0
5 0 1 1 0 0 0 1 0 1 0 0 1
6 0 1 1 0 0 0 1 0 1 1 0 0
OqeBHAHO,
51 = ( ). 52 = ( ). (200 ) 5 4 = ( ) .
S3= 010 ,
003
(300 ) 5 12 = G ) , 5 1S =C H ).
S5= 010 ,
002 102
MaTpRna P = Ph HMeeT CJIeAYIOI.I.(HH BRA:
1 2 3 4 5 6
-
1 2,42 0,53 0,58 0,75 0,33 0,33
2 0,58 2,42 0,92 0,58 0,5 0
3 0,58 0,92 1,58 0,92 0,33 0,67
4 0,75 0,58 0,92 1,75 0,33 0,67
5 0,33 0,5 0,33 0,33 2 1,5
6 0,33 0 0,67 0,67 1 ,5 - 1,83
HanpHMep,3JIeMeHT P23 = ph + ph + P 3 = + +
121
1IIr-. .... _ ... ,...
I
+ '4 = 0,92, TaK KaK Ha OO'beKTaX 2 H 3 HMelOT O,!{HHaKOBbIe
3HalleHH5I TOJIbKO npH3HaKH Xl, X 2 H X 4 , npHlIeM lIaCTOCTH 3THX
3HalleHHH paBHbI COOTBeTCTBeHHO 3, 3 H 4.
MaTpH a XXT = xkxI HMeeT CJIe,!{YIOInHH BH,l{:
1
5
2
2
5
3
2
3
5
4
2
2
3
5
5
1
1
1
1
5
6
1
o
2
2
4
5
1
2
3
4
5
6
3,l{eCb B (i, j)-H 51l1eHKe HaXO,!{HTC5I KOJIHlIeCTBO npH3HaKOB
Xk, 3HaqeHH5I KOTOpbIX COBna,l{aIOT Ha OO'beKTaX i H j, HJIH, qTO
TO JKe CaMOe, KOJIHlIeCTBO CTOJIOnOB TaOJI. 3.2, B KOTOpbIX
i-H H j-H SJIeMeHTbI paBHbI e,!{HHHl.(e.
BbIlIHCJIHM A, H Z ,!{JI5I MaTpHl.(bI P. He 3aoy,!{eM npo TPH-
BHaJIbHOe «napa3HTHoe» perneHHe: coocTBeHHbIH BeKTOp C no-
CTOHHHbIMH KOMnOHeHTaMH Z = ;6 (1, ..., 1), TaKoH, qTO
ZTZ = 1, cooTBeTcTByeT coocTBeHHoMY lIHCJIY A == 5.
MaTpHII.a P - 5 ZZT = (Pi} - ) HMeeT CJIe.n:YlOIII.HH BH.n::
1 2 3 4 5 6
1 1,59 -0, 25 -0,25 -0 08 -0,50 -0, 50
,
2 -0,25 1 ,59 0,09 -0,25 -0, 33 -0, 83
3 -0 ,25 0,09 0,75 0,09 -0,5 -0, 1 7
4 -0 08 -0 25 0,09 0,92 -0 5 -0 17
, , , ,
5 -05 -0 33 -05 -05 1 , 17 0,67
, , , ,
6 -0,5 -0,83 -0, 1 7 -0,17 0,67 1 ,0
HallHeM HTepaTHBHbIH npOl.(ecc (0.15) ,!{JI5I STOH MaTpHl.(bI.
MomHo JIH HatIaTb C Z (0) = (1, ..., I)? HeT, TaK aK BTOpOH
coocTBeHHbIH BeKTop ,l{OJI:>KeH ObITb oprOrOHaJIeH nepBoMY, MbI
,l{OJIjf{HbI paccMaTpHBaTb TOJIbKO BeKTopbI, KOTopble opToro-
1
HaJIbHbI Zl = -V6 (1, ...,1) H, CJIe.n:OBaTeJIbHO, HMelOT HYJIeBYlO
CYMMY KOMnOHeHT.
PaccMoTpHM, HanpHMep, Z (0) = (1, 1, 1, 1,-1,-1).
Pe3YJIbTaTbI BbItlHCJIeHHH no cpopMYJIaM (0.15) CBe,l{eHbI B TaOJI.
3.3.
122
I
_J
TaoJlHu.a 3.3
-
lIla r t
AU)
z(t)
-
0 - (1, 1 , 1, -1 -1 -1,)
, ,
1 5,52 (0,39; 0,51 ; 0,21 ; -0,09; -0,48; -0,54)
2 2,67 (0,36; 0,51 ; 0,16; 0,04; -0,51 ; -0,57)
3 2,72 (0,34 ; 0,50; 0,16; 0,09; -0,52; -0 f7
,
4 2,73 (0,34; 0,49; 0,16; 0,11; -0,53; -0,57)
5 2,74 (0,34; 0,49; 0,17; 0,12; -0,54; -0,57)
6 \ 2,74 (0,33; 0,48; 0,17; 0,13; -0,54; -0,56 )
7 2,736 (0,33; 0,48; 0,17; 0,13; -0,54; -0,56)
)
nOCJIe rneCToro Illara 3HalleHM5I A (t) M z (t) CTaOHJIM3HpO-
BaJIHCb npH ,l{aHHOM ypoBHe TOqHOCTM, TaK tITO MO:>KHO npHH5ITb
A = 2,74 H Z == z (6). BeJIHlIHHa A == 2,74 COCTaBJI5IeT 39%
CYMMbI Bcex HeTpHBHaJIbHbIX coocTBeHHbIx tIHCeJI Tr (P) -
- n == 12-5=7. .QJI5I MHTepnpeTanHH nOJIYlIeHHoro cpaKTopa
u
OneHHM BeKTopbI ak Cpe,l{HMX 3HaqeHHH Z B COOTBeTcTBYlOrn HX
KJIaCCaX npH3HaKoB Xk: al == (0,32;- 0,32), a 2 = (- 0,26;
0,26), a3 == (0,23; 0,48; -- 0,31), a4 == (0,28; -0,55), as ==
== (-0,09; 0,33; -0,03).
KOMnoHeHTw 3THX BeKTopoB MorYT paccMaTpHBaTbC5I KaK
MacllITaobI H3MepeHM5I OT,l{eJIbHbIX rpa,l{anMH npH3HaKoB npH
BbltIHCJIeHHH 3HatIeHHH z. HanpHMep, Ha BTOpOM OO'beKTe
(+, -, 0, +, B) 3HalleHHe z eCTb CYMMa 0,32+0,26+0,48-
-0,03 = 1,31, ,l{eJIeHHa51 Ha Iv == 2,74, liTO H ,l{aeT Z2 = 0,48.
BecoBbIe OneHKH 3HallHMOCTH OT,l{eJIbHbIX npH3HaKoB ak=
= ak8 (z, x kS ), COCTaBJI5IlOrnHe lv, HMelOT CJIe,l{YlOrnHe BeJlH-
s
u
tIHHbI H pa3JIO}KeHM5I lIepe3 OneHKH 3HallMMOCTM HX 3HatIeHHH:
al == 0,63 == 0,315+0,315 (23% OT Iv);
a 2 = 0,40=0,20+0,20 (15%);
a3 == 0,63 == 0,11 + 0,23 + 0,29 (23%);
a4 == 0,92 == 0,31 + 0,61 (34)%;
as == 0,13 = 0,02+0,11+0 (5%).
KaK BH,l{HO, HaHOOJIee xapaKTepHbIM ,l{JI5I <jJaKTopa 51BJI5IeTC5I
BTopoe 3HatIeHHe (-) npH3HaKa x 4 , KOTopoe ,l{eHCTBHTeJIbHO
BbI,l{eJI5IeT OTpHnaTeJIbHbIe KOMnOHeHTbI z. nepBbIH Ii TpeTHH
npH3HaKH TaK:>Ke xapaKTepH3YIOT cpaKTop z; HX cYMMapHoe C
x 4 B03,l{eHcTBHe COCTaBJI5IeT 80% BeJIHqHHbI Iv = 2,74.
123
-----
--
MaTpHua OCTaTOqHbIX CB5I3eH P - "'ZZT HMeeT CJIe.L{YIO HH
BH.L{:
1 ,29 -0,68 -0,40 -0,20 0 0,01
-0,68 0,96 -0,13 -0 42 0,38 -0,09
,
-0,40 -0,13 0,67 0,03 -0, 24 0,09
-0,20 -0,42 0,03 0,87 -0,31 0,03
0 0,38 -0 24 -0 31 0,36 -0 17
, , ,
0,01 -0,09 0,09 0,03 -0, 1 7 0,13
MaKCHMaJIbHOe cOoCTBeHHOe qHCJIO STOM MaTpHUbI "'2 =
- 1 ,927 H cooTBeTcTBYIOInHH coocTBeHHbIH BeKTop Z2 =
= (0,72; -0,64;- 0,12; 0,17;- 0,18; 0,05) BbIqHCJI5IIOTC5I C
nOMOInblO Toro }Ke npouecca. nOJIyqeHHbIH <paKTop yqHTbIBaeT
"'2/7 = 27,5% cYMMapHoH BeJIHqHHbI Tr (P), TaK qTO ooa no-
CTpoeHHbIX <paKTopa yqHTbIBalOT B oo eM 66,5 % BeJIHqHHbI
KpHTepHH (3.10).
AHaJIOrHqHbIM oopa30M BbIqHCJI5IeTCH MaKCHMaJIbHOe COo-
CTBeHHoe qHCJIO 14,2 ,nJIH M8TpHUbI XXT = X k X C
k
cooTBeTcTBYIO HM coocTBeHHbIM BeKTopoM Zl = (0,37; 0,38;
0,48; 0,45; 0,36, 0,40), KOTOpbIH ,nOBOJIbHO nOXOJK Ha TpHBHaJIb-
HbIH coocTBeHHbIH BeKTop U = (0,41, ..., 0,41) MaTpHUbI P.
CoocTBeHHbIH BeKTop Z2 = (-0,24; -0,44; -0,24; -0,16;
0,57; 0,58), cooTBeTcTBYIO HH coocTBeHHoMY 3HatleHHIO "'2 =
= 7,17, C TOqHOCTblO .no 3HaKa OJIH30K K nepBoMY HeTpH-
BHaJIbHoMY coocTBeHHoMY BeKTopy MaTpHUbI P H, CJIe,lJ.OBaTe-
JIbHO, HMeeT aHaJIOrHqHYIO HHTepnpeTaUHIO.
9TO HJIJIIOCTpHpyeT ,lJ.OCTaTOqHO BbICOKYIO YCTOHtlHBOCTb
pe3YJIbT8TOB, nOJIyqaeMbIX npM HCnOJIb30BaHHH H,lJ.eHHO OJIH3-
KHX MeTo,nOB aHaJIH3a ,lJ.aHHbIX, ,na}Ke npH CTOJIb «nJIOXO CTPYK-
TypHpoBaHHOH» HH<popMaUHH, KaK TaOJI. 3.1.
3.1.2. 3a8atta nOCTpOeHUJl flOMUHallbH020
tjJaKTopa-npoeKTopa
no aHaJIOrHH C 3a,natlaMH KOHCTpYHpOBaHH5I KOJIHqeCT-
BeHHbIX <t>aKTopOB Z MO:>KHO paCCMaTpHBaTb 3a,lJ.aqH KOHCTPYH-
pOBaHHH HOM HaJIbHbIX <t>aKTopOB B BH,lJ.e 6YJIeBCKHX MaTpHU
Z pa3MepHOCTW N x m, 3a,lJ.aHHbIX B O,lJ.HOH H3 <pOpM A, B, B
(c. 112). PaccMoTpHM 3,lJ.eCb 3a,lJ.aqy nOCTpOeHH5I HOMHHaJIbHOrO
<t>8KTOp8 - npoeKTopa.
By,neM CqHTaTb, tlTO HCKOMaH oYJIeBCKa51 MaTpHua Z HMe-
eT <t>OPMY A, T. e. xapaKTepH3yeT pa30HeHHe R = {Rl'...'
R m} MHO:>KeCTB8 9l Ha m KJIaCCOB. 3a,naqa COCTOHT B TOM, qTo6bI
124
..,..--
HaRTH TaKYIO MarpHUY Z, Ha KOTOpOH .L{OCTHraeTC MHHHMYM
nOKa3aTeJI5I
n n
d(X k , PZXh)= IXk-PZXkI2=IX-PzXI2. (3.13)
k= 1 k= 1
9Ta 3a.L{alla HMeeT CMbICJI H B CJIYlIae, KOf.L{a lIaCTb CTOJI6UOB
MaTpHlI.bI X 3a.L{aeT KOJIHlIeCTBeHHbIe npH3HaKH.
PaCCMOTpHM MaTpHUY Pz X, KOTopaH HMeeT pa3MepHOCTb
N X M. npH i E Rs cooTBeTcTBYIOIna51 i-51 KOMnOHeHTa k-
fO cToJI6ua Pz Xk STOH MaTpHUhI paBHa Cpe.L{HeMY 3HalleHHIO
x: .npH3HaKa Xk Ha 06'beKTaX KJIaCCa Rs (npH 6YJIeBCKOM Xk
3Ta BeJIHlIHHa paBHa YCJIOBHOH Bep05lTHOCTH TOfO, liTO 06'beKT
1I3 R s CO.L{ep)KHTCH B MHomeCTBe, BbI.L{eJI5Iel\10M Xk, (CM. c. 119).
OTclO.L{a CJIe.L{yeT, liTO
M m k -k 2
IX-PzXI2= (Xi-Xs).
k = 1 s= 1 i€R
s
9Ta BeJIHlIHHa HMeeT CMbICJI Cpe.L{HeH .L{HCnepCHH npH3HaKOB
B KJIaCCaX Rs. ,[{eHCTBHTeJIbHO, .L{HCnepCH5I Xk B MHO:>KeCTBe
Rs onpe.L{eJI5IeTC5I cpOPMYJIOH
2 1 ( k -k ) 2
O'ks =- Xi-X s ,
N s .
t€R
8
TaK liTO Cpe.L{H5IH HCnepCH5I Bcex CTOJI6UOB
M
cr: = I (x - x :t
MN s k 1 i€Rs
TaKHM oopa30M,
m m
IX-PzXI2=- MNsGi=NM PsG:,
s= 1 s= 1
(3. 14)
f,l{e Ps = N sf N - ,l{OJI5I 06'beKTOB S-fO KJIaCCa R.
,QOKa3aHO, liTO 3a,l{alla KOHCTpYHpOBaHH5I HOMHHaJIbHOrO
cpaKTOpa-npOeKTOpa - STO 3a,l{alla 0 nOCTpOeHHH pa36HeHH5I
u v v u
C MHHHMaJIbHOH Cpe,l{HeH BHYTpHKJIaCCOBOH ,l{HCnepCHeH, KO-
TOpaH 51BJI5IeTC5I KJIaCCHtleCKOH B aHaJIH3e KOJIHlIeCTBeHHbIX
,l{aHHbIX.
PaCCMOTpHM Tenepb HHTepnpeTaUHIO nOKa3aTeJI5I (3.13)
B TepMHHax 06'beKTOB H CBH3eH Me:>K,l{Y HHMH. BbIpa3HB (3.13)
Qepe3 CJIe,l{ COOTBeTCTBYIO ero npOH3Be,l{eHH5I MaTpHU H pac-
125
""""""-
-.c:-
KpbIB 3aTeM CK06KH, nOJIYQ]IM, tlTO MHHHMH3a H5I (3.13) 3KBH-
BaJIeHTHa MaKCHMH3aUHH nOKa3aTeJI5I
Tr (XT Pz X),
(3.15)
u
KOTOpbIH paBeH
Tr (Pz XXT) = Pij xf x1 ,
i, i k
P u 1 · . ER
r,l{e Pij - 3JIeMeHT MaTpHnbI z, paBHbIH N s .L{JI5I t, J 8
(s = 1,..., m) If HYJIIO npM i E Rs, j E Rt .L{JI5I S =f:= t. BeJII-p.IH-
Ha aij = x1x' - 3JIeMeHT MaTpHUbI XXT. npH 6YJIeBCKHX
k
n
cToJI6nax XXT = XkXI, H aij paBHo KOJIHqeCTBY npH3Ha-
k=l
KOB, no KOTOpbIM 06'beKTbI i H j COBna.L{aIOT. TaKHM 06pa30M,
m 1
Tr (XT Pz X) = N ail' (3.16)
I s.. R
s= I,JE 8
9TOT KpHTepHH OneHHBaeT Cpe.L{HIOIO CYMMapHYIO CBSl3b B
KJIaCCaX pa36HeHH5I R. npHMeHHTeJIbHO K npOH3BOJIbHbIM MaT-
PHUaM CB5I3H nOKa3aTeJIb (3.16) KaK KpHTepHH KaqeCTBa pa3-
6HeHH5I 6bIJI npe.L{JIO}KeH H3 3BpHCTHQeCKHX co06pa}KeHHH H
3KcnepHMeHTaJIbHO 06ocHoBaH 3. M. 5paBepMaHoM H ,l{p. [13].
Me TO.L{bI nOHCKa OnTHMaJIbHOrO B CMhICJIe (3.16) (a CJIe.L{O-
BaTeJIbHO, H B CMbICJIe (3.14)) pa36HeHH5I 6Y,l{YT paccMaTpH-
BaTbC5I B 5.1.
HHTepnpeTanHIO KOHCTpYHpyeMoro HOMHHaJIbHOrO cpaKTo-
pa MO}KHO npOBo,l{HTb Ha OCHOBe Q10PMYJI (3.14) H (3.16). 3Ha-
qHMOCTb cToJI6ua Xk OTHOCHTeJIbHO R H3Mep5leTC5I Cpe,l{HeH
.L{HCnepCHeH
m
an = p s a:k
s= 1
(3. 1 7)
u u u
HJIH BeJIHqHHOH Cpe,l{HeH CYMMapHOH CB5I3H
m m Nk2
k = xl x, = ,
s = 1 N s i, j ER s = 1 N s
s
(3. 18)
nOCKOJIbKY CYMMbI 3THX BeJIHqHH ,l{alOT (3.14) H (3.16). npH
3TOM CJIe.L{yeT HMeTh B BH.L{Y, qTO 3HaqHMOCTb x TeM BbIllIe, qeM
60JIbIlle k H qeM MeHbllle ak.
126
j
.....--...,.
B TOM CJIYlIae, KOr.L{a OTKJIOHeHHe (3.13) cpaKTopa Z OT X
BeJIHKO, B03HHKaeT npOOJIeMa nocTpOeHH5I CJIe.L{YIO HX <paK-
TOpOB. ,[{JI5I ee perneHH5I MO:>KHO BOCnOJIb30BaTbC5I CTaH.L{apTHbIM
npHeMoM nepexo.L{a K OCTaTOtIHbIM MaTpHuaM. HY)KHO TOJIbKO
HMeTb B BH.L{Y, liTO CTOJIOUbI MaTpHUbI Z .L{OJI:>KHbI ObITb HOpMHpO-
BaHHbIMH, eCJIH MbI XOTHM, 11 TOObI OHa HrpaJIa pOJIb MaTpHUbl
COOCTBeHHbIX BeKTopOB Zp (KOTOpa5l, KOHellHO, OTJIHlIaeTC5I
OT Z, nOCKOJIbKY He 005l3aHa ObITb MaTpHueii HOMHHaJIbHOrO
npH3HaKa). n03TOMY BMeCTO N-MepHoro OYJIeBCKOro CTOJIOua
zS, BbI.L{eJI5IIO erO KJIaCC Rs, CJIe.L{yeT paCCMaTpHBaTb HOpMHpO-
BaHHbIH CTOJIOeu ZS = zSIVN"s, a BMeCTO Z - MaTpHUY I,
--' ,...,;
COCTaBJIeHHYIO H3 zs. Tor.L{a, OtIe BH.L{H 0 , Pz = ZZT H BeJIHtIH-
Ha (3.15) paBHa Tr (XiZ.zTX) = Tr (ZTxXiZ). ,[{JI5I nOCT-
pOeHH5I CJIe.L{YIOInero <jJaKTopa no aHaJIOrHH C aHaJIH30M r JIaB-
HbIX KOMnOHeHT MaTpH[(hI XX T CJIe.L{yeT paccMaTpHBaTb MaT-
PHUY OCTaTOtIHbIX CB5I3eH XXT - ')../lZT, r.L{e A, - aHaJIOr
MaKCHMaJIbHOrO coocTBeHHoro lIHCJIa, paBHbIH Tr (ZTXXTij,
T. e. KpHTepHIO KalleCTBa KJIaCcHcpHKaUHH R (3.16).
I1HbIMH CJIOBaMH, ):{JI5I nocTpOeHH5I CJle.L{YIO ero HOMHHaJIh-
Horo cpaKTopa-npoeKTopa paccMaTpHBaeTcH MaTpHlI.a OCTaTOq-
HbIX CB5I3eH Me:>K.L{Y OOoeKTaMH ail - Arij, nOJIYlIeHHaH BblqH-
TaHHeM BeJIHlIHH AI N s H3 nOKa3aTeJIeH CBH3H ail Me:>K.L{Y 000"
eKTaMH, CO.L{ep)Ka HMHC5I B O.L{HOM H TOM :>Ke KJIaCCe Rs nepBoro
cpaKTopa. 3HalleHHe AIN s MO)KHO TpaKTOBaTb KaK BeJIHlIHHY
CB5I3H Me:>K.L{Y OO'beKTaMH, YlITeHHYIO nepBbIM cpaKTopoM, TaK
liTO CB5I3H Me:>K.L{Y OO'beKTaMH, nonaBIlIHMH B pa3HbIe KJIaCCbI pa3-
6HeHH5I R, Booo e He MeH5IIOTC5I. BeJlHlIHHa A xapaKTepH3yeT
TaK:>Ke 3HallHMOCTb, Bec <t>aKTopa OTHOCHTeJIbHO HCXO.L{HbIX .L{aH-
HbIX: lIeM OHa OOJIbrne, TeM MeHhrne BeJIHlIHHa OCTaTOllHOH no-
rpernHocTH (3.13).
QCTaeTC5I paccMoTpeTb TOJIhKO B03MO)KHOCTb KOHCTPYHPO-
BaHH5I <paKTopa-npoeKTopa B .L{PyrHx <jJopMax. TIpH nOCTpoe-
HHH cpaKTopa B BH.L{e oYJIeBCKOro CTOJIO[(a 3a.L{alla MaKCHMH3a-
lI.HH (3.15) 3KBHBaJIeHTHa 3a.L{atIe OTbICKaHH5I TaKoro nO.L{MHO-
JKeCTBa OO'beKTOB S £;; 91, Ha KOTOpOM .L{OCTHraeT MaKCHMYMa
Cpe.L{H5IH CB5I3b
f (8) = I I
aij.
i, j€S
(3. 19)
9TOT nOKa3aTeJIb nOJIHOCThlO aHaJIOrHQeH (3.16) H xapaKTe-
pH3yeT Cpe.L{HIOIO cYMMapHYIO CB5I3b B O.L{HOM H3 KJIaCCOB; Bce
paHee CKa3aHHoe 0 CMbICJIe H HHTepnpeTal(HH 3a.L{aQH C Hecy-
ecTBeHHoH MO.L{HcpHKaUHeH nepeHocHTcSI Ha 3TOT CJIYlIaH.
127
"
npH nOCTpOeHHH CJIe.L{YIOInHX <t>aKTOpOB <}>OpMbI B BbItIHTa-
1
eMaH BeJIHlIHHa AI I 8 I = f (8)1 I 8 I = 8'2 ail paBHa cpe.L{-
I 1,IES
HeM CBH3H Me>K.L{Y oO'beKTaMH BbI.L{eJIeHHOrO nO.L{MHO>KeCTBa S.
B03MO>KHbI .L{Ba nYTH nOCTpoeHHH CJIe.L{YIO Hx <paKTopOB.
COrJIaCHO nepBoMY H3 HHX Ha KaJK.L{bIH CJIe.L{YIO HH HCKOMbIlI
6YJIeBCKHH cToJI6en HaKJIa.L{wBaeTCH Tpe60BaHHe opTOrOHaJIb-
HOCTH npe.L{bI.L{Y HM cToJI6naM, TaK liTO OKOHlIaTeJIbHbIH pe3YJIb-
TaT .L{aCT pa36HeHHe MHOLKeCTBa in - <t>aKTOp BH.L{a A. TaKOH
nponecc MO:>KeT TpaKTOBaTbCH, KaK npH6JIH:>KeHHbIM MeTO.L{
pemeHHH 3a.L{allH nOCTpoeHHSI OnTHMaJIbHOrO B CMbICJIe (3.16)
pa36HeHHH C nOMOInblO pemeHHH BO MHorHX OTHOllIeHHHX 00-
JIee npocTbIX 3a.L{atl BbI.L{eJIeHHH nO.L{MHO:>KeCTB.
BTOpOH nYTb COCTOHT B TOM, 1JT06bI He HaKJIa.L{bIBaTb Ha no-
JIYlIaeMbIe cpaKTopbI-CTOJI6nbI HHKaKHX OrpaHH1JeHHH, TaK 1JTO
pe3YJIbTaT MOLKHO TpaKTOBaTb KaK HOMHHaJIbHbIR <t>aKTOp B
<t>opMe B.
3.1.3. 3aaatla nocrpoeHUR H,O.MUHaAbH020
ljJaKTo pa- npoe KtftlU
5Y,lI.eM C1JHTaTb, 1JTO Hw.eTCH 6YJIeBCKaH MaTpH-
na Z pa3MepHocTH N X m, xapaKTepH3YIO aH pa36HeHHe
R = {Rl' ..., Rm} MHOLKeCTBa 06'beKTOB Ha m KJIaCCOB (CTOJI-
oen zS BbI.L{eJIHeT KJIaCC Rs), KOTopaH MHHHMH3HpyeT cYMMap-
HYIO norpemHocTb
d(Z, PkZ) = IZ-PhZI2 (3.20)
k k
npe.L{CTaBJIeHH5I Z B L (X k ). PemeHHeM ,STOR 3a.L{a1JH H HBJIH-
eTCH HOMHHaJIhHbIH cpaKTop-npOeKlI.HH. OlleBH.L{HO, BBH,l(Y (n.17)
I Z-P k Z 1 2 = Tr [ZT -ZT Ph) (Z-P k Z)] =
k k
= n Tr (ZT Z)- Tr (ZT PZ),
n
r.L{e P = Ph -MaTpHna cBH3eH Me)K,lI.Y 06'beKTaMH, nOJIYlIae-
k=l
MaH CYMMHpoBaHHeM BeJIHlIHH, o6paTHbIx tlHCJIeHHOCTHM KJIaC-
COB, KOTOpbIM 'npHHa,ll.JIeLKaT cpaBHHBaeMbIe 06'beKTbI (c. 118).
TaK o KaK ZTZ - ,lI.HarOHaJIbHaH MaTpHna lIHCJIeHHOCTeM
N s KJIaCCOB pa36HeHH5I R, TO Tr (ZTZ) = N, II, CJIe.L{OBaTeJIb-
HO, 3a,ll.a1Ja 0 cpaKTope-npoeKnHH SKBHBaJIeHTHa 3a.L{a1Je nOCT-
128
-I
.;
I
I
1
--
poeHH5I 6YJIeBCKOH MaTpHUhI Z BH)\a A, MaKcHMH3HpYIO eH
BeJIHqHHY
Tr (ZT PZ) = Tr (PZZT) = (P, ZZT),
(3.21 )
r)\e nOC;Ie)\Hee BbIpa:>KeHHe HMeeT CMhICJI CKaJl5lpHOrO npoH3Be..
,neHH5I MaTpHQ P H ZIT, paccMaTpMBaeMbIX KaK N2-MepHbIe BeK-
TOpbI. YlIHTbIBa5l, liTO Z ZT - 3TO 6YJIeBCKa51 (I,O)-MaTpHQa
(riJ) pa30HeHH5I R, nOJIYlIaeM, liTO (3.21) eCTb CYMMa CB5I3eH
piJ BHYTPH KJIaCCOB pa30HeHH5I R:
N m
Ir (ZT PZ) = Pij'ij = Pij- (3.22)
;, / = 1 s = 1 i, jER 8
C )\PyroH CTOpOHbI, 3a)\atJa MaKCHMH3a[(HH (3,21) aHaJIOrHlI.
Ha 3a,l{alle nocTpOeHH5I coocTBeHHhIx BeKTopOB MaTpH[(bI P, co.
oTBeTcTBYIO HX MaKCHMaJIbHhIM coocTBeHHbIM 3HalleHH5IM. 3Ta
aHaJIOrH5I B ,l{aHHOM CJIytJae npocTHpaeTC5I ,l{OCTaTOllHO ,l{aJIeKO,
TaK KaK npH MaTpH[(aX X k BH,l{a A e,nHHHlIHbIH BeKTOp U =
= (1, ..., 1) npHHa,l{J1emHT MHO)KeCTBY TpHBHaJlbHbIX COOCT"
BeHHbIX BeKTopoB n L (X k ) MaTpH[(bI P. B TO )Ke BpeM5I STOT
k
BeKTOp npeACTaBJI5IeT MaTpH[(Y BH,l{a A, TaK KaK 3a,l{aeT pa3-
oHeHHe, COCTOH ee H3 e,l{HHCTBeHHoro KJIaCCa, H n03TOMY ,l{O-
nYCTHM KaK perneHHe 3a,l{atJH 0 HOMHHaJIbHOM <paKTope.npoeK.
[(HH. CJle,l{OBaTeJIbHO, KaK H npH KOJIHlIeCTBeHHOM aHaJIH3e.
-- 1
B03HHKaeT TpHBHaJIbHOe perneHHe u = -V - U C coocTBeHHbIM
n
3HaqeHHeM A, = n = Pij/ N. n03TOMY perneHHe 3a,l{aQH KOH-
CTpYHpoBaHH5I <t>aKTopa-npOeK[(HH CJIe,l{yeT HatJHHaTb C MaT.,
PH[(bI OCTaTOllHbIX CB5I3eH, nOJIytJaeMOH H3 P BbIlIHTaHHeM MaT.
,..., ,....,
PH[(bl napa3HTHbIX CB5I3eH AUUT, COCT05lIneH H3 O,l{HHaKOBbIX
SJIeMeHTOB Pij/ N2 = n/ N, paBHbIX Cpe,l{HeH CB5I3H Me:>K,l{Y 000"
eKTaMH.. BnpOlleM, 3TOT BbIBO,l{ MO)KeT ObITb c,neJIaH H H3 Heno..
cpe,nCTBeHHOf() aHaJIH3a KpHTepH5I (3.22): MaKCHMYM CYMMbl
CBH3.eH B CHJIY HX HeOTpH[(aTeJIbHOCTH ,l{OCTHraeTC5I oooe,l{HHe--
HHeM Bcex oooeKToB B O,l{HH KJIacc.
TaKHM 06pa30M, nOCJIe 3JIHMHHHpOBaHHlI napa3HTHoH cpe.n....
HeR CB5I3H MbI npHXO,l{HM K 3a,l{atJe 00 OTbICKaHHH pa30HeHH5I R,.
u
MaKCHMH3HpYIO ero CYMMY BHYTpeHHHx CB5I3eH C BbIlITeHHbIM;
noporOBblM 3HalleHHeM
m
(Pij- a).
s= 1 i,/c..Rs
(3.23)
3Ta 3a,l{atJa OY,l{eT ,neTaJIbHO aHaJlH3HpOBaTbCH B 9 5.1.
Q 3 . 174
129
. - y
-----.
nonbITaeMCH BbIpa3HTb KpHTepHH (3.21) qepe3 npH3HaKH.
I{JISI 3Toro 3aMeTHM, liTO
Tr (ZT PZ) = Tr «xL'xk)-l xI ZZT X k ).
k
,[{HarOHaJIbHbIe 3JIeMeHTbl MaTpHnbI XI ZZT X h paBHbI
x ks r X ks- 'ATk l
. i J . - 1 st,
. . & 1 t
"d
r,!{e Nk# = I Rk n Rt 1- lIHCJIeHHOCTb nepeCelleHH5I s-ro KJIaC-
s. ,
C8 pa36HeH 51 no k-MY npH3HaKY C t-M KJI8CCOM R.
CJIe,!{oBaTeJIbHO,
N kl
Tr ((xIx , J- 1 xI ZZT X , J = N ;= .5: .
s N s t s,t N s
u
QTO nponopnHOH8JIbHO cpe,!{HeH OJIe np8BHJIbHMX npe,!{CKa-
38HHH R npH nponopnHOH8JIbHOM nporH03e H8 OCHOBe X k .
TaKHM 06p830M,
n Nkl
Tr (ZT PZ) = - (3.24)
k= 1 s, t N s
.
CYMMapHa51 OneHKa npaBHJIbHOCTH nponopUHOHaJIbHOrO npo-
rH03a KJI8CCOB R no npH3HaKaM Xk, p8ccMaTpHBaeMbIM He3a-
BHCHMO ,!{pyr OT ,!{pyra. .
,[lJIH M8TpHUbI OCTaTOllHbIX CB5I3eH P - n/ N (3,!{eCb, KaK
!'I BCIO,!{Y ,lJ.8JIee, BbIqHTaHHe QHCJIa H3 M8Tp H nbI nOHHMaeTC5I KaK
bIlIHTaHHe H3 Bcex ee 3JIeMeHTOB) BeJIHlIHHa KpHTepHH p8BHa:
Tr ( ZT ( P - ) z) = Tr (ZT PZ) - Tr (ZT Z) =
=: Tr (ZT PZ)-n,
T. e. BeJIHlIHHe (3.22), YMeHbIlleHHoH Ha n.
AH8JIOrHQHO OneHKa 3H8QHMOCTH k-ro npH3H8Ka ,!{JISI
cpaKTopa-npoeKUHH R BbIlIHCJIHeTCH I{aK ee BKJIa,ll B
Tr (ZT ( P - ; ) Z), qTO' )J.aeT
N kl
st
Pk= -I.
s. t N:
(3.25)
- 3,!{ecb e,!{HHHna BbIlIHTaeTC5I KaK cooTBeTcTBYIO ee TPHBH-
8.nbHoe co6cTBeHHoe lIHCJIO Ph. _
BonpOCbI nOCTpOeHH5I CJIe,!{YIO HX <paKTOPOB, a TaK)Ke <paK-
TOpOB B q)opMe E H B peIllaIOTCH 3,!{eCb nOJIHOCTblO 8HaJIOfHQHO
t30
TOMY, K8K 3TO ,!{eJIaeTC5I npH OTbICK8HHH cpaKTOpOB-npOeKTOpOB
(CM. TaKJKe 4.1)
CJIe,!{yeT 3aMeTHTb T8K>Ke, qTO B CJIyqae CMeIll8HHbIX ,!{8HHbIX,
Kor,!{8 HMelOTCSI KOJIHqeCTBeHHbIe npH3H8KH, KpHTepHH (3.13)
II (3.20) TaK)Ke MorYT npHMeH5ITbC5I. npH 3TOM npOeKTOpbI
Ph ',!{JI5I KOJIHqeCTBeHHbIX CTOJIOnOB Xk HMelOT BH,!{
( Xk Xk )
P _ I I
It - , I Xk 1 2 '
KOTOpbIH He 3aBHCHT OT M8CllITaoa H3MepeHH5I x k . aTO BblroAHO
OTJIHqaeT KpHTepHH (3.20) OT (3.13), npH KOTOpOM B03HHKaeT
MaTpHna CB5I3eH XXT, 3aBHC5I a51 OT M8clIlTaooB H3MepeHH5I
Xk H, CJIe,!{OBaTeJIbHO, npHBo,!{SI 851 K p83HbIM peI.I1eHH5IM npH
pa3HbIx CTpYKTypax Mac1IlT800B.
3.2. PEfPECCHSI no HOMHHAJlbHbIM
nPH3HAKAM
3.2.1. n po6AeMa onucanUJl npu3naKoB
nYCTb HMeeTC5I CHCTeMa npH3H8KOB-OTOOpa)Ke-
HHH Xk: m C (k = I,...,n). BC5IKOH qHCJIOBOH <PYHKlI.HH
n aprYMeHToB cp (Xl, ...., x n ) MO)KHO nOCT8BHTb B COOTBeTCTBHe
HOBbfH npH3HaK cp: m -+ C no npaBHJIY cp (i) = cp (Xl (i),
..., x n (i)). MHO)KeCTBO npH3HaKoB, nOJIyqaeMbIX H3 ,!{8HHOH
T80JIHlI.bI 00'beKT-npH3HaK X = (Xl, ..., x n ) C nOMO blO <PYHK-
lI.HH <p 3a,!{aHHOrO MHO)KeCTB8 n-MeCTHbIX cf>YHKUHH <D, oy,!{eM
Oq03H8qaTb <I> (X). npOCTeHI.I1He npHMepbI <PYHKlI. HH cp: cp (Xl,
.. ., X n ) = Xk - BbIOOp npH3HaK8 Xk; <p (Xl,..., X n ) = CXIX I +
+.. .+CX n X n -JIHHeMHa51 KOMOHHalI.H5I npH3H8KOB (Cpe,!{HeB3Be-
I.I1eHHbIH npH3H8K) H T. n. ECJIH npH3HaK y: m C npHHa,!{JIe-
jl{HT MHO)KeCTBY <1> (X), oy,!{eM rOBOpHTb, qTO OH HMeeT <D-
OnUCaflUe npH3HaKaMH Xk - CPYHKlI. HIO cp (Xl,..., Xn), ,!{JI5I KOTO-
pOM Y = cpo <1>YHKlI.H5I <p (Xl, ..., X n ) ,!{8eT nOJIHOe OnHC8HHe
npH3H8Ka Y B TepMHHaX npH3H8KOB Xl, ..., X n . B qaCTHOCTH,
3Ha51 Xl (i), ..., x n (i), MO>KHO O,!{H03HatlHO nporH03HpOBaTb
3HaqeHHe y (i): OHO paBHO y (i) = cp (Xl (i), ..., x n (i)). B
3TOM CMbICJIe npo npH3HaK, HMelO HH <D-onHcaHHe, MO)KHO ro-
BOpHTb, tlTO OH <1>-npo2flo3upyeM no npH3H8K8M Xl,..., x n , 8 ero
<D-onHcaHHe H83bIBaTb <D-npeouKmopOM. H8HOOJIee ynoTpe-
OHTeJIbHbI ,!{B8 MHO)KeCTB8 <1>. O.1\HO H3 HHX - MH01KeCTBO Bcex
JIHHeHHbIX <PYHKlI.HH BH,!{8 cp (Xl, . . ., x n ) = (a, x) = a kXk
k
(a - npOH3BOJIbHbIH qHCJIOBOH BeKTOp), KOTopoe oy,!{eM 000-
5*
131
--
---......-
3HallaTb L. npo npH3HaK H3 L (X) OYAeM rOBOpHTb, tITO oil
HMeeT JIHHeHHOe OnHCaHHe npH3HaKaMH Xl, ..., X n . BTOpOe-
MHO:>KeCTBO BCeX B03MO:>KHbIX OAH03HallHbIX <PYHKnHH OT n ap-
rYMeHTOB, KOTOpOe MbI OYAeM 0003HallaTb M. MHO:>KeCTBO
M (X) HClIepnbIBaeT KJIaCC npH3HaKOB, HMelO HX OAH03HallHOe
OnHCaHHe B TepMHH8X npH3HaKOB Xl, ..., X n .
B TOM CJIYlIae, KOrAa MHO:>KeCTBO <D COCTOHT H3 BCeB03MO:>K-
HbIX JIHHeHHbIX KOMoHHau.HH KOHellHOrO MHO:>KeCTBa CPYHKnHH
'PI, <p2,..., c:pP, MHO/KeCTBO <D (X) cOBnaAaeT C MHO>KeCTBOM
L (cp (X)), rAe <p (X) - CHCTeMa npH3HaKOB <pI,..., <pp. I1Halle
rOBOpH, AJI5I nOJIYlIeHHSI <D (X) AOCTaTOllHO nepeHTH K HOBOH
CHCTeMe npH3HaKOB c:p (X), npeOOpa3YSI HCXOAHbIe npH3HaKH-
CTOJI0 nbI Xk B npH3HaKH-CTOJIOUbI yl = cpl (X , ..., X n ); ...;
yP = <pP (Xl,.. .X n ), nOCJIe lIero Y:>Ke MO}KHO OrpaHHlIHTbC5I
TOJIbKO JIHHeHHbIMH OnHCaHH5Il\1H nOJIYlIeHHOH TaOJIHnbI. 3THM
qaCTO nOJIb3YIOTCSI npH paClIeTaX, nOCKOJIbKY HCCJIeAOBaTeJIb
OObIlIHO HMeeT B BHAY JIHIllb KOHellHbIH CnHCOK npeOOpa30BaHHH
<pk, BKJIlOlIaIO HH CTaHAa pTHbIe a pH<pMeTHlIeCKHe, JIOrHlIeCKHe
H aJIreOpaHlIeCKHe onepanHH (B <pH3HlIeCKHX HCCJIeAOBaHH5IX
npHXOAHTC5I paCCMaTpHBaTb TaK:>Ke TpHrOHOMeTpHlIeCKHe
<PYHKnlfH). HanpHMep, Hap5IAY C npH3HaKaMH Xl - «OO'beM
<POHAOB» 1I X2 - «lIHCJIeHHOCTb paOOTaIOIUHX» J(JI5I CHCTeMbI npeA-
v
npH5ITHH, HCCJIeAOBaTeJIb MO:>KeT paCCMaTpHBaTb npH3HaK
X I /X 2 - «cpOHAOBOOPY:>KeHHOCTb». BonpOCbI BbIoopa oa3Hca
<pI, ..., tpP, 06ecnellHBalC ero E03MO:>KHCCTb JIHHeHHOro onHca-
HH5I HHTepeCYIO HX Hac npH3HaKOB npH3HaKaMH CHCTeMbI
X, nO:>KaJIyfi, HaHOOJIee TpYAHble H MaJIOHCCJIeAOBaHHbIe BO-
npOCbI COBpeMeHHOH TeopHH aHaJIH3a AaHHbIX.
B KOHTeKCTe aHaJIH3a KalleCTBeHHbIX npH3HaKOB npOOJIeMa
v
a 3HaQHTeJIbHOH Mepe ynpor.u.aeTC5I 3a ClIeT HCnOJIb30BaHHSI
HXOTOMHlIeCKoro npeACTaBJIeHH5I npH3HaKOB.
llYCTb X k - OYJIeBCKa5I TaOJIHna AJI5I HOMHHaJIbHOrO npH-
aHaKa Xk: in C, HMelOr.u.ero m 3HaqeHHH. llpOH3BOJIbHa51
OAH03HallHa51 <PYHKnH5I cp (x k ) CTaBHT B COOTBeTCTBHe Ka:>KAOMY
3HalleHHIO S lIHCJIO as, a COOTBeTCTBYIO HH npH3HaK c:p AaeT
cp (i) = as, eCJIH Xk (i) = s (i Em). 3TO 03HallaeT, liTO BeK-
TOp <p (i) (i E m) cOBnaAaeT C BeKTopOM X = Xha, rAe a =
= (aI' ..., am). MbI OKa3aJIH, liTO M (X h ) = L (Xli)' npH"
tIeM AJI5I BC5IKOrO X = Xha BeKTOp a 3aAaeT cOBoKynHocTb 3H&a..
lIeHHH COOTBeTCTBYIO ero CPYHKnHOHaJIbHO npeoopa30BaHHoro
npH3HaKa cp (x k ). TaKHM oopa30M, JIHHeHHbIe onHCaHH5I X k
lIClIepnbIBalOT KJIaCC Bcex onHcaHHH AJI5I HOMHHaJIbHOrO npH3Ha'"
Ka x k . 9TO OOCT05lTeJIbCTBO MO:>KHO HCnOJIb30BaTb AJI5I JIHHeH-
u u
lIoro nOCTpOeHH5I HeJIHHeHHbIX onHcaHHH KOJIHlIeCTBeHHoro
132
'
!
.."l
t
-
npH3HaKa: nepeBeCTH era B ,l{HXOTOMHlIeCKYIO cpOPMY X k , noC..
JIe 1Jero paCCMOTpeTb L (X h ).
KOHeIlHO, nepeBO,l{ KO.lIHqeCTBeHHoro npH3HaKa B KaqeCTBeH"
HYIO <POPMY COnp5ljl{eH C nOTepeH HHcpopManHH, O,l{HaKO 3Ta no-
Tep51 ,B03Me aeTC5I B03MO}KHOCTblO paCCMOTpeHH5I pa3.lIHlIHbIX
onHcaHHM, ,l{aBaeMbIX «orpYO.lIeHHbIM» npH3HaKOM, KaK .lIHHeii-
HbIX. B03HHKalO aSl a.lIbTepHaTHBa: .lIHHeHHbIe onHcaHHH Ha
QCHOBe KO.lIHlleCTBeHHoro npH3HaKa H.lIH BceB03MO:>KHbIe onHca-
HHSI Ha OCHOBe ero orPYO.lIeHHOrO BapHaHTa - B oo eM BH,l{e
He HCCJIe,l{OBaJIaCb. 3aMeTHM, qTO K. nHpCOH, O,l{HH H3 C03,l{aTe-
.lIeH MaTeMaTHqeCKOH CTaTHCTHKH, paCCMaTpHBaJI HOMHHa.lIb-
HbIe npH3HaKH KaK KaTeropH30BaHHbIe, orpYO.lIeHHbIe KO.lIH-
lIeCTBeHHbIe. KpHTepHeM pa3YMHOCTH MO,l{e.lIH aHa.lIH3a Kaqe-
CTBeHHbIX npH3HaKOB OH CqHTaJI B03MO:>KHOCTb ,l{OKa3aTe.lIbCTB8
Toro, qTO npH CTpeM.lIeHHH K HY.lIIO HHTepBa.lIOB pa30HeHH5I
u
OCH 3HalleHHH KO.lIHqeCTBeHHoro npH3HaKa Ha KaqeCTBeHHble
u
rpa,l{anHH, napaMeTpbI KaqeCTBeHHOH MO,l{e.lIH CTpeM5ITC5I K na-
paMeTpaM COOTBeTCTBYIO eM KO.lIHqeCTBeHHOH MO,l{e.lIH. Ha
3TOM OCHOBe, B qaCTHOCTH, K. nHpCOH npe,l{.lIO:>KHJI CBOM K03cp-
cpHnHeHT COnpSl)KeHHOCTH HOMHHa.lIbHbIX npH3HaKOB (STOT
KosqxpHnHeHT OY,l{eT paCCMOTpeH B n. 3.2.5). 3Ta H,l{e51 paCCMaT-
pHBa.lIaCb TaK}Ke ,l{.lI5I perpeCCHOHHoro aHa.lIH3a [62].
KOHeqHO, ,l{.lI5I BceH TaO.lIHnbI X M (X) =1= L (X) ,l{aLKe npH
,l{HXOTOMHqeCKHX npH3HaKax Xk; HanpHMep, sin (Xl +... + x n )
He MO)KeT ObITb .lIHHeHHO BbIpa:>KeH lIepe3 Xl, ..., x n . O,l{HaKO C
yqeTOM TOrO, qTO B qHC.lIO aHa.lIH3HpyeMbIx npH3HaKOB MOJKHO
BK.lIlOIIHTb npOH3BO.lIbHbIe <PYHKnHH HCXO,l{HOH TaO.lIHnbI, B03-
MO)KHOCTb paCCMOTpeHH5I pa3.lIHqHbIX npeOOpa30BaHHH KaLK-
):(oro H3 npH3HaKOB Xk B OT,l{e.lIbHOCTH KaK .lIHHeHHbIX BeCbM8
cy ecTBeHHO YBe.lIHqHBaeT OOraTCTBO npOCTbIX OnHCaHHH.
3Ta H,l{eH OY,l{eT OO.lIee nO,l{pOOHO pa3BHTa B n. 3.2.8.
PaCCMOTpHM HeKOTopbIH BbIXO,l{HOii nOKa3aTe.lIb y, KOTO'"
phIH ,l{O.lI:>KeH ObITb onHcaH npH3HaKaMH Xl, ..., X n C nOMO blO
,l{OnYCTHMbIX npeOOpa30BaHHM cP E <D. ECJIH y $:. <D (X), TO B Ka-
qeCTBe npHO.lIH)KeHHOrO OnHCaHHSI Y C.lIe,l{yeT BbIOpaTb O.lIHLKaii",
mee K HeMY <D-onHcaHHe. BbIOOp B KaqeCTBe MeTpHKH paCCTOSl-
HHH (n.2) (KBa,l{paT eBK.lIH,l{OBCKOH MeTpHKH) npHBO,l{HT K 3a,l{a-
qe
min I y -cp (X) 1 2 .
(1)6 <J)
(3.26)
PerneHHe STOM 3a,l{aqH y = cp (X) OY,l{eM Ha3bIBaTb <1>-
pe2peccuet1 npH3HaKa Y no npH3HaKaM Xl,..., x n . B TOM C.lIyqae,
KOr,l{a <D 51B.lIHeTCH .lIHHeHHbIM npOCTpaHCTBOM, cOBoKynHocTb
133
..,.........
BeKTopOB <D (X) TaK>Ke SIBJI5IeTC5I JIMHeHHbIM npOCTpaHCTBOM
(CYMMHpOBaHHe BeKTOpOB COOTBeTcTByeT CYMMHpOBaHHIO no-
pO}K alO HX HX <PYHKnHM; YMHO>KeHHe BeKTopOB aa tlHCJIa CO-
OTBeTcTByeT YMHO>KeHHIO CPYHKnHH Ha tlHCJIa). Tor,l{a 3a,l{atla
(3.26) eCTb He liTO HHoe, KaK 3a,l{alla 0 npoeKTHpOBaHHH BeKTopa
Y Ha npOCTpaHCTBO <1> (X)
1
" 1 2
min I y - y .
u€4> (X) .
06 tlHO paCCMaTpHBalOT ABa THna 3a,l{all pe peCCHOHHoro
aHaJIH3a: 1) KOr,l{a <I> (X) = M (X) (perpeCCH5I nepBoro pO,l{a,
Cpe,lJ.HSlH KBa.npaTHlIeCKa51 perpeCCH5I); 2) Kor,l{a <I> (X) =
= L (X) (perpeCCH5I BToporo pO,l{a, JIHHeMHaH perpeCCH5I).
(3.27)
3.2.2. MooeAu pe2peCCUOH,H,020 aHaAusa
PerneHHe 3a,l{allH nocTpOeHH5I cpe,l{HeM KBa,l{pa-
THqeCKOH perpeccHH nOJIYlIaeTCSI CJIeAYIOInHM oopa30M. Pac-
CMOTpHM pa30HeHHSI Rk MHO>KeCTBa in no OT,l{eJIbHbIM npH3Ha-
KaM xk(k = 1, ..., n) H HX nepecelleHHe R = RI...Rn. Ka}l{-
AbIH KJIaCC Rs pa30HeHHSI R COCTOHT H3 OO'beKTOB, xapaKTepH-
ayeMbIx KOHKpeTHoH KOMoHHanHeH S = (Sl' ..., sn) 3HalleHHH
npH3HaKOB Xl,..., x n H, CJIe,l{OBaTeJIbHO, He pa3JIHlIHMbIX no
3THM npH3HaKaM, a TaK>Ke H no JIIOObIM onHcaHHSlM Ha HX OC-
HOBe. PerpeCCH5I npH3HaKa Y no Xl, ..., x n - 3TO npH3HaK
y, npHHHMalO HH Ha OO'beKTaX KJIaCCa Rs OAHO H TO )Ke 3Hatle-
1
HHe Y8 = N Yi - BHYTpHKJIaCCOBOe Cpe,l{Hee. B qaCTHOCTH,
8 1ERs
eCJIH BCe R. 0,lJ.H03JIeMeHTHbI, T. e. Ka>K,l{bIH OO'beKT XapaKTepH-
3yeTcH CBOeH, OTJIHlIHOH OT ,l{PyrHx, KOMOUHanHeH S, TO Y = y.
B TOM CJIYtlae, KOr,l{a lIHCJIO npH3HaKOB ,l{OCTaTOllHO BeJIHKO,
KaK npaBHJIO, HMeeT MeCTO HMeHHO 3Ta cHTyanHSI. JIIOOOH
npH3HaK Y TOr,l{a HMeeT OnHCaHHe B TepMHHaX Xl, ...., Xn, HO
OHO CJIHlIIKOM HH,l{HBH,l{yaJIH3HpOBaHHO (,l{JISI Ka>K,l{OrO OO'beK-
Ta - CBoe) If He Y,l{OBJIeTBOp5leT TpeOOBaHHIO npOCTOTbI. HeT
v .
OCHOBaHHH O>KH,l{aTb, tlTO 3TO onHcaHHe cnpaBeAJIHBO KaK
3JIeMeHT TeopHH, T. e. AJISI JIIOObIX OO'beKTOB. n03TOMY cpe,l{-
H5IH KBaApaTHtleCKa51 perpeCCH5I MaJIO npHMeHSleTC5I Ha npaK-
THKe. ,[(JIH ee IfCnOJIb30BaHHSI Heooxo,nHMO: 1) JIHOO HaJIHlIHe
anpHopHoro 3HaooSl aHaJIHTHlIeCKoro BbIpa>KeHHSI pacnpe,l{e-
JIeHHH (y, Xl, ..., Xn), Ha OCHOBe KOTOpOrO MOLKHO HaHTH aHa-
JIHTHlIeCKH YCJIOBHOe paCnpe,l{eJIeHHe y npH 3a,l{aHHbIX Xl,
134
--
..., x n ; MaTeMaTHlIeCKOe OJKH,l{aHHe 3TOrO paCnpe,l{eJleHHH H 3a-
,l.{aeT if = cp (Xl, ..., Xn), r,l{e cp- aHaJlHTHlI CKaH <PYHKUHH, BH,l{
KOTOpOH: MO)Ket ObITb HCnOJlb30BaH ,l{JlSI HHTepnpeTaUHH; 2) JIHOO
HaJlHlIHe ,l{OCTaTOllHO KpynHbIx KJlaCCOB Rs (T. e. 60JIblIlHX 3Ha-
tJeHH'H N s) B nepecelleHHH pa30HeHHH R, TaK liTO COOTBeTCTBYIO-
!nHe KOMOHHan:HH S = (Sl' ..., Sn) CaMH MOrYT HCTOJIKOBbIB8TbCH
u
KaK YCTOHlIHBbIe CB5I3H Mem,l{Y 3HalleHHSlMH npH3HaKOB.
PelIleHHeM 3a,l{atIH JlHHeHHOH perpeCCHH 51BJlSleTCH B COOT-
BeTCTBHH c",fI.7 npOeKUH5I Pxy. TaKHM OOpa30M, JIHHeHHaH
perpeCCHSI y BeKTOpa y no npH3HaKaM Xl, ..., x n BbIpa}KaeTCH
CPOPMYJIOH
"
y=X (XTX)-l XT y=XSx: 1 XT Y= Xa,
(3.28)
r):(e
a = Si 1 XT y = (X TX)-l XT y-
(3.29)
BeKTOp K09<M>HUHeHToB as npH CTOJlOuaX Xk MaTpHUbI X B JIH-
HeHHOM npe,l{CTaBJleHHH Y tIepe3 X. CJle,l{yeT 3aMeTHTb, liTO
CTOJlOUbI Xk MaTpHUbI X B oo eM cJlYtIae MorYT OKa3aTbCH JIH-
HeHHO-3aBHCHMbIMH, TaK liTO MaTpHua Sx = XTX OY,l{eT Bbl-
pO}K,l{eHHOH. B 9TOM cJlYtIae no,l{ MaTpHueH Si 1 CJle,l{yeT nOHH-
MaTb OOOOIneHHYIO 06paTHYIO MaTpHUY ,l{JlSI SX (CM. n. 10).
3HaqeHHH ak Ko9<fxPHUHeHToB npH Xk B cpopMYJIe (3.29)
oTpa:malOT CBSl3b y (a 3HatIHT, H y) C Xk H MorYT HCnOJIb30BaTb-
CSI ,l{JISI HHTepnpeTaUHH nOJlYlIeHHoro pemeHHSI. B tIaCTHOCTH, HX
0.nH30cTb K HYJlIO CBH,l{eTeJlbCTByeT 00 O:rCYTCTBHH (JlHHeHHoii)
3aBHCHMOCTH y H x.
Pa3HHua y H X,oueHeHHaH paccToSlHHeM (3.27), Ha3bIBaeTCH
ocmamotf,H,OU aucnepcueu perpeccHH. ECJlH ORa OJIH3Ka K HYJIIO,
'"
TO Y HBJIHeTCSI ,ltOCTaTOllHO xopOlIleH annpOKCHManHeH y. l{JIH
oueHKH KatIeCTBa perpeCCHH Y,l{OOHO nOJlb30BaTbCSI OTHOCHTeJIb-
HbIM nOKa3aTeJIeM, He 3aBHC5IlUHM OT ,nJlHH paCCMaTpHBaeMblX
BeKTopOB:
2 I A 1 2 I '" 1 2
D= IYI-Y-Y =1- y-y .
1 Y 1 2 1 y 1 2
(3.30)
nOKa3aTeJlb D Ha3bIBaeTCSI Ko9<fxPHUHeHTOM aemepMUH,a uu.
OH H3MeHSleTC5I B npe,l{eJlax OT 0 ,l{O 1. BJIH30CTb D K 1 C:BH,lte-
TeJlhCTByeT 0 TOM, liTO HeOOnSlCHeHHaSl (OCTaTOllHaSl).,ltHCnep-
A
CMS! I Y - Y (2 MaJla, T. e. Ha:H:,l{eHHoe onHcaHHe AOCTaTOqHO xopo-
lIlO annpOKcHMHpyeT y. B lIaCTHOCTH, D = 1 Tor,l{a H TOJlbKO
135
-,
iJIZI ......
'I
,
'",
A
TOr,l{a, KOf,l{a y = y, TaK liTO HCKOMOe L-OnHCaHHe Onpe,l{e-
JISleTC5I CPOPMYJIOH (3.28). ECJIH )Ke D = 0, TO I y -- if I =
= I Y I, T. e. y = 0, TaK qTO JIHHeHHOro onHCaHH5I y B Tep-
MHHax Xk He cyr.u.eCTByeT.
B CHJIY Toro qTO Y H Y - Y opTOrOHaJIbHbI, no TeopeMe
nHcparopa CnpaBe,l{JIHBO paBeHCTBO I if I + I y - 'if I = I Y I ·
OHO n03BOJIHeT BbIpa3HTb D B <popMe
I 1 2
D = y (3.31)
I y 1 2
OTHomeHHH ,l{JIHHM npoeKnHH BeKTopa (06oSlCHeHHOM ,l{Hcnep-
CHH y) K ero COOCTBeHHOM ,l{JIHHe (nOJIHOM ,l{HCnepCHH y), KOTO-
poe, KaK H3BeCTHO, paBHO KBa,l{paTY KocHHyca yrJIa Me:>K,l{Y
y H if. 9TO 03HaqaeT, qTO npH neHTpHpOBaHHbIX Y H if BeJIHlIHHa
VD eCTb K09<P<P H n H eHT KOppeJI5In HH Me>K,l{Y Y H Y = PxY.
B CJIyqae perpeCCHH nepBoro pO,l{a OTHOlIIeHHe (3.31) Ha3bIBaeT-
C5I Koppellfll1uOHHblM omHOUleHueM. ,[{JI5I JIHHeHHOH perpeccHH
K09c}xP H n H eHT KOppeJI5InHH VD Ha3bIBaeTC5I K09c}xP H n H eHToM
MHO:JICeCm8eHHOU KOppellJll1UU Y H Xl, ..., x n H 0003HaqaeTC5I
V I A 1 2
P o . X 1 X n = y .
· · · 1 y 1 2
BeJIHqHHa p MeH5IeTC5I OT - 1 ,l{0 + 1 H xapaKTepH3yeT B03-
MO:>KHOCTH JIHHeHHOro onHCaHH5I Y B TepMHHax x k .
CTaH,l{apTHa51 CTaTHCTHqeCKa51 MO,l{eJIb JIHHeHHOH perpeCCHH
n03BOJI5IeT YTOqHHTb CMbICJI OneHOK BeJIHqHH I y - y 12, ak,
D, P H np. B COOTBeTCTBHH C 3TOM MO,l{eJIblO
(3.32 )
n
Y = ak Xk+ , (3.33)
k=l
r,ne BeKTop = (61' ..., n) - CJ1YQaHHbIH; ero K01\'fnOHeHTbI
i He3aBHCHMbI H HOpMaJ1bHO pacnpe eJIeHbI C HYJIeBbIM MaTe-
MaTHqeCKHM O)KH,naHHeM H O,l{HHaKOBOM ,nHcnepCHeH 0'2. BeK-
TOpbI Xk - 3a,l{aHHbIe.
OObIqHO B <POPMYJIY (3.33) BKJIlOqalOT BeKTOp nOCTOHHHbIX
(CBOOO,l{HbIX) qJIeHOB ao. MbI OOXO,l{HMC5I oe3 Hero, nOCKOJIbKY
MO>KHO CqHTaTb, qTO U = (1, ...,1) BKJIlOqeH B TaOJIHny X B
KaqeCTBe O,l{HOro H3 BeKTopOB x k .
CKa3aHHoe 03HaqaeT, qTO KOMnOHeHTbI BeKTopa Y TaK>Ke HOp-
MaJIbHO pacnpe,l{eJIeHbI C BeKTopOM Cpe,l{HHX 3HaqeHHH akxk H
k
,IlHcnepCH5IMH 0'2, npHqeM He3aBHCHMbI ,l{Pyr OT ,l{pyra. ,.[{pyrHMH
136
--
cnoBaMH, KOBapHaUHOHHaH MaTpHua pacnpe eneHHH y paBHa
a 2 E.
B 3THX YCJIOBHSlX BeKTop (3.29) a = Sx: 1 XTy SlBJIHeTCH He.,
CMemeHHOH oneHKOH BeKTopa MO eJIbHbIX KosqX}lHnHeHTOB a*,
npHqeM ero KOMnOHeHTbI HMelOT MHHHMaJIbHbIe HcnepCHH cpe-
,nH Bcex HeCMe eHHbIX JIHHeHHbIX OneHOK a*. Pacnpe eJIeHHe
a HOpMaJIbHO, a ero KOBapHanHOHHaSl MaTpHna paBHa a2SX1.
Be.TlHtIl1Ha I -=- 12 aeT HeCMeIl.\eHHYIO OUeHKY .n;JUI .n;Hcnep-
ClUJ. 0'2, npHtIeM OTHOIlIeHHe I Y_/1 2 HMeeT pacnpe.n;eJleHHe "1. 2 c
(J
N --n CTeneHHMH CBOOO bI, He3aBHCHMoe OT pacnpe eneHH
a. OTCIO,l{a CJIe,l{yeT, qTO BeJIHqHHa
t1l,=
(ak- ak)/(N -'-:n)
{I Y -II 1 2 [Si 1 )kk} 1/2
(3.34)
d U
KaK OTHOIlleHHe HOpMHpOBaHHOfi HopMaJIbHOH BeJIHllMHbI K KOp-
HIO KBa,l{paTHOMY OT HeSaBHCHMOH OT Hee BeJIHqHHbI x 2 /(N-n)
nO,l{qHHHeTCH t-pacnpe,l{eJIeHHIO CTblO;teHTa. 9TO n03BOJIHeT.
CTpOHTb ,l{OBepHTeJIbHbIe HHTepBaJIbI ):{JIH aZ HJIH npOBepHTb
rHnOTe3bI 0 ero 3HaqeHHSlX (B qaCTHOCTH, rHnOTe3Y aZ = 0).
BeJIHqHHY OCT8TOqHOH ,l{HCnepCHH d = J y - ""y 1 2 MO:>KHO
CJIe,l{YIO HM oopa30M HCnOJIb30BaTb JISI npOBepKH JIIOObIX
JIHHeMHbIX rHnOTe3 00 a*. nYCTb HMeeTCH rHnOTe3a 0 3Halle-
HHHX a*, BbIpa}KeHHaH ypa:J3HeHHeM Ha* = b, r,l{e H H b-
HeKOTopbIe 3a,l{aHHbIe MaTpHna H BeKTOp. B qaCTHOCTH, B03-
MO:>KHa rHnOTe3a as = a;+l = ... = a;+q = a. 0003HaqHM
qepe3 y' TOT BeKTOp Xa, KOTOpbIR MHHHMH3HpyeT Iy - Xal 2 npH
OrpaHHqeHHH Ha = b (ero MO)KHO HaRTH MeTO,l{OM MH01KHTeJIeH
JIarpaH}Ka), 8 OCTaTOqHYIO HcnepCHIO 'y - y' 1 2 - qepe3
d'. Tor,l{a HcnepcHoHHoe OTHOIIleHHe
F = (d' -d) / d
m (N-m) ,
r e m - paHr MaTpHnbI H, HMeeT pacnpe eJIeHHe <PHIllepa C m
H N - m CTeneHHMH CBOOO bI [87]. CTaTHCTHQeCKaH 3HaqH-
MOCTb STOM BeJIHqHHbI npHBO,l{HT K OTKJIOHeHHIO rHnOTe3bl
Ha* = b.
BnpOqeM, OTHOlIleHHSI BH,l{a (3.35) MorYT ObITb HCnOJIb30Ba-
HbI H He3aBHCHMO OT CTaTHCTHqeCKHX MO,l{eJIeii KaK nOKa38TeJIH
cy ecTBeHHOCTH rHnOTe3.
(3.35 )
131
'!""'1 .-___
3.2.3. PezpeccuR /<'OAuttecTBel-lflOZO npUsna1<-a
no HOlYlUnaAbflblAt (oucnepcuoHliblU anaAu3)
PaccMoTpHM CHaqaJIa CJIyqaM, Kor,l{a KOJIHqeCT-
BeHHbIH npH3HaK y npoeKTHpyeTC5I Ha 6YJIeBcKYIO MaTpHUY
X e,l{HHCTBeHHoro HOMHHaJIbHOrO npH3HaKa, xapaKTepH3ye-
Moro pa30HeHHeM R = {RIP..' Rm} MHO)KeCTBa m. KaK Y>Ke
OTMeqanOCb, B 3TOM cnyqae pxY = if = Xa eCTb BeKTOp BHYT-
. 1
pHKJIaCCOBbIX cpe,nHHX, TaK J.ITO as = N Yi - Cpe,l{Hee
s l E R8
npH3HaK8 y BHYTPH KJIaCC8 Rs.
COOTBeTCTBYIO aH CTaTHCTHqeCKa51 MO,l{eJIb eCTb
y (i) = a; + i ,l{JI5I i E Rs. (3.36)
t
t
..
PaCCMOTpHM rHnOTe3Y 0 TOM, qTO Bce a; paBHbI O,l{HOH H TOM
>Ke BeJIHqHHe a*. I1H8qe rOBOp5l, rHnOTe3a YTBep:>K,l{aeT, qTO
,l{nH ,nOCTaTOqHO a,l{eKBaTHoro onHCaHH5I y HeT Heooxo,nHMOCTH
B HCDORb30BaHHH npH3HaKa X, ,l{OCTaTOqHO BOCnOJIb30B8TbCH
ODHc8HHeM, ,naBaeMbIM npH3HaKoM u = (1, ..., I)T, KOTOphIH
CT8BHT B COOTBeTCTBHe BceM OO'beKTaM O,l{HY H TY 1Ke rp8,naUHIO.
0603HaQHM Qepe3 y' = P aY = BeKTop cpe)J.HHx 7 Y i. OqeBH)J.HO,
A "
, + '
y-y =y - Y y-y,
npHqeM BeKTOp Y - if opTorOHaneH BeKTOpy Y - y', KaK H JIIO-
- A
OMY ,npyrOMY 3ReMeHTY npOCTpaHCTBa L (X) (y - y' E
f L (X), TaK KaK u E L (X) H, CJIe,l{OBaTeJIbHO, y' = P u y E
E L (X), a TOr,l{a if y' KaK pa3HOCTb 3JIeMeHTOB L npHH8,l{-
JIe}KHT L). Cne,l{OBaTeJIbHO, MO)KHO BOCnOJIb30BaTbC5I TeopeMOH
nHcparopa:
d'=ly-y'12=Iy-yJ2+lu - y'12=d+/Y-Y'12. (3.37)
<1>opMYRa (3.37) ,l{aeT pa30HeHHe ,nHcnepcHH d' npH3HaKa
y (oTKRoHeHH5I OT cpeAHero) Ha BHYTpHKRaccoBYIO H Me}KKRaC-
).:
COBYIO: d = Iy - Y 12_ CYMMa ,l{HCnepCHH Y BHYTPH KJI aCCOB
"
Rs, Iy - y'12 - ,l{HcnepCH5I cpe,nHHX BHYTpHKJIaCCOBbIX 3Ha-
geHHH (Me)KKJIaCCOBa51 COCTaBJI5IIOIna51 oo eH ,l{HCnepCHH d').
KpHTepHH (3.35) n03BOJI5IeT npoBepHTb cTaTHCTHqecKYio 3Ha-
qHMOCTb rHnOTe3bI 0 paBeHCTBe BHYTpHKJIaCCOBbIX Cpe,l{HHx.
nORyqeHHe BbIBO,l{OB Ha OCHOBe pa30HeHH5I CYMMapHOH ,nHC-
nepCHH Ha COCTaBJISIIO He, OTBeqalOInHe pa3HbIM CTopOHaM
HBReHHH, npHH5ITO Ha3bIBaTb oucnepCUOflnblM anallU30M [42].
J38
TIpHMepbI HCnOJIb30BaHH5I ,!{HCnepCHOHHOrO aHaJIH3a Ha OC-
HOBe <t>OPMYJIbI (3.37) ,!{aHbI B [87]. .
OpH perpeccHH KOJIHqeCTBeHHoro npH3HaKa no MHorHM HO-
MHHaJIbHI>Kvf npH3HaKaM TaK:>Ke B03MO:>KHO pa3JIOLKeHHe oo eH
,nHcriepcMH no opTOrOHaJIbHbIM KOMnOHeHTaM. !{JI5I oOJIerqe-
HH5I 3TOH B03MO:>KHOCTH B TaOJIHUY qO'beKT-npH3HaK BKJIIOtIalOT
nO,!{TaOJIHUbI, cooTBeTcTBYIO He pa3J1HlIHbIM nepeCetIeHH5IM.
TaK, MO:>KHO paccMaTpHBaTb HepapXHqecKYIO KJIaCcH<pHKaUHIO
H8 OCHOBe pa30HeHHH R 1 , ..., Rn, COOTBeTCTBYIO HX ,!{aHHbIM
npH3HaKaM. HanOMHHM, qTO HepapXHtIeCKaH KJIaCCH HKaUHH
3a,!{aeTCH cOBoKynHocTblO pa30HeHHH Rl, R 1 R2, ..., R 1 R 2 .. .Rn.
,[{JI5I npOBe,!{eHHH ,!{HcnepCHOHHoro aHaJIH3a no HepapXHtIecKOH
cxeMe TaOJIHua X COCTaBJI5IeTC5I H3 nO,!{TaOJIHU Xl' X 12 , X 123 ,
..., X I ... n , COOTBeTCTBYIO HX pa36HeHH5IM HepapXHt!eCKOH
KJIaCCH HKaUHH. 0003HallHM npOeKUHH Y Ha nO,!{npOCTpaHCT-
Ba, COOTBeTCTBYIOU(He 3THM nO,!{TaOJIHuaM, qepeS Yl' Yla,...,
UI' '.n. OtIeBH,!{HO,
y-y' = (Y- YI ... n) + (Yl ." n -YI... n-l) +
+ (YI ... n-l -YI ... n-2) +... + (YI -y').
MO:>KHO nOKa3aTb, tITO YI". 'k = P X1 ... 11. YI". 'k+l (k = 1, ...,
n - 1), eCJIH BCnOMHHTb, liTO YI... 'k BeKTOp Cpe,!{HHX 3HaqeHHH
BeKTOpa Y no KJIaCCaM pa30HeHH5I R I .. .Rk. B CHJIY Toro tlTO
L (Xl) L (X I2 ) c: ... L (X I ... n ) (3aMeTHM, KCTaTH, qTO
i (Xl.. .n) = M (X)), pa3HOCTb Y - YI...n oy,!{eT OpTOrOHaJIbHa
BCeM OCTaJIbHbIM pa3HOCT5IM (KaK 3JIeMeHTaM npOCTpaHCTBa
L (Xl... n», pa3HOCTb YI".n - YI ... (n-l) - BCeM nOCJIe,!{ylO-
HM pa3HOCT5IM (KaK 3JIeMeHTaM npOCTpaHCTBa b (Xl... en-I»)
H T. ,!{., TaK tlTO npHMeHHMa TeopeMa nHQJaropa:
I y-y' 1 2 = I Y-Yl ... n (2+ 1Yl... n -Yl ... (n-l) 1 2 +
+...+ IYI-y'12.
Ka>K,!{oe CJIaraeMOe I YI".. 'k - Yll.. 'k-l Ja 3Toro pa3JIO>Ke-
HH5I BbI,lI.eJIHeT 3<MJeKT BJIHHHHH npH3HaKa Xk, eCJIH BJIHHHHe
1 k-l - .
X , ..., x YiKe YllTeHO. .
B ,!{PyrHx cxeMax nOCTpOeHH5I TaOJIHUbI X C nOMOInblO B3H-
TH5I pa3JIHlIHbIX nepeCetIeHHH nO,!{MHO:>KeCTB pa30HeHHH Rl,...,
Rn pa3JI01KeHHe OOmeH ,!{HcnepCHH MO>KeT OKaSaTbC5I He CTOJIb
npOCTbIM, H B 06 eM CJIYtIae npHXO,!{HTC5I OrpaHHqHBaTbCH
-TaKHMH HCXOJtHblMH npH3HaKaMH, B3aHMO,ll.eHCTBHe KOTOpbIX
. ,npyr C J\pyroM OrpaHHlleHO. TaK, npH ,!{HcnepCHOHHOM aHa-
JIH3e ,!{aHHbIX no ,nB)'M npH3HaKaM Xl, x 2 C yqeTOM BSaHMOJ1e -
CTBHH paCCMaTpHBaeTC5I TaOJIHua X, COCTOH a51 HS Xl' XI H
139
-.,
-
f:
t
.
?
X 12 ' nplilIeM .L{JI5I OOJIerlIeHM5I aHaJIM3a TpeoyeTcH, II TO 6bI pa3-
OHeHHSI R 1 Ii R2 ObIJIH He3aBHCliMbI CT3THCTHIIeCKH (9TO 06bIIIHO
.L{OCTMraeTC5I TeM, lITO BO BCeX KJIaCcaX R 1 H R 2 <pHKcHpyeTcSI
O.L{HHaKOBOe IIHCJIO HaOJIIO.L{eHHH), TOr.L{a, MOjJ{HO npOBepHTb,
9JIeMeHTW pa3JIOjJ{eHH5I
y-y' = (Yl- y') + (Y2-Y') + (Y-YI-Y2 -1- y')
OpTOrOHaJIbHbI, H Ha HX OCHOBe MO)KeT ObITb BbInOJIHeH ,nHcnep-
CHOHHbIH aHaJIH3. nO.L{p06HOCTH CM., HanpMMep, B [42, 87].
OTMeTHM, lITO B .L{licnepCHOHHOM aHaJIH3e ueHTp T5I:>KeCTH
nepeHeceH C 06 eH np06JIeMbI onHCaHM5I Ha npOOJIeMY BbI5ICHe-
HH5I, KaKHMH B3aHMo.L{eHCTBM5IMH MeJK.L{Y nepeMeHHbIMH OOn-
51CH5IeTC5I .L{MCnepCH5I 'Y - Y" C nOMo blO nO.L{xo.L{51 ero nJIa-
Ha npoBe,neHH5I HaOJIIO.L{eHMH.
3.2.4. Pezpeccufl nOMunaAbflblX npu3HaKo8
ECJIH Y - HOMMHaJIbHbIH npH3HaK, TO .L{JI5I ero
3a.L{aHli51 B BH.L{e KOJIHlIeCTBeHHOH Ta6JIHUbl 06'beKT-npH3HaK
Tpe6yeTC5I 6YJIeBCKa51 MaTpHua Y, a He BeKTOp. 1JTO CIIHTaTb
perpeCCHeH MaTpHUbI Y Ha npOCTpaHCTBO L (X)?
nepBoe, lITO npHXO.L{HT B rOJIOBY,- 3TO HCKaTb TaKYIO MaT-
PH[(Y Y CO CTOJIO[(aMH H3 L (X), KOTOpa51 ObI HaHJIYIIIIlHM 06-
pa30M annpOKCHMHpOBaJIa MaTpliUY Y, T. e. MHHHMH3HpOBaJIa
paCCT05lHHe
d(Y,Y)=ly-yf2. (3.38)
,. mz .....
OlIeBH.L{Ho, I Y - YI 2 = I yt - y t l2, TaK QTO np06JIeMa
t= 1
paCna.L{aeTC5I Ha 3a.L{alIH annpOKCHMaUMH Ka1K.L{oro OT.L{enbHoro
CTOJIOua y t B L (X). PeIIleHH5IMH 3THX 3a.L{aq 51BJI5IIOTC5I npo-
eKUHH yt = Px yt, TaK lITO .V = Px Y.
TaKliM oopa30M, perneHHe 3a.L{aQH MMHHMM3aUMH (3.38)
nO.L{pa3YMeBaeT npoeKTMpoBaHlie MaTpHI.I.bI Y KaK cOBoKynHo-
CTH 6YJIeBCKHX CTOJI6uOB yt Ha L (X). npH 3TOM Y = X X
(XT X)-l XT Y = XA, r.L{e A = (XT X)-l XT Y - MaTpHua
co cTon6uaMH at = (XT X)-l XTyt K03ctx1>HUHeHToB a JIHHeii..
'" ml t .
Horo npe,ncTaBJIeHH5I yt = as X S . K aK Y1Ke OTMeqaJIOCb
, s=:1
eCJIH CTOJIOUbl X S opTOrOHaJIbHbI APyr ,npyry, TO a eCTb cpel{-
Hee SHaqeHHe y' B KJIaCCe Rs npH3HaKa X, T. e. YCJIOBHaH
Bep05lTHOCTb p (tIs) = 8t. .
\40
KalleCTBO annpOKCHMalIJIH XapaKTepH3yeTC5I BeJIHlIHHOH
I Y -PX Y 1 2 = Tr (YT Y)- Tr (yr Px V), (3.39)
npHlIeM HeTpYAHO BHAeTb
N: t
Tr (yrY)=N H Tr (yr Px Y) =Tr (Px yyr) =
N s .
s, t
TaK liTO OCTaTOqHa51 AHcnepcli51 (3.39) MO)KeT ObITb BbIpa)KeH3
Qepe3 COBMeCTHoe pacnpeAeJIeHHe X H Y CJIeAYIO HM oopa30M:
IY- P x YI 2=N ( 1- N;t ) =Na, (3.40)
s, t s.
rAe ex - CpeAH5IH AOJI5I OIliHOOK nponop HOHaJIbHOrO nporHO"
3a (2.66). MbI BHAHM, KaK cOBnaAalOT nOKa3aTeJIH KalleCTBa
onHCaHH5I, OCHOBaHHbIe Ha, Ka3aJIOCb ObI, COBeprneHHO pa3JIHlI..
HbIX npeACTaBJIeHH5IX.
N2
BeJIHlIHHa XapaKTepH3yeT BKJIaA s...ro 3HalleHH5I npH"
t N s .
3HaKa X B Tr (YTPX Y) H MO)KeT paCCMaTpHBaTbC5I KaK XapaK-
TepHCTHKa ero HHcpopMaTHBHOCTH. AHaJIOrHlIHO BeJIHlIHHa
N2
xapaKTepH3yeT HHcpopMaTHBHOCTb X AJI5I onHcaHHSI
5 N s .
t-ro 3HalleHH5I Y B CHJIY MOAeJIH (3.38).
B03MO)KHa APyra51 nOCTaHOBKa 3a,l{allH onHCaHH5I Y B Tep..
MHHax X, npH KOTOpOH npoH3BOAHTC5I annpOKCHMaUHH He
OTAeJIbHbIX CTOJIOnOB yt, a HX JIHHeHHOM KOMOHHanHH y =
= btyt == Vb, rAe b == (b I , .., b ms )' KaK onHcaHO B n. 1.1.4,
t
MO)KeT paccMaTpHBaTbC5I KaK BeKTop MacrnTaOHPYIO HX MHO.
)KHTeJIeM HJIH lIHCJIOBbIX OneHOK, MemO1\, 3HaqeHHM t. B KaqeCT.
Be onHCaHH5I 3TOH KOMOHHanliH B L (X) BbIcTynaeT oJIH)Kaii..
rnHM K HeM BeKTop if == Px y. BeKTOp b eCTeCTBeHHO BbIOHOOI
paTb TaKHM oopa30M, lITOObI OH OOeCnetIHBaJI HaHJIYlIrnee onH..
caHlie, T. e. MHHHMH3HpOBaJI OTKJIOHeHHe
I y- 11 1 2 = I Yb-P x Yb 1 2 = b T YT Yb-b T YT Px Yb (3.41)
BeKTopa y OT ero npoeKnHH Ha L (X). nOCKOJIbKY 3TOT nOKa-
3aTeJIb AOCTHraeT MHHHMYMa npH HYJleBOM b, orpaHHlIHM KJIaCC
BeKTopOB, HCnOJIb3yeMbIx AJI5I MeTOK, BeKTopaMH cpHKcHpoBaH-
HOM AJIHHbI, T. e. nOTpeoyeM
IbI 2 =b T b== 1, (3.42)
If npH .9TOM OrpaHHqeHHH oYAeM MHHHMH3HpOBaTb (3.41).
141
--
\
"'\'w.
.
...
PeI.IleHMe 3a.natIM MOmHO nOJIYtIMTb, npMMeH5I5I KJIaCCMtIeC-
KHH MeTO.L{ MHO>KMTeJIeH JI8rpaH>K8, COrJI8CHO KOTOPOMY onTH-
MaJIbHbIH BeKTOp b 06pa aeT B 0 npOH3BO.L{HYIO <PYHKUHH JIar-
paHma 3a.L{atIH (3.41), (3.42):
<D (b) = bTyTYb - bTYTPX Yb + Iv (1 - bTb).
npO.L{MqxpepeHIIJlpyeM <D no b COfJIaCHO npaBHJIY (n.19)
H npHp8BH5IeM pe3YJIbT8T HYJIIO:
oeD =2YTYb-2YTP x Y-2Ab=0,
ob
TaK liTO
yr (E - Px) Y b -== 'Ab,
(3.43)
npHtIeM
Iv = b T YT Yb -b T yr Px Yb = I y-y r\ (3.44)
liTO nOJIytIaeTC5I YMHO)KeHHeM (3.43) Ha b T CJIeBa C YlIeToM (3.42)
_ 3TO nOKa3bIBaeT, tITO b 51BJISleTC5I c06CTBeHHbIM BeKTopOM
MaTpHUbI YT(E - Px) Y, COOTBeTCTBYIO HM ee MMHHM8JIbHOMY
-c06CTBeHHOMY lIMCJIY (3.44). YMHO)K8H (3.43) CJIeBa H8 Y, no-
JIYtIHM TaK>Ke
YYT (E .:.- Px) Y = AY, (3.45)
T8K liTO npoeUMpyeMa51 JIMHeHHa51 KOMOI-lHaUHSI Y = Yb-
-3TO CQOCTBeHHbIH BeKTOp M8TpHUbI yyr (E -- Px), COOTBeTCTBY-
IO MH TOMY :me A.
3aMeTHM Tenepb, liTO CJIe.L{ M8TpMUbI yr (E - Px) Y, P8B-
HbIH B CHJIY (3.40) Na, x8paKTepM3yeT K811eCTBO onHcaHHH
Y B L (X), T. e. oCTaTollHYIO .L{HCnepCHIO IY -Px Y12. nOCKOJIb-
KY CJIeJ( MaTpHUbI p8BeH cYMMe ee c06cTBeHHbIx tlMCeJI, MbI
-BM.L{HM, liTO BeJIHtlHHa A xapaKTepH3yeT TY MHHMMaJIbHYIO lIaCTb
0I.IlH6KH nponopUHOH8JIbHOrO nporH03a Y no X, KOTOp8H npH-
-XOJ(HTC5I H8 onHcaHHe Y C nOMO blO Px y. 31'0 Henocpe,lJ.CTBeB-
-HO BbITeK8eT H3 <POPMYJI (3.44) II (3.43).
AH8JIOrHtIHbIM 06pa30M C YlIeTOM p8BeHcTB8
In
fY_ P x Y J2=Na = Ak,
k== 1
(3.46)
}
t
r.L{e m = min (nil' m 2 ) - paHr MaTpHIIbI yr (E - Px) Y, HB-
TepnpeTMpYIOTCSI OCT8JIbHbIe coocTBeHHbIe tlHCJI8 An (k < m)
3TOM M2lTPHUbI.. CooTBeTcTBYIOIIlHe HM coocTBeHHbIe BeKTopbI
yk = Yb k 06p83YIOT opTOrOHaJIbHylO cHcTeMY BeKTopoB, no-
CJIe,nOB8TeJIbHO - CHHM8 HX BCID OWHOKY (3.46), npMtleM 3Ta
'\.
.142
CHCTeMa Opr8HH30BaH8 TaKHM OOp830M, liTO H8 K8)K,l{bIH CJIe-
.llYlOllJJ1H BeKTOp yk npHXO,l{HTCSI MHHHM8JIbHO B03MO}KHa51
qaCTb OIIlHOKH (3.46) npH- YCJIOBHH ero opTorooaJIbHOCTH npe-
)lbI,l{Y HM BeKTop8M. 31'0 OTJIHlIaeT CHCTeMY {yk} OT HCXO HOH
CHCTeMbI opTOrOHaJIbHbIX BeKTopoB {yt}.
<1>opMYJIa (3.44) n03BOJISleT BbIpa3HTb A, lIepe3 3JIeMeHTbI
MaTpHn bI yr (E - Px) Y, T. e. tlepe3 COBMeCTHoe p8Cnpe,l{eJIe-
HHe X H Y:
( N ) 2 N st N st '
A. = N s . - N bt - b t bt, N . (3.47)
s. t s. t::/=t' s s.
3TO pa3JIO:>KeHHe n03BOJISleT 8HaJIH3HpOBaTb BKJIa,l{bI OT,Z(eJIb-
HbIX 3HalleHHH S H i, nap (5, i) H (i, I'), 8 TaK)Ke TpoeK (5, i, i/)
B BeJIH1JHHY A. H8npHMep, BKJIa,Z( KOMOHHanHH 3HaqeHHH S H
t paBeH
( N s . - N s ' ) bl-bt Nst N st ' b t "
N s . N s . t'::/=t
3.2.5. KaH,OHut/,eCK,Uu anaAU3 flOMunaAbflblX
npU3HaK,08
B p8ccMoTpeHHoH 3a,l{8Qe onHCaHH5I JIHHeHHOH
KOMOHHanHH y = Yb B L (X) HCnOJIb30BaJIOCb orpaHHlIeHHe
(3.42), <PHKcHPYlOrnee B KaqeCTBe p8BHonpaBHbIx ,l{onycTHMbIX
npeTeH,l{eHToB BceB03MO}KHhIe CTPYKTYPbI BeCOBbIX K03qxp H n H -
eHTOB b t ,l{JISI OT,l{eJIbHbIX 3H8qeHHH npH3HaK8 Y. npH 3TOM
caM BeKTop y = Yb HrpaeT no,Z(qHHeHHYIO pOJIb, 51BJI5I5ICb pe-
3YJIbT8TOM npeJIOMJIeHHR b nOCpe,l{CTBoM M8TpHnbI Y. B03-
MO:>KeH HHOH 8KneHT B nOCTaHOBKe 38,l{allH onHC8HHSI, Kor,l{a
BeKTopbI y E L (Y) caMH paccMaTpHBalOTcSI KaK paBHonpaB-
HbIe OO'beKTbI onHC8HHSI, H, CJIe,l{OB8TeJIbHO, nOHCK pelIleHH5I
38,l{allH ,l{OJI)KeH BeCTHCb B MHO)KeCTBe BeKTopoB Y o,l{HHaKOBoH
,l{JIHHbI. TaKoH 8KneHT npHBO,l{HT K 3a,l{alle MHHHMH3anHH (3.41)
npH YCJIOBHH
yT y=bTYTYb= 1. (3.48)
B CHJIY (3.48) MHHHMH3anH5I (3.41) 3KBHB8JIeHTH8 MaKCH-
MH3anHH
b T yr Px Vb.
(3.49)
,.QJIH peweHHH 38.L{8l1H M8KCHMH3aUHH (3.49) npH YCJIOBHH
(3.48) On5ITb BOCnOJIb3yeMcSI YCJIOBHeM paBeHcTBa HYJIIO npo-
143
,.....,. .
H3BO,nHOH <PYHKnHH JIarpaH}I{a
<I> = b T yr Px Yb +A (l-b T yr Vb),
liTO ,naeT
yr Px Yb = AYT Vb,
(3.50)
npHlIeM Iv = bTyrpx Vb, .1ITO nOJIYlIaeTC5I YMHb>KeHHeM (3.50)
Ha b T CJIeBa.
YMHO:>Ka51 (3.50) CJIeBa Ha Y (yr Y)-l, nOJIYtIHM ,nJI5I
y = Yb
Py Px Y = AY,
(3.51)
oTKy,na CJIe yeT, liTO HCKOMbIH BeKTop Y RBJIHeTC5I coocTBeH-
HbIM BeKTopoM MaTpHnbI PyPX' T. e. KaHOHHtIeCKHM BeKTopoM
,nJI5I X II y, cooTBeTcTBYIO HM MaKCHMaJIbHOMY COOCTBeHHO-
MY lIHCJIY Iv (CM. pa3,neJI. n.9).
B COOTBeTCTBHH C n.9 BeKTOp MaCIIITaOHPYIO HX MHO HTe.
JIeM b 51BJI5IeTC5I COOCTBeHHbIM BeKTopOM MaTpHnbI BA, r,ne
B = (VrV)-l yrx, A = (XTX)-lXry, COOTBeTCTBYIO HM TO.
MY 1Ke MaKCHMaJIbHOMY COOCTBeHHOMY IIHCJIY. npOeKnH5I X =
= Px Y BeKTopa Y eCTb COOCTBeHHbIH aeKTOp MaTpllnbI PXPy,
npHlIeM X = Xa, r,ne a - COOCTBeHHbIH BeKTOp MaTpHnbI AB,
ooa cooTBeTcTBYIOT O,nHOMY H TOMY 1Ke coocTBeHHoMY lIHCJIY Iv.
TaK KaK X H Y HMelOT <POPMY A, TO L (X) n L (Y) CO"
,nepjl{HT HeHYJIeBbIe 3JIeMeHTbI. B qaCTHOCTH, KaK OTMellaJIOCb,
u = (1, ..., 1) E L (X) n L (Y). 3TO napa3HTHoe perneHHe
onpe,neJI5IeTCH <pOpMOM 3aAaHH5I npH3HaKoB H He ,l{OJI}KHO pac-
CMaTpHBaTbC5I, KaK H COOTBeTCTBYIOInee COOCTBeHHoe lIHC'"
JIO 1.
PaccMoTpHM HeTpHBHaJIbHWe KaHOHHlIeCKHe BeKTopM Y =
<= Vb H x = Xa tIHCJIOBbIX 3HatIeHHH npH3HaKoB Ha OO'beKTaX,
kOTopbIe OnTHMaJIbHbI C TOllKH 3peHH5I KpHTepH5I (3.41). KaK
YLKe OTMetIaJIOCb, cooTBeTcTBYIO He lIHCJIOBbIe 3HalleHH5I as II
b t MorYT paccMaTpHBaTbC5I KaK lIHCJIO:9bIe MeTKH 3HalleHHM npH-
aHaKa. Hx ocooeHHocTb COCTORT B TOM, tITO ,l{BYMepHoe pacnpe-
):{eJIeHHe BeKTopOB X H Y aCHMnTOTHtIeCKH no N HOpMaJIbHO, TaK
1ITO pacnpe,l{eJIeHHe Pst = N st / N HX COBMeCTHbIX 8HalleHHH
(as, b t ) nOJIYtIaeTC5I H3 HOpMaJIbHOrO pacnpe,l{eJIeHH5I COOTBeT-
CTBYIO HM HHTepBaJIHpOBaHHeM HenpepbIBHbIX oceM [42].
3Ta ocooeHHocTb ooecnellHBaeT B03MO)KHOCTb npHMeHeHH5I K
X H Y CTaH,l{apTHbIx MeTO,l{OB MaTeMaTHtIeCKOM CTaTHCTHKH, KO-
TopbIe xoporno OoocHoBaHbI HMeHHO ,l{JI5I HOpMaJIbHOrO pacnpe-
,neJIeHHH.
I
I
J
144
--,
PaCCMOTpHM Tenepb CMbICJI neKTopOB a H b KaK COOCTBeIt..
HbIX .L{JI5I MaTpH AB H BA, npHqeM B CHJIY CPOPMYJIbI (n. 34)
a = Ab/V A , b = Ba/V . (3.52)
TIpOH3BOJIbHbIH (8, t)-H3JIeMeHT MaTpHUbI A eCTb Nst/N s ., a
(8, t)-H 3JIeMeHT MaTpH bI B paBeH N st/ N. t, n03TOMY cpOPMYJIbI
(3.52) MorYT ObITb nepenHcaHbI B CJIe.L{YIO eM BH.L{e:
as = I N st bt/V A., b t = I N st as/V A.. (3.53)
t N s . s N. t
PaccMoTpHM CJIe.L{YIO YIO HHTepnpeTanHIO 3THX COOTHOIlIe-
HHH. ,[{JI5I s-ro 3HaqeHH5I X paCCMOTpHM pacnpe.L{eJIeHHe Pt ==
= Z't BepoHTHOCTeH: 3HaqeHHH: Y, a )J.JHI t-ro 3HaqeHHH Y-
s.
pacnpe,ll.eJIeHHe p; == N st / N. t Bep05lTHOCTeH 3HaqeHHH x. no-
TpeoyeM, qTOObI MeTKa as paBHSlJIaCb Cpe.L{HeMY 3HaqeHHIO
(neHTpy T5I}KeCTH) MeTOK Y, T. e. as == Ptbt, a i-SI MeTKa
t
Y paBHSlJIaCb Cpe,ll.HeMY 3HaQeHHIO MeTOK X, T. e. b t == p as.
KaK CJIe,nyeT H3 (3.53), 3TO B03MO}KHO TOJIbKO npH TpHBHaJIb-
HOM coocTBeHHoM qHCJIe J..., == 1. Ho B CHJIY Tex )Ke COOTHOIlle-
HI1H MO}KHO YTBep}K,lI.aTb, qTO KaHOHHqeCKHe MeTKH nponopnH-
OHaJIbHbI neHTpaM T5I)KeCTH, npHqeM MaKCHMaJIbHOe 3Haqe-
HHe J..., OOeCneqHBaeT HaHMeHbIllee OTKJIOHeHHe K03 HnHeHTa
nponopnHOHaJIbHOCTH 1 /V J..., OT e,ll.HHHnbI. CKa3aHHoe n03BOJISleT
TpaKTOBaTb BeJIHqHHbI as H b t KaK KOOp,ll.HHaTbI O.L{HOH H TOR
e OCH, qTO n03BOJISleT KO HqeCTBeHHO COOTHeCTH 3HaqeHHSI
.L{BYX pa3JIHqHbIX HOMHHaJIbHbIX npH3HaKOB H TeM caMbIM 00-
JIerqHTb HHTepnpeTanHIO [116, 152].
<I>OPMYJlbI (3.53) n03BOJI5IIOT TaK}Ke TpaKTOBaTb paccMoT-
peHHbIH MeTO,ll. onHCaHH5I KaK aHaJIOr MeTO,ll.a r JIaBHbIX KOMno-
HeHT ,lI.JISI TaOJIHnbI COnpSl}KeHHOCTH B HCKpHBJIeHHbIX npocT-
paHCTBax. ,[{JISI 3Toro BMeCTO OOblqHOrO CKaJI5IpHOrO npoH3Be-
eHHSI paccMoTpHM ,lI.JISI m1"MepHbIX BeKTopoB a, a' CKaJI5IpHOe
npoH3Be)J.eHHe a X a' = a'TSx1a' = N l a,a;, a )J.JHI mi-
s s.
MepHblx BeKTopoB b, b ' CKaJI5IpHOe npOH3Be,ll.eHHe
bx b'=bTSy1b= I btbt.
t Nt
¥lHbIMH CJIOBaMH, npHnHlIleM BceM Koop,ll.HHaTaM Beca, 06.
paTHbIe qHCJIeHHOCT5IM COOTBeTCTBYIO HX KJIaCCOB. B 3THX
npocTpaHcTBax npHMeHeHHe MaTpHI.I.hI S == XTY = (N st )-
145
.......
...............
Ta6JIH[(bI COnp5l)KeHHOCTH - K BeKTopaM a M b .L{aeT Sx b=
== (yry)-l Sb H aT X S = aTS (XTX)-l. AHaJIOfHlJHO nOJIY-
qHM SxS T = (YTY)-l SST, STXS = STS (XTX)-l.
Co6CTBeHHbIH BeKTOp a MaTpH[(bI ST X S B nCKpHBJIeHHOM
npOCTpaHCTBe Y.L{OBJIeTBOpHeT paBeHCTBY
'Aa= ST X S X a=(YT y)-t ST (XT X)-l Sa = BAa,
KOTopoe nOKa3bIBaeT, liTO a - COOCTBeHHbIH BeKTOp MaTpH[(bI
BA B OObIQHOM npOCTpaHCTBe. AHaJIOrHqHO COOCTBeHHbIH BeK-
TOp MaTpH[(bI Sx ST B HCKpHBJIeHHOM npOCTpaHCTBe COBna,naeT
C COOCTBeHHbIM BeKTopOM MaTpH[(bl AB. TaKHM oopa30M, BeK-
TOpbI MeTOK a, b MO:>KHO paCCMaTpHBaTb KaK pe3YJIbTaT npH-
MeHeHHH MeTO,l{a rJIaBHbIX KOMnOHeHT K MaTpH[(e S B HCKpHB-
v v
JIeHHhIX npOCTpaHCTBax, nOpO:>K,l{eHHbIX STOH MaTpH[(eH.
COBMeCTHoe paCCMOTpeHHe CKaJI5IpHbIX npOH3Be,l{eHHH X B
npOCTpaHCTBax CTpOK H CTOJIO[(OB MaTpH[(bI COnpR:>KeHHOCTH
npHBO,l{HT K CJIe,nYIO HM paCCTOHHH5IM Me:>K,l{Y ee CTpOKaMH H
CTOJIOuaMH:
d(s, s')= 1 ( Nst
t N.t N s .
d(t, t')= ( Nst
s N s . N.t
N s ' t ) 1 ,
N s ' .
N st \ I
N. t , )
(3.54)
TIpH 3TOM B KalleCTBe KOop,l{HHaT OO'beKTOB-CTpOK s pac-
cMaTpHBaIOTC5I 3JIeM eHTbI YCJIOBHOrO pacnpe,l{e.neHHH Pt =
= N st / N s . H Me:>K,l{Y HHMH BbIlIHCJIHeTC5I paCCTOHHHe B HCKPHB-,
JIeHHOM npocTpaHcTBe; B KatIeCTBe KOOp,l{HHaT OO'beKTOB-CTOJIO"
[(OB t paCCMaTpHBalOTCH YCJIOBHbIe BepOHTHOCTH P =
= N st/ N. t H paCCT05lHHe BbIlIHCJIHeTCH Me:>K,l{Y 3THMH BeKTopaMH
B HCKpHBJIeHHOM npocTpaHcTBe.
PaccToHHHR (3.54) npHBJIeKaTeJIbHbI C HHTYHTHBHOH TOllKH
3peHHH, nOCKOJIbKY OTO:>K,l{eCTBJI5IIOT 3HaqeHHH npH3HaKoB C
O,l{HHaKOBbIMH YCJIOBHbIMH pacnpe,l{eJIeHHHMH; B 3TOM CMbICJIe
OHH OCHOBaHbI Ha yqeTe CTPYKTYPbI, a He OO'beMa pacnpe,l{eJIe-
HHH. MO)KHO ,l{OKa3aTb TaK)Ke [116], tITO eCJIH YCJIOBHbIe pacnpe-
,l{eJIeHHH CTpOK S H S' O,l{HHaKOBbI, TO OO'be,l{HHeHHe 3THX CTpOK
B O,l{HY CTPOKY TaOJIH[(bI COnp5l)KeHHOCTH (CYMMHpOBaHHeM 3JIe-
MeHTOB) He H3MeHHT CHCTeMY paCCT05lHHH Me:>K,l{Y CTpOKaMH
(TO :>Ke BepHO H ,l{JI5I CTOJIO[(OB).
IlpHBe,l{eHHbIe COOOpa)KeHH5I ,l{JIH MeTO,l{a (3.53) ObIJIH pa3BH-
TbI)I(. BaH3eKpH [116], KOTOpbIH Ha3BaJI ,l{aHHbI:H MeTO,n Memo-
iJOM KoppecnOflaefl'1ut1 (CM. TaKJKe [152]).
146
..
v
,[{JISI O eHKH 3HaQHMOCTH OT,!{eJIbHbIX 3HaQeHHH npH3HaKOB,
TaK }Ke KaK H B npe,!{bI,!{Y HX CJIyqa5lX, paCCMOTpHM HX BKJIa-
,lJ.bI B BeJIHqHHY
A.=bT)'TPX ),b= N: t br.
s. t Nso
(3.55)
I1cxo,!{51 H3 3TOrO 3HaqHMOCTb t-ro 3HaqeHHSI npH3HaKa Y
MO)l{eT ObITb OneHeHa CJIaraeMbI f '!.J.v: t bl / N s ., a 3HaqHMOCTb
s
s-ro 3HaqeHHSI X - BKJIa,!{OM Ns br/ N s. .
t
B TOM CJIyqae, Kor,!{a 3HaqeHHe A He CJIHlllKOM BeJIHKO no
cpaBHeHHIO C CYMMOii Bcex COOCTBeHHbIX qHCeJI Tr (PXPy) . l,
CJIe,!{yeT nepeXO,!{HTb K nOHCRY CJIe YIOI.1JJ X KaHOHHqeC HX BeK-
o
TOpOB ,!{O Tex nop, TIOKa HX CYMMapHbIH Bec He ,!{OCTHrHeT :me-
JIaeMOH ,!{OJIH BeJIHqHHbI Tr (PxPy) - 1. OOblQHO ,!{JIH HHTep-
npeTanHH ,!{OCTaTOqHO HMeTb O,!{HH-,!{Ba aKTopa C TeM, qTOOW
MomHO ObIJIO BH3yaJIbHO npe,!{cTaBHTb COOTHOlIleHHe 3HaqeHHH
B aKTopHoM npocTpaHcTBe [152J.
nYCTb m = min (m 1 , m 2 ), Tor,!{a ,!{JI5I MaTpHn X H Y HMe-
eTCH m - 1 HeTpHBHaJIbHOe KaHOHHqeCKOe qHCJIO A1 Az
... Am-l o. CYMMa 3THX qECeJI MO)Ke:r HCnOJIb30BaTbCH
,!{JISI xapaKTepHcTHKH nOJIHOH CBSl3H Me)K,!{Y X H Y. HanoMHHM,
qTO 3HaqeHH5I V As KaHOHHqeCKHX KOppeJIHnHH xapaKTepH3YIOT
CTeneHb KOppeJIHUHH Me>K,!{Y BeKTopaMH OpTOHOpMHpoBaHHhIx
oa3HcoB XS H yS npocTpaHcTB L (X) H L (Y), HaH60Jlee opHeHTH-
pOBaHHhIx ,!{Pyr K ,!{Pyry. ECJIH As = 0, TaK QTO A 1 =... =
= A m -l = 0, TO npOCTpaHCTBa L (X) H L (Y) O,!{HOMepHbI
H COBna,!{aIOT CO CBOHM nepeCeqeHHeM L (X) n L (Y). B 3TOM
CJIyqae MaTpHna PXPy HMeeT e,!{HHCTBeHHoe HeHYJIeBOe co6-
CTBeHHoe qHCJIO A, = 1 C COOTBeTCTBYIO HM co6cTBeHHbIM BeK-
..- -
TOpOM U = (1, ..., l)/V N H B CHJIY H3JIO)KeHHOrO B 0.9 HMe-
eT BH,!{ -PXPy = auT, TaK qTO ,!{JIH BCHKoro 9JIeMeHTa Pij 3TOH
MaTpHlI.bI HMeeT MeCTO paBeHCTBO
1
Pij =-.
. N
Ho c ,!{pyroii CTOpOHbI, no onpe,ll.eJleHHIO MaTpHlI.bI PXPy
ee 3JIeMeHTbI Pij npH x (i) = s, Y (j) = t paBHbI N st/N s. N. t, TaK
qTO nOJIytIaeM ,!{JI5I JIIP6blX S H t paBeHCTBO
N st 1
=-,
N B . N. t N
J47
\
-- -'"
KOTopoe nOKa3bIBaeT, tITO npH3HaKH X H Y CTaTHCTHlIeCKH He-
3aBHCMMbI. 3TO 03HatIaeT, liTO As = 0 TOr,!{a H TOJIbKO TOr,!{a,
KOr a npM3HaKH He3aBHCHMU.
MaKCMMaJIbHOe 3HatIeHHe As = m - 1 ,!{OCTHraeTC5I npH
Al =.. . = Am-l = 1, T. e. npH COBna,!{eHHH BeKTopoB X S C y5, TaK
tITO L (X) COBna,!{aeT C L (Y). nOCKOJIbKY ,!{JI5I CJIYlIa51 pa30He-
HHM L (X) COBna,!{aeT C M (X), L (Y) c M (Y), TO 3TO 03HatIaeT,
liTO COBna,!{aIOT pa30MeHH5I, 3a,!{aBaeMbIe npH3HaKaMH. I1HbIMM
CJIOBaMH, Al =... =A m - 1 = 1 Tor,!{a H TOJIbKO Tor,!{a, Kor,!{a MaT,.
PH[(bI X H Y paBHbI C TOllHOCTblO ,!{O nepeCTaHOBKH CTOJIO[(OB.
m-l
BoooI..I.(e, CYMMa I, As COBna,LlaeT C <D 2 (2.57), TaK KaK
s= 1
m-t N
'" A.s =Tr{P x Py) -1 = Pii-1 = N st st 1 =<1>2,
Ns.N. t
5=1 I s,t
3TO ,!{aeT pa3JIo*eHHe CTaH,!{apTHoro nOKa3aTeJI5I KBa,!{pa-
THlIeCKOM COnp5l:>KeHHOCTH <1>2 Ha BeJIHlIHHbI As, YlIHTbIBaeMbIe
COOTBeTCTBYIOI..I.(RMH KaHOHHlIeCKHMH BeKTopaMH. Bep05lTHO,
nepBbIM, KTO OOpaTHJI BHHMaHHe Ha 3TO pa3JIOJKeHHe, ObIJI
<I>. JI3HKacTep [150].
3aMeTHM TaK:>Ke, tITO B 3THX TepMHHax K03qxpH[(HeHT Kpa-
Mepa (2.58) eCTb He liTO HHoe, KaK cpe,!{Hee 3HalleHMe COOCTBeH-
HbIX lIHCeJI As:
I
I
r
I
I
I
!
I
I
I
K 2 = 1 As.
m-l
5
K. nHpCOH nOKa3aJI, tITO eCJIH X H Y nOJIYtIeHbI B pe3YJlb-
u
TaTe HHTepBaJIHpOBaHH5I oceR ,!{BYMepHoro HOpMaJIbHOrO pa3-
OHeHH5I pacnpe,!{eJIeHH5I C K03<p<PH[(HeHTOM KOppeJl5l[(HM p, TO
npM YMeHblIleHHH ,!{JIHHbI HHTepBaJIOB BeJIMlIHHa <1>2 CXO,!{HTC5I
K p/(l - p), TaK KaK KaHOHMtIeCKHe lIMCJIa ,!{JIH HOpMaJIbHOrO
pacnpe,!{eJleHM5I oopa3YIOT reOMeTpHlIeCKYIO nporpeCCMIO
p, p2, p3, ..., CYMMa qJIeHOB KOTOpOM H ,!{aeT p/(l - p).. n03TO-
MY B KalleCTBe nOKa3aTeJI5I CB5I3H X H Y K. nHpCOH npe,!{JIO*HJI
HcnOJIb30BaTb BeJIHtIHHY
IT2 = X 2 (3.56)
N+X 2
KOTopaH CXO,!{MTC5I K p2 npH YMeHblIleHMH HHTepBaJIOB pa30HeHH5I
HCXO,!{HbIX HenpepbIBHbIX oceM. nOKa3aTeJIb nHpcoHa IT,
KaK H <1>, paBeH HYJIIO Tor,!{a H TOJIbKO Tor,!{a, Kor,!{a npH3HaKH
CTaTHCTHlIeCKH He3aBHC HMbI. MaKCH MaJIbHOe ero 3HaqeHHe npH
$2 = m - 1 paBHO V m /(m + 1) H, CJIe,!{OBaTeJlbHO, He ,!{O-
CTMraeT e,!{MHH[(bI ,!{a:me npH paBeHCTBe X H Y. t
148
t"'ft";.,...
3.2.6. 3a8atlu pezpeccuu no cucre.Me
HOMUflaA bflblX np U3flaK08
nYCTb HOMHH8JIbHbIe npM3H8KH Y H Xl,..., x n , 3a-
]lalOIIIJle pa30HeHH5I S = {St}, R 1 = {R:}, ..., Rn = {R:} H8
MHO>KeCTBe OO'beKTOB, npe,!{CT8BJIeHbI 6YJIeBCKHMH M8TpHUaMH
Y, Xl' ..., X n p83MepHOCTH NXm, Nxml,..., Nxm n COOT-
BeTCTBeHHO.
KaK YTOqHHTb npOOJIeMY onHCaHH5I npH3H8Ka y npH3H8K8MH
Xl, ... x n B TepMHHax 3THX M8TPHQ?
P83BMBa51 H,!{eH npe,!{bI,!{YInHX p83,!{eJIOB, npH,!{eM K 38,!{aQe
MHHHMH3aUHH CYMM8pHoro OTKJIOHeHH5I
d (Yb, Ph Yb) = I Yb -Ph Yb 1 2 (3.57)
k k
no BceM BeKTop8M b, Y,!{OBJIeTBOp5lIOInHM paBeHCTBY
bTb = 1
(3.58)
HJIH
b T yTYb=l. .(3.59)
3,!{ecb BeKTOp b = (b t ) - 3TO BeKTOp T8KHX lIHCJIOBbIX Me-
TOK b t 3HalleHHH npH3HaKa y, npH KOTOpbIX tlHCJIOBOH BeKTOp
y = Vb, CT8B51I HH B COOTBeTCTBHe OO'beKT8M i MeTKH b t 3Ha-
qeHHH y (i) = b t B cpe,!{HeM OJIH>Ke Bcero K npocTpaHcTB8M
L (Xl)' ..., L (X n ), T. e. K npOeK[(H5IM Xh, = Ph Vb, HMeIOIIIJIM
BH,!{ Xh, = Xh,ah" r,!{e BeKTopbI ah CYTb ah, = (XIXh)-l XIYb.
3aMeTHM, liTO
I Yb-PhYb 1 2 = Tr [(Yb-Ph, Yb)T (Yb-Ph, Yb)] =
k k
=nbTfTYb-bTyT PYb, (3.60)
r,Lle P = Ph, - Y>Ke paccMoTpeHH851 B 3.1 M8TpHua CB5I3H
k
MeLK,!{Y OO'beKTaMH. nOJIYlIeHHoe BbIpa>KeHHe nOKa3bIBaeT,
liTO 8H8JIH3 perneHH5I 38,!{allH MHHHMH3aQMH (3.57) nOJIHOCTblO
8H8JIOrHtleH aHaJIH3Y 3a,!{811H MHHHMH3a[(HH (3.41), TOJIbKO
BMeCTO Px 3,!{eCb cpHrYPHpyeT cYMMapH851 MaTp H U8 P =
= (Pij).
T8KHM oopa30M, MO>KHO YTBep}l{,!{aTb, liTO perneHHe 3a,!{atIH
MHHHMH38UHH (3.57) npH orpaHHlIeHHH (3.58) 38,!{8eTCH c06-
CTBeHHbIMH BeKTop8MH M8TPH[(bI YYT (n E - P) (aH8JIOrHtI.
HO (3.45) ), 8 npH orp8HHlIeHHH (3.59) - COOCTBeHHbIMH BeK-
TopaMH M8TpHUbI PyP (aH8JIOrHtlHO (3.51».
149
.....
3JIeMeHTbI 3THX MaTpHU MOrYT 6bITb OXapaKTepH30BaHbI CJIe..
,nYIO HM 06pa30M. ,aJI5I i ESt, r,ne S = {St} - pa36HeHHe no
npH3HaKY Y, (i, j)-if 3JIeMeHT PyP paBeH cpe,nHeH CBH3H 000-
eKTa j C 06'beKTaMH KJIaCCa St, T. e. I, PZj/ Nt. AHaJlOrHqHO
, lESt
npH i ESt, JESt, (i, j)-it 3JIeMeHT MaTpHUbI yyr (n E - P)
. paBeH 'nett' -- I, pjz, r,ne ett' - 3JIeMeH.T e,nHHHtIHOH
I ESt
-MaTpHlI.bI E.
3HalleHH5I nOKa3aTeJIH (3.57) xapaKTepH3YIOTC5I COOTBeTCT-
BYIOInHMH coocTBeHHbIMH tIHCJIaMH 3THX MaTpHU, TaK qTO,
KaK H B npe,nbI,ny eM nYHKTe, CYMMbI co6cTBeHHbIx tJ:HCeJI,
T. e. Tr (PyP) H Tr (YYT (nE - P» MorYT paccMaTpHB8TbCH
. KaK xapaKTepHcTHKH CB5I3H npH3H8K8 Y C CHCTeMOH Xl' ...,
Xn.
,[(JISI MaTpHUbI PyP CYMM8 ee COOCTBeHHhIX HeTpHBHaJIbHbIX
lIHCeJI paBHa B CHJIY ,nOKa3aHHOrO H8 c. 148
Tr (Pv P)-n = (Tr (Pv Ph)-I)= cp2 (Y, X k ), (3.61)
k k
r,ne <1>2 - CTaH,napTHbIH nOKa3aTeJIb KB8,np8THqeCKOH conp5l-
)KeHHOCTH (2.57). 3To 03HaqaeT, tlTO C TOtIKH 3peHHH KpHTe-
pHH (3.57) B KaqeCTBe MepbI CB5I3H npH3HaKa H cOBoKynHocTH
npH3H8KOB MO)KeT HCnOJIb30BaTbCH CYMM8 non8pHMX CBSl3eH
(3.61) .
C ,npyroH CTOpOHbI, B TepMHH8X 3JIeMeHTOB MaTpHUhI Py P
m m 1
Tr{PyP)-n= : n= N; plJ-n,)
t= t !,lES t t= t J.IES,
(3.62)
T. e. CBSl3b (3.61) npH3HaKa Y C CHCTeMOH Xl' ..., X n Onpe,lJ.e-
JIHeTCH CYMMapHOH Cpe,lJ.HeH CBH3bID 06 eKToB no npH3H8KaM
X k B pa36HeHHH S. 3,necb nOJIH8H aH8JIOrHSI C 38,naQeH nocTpoe-
HHH <l>aKTopa-npoeKTopa, TOJIbKO Y He KOHCTpYHpyeTcH, a 33-
.naH. CJIe.nOBaTeJIbHO, KpuTepHH (3.16) 38,naQH nocTpoeHHH
<l>aKTop8'-npoeKTopa TaK1Ke M01KeT 6bITb .HHTepnpeTHpoBaH
KaK CYMMa Mep CBSl3H npH3H8KOB Xl' ..., X n C <i>8KTOpOM Z.
P8CCMOTpHM Tenepb C)'MMY COOCTBeHHblX QUCen MaTpHUhI
yyr (nE - P), X8p8KTepH3YIOInYIO, KCTaTH rOBOpH, C)'MMap-
HOe paCCT05lHHe
I Y -Ph Y pa=Tr(YYT (nE-P») = Tr(YYT (E-P k )=
k k .
150
_.A
-
= N (X (Y, X k ), (3.63)
k
r,ne a (Y, X k ) - nOKa3aTeJIb (2.66) CBH3H MeLK Y X k If Y,
QTO CJIe yeT H3 (3.46). C PyroH CTOpOHbI,
m
Tr (yyT (nE -P» = Nn - Pi/t (3.64)
t== 1 i, lESt
TaK liTO, OKa3bIBaeTCH, CYMMapHaH BHYTpeHHHH CBH3b no npH3-
HaK8M X h B pa30HeHHH S xapaKTepH3yeT Cpe)(HIOIO OIllHOKY
nponopUHOHaJIbHOrO nporH03a Y no K8)K)\OMY OT,l{eJIbHOMY
npH3HaKY X k .
npH aH8JIH3e KOHKpeTHoro COOCTBeHHoro BeKTopa MeTOK
b CJIe,l{yeT HMeTb B BH)\Y, liTO BKJIa,l{ k-ro npH3HaKa B BeJIH-
IIHHY COOTBeTCTBYIO ero COOCTBeHHoro 3HalleHH5I A, =
= bTYTPyPYb (HJIH A = bTYT (YYT (nE - P» Vb) one-
HHB8eTCH BeJIHIIHHOH Ah = bTYTPyPk Yb = bTYTPk Yb =
= yTXh' r,l{e y = Vb, Xk = PkY, CKaJIHpHOrO npOH3Be,l{eHHH
Yb Hero npoeKUHH Ha L (X k ) (HJlH BeJIHlIHHOH Ak =
=yT (YYT (E - Pk»y).
TaKHM oopa30M, KpHTepHH (3.57) npHBo,l{HT K npocTbIM Ii
HarJIH,l{HbIM KOHCTPYKUHHM, TaR liTO onHcaHHe Y B CHCTeM8
Xl' ..., X n Onpe,l{eJI5IeTCH TOJIbKO napHbIMH CBSl351MH BbIXO,l{HO-
ro nOK83aTeJISI Y C Ka}K,l{bIM H3 BXO,l{HbIX nOKa3aTeJIeH X k .
9T8 npOCTOTa CB5I3aHa TaK>Ke C CYIneCTBeHHbIM He,l{OCTaTKOM
u
paCCMOTpeHHOH MO,l{eJIH: COBepIlleHHO HenOHSlTHO, KaK HCnOJIb-
30BaTb npOeKUHH Xk = Ph Yb HJIM COOTBeTCTBYIOInHe BeKTOpbI
ak B Xk = Xkak ,l{JISI XapaKTepHCTHKH H, B lIaCTHOCTH, npo-
rH03a Y.
B03MO}KHOCTb nOJIYlIeHHH OTBeTa Ha nOCTaBJIeHHbIH BOnpOC
OOeCneIlHBaeTC5I ,l{PyroH MO,l{eJI, blO npH KOTOpOH B pOJIH nOKa-
u
3aTeJIH K811eCTBa OnHCaHHSI B cTynaeT KpHTepHH
, y - Ph Y 1 2 = I Y -py 1 2 ,
k
(3.65)
COrJIaCHO KOTOPOMY CTOJIOUbI yt MaTpHUbI Y npHOJIH1KeHHO OnH-
CbIB810TCH CYMMaMH npOeKUHH Phyt. JIlOoonbITHO, liTO no-
k
nbITKa BBe,l{eHHH B KpHTepHH (3.65) BeCOBbIX K09c}xPHUHeHTOB
ak npH3HaKOB Xk C neJIblO )\OnOJIHHTeJIbHOH MHHHMH3aUHH MO-
JIJ 1 <p H U H pOBaHHoro KpHTepHH I Y - CXkPk Y 1 2 3a ClIeT BbIOOpa
k
CXk HHlIero He ,naeT: MHHHMYM TaKoro MO,l{H<pHUHpOBaHHoro
KpHTepHSI ,nOCTHraeTC5I npH O,l{HHaKOBbIX Becax npH3HaKoB.
151
L
-
Cor JIaCHO (3.65)
J Y _P)l J2 = Ir «Y _py)T (Y -pY)) = Tr (E _P)2 YV t )::::
= (yyT, (P_ E)2), (3.66)
r,!{e nOCJIe,!{Hee BbIpa)KeHHe eCTb CKaJI5IpHOe npOH3Be,!{eHHe MaT-
pHI.{ Y YT H (P - E)2 KaK BeKTopOB N2-MepHoro npOCT-
paHCTBa.
HanOMHHM, liTO YYT eCTb oYJIeBCKa51 (1 ,O)-MaTpHl.{a pa30He-
HHSI S, HMelO aH (i, j)-M 3JIeMeHTOM e,!{HHHI.{Y Tor,!{a H TOJIbKO
Tor,!{a, Kor,!{a 3HalleHHSI npH3HaKa y Ha OO'beKTaX i H j COBna-
,!{aIOT. MaTpHl.{a B = (P - E)2 = p2 - 2P + E MOJKeT
paccMaTpHBaTbC5I KaK MaTpHQa CBSl3eH Me}K,!{Y OO'beKTaMH;
ee 3JIeMeHTbI b ij = Pil P,j - 2Pij + eij xapaKTepH3YIOT
I
CTeneHb nOXO)KeCTH cooTBeTcTBYIO HX CTpOK i H j MaTpHl.{bI
P. BeJIHlIHHa b ij OTpa}KaeT CB5I3b OO'beKTOB i, j C YlIeTOM HX
CBSl3eH C ,!{PyrHMH OO'beKTaMH.
n
nOCKOJIbKY p2 = PkP l , TO b ij BbIpa)KaeTC5I lIepe3 CYMMY
k.l= 1
(i, j)-x 3JIeMeHTOB MaTpHI.{ PkP l , KOTopbIe, KaK HeTpY,!{HO BH-
N kl
AeTb npH i E R , j E R: paBHbI k st l ' TaK liTO
N s Nt
Nkl
b i ) -2 p .. +e,.
= N k :' I] ]'
k. 1st
rAe R H R KJIaCCbI pa30HeHHH Rk, Rl, COAep}Ka He i H j
COOTBeTCTBeHHO. npeACTaBJIeHHe Pij lIepe3 lIaCTOTbI y*e OOCY}K..
,l{aJIOCb, a eij = 0 npH i *" j.
B CHJIY (3.66) BeJIHlIHHa OCTaTOllHOH ,!{HcnepCI1I1 pasHa:
(3.67)
m
I Y -PY 1 2 = b ij , (3.68)
t= 1i, JESt
T. e. CYMMapHOH CB5I3H b ij Me}KAY OO'beKTaMH B pa30HeHHH S.
3Ta CYMMapHa51 CBSl3b xapaKTepH3yeT KalleCTBO onHcaHHSI Y
lIepe3 X k H TeM caMbIM CBSl3b Me}K,!{Y Y H Xl, ..., x n B AaHHOH
MO,!{eJIH. OCTaBJISleM QHTaTeJIIO BbIBOA npeACTaBJIeHH5I 3TOM
BeJIHlIHHbI qepe3 lIaCTOTbI pacnpe,neJIeHHH (SRkRI) TpoeK npH-
3HaKOB Ha OCHOBe (3.67), (3.68).
B CHJIY (3.65) Ka)KAbIH CTOJIOeu yt MaTpHl.{bI Y annpoKcH-
MHpyeTcSI CYMMOH xl, rAe xk = X k a , a KOMnOHeHTbI a =
k
= (xt Xk)-l xI yt HMelOT CMbICJI YCJIOBHbIX BepOSlTHOCTeH t
152
.
.....<
npH COOTBeTCTBYIOInHX 3HaqeHHHX X k . 3TH CYMMbI Xha
k
u
MorYT HcnonbSOBaTbCH nH npe CKa3aHHH 3HaqeHHH npH3Ha-
Ka y no 3HaqeHHHM npH3HaKOB x k . A HMeHHO ecnH Ha 06'beKTe
peaJIH3yeTcH KOM6HHaQHH (8 1 , .. ., Sn) 3HaqeHHH 3THX npH3Ha-
KOB, TO HaH60JIee COOTBeTcTByeT (3.65) TO 3HaqeHHe t npH3-
HaKa y, Ha KOTOpOM OCTHraeTCH MaKCHMYM CYMMbI a h YCJIOB-
HbIX BepOHTHOCTeH t npH YCJiOBHH She k
KaK H B npe bI YInHX pa3 eJIaX, CYMMapHYIO XapaKTepHC-
THKY (3.65) MO)KHO pa3JIo:mHTb Ha COCTaBJIHIO He, nOJIYtlaeMbIe
npH paCCMOTpeHHH 3a aqH onHcaHHH He JIH MaTpH[(bI Y, a
JIH BeKTopOB y = Yb tlHCJIOBbIX MeTOK b i , BbI6HpaeMbIX H3
YCJIOBHH MHHHMH3a[(HH OTKJIOHeHHH
I Yb-PYb 1 2 = b T yT (P_E)2 Yb (3.69)
bTb = 1. (3.70)
AHaJIOrHqHO 3a aqe (3.41), (3.42) OTbICKaHHe TaKoro b CBO-
HTCH K nocTpoeHHIO c06cTBeHHoro BeKTopa MaTpH[(bI
yT (P- E)2 Y, cooTBeTcTBYIO ero ee MHHHMaJIbHoMY HeHY JIeBOMY
c06cTBeHHoMY tlHCJIY. npH 3TOM BeKTop y = Yb HBJIHeTC5I
c06cTBeHHbIM BeKTopoM MaTpH[(bI yyT (P - E)2, COOTBeT..
CTBYIO HM TOMY :me c06cTBeHHoMY qHCJIY.
B CHJIY KpHTepHH (3.69) B KaqeCTBe onHcaHHH BeKTopa y=
= Yb 6epeTcH Xk' r e X11,= Xha11" an = (XI X11,)-IX Yb.
k
3HaqHMOCTb npH3HaKa Xk nH onHcaHHH MOJKHO O[(eHHTb
ero BKJIa OM A11, = bTYT (Ph - E)(P - E) Yb B cYMMapHoe
c06cTBeHHoe 3HaqeHHe (3.69).
CYMMa Bcex cp6cTBeHHbIx qHCeJI MaTpH[(bI YYT (P -E)2,
T. e. ee CJIe , paBHbIH cYMMapHoH CB5I3H b ij B pa36HeHHH S
(3.68), xapaKTepH3yeT cYMMapHoe KaqeCTBO annpOKCHMa[(HH
npocTpaHcTBa Z (Y) cYMMaMH BeKTopoB H3 L (X k ).
npe CTaBJIHeT HHTepec TaK:me paccMoTpeHHe 3a aqH MHHH-
MH3aUHH (3.69) npH YCJIOBHH
bTyrYb = 1,
npH YCJIOBHH
(3.71)
KOTopoe BbI eJI5IeT B KaqeCTBe paBHonpaBHbIx npeTeH eHToB
,l{JIH onHcaHHH BceB03MO}KHbIe HOpMHpoBaHHbIe BeKTOpbI y =
= Vb. 9Ta 3a aqa aHaJIOrHqHO (3.41), (3.48) CBO HTCH K 3a aqe
nocTpoeHHH c06cTBeHHoro BeKTopa y MaTpH[(bI Py (P _E)2,
cooTBeTcTBYIO ero ee MHHHMaJIbHoMY c06cTBeHHoMY qHcny
Iv = byT (P - E)2Yb. 3aMeTHM, tlTO BeKTop U = (1, ...,1)
HBJI5IeTCH coocTBeHHbIM BeKTopoM 3TOM ?vlaTpH[(bI, COOTBeTCT-
BYIO HM HY JIeBoMY c06cTBeHHoMY QHCJIY.
\53
Mepa CB5I3H Y H Xl' ..., X n , nOpOLK,naeM851 9TOR 3a,ll.atJeii,
BhIpaJKaeTC5I CYMMOH COOCTBeHHbIX tJHCeJI MaTpHUbI Py (P-
- E)2, T. e. Cpe,ll.HeH CYMMapHOR CB5I3blO b ij B KJIaCCaX pa36He-
HH5I S:
m 1
Tr(P y (P-E)2) = (Py, (P-E)2) = Ii" bi}' (3.72)
t:c.: 1 t i, lESt
MephI CB5I3H (3.68) H (3.72), B03HHKalO He B 3aAaqaX MHHH-
-MH3aUHH (3.69), aHaJIOrHtJHbI nOKa3aTeJI5IM KatJeCTBa KJIaCCH-
<pHKaUHH, B03HHKalO HM H B 3a,nallax 0 <t>aKTope-npoeKUHH II
<paKTope-npoeKTope, C TOllHOCTblO ,no Mep CB5I3H MeLK,ny OO'beK-
TaMH. 9TO 03HallaeT, tJTO B03MO}KHa MOAHcpHKaUHSI 3a,naQ KOH-
CTpYHpoBaHH5I, npHBo,n5lInaH HMeHHO K 9THM KpHTepH5IM.
MbI He ,neJIaeM SToro, nOCKOJIbKY aJIrOpHTMbI peweHHH COOT-
BeTCTBYIO HX 3a,naq O,nHH H Te 1Ke (CM. 9 5.1).
3.2.7. HAA1OcTpaTu8HblU npuMep
npOHJIJIIOCTpHpyeM nOJIYtJeHHhIe pe3YJIbTaThI Ha
npHMepe TaOJI. 3.1, 3.2. PaccMoTpHM CHatJaJIa 3aAatJY onHca- .
HHH ,nnSl napbI HOMHHaJIbHbIX npH3HaKOB, npHlIeM B KatJeCTBe
X npHMeM X 3 , a B KatJeCTBe Y-X l - Tor,na MaTpHUbI SY= YYT,
P x , Py HMelOT CJIeAYIO HH BHA:
1 1 1 000
1 I I 0 0 0
S y = 1 1 I 0 0 0 P
000111 ' v=
000 1 1 I
o 0 0 1 I I
Px =
1/3 1/3 1/3 0 0 0
1/3 1/3 1/3 0 . 0 0
1/3 1/3 1/3 0 0 0
o 0 0 1/3 1/3 1/3
o 0 0 1/3 1/3 1/3
o 0 0 1/3 1/3 1/3
1/2 0 0 . 1/2 0 0
o I 0 0 0 0
o 0 1/3 0 1 /3 1/3
1/2 0 0 1/2 0 0 _
o 0 1/3 O. 1/3 1/3
o 0 1/3 0 1/3 1/3
OTCIOAa nonYtJaeM:
Pvx = Pv Px =
1/6 1/3 1/9 1/6;1/9 1/9
1/6 1/3 1/9 1/6'1/9 1/9
. 1/6 1/3 1/9 1/6 1/9 1/9
1/6 0 2/9 1/6 2/9 2/9
1/6 0 2/9 1/6 2/9 2/9
1/6 0 2/9 1/6 y 2/9 2/9
,
154
If.
...-'"
S1' -81' Px =
1/2 0 2/3 -1/2 --1/3 -1/3
112 0 2/3 - 1/2 -1/3 - i/3
1/2 0 2/3 -1/2 "-1/3 -1/3
--1/2 0 -2/3 1/2 1/3 1/3
-1/2 0 -2/3 1/2 1/3 1/3
-1/2 0 -2/3 1/2 1/3 1/3
- OqeBH,l{HO, paHr 3THX MaTpHl.( paBeH ,l{BYM - KOJIHqe TBY.
HeCOBna,l{alO HX CTPO}{, H OHH HMelOT no ,l(Ba opTOrOHaJIbHbIX
co6CTBeHHbIX BeKTopa. ,QJI5I OTbICKaHH5I COOCTBeHHbIX BeKTopOB
y = Vb, x = Xa MaTpHlI.bI pYX MbI B COOTBeTCTBHH C TeopHeH,
H3JIO}KeHHOH B n. 3.2:5, paccMoTpHM MaTpHny S =
=Sj 1 yTXSi 1 XT Y, COOCTBeHHbIMH BeKTopaMH KOTOpOH 51BJI5IIOT-
C5I a H b. 3Ta MaTpHna HMeeT pa3MepHocTb 2 x 2, a ee HCnOJIb30-
BaHHe cy ecTBeHHO ynpo aeT BbIqHCJIeHH5I. HeTpY,l{HO BH,l{eTb,
qTO
.
-..!... 0 0
0 2
S = S1 1 8 13 S;- 1 S31 = ell) 010
X
0 1 102
-.. o o
3
3
'1 1) 11 7
--
X ( = 18 18
7 11
--
18 18
MaTpHny S MO}KHO nOJIyqHTb H3 pYX, OCTaBHB B HeM TOJIbKO.
u
ABe pa3JIHqalOI.l\HeCH CTpOKH H CYMMHpoBaB B HeH KOMnOHeHTbI
C HOMepaMH paBHbIx CTpOK. TpHBHaJIbHoMY coocTBeHHOMY
qUCJIY A, = 1 MaTpHnbI S COOTBeTcTByeT HOpMHpoBaHHbIH coo.
( 1 1 ) ,..,
CTBeHHblH BeKTOp b = 1/2 ' 1/2 (npH 3TOM bTb = 1). OCTaeTcSI
paccMoTpeTb MaTpHny OCTaTOqHbIX CB5I3eH, nOJIyqaeMYIO H3
3JIeMeHTOB S BbIqHTaHHeM 1/2:
-' -'
S-b b T =
1
9
1
-'-
9
1
--
9
1
9
nOCJ1e,l{H5I5I MaTpHna, OqeBH,l{Ho,. HMeeT B KaqeCTBe COOCT-
BeHHoro BeKTop. C o,l(HHaKOBbIMH KOMnOHeHTaMH pa3Horo 3HaKa
155
b = (1, - 1) C co6cTBeHHbIM tIifCJIOM A. = (paBHbIM CJI '
Y MaTpHU S H P XY 3a BbItleTOM e,nHHHQbI). CooTBeTcTBYIOIII.HH
BeKTop y = Yb == Xlb = (1, 1, 1, -1, -1, -1). Co6cTBeH-
u u
HbIH BeKTOp x MO)KHO HaHTH KaK npOeK[(HIO BeKTopa Y Ha
1
L (X) == L (X 3 ). CJIe OBaTeJIbHO, X = Px Y == (0,1, - 3,0,
1 1 ) u
- 3 ' - 3 · .;:TI0 nOKa3bIBaeT, qTO BeKTOp BeCOB 3HaqeHHH npH-
1
3HaKa X = X 3 paBeH a = (0,1, - 3)' qTO COOTBeTcTByeT HX
[(eHHOCTH ,!{JI5I onHCaHH5I Y == Xl.
,[{eiiCTBHTeJIbHO, nepBoe 3HaqeHHe a npH3HaKa x 3 6eCnOJIe3-
HO ,!{JISI nporH03a npH3HaKa Xl, TaK KaK npH X3 = a Xl npHHHMa-
eT 3HaqeHH5I nJIIOC H MHHYC C O,!{HHaKOBOH Bep05lTHOCTblO. Ha-
npOTHB, 3HaqeHHe X3 = 6 0 H03HaqHO npe,!{CKa3bIBaeT Xl =
= + H B ,!{BYX CJIyq.g5lX H3 Tpex ,!{JI5I X3 = B nOJIyqaeM Xl = -.
3aMeTHM TaK}Ke, qTO x = Px Y Ha caMOM ,!{eJIe eCTb He qTO
}fHOe, KaK BeKTOp cpe,!{HHX 3HaqeHHH npH3HaKa Xl B KJIaCCaX
06'beKTOB, O HHaKOBbIX no X3. aTO COOTBeTcTByeT TeopHH:
u
AJl5I HOMHHaJIbHbIX npH3HaKOB HX JIHHeHHa51 perpeCCH5I paB-
u u
Ha cpe HeH KBa,!{paTHqeCKoH.
2
BeJIHqHHa A, = '9 3a aeT 3HaqeHHe nOKa3aTeJI5I CB5I3H (X
Me)K)J.Y X H Y. BeJIHtIHHa X 2 = Na = 6 · = : xapaKTepH-
3yeT CBSl3b Me)K y X Ii Y. no HeM MO)KHO npoBepHTb rHnoTe3Y
o cTaTHCTHqeCKOH He3aBHCHMOCTH X H Y. ECJIH 6bI paccMaTpH-
aaeMaSl BbI60pKa Y OBJIeTBOp5lJIa CTaTHCTHqeCKHM npe nOJIO-
4
>KeHHSlM, TO 3HaqeHHe X 2 == 3 npH (3-1) (2-1) = 2 CTeneH5IX
cBo60 bI nOKa3bIBaeT, qTO npHHSlTHe rHnOTe3bI He3aBHCHMOCTH
B nOJIOBHHe CJIyqaeB He npOTHBOpeqHT aHHbIM. ,[{JI5I cpaBHe..
HH5I BblqHCJIHM BeJIHqHHY X 2 no <popMYJIe (2.55):
X2=6I (P -P8'P.t)2 =6
s,t Ps.p.t
( 1 1 1 ) 2
-- -.. . --
6 2 3
1 1
-.-..
2 3
( I 1 I ) 2 ( I I 1 ) 2
+_ 6- 2 '6 + 6-2'2 +
1 1 1 1
-. .---.
2 6 2 2
156
--'"
( J... _ . J... ) 2 ( 1.._ 1- . 1- ) ' ( 0 _ ..!... . J..., ) 2
623 + 322 + 26
+ 1 1 1 1 1 1
-.- -.- ---.-
2 3 2 2 2 6
( 1 4 4 1 ) 4
=60+ 12 + 144 +0+ 144 +12 =3.
CnaraeMble X 2 BblnHcaHbI B nOp5l,l{Ke n05lBJIeHHH KOM6HHa-
II.lui (s, t) B Ta6JI. 3.1; nOCJIe,l{Hee CJIaraeMOe cooTBeTcTByeT KOM-
6HHanHH (6, -), He cO,l{epJKa eHC5I B Ta6JIHne.
PaCCMOTpHM Tenepb perneHHe HeCHMMeTpHlIHOH 3a,naQH
perpeCCHOHHoro aHaJIH3a (3.41), (3.42). MaTpHna SYP x TaK}Ke
HMeeT Bcero ,l{Be pa3JIHlIalO HeC5I CTpOKH, H ee c06cTBeHHbIe
lIHCJIa H BeKTopbI TaKJKe MorYT 6bITb HaH,neHbI 1f3 aHaJIH3a MaT"
pHII.bl T, nOJIYlIeHHOH Y,l{aJIeHHeM O,l{HHaKOBbIX CTpOK C nOCJIe-
,l{YIO HM CYMMHpOBaHHeM COOTBeTCTBYIO HX KOMnOHeHT:
7 7
- --. =. _ ( 7 -7).
T= 6 6
7 7 6 -7 7
-- -
6 6
Y STOB MaTpHnbI Bcero O,l{HH c06cTBeHHbIH BeKTop (c O,nH-
HaKOBbIMH no BeJIHlIHHe, HO npoTHBOnOJIOJKHbIMH no 3HaKY KOM-
7
nOHeHTaMH b = (1, -1)); c06cTBeHHoe lIHCJIO Iv = "3. CooTBeT-
CTBeHHOSTOMY HCKoMbIBBeKToPY = Yb = (1,1,1,-1,-1,-1).
1 1 1 )
Ero npO eK n H 5I x = Px Y = (0, 1, - 3' 0, - 3 ' - 3
HMeeT BeKTop BeCOBbIX K03cpqmII.HeHTOB a = (0,1, - i), TaK
liTO perneHHe B ,l{aHHOM CJIYlIae nOJIHOCTblO COBna,l{aeT C npe-
,l{bI,l{Y HM .
7 7
BeJIHlIHHa nOKa3aTeJI5I CB5I3H (3.40) paBHa 3' TaK liTO 18 =
= ex (01lIH6Ka nponopnHOHaJIbHOrO nporH03a Y no X). OrnH6Ka
nponopnHOHaJIbHOrO nporHo3a Y Ha OCHOBe ero co6CTBeH-
Horo pacnpe,l{eneHH5I
2 ( 1 ) 2 ( 1 ) 2 1
6(Y)= 1- p.t= 1-"2 - 2" =2"'
157
n03TOMY K09<pqnH.I.HeHT (2.68) w paBeH:
1 7
-- -
2 18
1 .
2
w=
2
=-,
9
qro COB'na.n;aeT co 3HaqeHHeM <1>2= .
TaKOe COBna,lteHHe OOnHCHSIeTCH TeM, qTO AnH ,ltHXOTOMH-
yeeKHX npH3HaKOB X, Y ooe MO,ltenH perpeccHoHHoro aHaJIH3a
,ltalOT OJtHHaKOBoe perneHHe, B qaCTHOCTH W == X 2 j N = <1>2.
Tenepb paCCMOTpHM perpeCCHIO npH3HaKa Y = Xl no OC-
5
TaJIbHbIM BeKTopaM Taon. 3.1. MaTpHlI.a P = Ph HMeeT
k=2
CJle,ltYIO HM BHA:
2,09 0,25
0,25 2,09
0,25 0,58
0,75 0,58
0,33 0,5
0,33 °
P=
TaK lITO
0,25
0,58
1,25
0,92
0,33
0,67
0,75
0,58
0,92
1,42
o
0,33
0,33
0,5
0,33
o
1,67
1,17
0,33
o
0,67
0,33
1,17
1,5
0,86 0,97 0,69 0,75 0,39 0,33
0,86 0,97 0,69 0,75 0,39 0,33
0,86 0,97 0,69 0,75 0,39 0,33
0,47 0,36 0,64 0,58 0,94 1,00 ·
0,47 0,36 0,64 0,58 0,94 1,00
0,47 0,36 0,64 0,58 0,94 1,00
Py P =
CHTyall.HSI C coocTBeHHbIMH BeKTopaMH 3,lteCb T3 }Ke, qTO H
npe}K,lte. Hx Bcero ,ltBa, TaK KaK pa3nHqHbI TonbKO ,ltBe CTpOKH.
CYMMbI 3neMeHTOB Ka}K,ltOrO CTOnOll.a B 3TOM MaTpHlI.e O,ltHHaKO-
BbI; CJIe,ltOBaTeJIbHO, BeKTOp C eAHHHqHbIMH KOMnOHeHTaMH
SlBn eTCSI ,ltnSl Hee coocTBeHHbIM (B COOTBeTCTBHH C npe,ltnH-
caHHSlMH TeopHH). BTOpOH coocTBeHHbIM BeKTop AOn}KeH ObITb
opToroHaneH nepBoMY, T. e. HMeTb HyneBoe cpe,ltHee. npH 3TOM
nepBbIe TPH KOMnOHeHTbI BeKTopa ,lton}KHbI COBna,ltaTb, TaK KaK
COBna,ltaIOT COOTBeTCTBYlOrnHe CTpOKH B PyP. no TOM }Ke npH-
tIHHe ,lton)KHbl cOBna,ltaTb H nocne,ltHHe ero TPH KOMnOHeHTbI.
TaKHM oopa30M, e.L{HHCTBeHHbIH HeTpHBHanbHbIH coocTBeH-
HbIH BeK!Op MaTpHlI.hl PyP paBeH Y = (1,1,1,- 1, -1,-1), a
COOTBeTCTBYlOrnee cooCTBeHHoe tIHCnO A = 1 ,05, B qeM MO)KHO
158
-.....
y6e HTbC5I, YMHO)Ka51 PVP Ha y. 9TO H eCTb Cpe,l{HSl5I c.YMMap-
HafI BHYTpeHH5I5I CB5I3b (3.86) B pa30HeHHH S = {I - 2 - 3,
4-5-6} no npH3HaKY Xl (3a BbIlIeTOM n = 4). -
,UeHCTBHTeJIbHO, CYMMa 9JIeMeHTOB KJIeTKH H3 nepBbIX
TpeX CTpOK H CTOJIOnOB MaTpHnbI PyP paBHa 3 (0,86 + 0,97 +
+ 0,69) = 3.2,52, TaK liTO Cpe H5I5I CYMMapHa5I CB5I3b B KJIaC-
ce OO'beKTOB {1,2,3} paBHa 2,52. AHaJIOrHlIHO Cpe,l{HSlH CYM-
MapHaH CB5I3b B KJIaCCe {4,5,6} paBHa: 0,58+0,94 + 1,00 =
= 2,52, TaK qTO a = 2,52 + 2,52 - 4 = 1 ;04, liTO COBna,l{a-
eT CAB npe,l{eJIax TOllHOCTH BbIlIHCJIeHHH.
PaCCMOTpHM Tenepb Bonpoc 0 BeCOBbIX K09<MUlnHeHTaX
npH3HaKoB Xk (k = 2,3,4,5) OTHOCHTeJIbHO Xl. BeKTOp Xh, =
= PkY HMeeT KOMnOHeHTaMH Beca aks CBOHX 3HaqeHHH s. BeK-
TOp BeCOB ak = (aks) BbIlIHCJI5IeTC5I no CTaH,lI.apTHOH $opMYJIe
(3.1) :
ah, = Sk'l XI y.
HMeeM a 2 = (- , ), aa = (0, I, - ), a, = ( , 1)
a 6 = (- , I, 0). 9TO nOKa3b1BaeT, B qaCTHOCTH, qTO BTOpbIe;
3HaqeHH5I npH3HaKOB x 3 , x4, x5 0,l{H03HallHO nporH03HpYIOT co-
OTBeTCTBYIO He 3HalleHH5I Xl (0 lIeM CBH,l{eTeJIbCTBYIOT e,l{HHHlI-
Hble BeJIHqHHbI KOMnOHeHT aks).
r' OO HH BeC k-ro npH3HaKa onpe,l{eJI5IeTC5I ero BKJIa,l{OM B CYM-
MapHoe 3HalleHHe (3.70) A = yTPy Py, T. e. BeJIHlIHHOH
Ak = yT Pv P k Y = yT Pv X h ah = yT X k ah, = yT Xli,
liTO COBna,l{aeT C (3.42), TOJIbKO 3,l{eCb BMeCTO $aKTopa z pac-
cMaTpHBaeTcH BeKTop y. TaKHM oopa30M, CHallaJIa nOJIYlIaeM
,l{JI5I Xh, = Xh,ah, CJIe,l{YIO He BbIpaLKeHH5I:
X2 = ( --+ ' t. , -i- ' -+, - +, - +) ;
X 3 = ( 0, J, - , 0, - , - ) ;
3 3 3
X 4 = ( J.... -1 -1 ) .
2' 2' 2' 2' , ,
X s = ( l 0 - - ° - ) .
" 3' 3" 3
2 - 4
Tenepb BbIlIHCJI5IeM An = yTXk: A 2 = '3' A3 == 3' A4 = 3,
..
. 159
4
Ao = 3. TaKHM oopa30M, BKJIa npH3H3Ka X4 B J..., COCTaBJIHeT
nOQTH CTOJIbKO }Ke, CKOJIbKQ BKJI3)t OCTaJIbHhIX npH3H3KOB.
HaHMeHee HHcpopMaTHBeH npH3HaK x 2 . 3ro COOTBeTcTByeT HH..
TYHTHBHbIM npe CTaBJIeHH5IM: 3HaHHe 3HalleHH5I x 2 ,!{aeT npa..
BHJIbHbIH OTBeT 0 3HalleHHH Xl JIHilib B ,!{BYX CJIYlIa5lX H3 TpeX,
TOr,!{a KaK 3HaHHe X 4 npHBO,!{HT K e HHCTBeHHOMY BapHaHTY
OllIH6KH (npH X 4 = + npe CKa3aTb Xl = +, KOr)ta Ha CaMOM
,!{eJIe Xl = -), ,na H TO He 60JIee lIeM B O HOM CJIYlIae H3 lIeTbI-
pex.
AHaJIOrHQHbIe pe3YJIbTaTbI nOJIYlIaTC5I H npH aHaJIH3e co6-
CTBeHHbIX BeKTopoB MaTpH[(bI YYTP, TaK KaK yyr OTJIHlIaeTC5I
1
OT Py TOJIbKO nOCT05lHHbIM MHO)f{HTeJIeM 3 .
PaccMOTpHM Tenepb MO eJIH onHCaHH5I, npe,!{Ha3HatleHHbIe
JI5I npe cKa3aHH5I 3HaQeHHH BeKTopoB y = Yb Ha OCHOBe HX
npe CTaBJIeHH5I B BH,!{e PkY = Xkak COrJIaCHO KpHTepHIO
k k
(3.65). ,aJI5I 3Toro Heo6xo HMO HaHTH co6cTBeHHbIH BeKTOp
y = Yb MaTpH[(bI yyr (P - E)2, cooTBeTcTBYIO HH ee MH-
HHMaJIbHOMY co6cTBeHHoMY lIHCJIY, HJIH, tlTO TO JKe caMoe,
HaHTH co6CTBeHHbIH BeKTOp b MaTpH[(bI yr (P - E)2Y, COOT-
BeTCTBYIOInHH TOMY )Ke co6CTBeHHOMY lIHCJIY. MaTpHI.la (P-
- E)2 HMeeT CJIe,!{YIO HH BH :
..
2,09 1,29 1,5 1,6 1,17 1,32
1,29 2,17 1,54 1,6 1,15 1,25
(P-E)2= 1,5 1,54 1,87 1,36 1,46 1,27
1,62 1,60 1,36 2,03 1,23 1,17
1,17 1,15 1,46 1,23 2,29 1,70
1,32 1,25 1,27 1,17 1,70 2,)29
3 M8TPHlI.3 S = yr (P - E)2 Y nOJIYlIaeTC5I H3 Hee CYMMHPO-
BaHHeM 3JIeMeHTOB KJIeTOK 3 X 3, nOJIYlIaeMbIX pa3 eJIeHHel\1
CTpOK H CTOJI6[(OB MaTpH[(hI (P - E)2 «nocpe HHe», B COOT-
BeTCTBHH C pa36HeHHeM no y, T. e. Xl:
s = yT (P_ E)2 Y = ( 14,8 12,2 ) .
12,2 1 4,.8 .
160
I
-
OqeBH,l{HO, BeKTOp U = (1,1) 51BJIHeTCH coocTBeHHblM AJIH
S, npHqeM eMY COOTBeTcTByeT coocTBeHHoe qHCJIO A, = 27.
OCTaTOqHaH MaTpH a
S-A,UU T = ( 1,3 -1,3 ) .
-1,3 1,3
r e U = (1/ I/V2), HMeeT e,l{HHcTBeHHbIH coocTBeHHbIH BeK-
TOp b = (1, -1), cooTBeTcTBYIOInHH coocTBeHHoMY tlHCJIY
Ab = 2,6, KOTopoe HBJIHeTC5I lYIHHHMaJIbHbIM JtJHI S. JIerKo
BH,l{eTb, qTO ').;;6b T nOJIHOCTblO HCqepnbIBaeT OCT8roqHYIO MaT-
pHny.
TaKHM oopa30M, b = (1,-1) H Y = Yb = (1,1,1,-1,-1,
-1), TaK qTO peI.I1eHHe COBna,l{aeT C TeM, KOTopoe ObIJIO nOJIY-
qeHO npH aHaJIH3e MaTpHn Py P II Y yTp. 9TO OO'bHCHHeTcH TeM,
qTO COOCTBeHHhIe BeKTopbI MaTpHnbI Py, paBHbIe 1 (1,1,1-1,
-1,-1) H (1,1,1,1,1,1), B ,l{aHHOM CJIyqae HBJIHIOTCH coo-
CTBeHHbIMH ,l{JIH P, a coocTBeHHbIe BeKTopbI MaTpHn P II (E-
_P)2 COBna,l{aIOT BCer,l{a.
""
CJIe,l{OBaTeJIbHO, onHcaHHe y = X1&ak BeKTop8 y = Yb
k
xapaKTepH3yeTcH YLKe BbIqHCJIeHHbIMH BeKTopaMH ak H, CJIe-
,l{OBaTeJIbHO, HMeeT BH,l{
"" I! + I t + 3 1 3 1 4 4 I 5 5
Y=-3 Xt 3X2 X2-3X3+2Xt-X2-3Xl-f-X2,
r,l{e x: 0003HaqaeT S-H CTOJIOen nO,l{MaTpHUbI X k . Pa3JIHqHbIe
""
B03MOLKHbIe KOMoHHaUHH 3HaqeHHH Xk, Y H Y CBe,l{eHbI B
TaOJI. 3.4.
3Ta TaOJIHna HCHO nOKa3bIBaeT, qTO Ha OCHOBe TOJIbKO JIH-
HeHHbIX onHcaHHH oe3 yqeTa B3aHMO,l{eHCTBHH ,l{OOHTbCH XOPO-
illeH TOqHOCTH B CKOJIb-HHOY,l{b CJIO:>KHOM CJIytlae HeB03MO)KHO.
OcooeHHO HarJI5I,l{HO 3TO BH,l{HO npH nporH03e 3HatleHHH Y Ha
OO'beKTe 4: 3,l{eCb ,l{aJKe 3HaK paCqeTHOrO 3HaqeHHH Y He COBnaJI
CO 3HaKOM <jJaKTHlJeCKOrO (12-51 cTpoKa TaOJI. 3.4). 3TO npo-
H30I.I1JIO nOTOMY, . qTO 3TOT OO'beKT HBJI5IeTC5I HeTHnHqHbIM ,l{JIH
npH3H8KOB X2 H x 4 .
1 Bo06me, MaTpHlI.a Pk HMeeT B KatJeCTBe c06cTBeHHblx BeKTopoB,
CooTBeTcTBYIOll\HX e,nHHHQHOMY coocTBeHHoMY 3HatJeHHIO (a ApyrHx
HeHYJIeBblX cot5cT eHHblx QHCeJI onepaTopbl npoeKTHpOBaHHH He HMe-
lOT), BeKTopbl BH,na x = Xa, 3JIeMeHTbI KOTOpblX a, paBHbI 1 HJIH-1.
6 3SK. 1743
161
TaOJlHua 3.4
H2 x. x' x' Xl Y pacqeTHOe I g c\laKTH"
qeCKOe
1 1 1 5 I
1 + 0 + a --,+1+---=-
3 236
2 + 6 a - + l-l- = - I
I 1 1 1
3 1- B - a -----1--=-2 -1
3 3 3
I I 1 1 2
4 + B - 0 - - 1 +1 - -
- ..-- - --
3 3 3
5
o
a
+
1 1 1 3
-+1+---=:-
3 2 3 2
115
6 - 0 + 6 - + 1 +- + 1 = 2 -
326
1 1
7 - 0 - a -+1-1-- =0
3 3
8
+
a
B
9 / +
10/+
II / -
+
B
a
!
-1
B
B
6
+
B
12
- I a
a
+
1 J, 1 1 1
---+---=-
3 3' 2 3 6
1 1 1 J
--+0+-+0=-
3 2 6
1 1 5
- 3-3- 1 +0=- 3
1 1 5
3+ 1 +2+ 0 =1 6
1 1 1 1
3+ 0 +2-3=2
1
t
-I
1
-1
1 2
13 - a - B - +0-1 +0=--
3 3
14
+
o
1
+ a
117
--+0+-+1=-
3 2 6
3.2.8. KOflCTPyupoBaHue neAuneunblX
onucanuu
npe,l{bI,l{y ee H3JIO}KeHHe nOKa3aJIO, qTO JIHHeH-
HbIe MO,l{eJIH n03BOJf5llOT yqeCTb He DOJIee qeM napHoe B3aHMO-
,l{eHCTBHe npH3HaKOB. B TO me BpeM5I HCnOJIb30BaHHe nepe-
162
I
I
I
I
I
.,4
ceqeHHH HOMHHaJIbHbIX npH3HaKOB, T. e. npOM3Be.L{eHMH COOT-
BeTCTBYIO MX OYJIeBCKMX CTOJIOU.OB, n03BOJI5IeT MaKCHMaJIbHO
paCllIHPHTb B03MOLKHOCTH JIHHeHHOrO OnHCaHH5I. B lIaCTHOCTH,
nepeCelleHHe BCeX npH3HaKOB nOpOm.L{aeT BCeB03MO:>KHbIe OnH-
caHH5I, KOTOpbIe MOrYT OhITb C.L{eJIaHbI Ha HX OCHOBe, KaK
JIHHeHHble. Ka3aJIOCb 6bI, 3TO B npHHnHne pernaeT Bce npoo-
JIeMbI, CB5I3aHHhle C onHcaHHeM. O.L{HaKO 3TO He TaK.
4eM CJIOLKHee onMcaHHe, TeM MeHhrne KJIaCC Y.L{OBJIeTBOp5l-
IOII.(HX eMY OO'beKTOB, TeM MeHee OO He 3aKOHOMepHOCTH OHO
,naeT, B npe.L{eJIe BblpOm.L{a5lCb B lIaCTHbIe YTBep)K.L{eHM5I THna
«.L{aHHOMY OO'beKTY npHcy a .L{aHHa51 ocooeHHOCTb». n09TOMY
YBeJIHlIeHHe CJIO)KHOCTH onHCaHH5I .L{OJIJKHO KOMneHCHpOBaTbC5I
cy ecTBeHHbIM pOCTOM OO'beMa BbIOOpKM, no KOTOpOH .L{eJIaeTC5I
3TO onHcaHHe [16].
B YCJIOBH5IX HKcHpoBaHHoH cOBoKynHocTH oODeKToB B03-
HHKaeT npOOJIeMa BbIoopa .L{OCTaTOLJHO HH opMaTMBHbIX H B TO
)Ke BpeM5I .L{OCTaTOllHO npOCTbIX onMcaHMH.
,aJI5I YTOllHeHH5I BbICKa3aHHbIX nOJIOJKeHHH paCCMOTpHM npo-
OJIeMY onHCaHH5I cOBoKynHocTH Y = {y1, ..., ym} BbIXO.L{HbIX
npH3HaKOB .L{aHHOrO MHOJKeCTBa OO'beKTOB m B TepMHHaX
BXO.L{HbIX npH3HaKOB X = {Xl, ..., X n } (BCe paCCMaTpHBae-
MbIe npH3HaKH HOMHHaJIbHbIe). 0003HallHM nepeCelleHH5I pa3-
OMeHMH MHOJKeCTB X H Y qepe3 n X H n Y COOTBeTCTBeHHO.
KaJK.L{bIH KJIaCC pa30MeHM5I n X XapaKTepH3yeTcH HeKOTO-
U If!:. U ( 1 n ) U 1
pOH KOMuHHanMeH S = S, ..., S 3HaqeHHH npH3HaKOB X ,...,
X'1. AHaJIorHqHO KJIaCChI pa30HeHH5I n Y XapaKTepH3YlOTCH
KOMOHHanH5IMH t = (t 1 , ..., t m ) 3HaqeHHH npH3HaKOB y1,...,
ym. BCeB03MO)KHbIe OnMCaHM5I H3 M (X) = L (n X) COOTBeT-
CTBYIOT TeM rpynnHpoBKaM, KOTOpbIe BKJIIOLJalOT n X, T. e.
nOJIYQaIOTC5I OO'be.L{HHeHHeM HeKOTC1pbIX KJIaCCOB n X. AHa-
JIOrHlIHO M (Y) OOpa3YIOT pa36HeHH5I, nOJIYQeHHbIe YKpynHe-
HHeM pa30HeHH5I n Y.
BC5IKOe pa36HeHHe REM (X) 51BJI5IeTC5I THnOJIOrHeH D
TepMHHax c. 13, TaK KaK Kam,l(bIH ero KJIaCC onHcaH B TepMH-
Hax npH3HaKOB X. ,[{eHCTBHTeJIbHO, Ka:>K,l{bIH KJIaCC pa30He-
HH5I R 51BJI5IeTC5I 06'be.L{HHeHHeM HeKOTopbIX KJIaCCOB nepece-
lIeHH5I n X, KOTopbIe HMelOT BH,l{ Rs npH nO,l{XO,l{51InHX KOMOH-
HanHHX S = (Sl" ..., sn) 3HaQeHHH npH3HaKOB Xl, ..., x n .
COBoKynHocTb COOTBeTCTBYIO HX KOMOHHau.HH Sl, 82' ..., Sp,
Onpe,l{eJISlIO HX ,l{aHHbIH KJIaCC pa30HeHHH R, II 51BJI5IeTCH ero
onHcaHHeM. Y,l{OOHO npe,l{CTaBJI5ITb 3TY cOBoKynHocTb B Tep-
MHHax aJIreOpbI JIOrHKH. Ka:>K,l{a51 KOMOHHanHSI S = (8 1 ,...,
Sn) Onpe,l{eJI5IeT KJIaCC R s, KOTOpbIH BbI,l{eJI5IeTC5I TaK:>Ke YT-
Bep:>K,l{eHHeM «MHO)KeCTBO OO'beKTOB, Y,l{OBJIeTBOpSlIO HX 3Ha-
6*
163
-
qeHH5IM SI H S2 H ... H Sn», T. e. Rs CO.L{ep>KHT Te H TOJIbKO Te 06'b-
eKTbI, KOTOpbIe .L{eJIalOT HCTHHHbIM BhICKa3bIBaHHe (TOllHee,
n pe,nH KaT) SI /\ S2 /\ ... /\ Sn - KOH,' lOflKl{UIO EJlleMeflmapHblX
8bLCKa8b18QHUU Sk (HanOMHHM, qTO OnepalIJI5I KOH'bIOHKnHH 1\
COOTBeTcTByeT JIOrHqeCKOH CB5I3Ke «H»). TaKHM 06pa30M, 06'b-
e,l{HHeHHe RSI U ... U Rs KJIaCCOB pa36HeHH5I n x Onpe,l{e-
JI5IeTC5I aU3?JlOflKl{UeU SI V PS2 V... v Sp COOTBeTCTBYIO HX KOHn-
IOHKnHH (T. e. CJIO>KHbIM BbICKa3bIBaHHeM «SI HJIH S2 JIJIH ...
HJIH Sp) B TOM CMbICJIe, qTO OHO COBna,l{aeT C MHO>KeCTBOM 06'b-
eKTOB, Ha KOTOpbIX STa ,nH3'bIOHKnH5I HCTHHHa. TaKHM o6pa-
30M, OnHCaHHe KJIaCCa pa36HeHH5I R 3a,l{aeTC5I Ha 513bIKe aJIre6-
pbI JIOrHKH B BH,ne ,l{H3'bIOHKnHH HeCKOJIbKHX KOH'bIOHKnHH
SJIeMeHTapHbIX npe,l{HKaTOB Sk, XapaKTepH3YIO HX OT.L{eJIbHble
KJIaCCbI pa36HeHH5I no k-MY npH3HaKY. 3aMeTHM, liTO KOH'bIOHK-
UHH, BXO,l{51 He B COCTaB TaKoro COCTaBHoro npe,l{HKaTa, He
06513aTeJIbHO CO,l{ep}l{aT 3HaqeHH5I Bcex npH3HaKOB x k . TaK,
KOH'bIOHKnH5I sls2 onpe,l{eJI5IeT MHO>KeCTBO 06'beKTOB, HMelO-
Il\HX 3HaqeHH5I: nepBoro npH3HaKa - SI, BToporo - S2, a
OCTaJIbHbIX - KaKHe yrO,l{HO.
Oc06eHHO Y,l{OOeH 513bIK aJIre6pbI JIOrHKH npH ,l\HXOTOMH-
qeCKOM npe,l{CTaBJIeHHH HCXO,l{HbIX npH3HaKOB. B STOM CJIyqae
JtJIH K8>KJtoro npH3H8K8 Xk M01KHO paCCMaTpHBaTb Bcero O.L{HH
v v
SJIeMeHTapHbIH npe,lJ.HK8T, BbI,l{eJI5IIO HH O,l{HH KJIaCC COOTBeT-
CTBYIOw.ero pa36HeHH5I; BTOpOH: KJIaCC BbI,l{eJI5IeTC5I C nOMOInblO
,
OTPHU8HH5I. TaKHM 06pa30M, B KOH'bIOHKnH5IX S MorYT HCnOJIb-
SOB8TbC5I He TOJIbKO npe,l{HKaTbI Sk, HO H HX OTpHnaHH5I Sk,
KOTopbIe HCTHHHbI Ha 06'beKTaX, He BXO,l{51III.HX B R k k .
s
3aMeTHM, qTO B KaqeCTBe SJIeMeHTapHbIX BbICKa3bIBaHHH MO-
rYT paCCMaTpHBaTbC5I He TOJIbKO npe,l{HKaTbI BH,l{a Sk, YTBep:>K-
,naIOInHe, tITO 06'beKT HMeeT 3HaqeHHe Sk no k-MY npH3HaKY,
HO H YTBep:>K,l{eHH5I 60JIee CJIO>KHOrO xapaKTepa, .C yqeTOM
CBOHCTB lIIKaJI ,l{aHHbIX npH3HaKOB, KaK STO onHcaHO B n. 1.1.4.
,[{JIH paHroBbIx npH3HaKoB B KaqeCTBe SJIeMeHTapHbIx BbI6H-
p810TCH YTBep1K,l{eHH5I «3HaqeHHe npH3HaKa He xY}l{e, qeM Sk»,
,lJ.JlH HHTepBaJIbHbIX - «3HaqeHHe npH3HaKa Ha CTOJIbKO-TO
STaJIOHHbIX paCCT05lHHH 60JIblIle, qeM Sk», ,nJI5I OTHOCHTeJIbHbIX-
«3HaqeHHe npH3HaKa BO CTOJIbKO-TO pa3 60JI, blIle qeM Sk».
H SJIeMeHTapH e BWCKa3 BaHHH MorYT KOHCTpYHpoBaTbc5I
no Mepe Ha,l{06HOCTH Henocpe,ncTBeHHo no HCXO.L{HOH Ta6JIHue
06'beKT-npH3HaK [57].
I10CJIe STOro He60JIbiliOrO oTcTynJleHH5I 0 B03MO>KHOCTH HC-
nOJIb30BaHH5I H3bIKa aJIre6pbI JJOrHKH, KOTOpbIH lIIHpOKO
npHMeH5IeTC5I JtJIH nOCTaHOBKH H perneHH5I 3a,l{aQ onHCaHH5I
164
[7,55 - 57, 63), nepeH.L{eM K CPOPMYJIHpOBKe npOOJIeMbI <pop-
MHpoBaHHH onHcaHHH B paMKax annpoKcHManHoHHWX KOH-
aenI.u 1iL
B COOTBeTCTBHH C n.3.2.1 B KalleCTBe onHcaHHH CHCTeMbI
Y B TepMHHax X CJIe.L{yeT paCCMaTpHBaTb TaKoe pa30HeHHe
REM (X), KOTopoe, 0Y.L{YlIH .L{OCTaTOllHO npOCTbIM, ObIJIO ObI
MaKCHMaJIbHO OJIH3KO K CHCTeMe Y. aTO pa30HeHHe R OY.L{eM
Ha3bIBaTb mUnO/102UeU no npU3HaKaM X omHocumellbHO npU3-
HaK,08 Y.
npH X = Y THnOJIOrHH R MO)KeT paCCMaTpHBaTbCH KaK
BHYTpeHHHH cpaKTOp CHCTe 1hI p330HeHHH, HeJIHHeHHbIH aHa-
JIor perneHHH C nOMO bIO KOMnOHeHTHoro aHaJIH3a. PaCCMOT-
peHHe THnOJIOrHH REM (X) OTHOCHTeJIbHO Y COBMeCTHO C TH-
nOJIOrHeH S E M (Y) OTHOCHTeJIbHO X MO)KeT paCCMaTpHBaTbCH
T3K)Ke K3K aHaJIOr pemeHHH 3a.L{allH K3HOHHlIeCKoro aHaJIH33
v
.L{BYX cOBoKynHocTeH npH3HaKOB.
,[(JIH YTOllHeHHH npOOJIeMbI nOCTpoeHHH THnOJIOrHH R E
E M (X) OTHOCHTeJIbHO Y HeooXO.L{HMO CCPOPMYJIHpOBaTb KPH-
TepHH npOCTOTbI pa30HeHHH, 3 TaK)Ke KpHTepHH HX OJIH30CTH
K CHCTeMaM npH3HaKOB.
B COOTBeTCTBHH C MaTepHaJIOM npe.L{bI.L{Y HX nYHKTOB KPH-
TepHH OJIH30CTH pa30HeHHH K CHCTeMe npH3HaKOB MO)KeT ObITb
C<P0PMYJIHpOBaH KaK B TepMHHax CBH3eH Me y OOoeKTaMH,
YlITeHHbIX B .L{aHHOM pa30HeHHH, TaK H B TepMHHax CBH3eH
SToro pa30HeHHH (KaK HOMHHaJIbHOrO npH3H3Ka) C npH3HaKa-
MH CHCTeMbI Y. B qaCTHOCTH, MO.L{eJIb (3.57) npHBo.L{HT K HC-
nOJIb30BaHHIO CYMMbI KosqxpHnHeHToB CBH3H BH.L{a (3.61) HJIH
(3.63) B KalleCTBe OneHKH OJIH30CTH pa30HeHHH K CHCTeMe
npH3HaKOB.
CJIO)KHOCTb pa30HeHHH eCTeCTBeHHO OneHHBaTb ero pac-
. CTOSlHHeM .L{O YHHBepCaJIbHOrO pa30HeHHH U, T. e. .L{HCnepCHeH,
() (R), sHTponHeH H (R), lIHCJIOM KJIaCCOB m H .L{PyrHMH no.L{oo-
HhIMH MepaMH Ha OCHOBe (1.5). .
O.L{HaKo YTOllHeHHH KpHTepHeB OJIH30CTH R K Y H CJIO)KHO-
CTH R, KaK npaBHJIO, .L{aJIeKO He .L{OCT3TOllHO H3-3a HeB03MO)K-
HOCTH nOJIHOrO nepeoopa Bcex BapHaHTOB B CKOJIb-HHOY.L{b pe-
aJIbHbIX CJIYlIaHX, npH HaJIHlIHH .L{eCHTKOB npH3HaKOB Ii COTeR
OOoeKTOB. HeOOXO.L{HMO pa3YMHWM oopa30M COKp3THTb OOneM
nepeoopa. KaK nOKa3aJI onbIT paClIeTOB, .L{OCTaTOllHO xopomo
HHTepnpeTHpyeMbIe pe3YJIbTaTbI .L{aeT CJIe.L{YIO HH cnocoo op-
raHH3anHH nepeoopa [22, 91].
OpraHH3yeM nponecc, Ha K3:>K.L{OM mare KOToporo paccMaT-
pHBaeTCH HeKoTopoe pa30HeHHe REM (X), a TaKJKe BceB03-
MOLKHbIe pa30HeHHH R:, Ka)K.L{Oe H3 KOTOpblX nOJIYlIaeTCH H3
165
-.; --
R pa36HeHHeM s-ro KJIaCCa R no npH3HaKY x k . OlleBH,l{HO,
R: TaKme npHHa.L{JIe}KHT M (X). Ha CJIe.L{YIO eM IlIare pac-
CMaTpHBaeTC5I TO H3 pa36HeHHH R:, KOTopoe, Y.L{OBJIeTBOp5l51
Tpe60BaHHSlM npOCTOTbI, B TO }Ke BpeM5I 51BJI5IeTC5I HaH60JIee
6JIH3KHM K Y no BbI6paHHOMY KpHTepHIO. Ha nepBOM IlIare
paCCMaTpHBaeTC5I pa36HeHHe U, COCT05l ee H3 e.L{HHCTBeH-
Horo KJIaCCa, BKJIlOlIalO ero Bce 06'beKTbI. nponecc 3aKaHtIM-
BaeTC5I Ha R, eCJIH Bce pa30HeHH5I R: He Y.L{OBJIeTBOp5lIOT ccpop-
MYJIHpOBaHHOMY KpHTepHIO npOCTOTW.
PHC. 3.1
,
I
3aMeTHM, tITO 3a ClIeT nornarOBOM opraHH3anHH nponecca
nepe60p 3.L{eCb COKpa eH TaK, liTO Ha Ka}K.L{OM mare .L{eJIaeTC5I
BbI60p H3 OTHOCHTeJIbHO MaJIOrO MHO}KeCTBa pa36HeHHH R:.
TaKHe cnoco6w OrpaHHQeHHH nepe60pa HOC5IT Ha3BaHHe aJI
ropHTMoB JIOKaJIbHOH onTHMH3anHH H 60JIee HO.L{po6HO 6 Y.L{YT ,
o6cY}K.L{aTbC5I B 5.1.
Pe3YJIbTaT npHMeHeHH5I BbImeonHcaHHoro nponecca npe,l{-
CTaBJI5IeT co6oH HepapXHIO THna H306pa.}KeHHoH Ha pHC. 3.1.
BepxHHH ypOBeHb HepapXHH HMeeT e,l{HHCTBeHHYIO BepmHHY
(KopeHb) H COOTBeTcTByeT HCXO,l{HOMY YHHBepCaJIbHOMY pa3-
6HeHHIO U. BepmHHbI CJIe.L{YIO ero YPOBH5I npe,l{CTaBJI5IIOT
co6oH KJIaCCbI 06'beKTOB, COOTBeTCTBYIO HX KOHKpeTHbIM 3Ha-
tIeHH5IM npH3HaKa Xk, T. e. KJIaCCbI R pa36HeHH5I Rk, BbI6paH-
Horo Ha nepBOM mare nponecca; HeKOTopbIe H3 3THX KJIaCCOB
Ha nOCJIe,l{YIO HX marax pa36HBaIOTC5I no ,l{PyrHM npH3HaKaM
CHCTeMbI X, H T. ,l{. TaK, Ha pHC. 3.1 BepmHHa, nOMeqeHHa51
CHMBOJIOM x:, npe,l{CTaBJISleT C060H nO,l{MHO)KeCTBO KJIaCCa
166
---
R , BbI,l{eJI5IeMOe t-M 3HaqeHHeM npH3HaKa X', T. e. MHO:>KeCTBO
R: n RL COOTBeTCTBYIO ee KOH'bIOHKIIJIH S /\ t.
B 06 eM cJIyqae nOJIyqaeMOe ,l{epeBO OTp3maeT npO[(eCC
nOCJIe,l{OBaTeJIbHbIX pa30HeHHH MHO:>KeCTB OO'beKTOB, COOTBeT-
CTBYIO HX OT.L{eJIbHbIM BeplIllfH3M .L{epeBa, Ha nOJ{KJIaCCbI, on-
peJ{eJI5IeMbIe KaKHM-JIHOO npH3HaKOM 1f3 X, npHqeM 3TM nOJ{-
KJIaCCbI npe.L{CTaBJIeHbI BepllIHHaMH CJIe.L{YIO ero YPOBHSI, CMe)K-
HbIMH C J{aHHOH. TynHKoBbIe (BlfC5IqMe) BepllIMHbI .L{epeBa npe.L{-
CTaBJI5IIOT COOOH KJIaCCbI OKOH1JaTeJIbHOrO pa30HeHH5I. Kam-
u u u u
J{bIH TaKOIf KJIaCC XapaKTepH3yeTcH KOH'bIOHKnHeH 3HaqeHMH
npH3HaKOB, COOTBeTCTBYIO MX BCeM BeplIlHHaM [(enH, COe.L{M-
H5IIOIneH J{aHHYIO TynHKOBYIO BeplllMHY C KOpHeM ,l{epeBa. TaK,
Ha plfC. 3.1 BepllIHHa a npe,l{CTaBJISleT KJIaCC OO'beKTOB, onpe-
,l{eJI5IeMbIH KOH'bIOHK[(HeH x n x n x , T. e. MHOjJ{eCTBO Tex
H TOJIbKO Tex OO'beKTOB, Ha KOTOpbIX npH3H3KH Xk, x', x m
HMelOT 3HaqeHH5I S, t, q COOTBeTCTBeHHO.
n03TOMY nO,l{ooHbIe aJIrOpHTMbI eCTeCTBeHHO Ha3bIBaTb Me-
TOJ{3MH uepapXUtleCKOU zpynnupo8KU HJIH nocIleoo8amellbflblx
paa6uefluu. 9TH MeTO.L{bI B TeqeHHe nOCJIe,l{HerO J{ecSlTMJIeTM5I
pa3BHBaJIHCb pSl,l{OM aBTopoB [22,64,66,88,89,91, 102, 103,
165], OTJIHqa5lCb YTOqHeHHeM KpHTepM5I CJIOmHOCTM, KpHTepM5I
[(eJIH, 06'beMOM nepeoopa Ha Ka:>K,l{OM 1I13re.
113 Bcero MHomeCTB3 aJIrOpHTMOB MeTO.L{a HepapXMqeCKOH
rpynnHpoBKH MbI OCTaHOBMMC5I nO.L{pOOHO TOJIbKO H3 aJIrOpMT-
Me [91], HCnOJIb3YIOIneM BbIllleonMcaHHbIH npo[(eCC JIOKaJIbHOrO
nepeoopa, qHCJIO KJI3CCOB KaK Mepy CJIO:>KHOCTH pa30HeHM5I,
CYMMY CBSl3eH W (2.68) pa30MeHH5I R C nplf3HaKaMH CHCTeMbI
X. B CHJlY aCHMMeTpHqHOCTH W npH OTblCKaHMH REM (X),
KOTopoe n03BOJI5IeT H3HJIyqlIlMM 06pa30M nporH03MpOBaTb 3Ha-
qeHM5I npM3HaKOB CHCTeMhI Y, paCCM3TpMB3eTC5I 3a,naQa onTH-
MM3a[(lfM w (R, S'). I1cnOJIb30BaHMe w (S', R) npeJ{nOJIa-
1 1
raeT nOCTpoeHHe TaKoro R, KOTopoe caMO HaHJIyqllIHM oopa-
30M nporH03HpyeTC5I no npM3HaKaM ClfCTeMhI Y. 3aMeTHM,
QTO
w(SI.R)=m- 1 a(SI.R).
I tj (R) I
w(R, S')=m- a,(R, SI)/B(SI).
I I
(3.73)
(3.74)
3TO 03HaqaeT, 1JTO MaKCHMH3a[(M5I (3.73) p3BHOCMJIbHa
MlfHlfMH3a[(HH OTHOCHTeJIbHOH BeJIHqHH cYMMapHoH OIIlHOKH
167
-,
npOnOplI.HOHaJIbHOrO npOrH03a ( (X (8', R)/B (R)) KJIaCCOB
R no yl. KpHTepHH (3.74), KOTOpbIH H 6YAeT paccMaTpHBaTb-
CH ,lJ.aJIee, aHaJIOrHqeH KpHTepHIO (3.63), TOJIbKO npHsHaKH y'
6epYTcH C BecaMH, 06paTHbIMH HX JtHcnepcHHM. OH xapaKTe-
pH3yeT KaqeCTBO nporH03a SHaqeHHH npH3HaKoB Y no KJIaC-
cH<pHKa HH R.
PaccMoTpHM npHpaw.eHHe nOKa3aTeJIH w (R, S') npH nepe-
xO,l{e OT TeKym.ero pasoHeHHH R K pasoHeHHIO R:, nOJIY4eH-
HOMY HS R pasJte.neHHeM Sera KJIacca no npH3HaKY x k . ,[(JIH
3Toro 0603HaqHM KJIaCCbI pasoHeHHH 8' HH,l{eKCOM t, KJIaCCbI
R - HHJteKCOM s, KJIaCCbI Rk no npH3HaKY Xk - HH,lJ.eKCOM
q. Tor,u.a N'sq eCTb qHCJIO 06'beKTOB Rs, HMelO HX OAHOBpe-
MeHHO t-e 3H8qeHHe y' H q-e SHaqeHHe x k . BeJIHqHH8
w(R 8')-1- I Nr..
, 14 - (8') s. t NN. s . '
TaK qTO npHpaw.eHHe A k = w (R:, 8') - w (R, S') paBHO:
lk = 1 [ Nlsq _ Nr.. ] . (3.75)
· (8 ' ) t. q NN .sq t NN. s .
KaK BH,lJ.HO, :k 3aBHCHT TOJIbKO OT pacnpeAeJIeHHH 8' B
S-M KJIaCCe R, H, CJIeAOBaTeJIbHO, JtJIH BblqHCJIeHHH HeT Heoo-
XOAHMOCTH paCCMaTpHBaTb KJIaCCbI pa30HeHHH R, OTJIHqHbIe
OT paS,lJ.eJIHeMOrO Ha ,naHHOM mare.
8THM CBOHCTBOM (npHpa eHHe onpeAeJI5IeTCH TOJIbKO H3Me-
HeHHeM pa3,l{eJIHeMOrO KJIaCca) 06JIa,na1OT H Bce OCTaJIbHbIe no-
Ka3aTeJIH CB5I3H, paCCMaTpHBaeMbIe B ,l{aHHOH KHHre (CM.
9 2.2, 4,2), qTO cy ecTBeHHO ynpo aeT HX HCnOJIb30BaHHe B
aJIrOpHTMax HepapXHqeCKOH rpynnHpOBKH. npH,HCnOJIb30Ba-
HHH nOKa3aTeJIeii BHAa (3.73) BeJIHqHHa B (R) HaXO,l{HTC5I B
3HaMeHaTeJIe H He ,l{aeT B03MO)KHOCTH BbIpa3HTb npHpa eHHe Il
CTOJIb npOCTbIM cnoc060M. BnpOqeM, 3TH nOKa3aTeJIH HMelOT
OrpaHHqeHHOe npHMeHeHHe, TaK KaK 06bIqHO TpeoyeTcH no
pa36HeHHIO R onpeAeJIHTb Y, 8 He Ha060pOT.
TaKHM oopa30M, Ha oo eM mare aJIrOpHTMa paCCMaTpHBa-
eTCH pa30HeHHe R, ,nJIH KOToporo no cpopMYJIe (3.75) onpe,l{e-
JIHIOTCH BeJIHqHHbI : = A k, xapaKTepHsyro He KaqeCTBO
I
YJIyqlIIeHH5I onHcaHHH npH3HaKOB CHCTeMbI Y B pe3YJIbTaTe
pa30HeHH5I s-ro KJIaCCa R no npH3HaKY x k . B KaqeCTBe HC-
XO,l{HOro .nJIH cJIe.nYlOmero mara pa36HeHHH BbI6HpaeTC5I TO
pa36HeHHe R:, KOTopoe MaKCHMH3HpyeT .
168
Bce BbP.IHCJIeHHH MO)KHO npOH3BO,l{HTb B TepMHH8X TaOJIHU
conp5l)KeHHOCTH pa30HeHH5I R C Ka:>K,l{bIM 'H3 npH3HaKOB yl.
ECJIM, HanpHMep, 3HaqeHH5IM yl COOTBeTCTBYIOT CTpOKH, 8
KJIaCCaM R - CT0J10l.(bI Ta6JIHl.(bI COnp5l:>KeHHOCTH, TO ,l{JIH
nepeXO,l{8 OT R K R: CJIe,l{yeT SaMeHHTb s-fl CTOJIOen TaOJIHI{bI
COnp5l:>KeHHOCTH (S'R) mk CTOJIOl.(aMH, K8:>K,ltbIH H3 KOTOphIX
xapaKTepH3yeT ,lJ.OJIH COOTBeTCTBYIOw.ero 3HaqeHHH Xk B KJI8C-
cax 8'. TIpH STOM 3JIeMeHTbI s-ro YAaJIHeMOrO CTOJIOl.(8 HBJISI-
IOTC5I MaprHHaJIbHbIMH 3H8qeHHHMH (CYMMaMH CTpOK) TOM
nO,l{TaOJIHl.(bI, KOTOpa51 ero 3aMeH5IeT. BbIqHCJIeHHe t1: k CBO-
,[lHTC5I K aHaJIM3Y sToii 3aMeH5IlOrneii nOATaOJIHl.(bI.
B pe3YJIbTaTe npMMeHeHH5I ,l{aHHOrO aJIrOpHTMa nOJIyqaeTC5I
,l{OBOJIbHO OOJIblIlOe KOJIHqeCTBO TynHKoBbIX KJIaCCOB, Ka:>KAbIH
M3 KOTOpbIX xapaKTepH3yeTC5I TOM KOHKpeTHOH KOMOHHal.(Heii
(KOH'bIOHKl.(Heii) 3H8qeHHH npH3HaKoB, KOTOpa51 ero BbIAeJIHJIa
B npol.(ecce nOCJIeAOBaTeJIbHbIX pa30HeHHH. ,[{JI5I ooeCneqeHHH
B03MO)KHOCTeii cOAep:>KaTeJIbHOrO aH8JIH3a HeooxoAHMO CBeCTH
STH KOMOHHal.(HH B HeOOJIblIlOe qHCJIO rpynn, Ka a51 H3 KO-
TOpbIX OYAeT xapaKTepH30BaTb MHO}KeCTBO OO'beKTOB, onpeAe-
u u u
JI5IeMbIX TOH HJIH MHOH BXO,l{5IrneH B ,l{aHHYlo rpynny KOH'bIOHK-
[(HeM. HHbIMH CJIOBaMH, OKOHqaTeJIbHOe pa30HeHHe OO'beKTOB
OYAeT COCT05lTb H3 KJIaCCOB, nOJIYQaeMbIX OO'beAHHeHHeM MHO-
:>KeCTB OO'beKTOB, COOTBeTCTBYlOrnHX HeKOTopbIM TynHKoBbIM
BeplIlHHaM ,l{epeBa. Ka)KAhIH TaKoii KJIaCC xapaKTepH3yeTcH
B TepMHHax OYJIeBCKOH 8JIreOphI ,l{H3'bIbHKl.(HeH Tex KOH'bIOHK-
[(HM, KOTopbIe OO'be,l{HHeHbI B AaHHoM KJIaCce.
nYCTb m - 3ap8Hee <pHKCHpOB8HHoe KOJIHqeCTBO KJIaC-
COB OKOHllaTeJIbHOrO pa36HeHHH. nOTpe6yeM, IITOObI OO'beAH-
HeHHe TynHKoBbIX BepllIHH AepeB8 B m rpynn npOHCXO,lJ.HJIO C
MHHHMaJIbHbIMH nOTep5lMH JtJIH K811eCTBa KJIaCCHcpHK8l.(HH no
cpaBHeHHIO C pa30HeHHeM, onpeAeJIeHHbIM TynHKoBbIMH Bep-
IIlHHaMH ,lJ.epeBa. HHbIMH CJIOBaMH, B KaqeCTBe KpHTepHH Ka-
qeCTBa OO'be,lJ.HHeHH5I nO-npe)KHeMY OY,l{eM paCCMaTpHBaTb CYM-
MY CB5I3eH (3.74).
OO'be,l{HHeHHe' npoBe,l{eM C nOMOInblO JIOKaJIbHO-OnTHM8JIb-
Horo aJirOpHTMa nOCJIe,lJ.OBaTeJIbHbIX nepeMerneHHH OO'beKTOB
(CM. n. 5.1.5). H8 oorneM 1I18re SToro aJIrOpHTM8 paCCMaTpHBa-
eTC5I KaKoe-JIH60 pa30HeHHe MHOLKeCTBa in H8 m KJIaCCOB, a
TalOKe BapHaHTbI nepeMew.eHHH nOP-MH01KeCTB8 OO'beKTOB,
BhI,lJ.eJI5IeMOrO q-H BeplIIHHoii ,nepeB8, B p83JIHIIHhIe KJI8CCbI R.
npHHHMaeTC5I TOT BapHaHT, KOTOpbIH ,lJ.aCT MaKCHMaJIbHOe npH-
parneHHe (3.75). 3TOT npol.(ecc nOCJIe,lJ.OBaTeJIbHO npHMeHHeTcH
K Ka>K,lJ.OH q-H TynHKoBoii BeplIlHHe B nop,HJtKe HX HYMepaUHH.
169
nOCJIe 3TOrO npoQecc nOBTOp5leTC5I CHatJaJIa, npHtJeM B KatJeCT-
Be HCXO,l{HOrO pa30MeHH5I OepeTC5I nOJIyqeHHbIH pe3YJIbTaT, H
TaK ,l{O Tex nop, nOKa (3.74) He nepeCTaHeT YBeJIHtJHBaTbC5I.
ECJIH 3HaqeHHH HeKoTopbIX npH3HaKoB Xk ynopR,l{OtJeHbI,
TO npou.ecc nepeMeIneHMH MO}KHO orpaHHtJHTb TaK, tJTo6bI OKOH-
tJaTeJIbHble KJIaCCbI CO,l{ep}KaJIH HHTepBaJIbI 3HatJeHHH 3THX npH-
3HaKOB, a He pa3p03HeHHbIe rpa,l{a[(HH.
BCR npo[(e.L{ypa nepeMeIneHMR onRTb MomeT 6bITb npoBe-
,l{eHa B TepMMHax Ta6JIH[( conpRmeHHoCTM Ka>K,l{OrO npM3HaKa
yl C pa36MeHHeM R MHomeCTBa m Ha. KJIaCCbI, cooTBeTcTBYIO-
He TynHKoBbIM BepI.I1HHaM ,nepeBa. KaK H paHee, npMMeM,
tJTO KJIaCCbI R (B ,l{aHHOM cJIytJae - TynHKoBble BepllIHHbI) CO-
oTBeTCTBYIOT cToJI6uaM Ta6JIH[(bI COnpR>KeHHOCTH, TaK tJTO
,l{JIR Bcex l3TM CTOJI6[(bI O,l{HM M Te :me. TorAa TeKy ee pa36Me-
u
HHe B npo[(eCce nOCJIeAOBaTeJIbHbIX nepeMe eHMH onpeAeJIR-
eTCH <paKTHtJeCKH pa36HeHHeM CTOJI6[(OB Ta6JIH[(bI COnpH>KeH-
HOCTH Ha m KJIaCCOB B COOTBeTCTBHH C pa36HeHHeM MHO>KeCT-
Ba TynHKoBbIX BeplIlHH. Ta6JIHu.a COnpR>KeHHOCTH S' C TeKY-
HM pa30HeHHeM R nOJIYtJaeTCR CYMMHpOBaHHeM CTOJI6[(OB
Ta6JIHUbI (SIR) BHYTPH Ka}K,l{OrO H3 3THX m KJIaCCOB. npHpa-
IneHHe nOKa3aTeJIH W (R, Sl) B pe3YJIbTaTe nepeHoca q-ro
cToJI6u.a H3 S-H rpynnbI B rpynny S' (s, S' = 1, ..., m) paBHO, KaK
JIerKO BHAeTb,
(Nts-Ntq)2
W ,-
-"
t N (N .s-N .q)
+ (N ts ' +N tq )2 _ Nt _
t N(N.s,+N. q ) t NN. s
_ Nl s '
t NN. s '
3Aecb, KaK H B (3.75), N tq tJHCJIO 06'beKTOB B q-if TynHKo-
BOR rpynne, HMelO HX t-e 3HatJeHHe npH3HaKa- y:; N tB =
= N tq -3JIeMeHTbI Ta6JIH[(bI conpH:meHHocTH (SIR), nOJIYtJae-
QEP s
MbIe CYMMHpOBaHHeM N tq BHYTPH s-ro H S' -ro KJIaCCOB CTOJIO-
nOB, onpe,neJIHIO HX R.
<PopMYJIa (3.76) 03HaQaeT, tJTO BeJIHtJHHa (3.74) saBHCHT
TOJIbKO OT Tex KJIaCCOB, Me>KAY KOTOpbIMH nepeMernaeTcH
q- if cToJI6eu.
B KatJeCTBe HatJaJIbHOrO pa30HeHHR MHO>KeCTBa TynHKoBbIX
BepI.I1HH Ha m KJIaCCOB YAo6HO 6paTb KaKOe-JIH60 H3 pa36He-
HHM, oopa3yeMblx 60JIee BbICOKHMH HpycaMH ,nepeBa.
B TOM cJIYtJae, KorAa cOBoKynHocTb npH3HaKoB Y MO>KeT
ObITb npeACTaBJIeHa O.L{HHM pa36HeHHeM S MHomeCTBa 06'beKTOB
(3.76)
170
(HanpHMep, eCJIH Y COCTOHT H3 e HHCTBeHHOrO npH3HaKa),
MO)KHO BOCnOJIb30BaTbCH pyrHM MeTO OM OO'beAHHeHHH TynH-
KOBbIX rpynn, OCHOBaHHbIM Ha H eHX n.2.2.2. npH 3TOM TY-
nHKOBbIe rpynnbI OYAYT OO'be HHeHbI B KJIaCCbI R tJ COOTBeTCT-
BylO He KJIaCCaM St pa30HeHH5I S. COOTBeTCTBHe TynHKoBoH
rpynnbI TOMY HJIH HHOMY KJIaccy St BbIHBJI5IeTCH C nOMO blO
,[leTepMHHHCTCKOrO peIllalO erO npaBHJIa. B qaCTHOCTH, HC-
nOJIb3Y51 Mepy (2.77), npHxo HM K CJIeAYIOIneMY KpHTepHIO
OTHeceHHH TynHKoBoM rpynnbI K KJIaccy R t: rpynna OTHOCHT-
C5I K TOMY KJIaccy R t (T. e. cooTBeTcTByeT S t), AJI5I KOToporo
MaKCHMaJIbHO npHpa eHHe AOJIH OO'beKTOB St B AaHHoH rpyn-
ne no cpaBHeHHIO C AOJIeM St BO BceM cOBoKynHocTH oOneK-
TOB ffi. OnHCaHH5I rpynn Rt B TepMHHax X npH 3TOM MO)KHO
paccMaTpHBaTb KaK onHCaHH5I KJIaCCOB St B TepMHHax X.
PaccMoTpHM npHMeHeHHe aJIrOpHTMa HepapXHqeCKOrO rpyn-
nHpoBaHHH )!JIH onHCaHH5I npH3HaKa Xl TaOJI. 3.1 C nOMO blO
OCTaJIbHbIX npH3HaKoB 3TOM TaOJIHII.bI. Ha nepBoM I.I1are pac-
cMaTpHBalOTcH BceB03MO)KHbIe BapHaHTbI pa30HeHH5I MHO)KeCT-
Ba OO'beKTOB no AaHHMM npH3HaKaM; BMoHpaeTcH pa30HeHHe
no npH3HaKY X4, AJIH KOToporo w = 0,5, qTO OOJIee qeM BABoe
npeBbII.I1aeT 3HaqeHHH w AJIH OCTaJIbHbIX x k . ,[{aJlee paCCMaT-
pHBalOTCH BceB03MO)KHbIe BapHaHTbI pa30HeHH5I KJIaCCOB R4
no OCTaBI.I1HMC5I npH3HaKaM, H JIyqI.I1HM npH3HaeTCH pa30He-
HHe nepBoro KJIaCCa {I,2,3,4} no Xli C w === 0,166, TaK 1ITO Te-
nepb oo03peBaeTcH pa30HeHHe {I, 2, 3-4, 5-6}. HaHJlyq-
IllHM AJI5I 3Toro pa30HeHHH BapHaHToM 51BJIHeTCH pa3 eJIeHHe
TpeTbero KJIaCCa no npH3HaKY x 3 C npHpaII(eHHeM w = 0,333.
nOJIyqeHHOe pa30HeHHe {I,2,3,4, 5-6} nOJIHOCTblO AeTepMH-
HHpyeT R l === {I-2-3, 4-5-6}, T. e. BKJIlOqeHO B Rl;
3HaqeHHe w = 1. OO'beAHHHM Tenepb nOJIyqeHHbIe KJIaCCbl
BABe rpynnhI. OqeBH HO, JIIOOOH H3 onHcaHHbIX MeTOAOB -
nepeMemeHHe HJIH AeTepMHHHCTCKHH nporH03 - AaeT OAHHa-
KOBbIM pe3YJIbTaT - pa30HeHHe {1-2-3, 4-5-6}, COBna..
alOInee C R l .
r .Ila8a 4
Sl3bIK MAT PHI( CBSl3H
B AHAJlH3E AAHHbIX
4.1. 3AAAqA KOHCTPYHPOBAHHSI 4»AKTOPOB
4.1.1. AnnpOKCUMatfUJl CUCTeMbl MaTputf
OTflOUleflURMU
B STOR rJIaBe MbI paccMoTpMM MO,l{eJIM KOHCTPYH-
u
pOBaHH5I H onHcaHHSI ,l{aHHbIX B TepMHH3x OTHOIlleHHH H MaT-
pHU CB5I3H (CM. 1.2).
nYCTb r l , ..., r n - KBa,npaTHbIe MaTpHl(bI p33MepHocTH
N X N, x3paKTepH3YIO He CB5I3H Me)K,ny OO'beKTaMH no K3)K-
,l{OMY H3 33,l{3HHbIX n npM3HaKoB. 3acpHKcHpyeM HeKoTopbIR
KJIaCC M3TPHU N X N, x3paKTepM3YIO MH npH3HaKH, ,l{0-
u
nYCTHMbIe B K3QeCTBe 3rperHpoB3HHbIX npe,l{cT3BJIeHHH ,l{3H-
HOM CHCTeMbI npH3H3KOB. H3npHMep, eCJIH npe,l{CTaBJI5IIO MH
cpaKTop KOHCTpYHpyeTc5I KaK paH)KHpOBaHHe, TO COCTOHT
H3 BceB03MO)KHbIX OYJIeBCKMX N X N M3TPHl(, COOTBeTCTBYIO-
HX BceM B03MO)KHbIM JIMHeHHbIM KBa3HnOp5l,l{K3M. ECJIH He-
OOXO,l{HMO CKOHCTpYHpoBaTb pa30HeHHe, TO MO)KeT COCTO-
HTb H3 BceB03MO)KHbIX OJIOllHO-,l{H3rOHaJIbHbIX MaTpHl(, CO-
OTBeTCTBYIO HX pa30HeHH5IM, npHtleM ero SJIeMeHTaMH MorYT
ObITb: 3) (1 ,O)-MaTpHQbI, 0) (A, O)-M3TpHUbI, B) (a, )-M3TpHl(bI, f
r) M3TpHl(bI C BeCOBbIMH Kos<p<pHQHeHT3MH As == 1/ N s H T. n.
B K311eCTBe paCCT05lHH5I MeJKAY CHCTeMOH M3TpHl( r l ,..., r n H
MaTpHueH r E HCnOJIb3yeM CYMMY
t
n n /
D(r)= d(r,rk)= Ir-rkl,2 (4.1)
k=1 k-1
BnOJIHe aHaJIOrHlIHYIO cYMMe OCTaTOllHbIX ,l{HcnepcHH cpaKTop-
Horo aHaJIH3a B TepMHHax TaOJIMl( OO'beKT-npH3HaK ( 3.1).
npH3HaK, 33,l{3BaeMbIH MaTpHl(eR r E , Ha KOTOpOH ,l{0-
CTHraeTC5I MHHHMYM D (r) no r E , oYAeM H33bIB3Tb npeocmae-
JlRJOI,J,{UM cjJaKmopoM CHCTeMbI r 1 , ..., r n B KJI3CCe (g (HJIM (£ - '
4>3KTOpOM).
172
PaCCMOTpHM npe CTaBJIeHHe (4.1) Qepe3 3JIeMeHTbI rb, rij
MaTpHl.( r k H r (k = 1, ..., n; i, j = 1, ..., N). HMeeM
n N N n
D(r)= (r -rij)2= «r )2+ri -2r riJ).
k=1 1./==1 1,'/==1 k=1
nepBbIe CJIaraeMbIe B 3TOM pa3JIO:>KeHHH onpe,!{eJIHIOTCH 3a,!{aH-
HbIMH MaTpHl.(aMH r k , H, CJIe,!{OBaTeJIbHO, nOCTOHHHbI. l103TOMY
MHHHMH3al.(HH D (r) 3KBHBaJIeHTHa MaKCHMH3al.(HH <PYHKl.(HH
N 1 n ( 1
G{r)= -;;- r J'f}- 2'1/)=
i./= 1 k=a 1
N
= (ai/'tJ-+r1i) , (4.2)
1,J 1
paBHOH - 2 D (r) 3a BbIlIeroM ynoMHHYTOH nOCTOHHHOH.
3,!{eCb
1 II
ail = - ' J- - (4.3)
n k==l
cpe,!{HHH CBH3b Mem,!{y OO'beKTaMH no BceM HCXO,!{HbIM npH-
3HaKaM. Cor JIaCHO onpe,!{eJIeHHIO ,!{JlHHbI I r 1 2 = r (CM. <pop-
-;:-/
MYJIY (n.I7»), HMeeM
N 1
G{r)= aiJ'Ij-TlrI2. (4.4)
1./ == t
PaccMoTpHM ,!{aJIee KOHKpeTHbIe Bap'HaHTbI <PYHKl.(HH G (r)
,!{JI5I HeKoTopbIX HaHOOJIee HHTepecHbIx c TOqKH 3peHHH KaqeCT-
BeHHoro aHaJIM3a CJIyqaeB. B 3TOM pa3,!{eJIe OrpaHHqHMCH
cJIyqaeIyf, Kor,!{a MHOJKeCTBO it COCTOHT H3 oYJIeBCKHX (1,0)-
MaTpHl.( pa3MepHocTH N X N, cooTBeTcTBYIO HX oHHapHbIM
OTHOlIleHH5IM Ha MHomeCTBe m.
B STOM CJIyqae rlj = rij H I r 1 2 = I P I, r,!{e p - OHHapHoe
OTHOlIleHHe, COOTBeTCTBYIO ee MaTpI1l.(e r : p = {(i, j) I rij= I}.
Tor,!{a B COOTBeTCTBHH c (4.2) H (4.4)
N
G (r) = (aij- a) rij = aij- a I p I, (4.5)
I, / = 1 (i, /)ep
1
r)te a = 2. TaKMM oopa30M, ,!{JIH BbIqHCJIeHHSI BeJIHqHHbl
G (r) B 3TOM CJIyqae HY*HO H3 Bcex CB5I3eii aij BbI1JeCTb nopo-
173
..
1
rOBoe 3HalleHHe a = "2' nOCJIe lIero CYMMHpoBaTb Bce CB5I3M
aiJ - no TeM napaM (i, j), KOTopbIe npHHa)J.JIe)KaT P (T. e.
no riJ = 1).
B Ba:>KHOM lIaCTHOM CJIyqae, Kor,l{a MHO:>KeCTBO COCTOHT
H3 cHMMeTpHqHbIX MaTpHU, oe3 KaKOH-JIHOO nOTepH 06 HOCTM
MO:>KHO ClIHTaTb MaTpHUY a = (aiJ) cHMMeTpHqHOH, TaK
KaK B npoTHBHOM cJIYlIae ,l{OCTaTOllHO npoH3BecTH cJIe,l{YIO ee
cHMMeTpH3YIO ee npeoopa30BaHHe:
, aiJ + a J "i
al.=
J 2 '
H BMeCTO a paccMaTpHBaTb cHMMeTpHlIHYIO MaTpHUY a'. 3a-
KOHHOCTb 3Toro npeoopa30BaHHH rapaHTHpyeTcH TeM, liTO npH
CHMMeTpHqHOM r
a/j 'ij + ah 'it = (all + a/i) 'iJ = (aiJ + aji) 'iJ = aiJ 'tJ+ aji 'Ji,
TaK liTO G (r) O,l{HO H TO :>Ke ,l{JI5I a H a' npH JIIOOOH CHMMeTpHq-
HOM r.
IlYCTb MaTpHua r xapaKTepH3yeT pa30MeHHe co CTPYKTY-
pOM (R, "), r,l{e R = {R 1 , ..., Rm}. Tor,l{a (4.5) MO:>KeT ObITb
nepenHcaHO CJIe,l{YIO MM 00pa30M:
G(R, x)= ( aij- ) = atJ-
(S,t)E'X lERs jER t 2 (S,t)E'X IE Rs/ER,
- N s Nt- (4.6)
2 (S,t)EX
3HalleHHe CPYHKUHM G ,l{JI5I pa30HeHM5I co CTPYKTYPOH (R, ,,)
u 1 If!:. ..
paBHO CYMMe CB5I3eH aij- 2 Me}K,l{Y Ou'beKTaMH t H J H3 KJIaC-
COB R s H R h CB5I3aHHbIX po CTpYKType ".
MbI nOJIYlIHJIH CJIe,l{YIO HM pe3YJIhTaT. 3a,l{alla annpOKCH- ,
MaUHH B TepMHHax npH3HaKOB (MMHHMM3a[(M5I (4.1))'-3KBHBa-
u
JIeHTHa 3a,l{alle aHaJIH3a CTPYKTYPbI MaTpHUbI CB5I3eH Me:>K,l{Y
OO'beKTaMH (MaKcHMH3aUH5I (4.5)).
9TOT pe3YJIbTaT BnOJIHe aHaJIOrHlIeH pe3YJIbTaTaM 9 3.1 00
3KBHBaJIeHTHOCTM 3a.L{all <paKTopHoro aHaJIJf3a H 3a.L{all aHa-
JIM3a CTPYKTYPbI MaTpH[( CB5I3H Mem.L{Y OO'beKTaMH. CJIe,nyeT
u
OTMeTHTb, qTO BbIlIMTaHHe nopora CYIneCTBeHHOCTH CB5I3eH B
9 3.1 OO'b5ICH5IJIOCb >KeJIaHMeM H30emaTb TpMBHaJIbHbIX perne-
HHM (coocTBeHHbIx BeKTopoB, COOTBeTCTBYIOInHX coocTBeH-
HOMY lIHCJIY, paBHoMY e.L{MHH[(e), Tor,na KaK 3.L{eCb HaJIHlIHe no-
pora cy ecTBeHHocTH Henocpe,l{CTBeHHo BbITeKaeT H3 <PYHKUHO-
174
HaJla (4.1). KpoMe Toro, 3.L{eCb ynpo aIOTC5I nocTpOeHH5I, CB5I-
u
3aHHbIe C annpOKCHMalIJleH B KJlaCCe CTPYKTypHbIX npH3Ha-
KOB.
B Ba)KHOM lIaCTHOM CJlYlIae, KOr.L{a is COCTOHT H3 MaTpH
HeynopSJ.L{OqeHHbIX pa36HeHHH (x cpHKcHpoBaHHo H COCTOHT H3
Bcex nap BHJJ.8 (S,5)), KpHTepHH (4.6) npHHHMaeT BH.L{ 1
m ( 1 m 1 m
G(R)= . a iJ -"2) = aij-T N:.
s= 1 I.}ERs s = t1dE:Rs s= 1
(4.7)
TaKHM 06pa30M, HOMHHaJlbHbIH npe.L{CTaBJI5IIO HH cpaKTOp
B 3TOM CJIYlIae COBna.L{aeT C pa36HeHHeM, .L{J151 KOToporo CYMMa
BHYTpeHHHX CB5I3eH (4.7) (c YlIeTOM nopora cy ecTBeHHocTH
a = ) MaKCHMaJIbHa.
1
CMblCJI BeJlHlIHHbl a = :2 KaK nopora cy ecTBeHHocTH HH.L{H-
BH.L{yaJIbHbIX CB5I3eH Me)K.L{Y 06'beKTaMH COCTOHT B TOM, tlTO
npH aij > a 06'beKTbI i H j BbIrO.L{HO nOMe aTb B O.L{HH KJIaCC,
a npH aij < a - HeBblrO.L{HO: B nepBOM CJlYlIae .L{06aBJleHHe
aij - a YBeJIHlIHBaeT (4.7), a BO BTOpOM - YMeHblIlaeT.
1
BTOp851 HHTepnpeTan H 5I BeJlHlIHHbI nopora a = '2 BbITeKaeT
H3 nOCJIe.L{HeH lIaCTH CPOPMY J1bl (4.7), KOTOpa51 npe.L{CTaBJI5IeT
G (R) B BH,ne ,l{BYX CJIaraeMblX. nepBoe H3 HHX xapaKTepH3yeT
CYMMY BHYTpeHHHx CB5I3eH aij B pa36HeHHH R, BTopoe - CTe-
neHb paBHOMepHOCTH 3Toro pa36HeHH5I. MaKCHMH3anH5I G (R)
Tpe6yeT MaKCHMH3anHH CYMMbI BHYTpeHHHX CB5I3eH ail H
s lt/ERs
MHHHMH3aUHH nOKa3aTeJI5I HepaBHOMepHOCTH e (R) =
= N2 - d (R) = N . I1HbIMH CJIOBaMH, G (R) eCTb CYMM8
s
.L{BYX HeCOBna,nalO HX KpHTepHeB, nepBbIH H3 KOTOpbIX Tpe-
6yeT MaKCHMH3HpOBaTb CYMMY BHYTpeHHHX CB5I3eii, 8 BTOpOH-
CTeneab paBHOMepHOCTH pacnpe,l{eJIeHH5I pa36HeHH5I R. BenH-
1
lIHHa a = 2 5IBJI5IeTC5I BeCOBbIM nOKa3aTeJIeM, COH3Mep5l1O HM
3TH KpHTepHH, H M01KeT HHTepnpeTHpOBaTbC5I, CJIe,nOBaTeJIbHO,
KaK nOKa3aTeJIb KOMnpOMHcca Me)K,l{Y Tpe6oBaHH5IMH MaKCH-
MH3anHH aij H MHHHMH3anHH e(R). To, liTO 3TH Tpe6oBa-
s It/ERs
1 3TOT KpHTepHH paCCMaTpHBaJlCH aBTopOM B paMKax TeopHH aHa-
JIH3a KaqeCTBeHHblX npH3HaKOB HatIHHaH C 1969 r. [65]. He,naBHo 06-
HaPY}I{HJIOCb, qTO paHee OH ObIJI paccMoTpeH C. PeHbe B MaJlOH3BeCT-
HoA pa60Te [159].
175
8H5I pa3JIHtlHbI, BH,l{HO, HanpHMep, H3 Toro, qTO npH nOJIOmH-
u
TeJIbHbIX ail CYMMa BHYTpeHHHX CB5I3eH ,l{OCTHraeT CBoero
MaKCHMYMa npH MaKCHMaJIbHO HepaBHOMepHOM pa30HeHHH,
TaK KaK, qeM OOJIblIIe KJIaCC Ra, TeM OOJIbllIe CYMMa CB5I3eH
YlaiJ. Ha06opoT, MHHHMYM e (R) ...(npH saAaHHOM m) .llOCTH-
WER,
raeTCH npH paBHOMepHOM pacnpeAMeHHH: N 1 = ... =
=Nm=N/m.
K aHaJIOfHqHOH SKCTpeMaJIbHOH 3a,l{aqe CBOAHTCH 3aAaqa
nOCTpOeHH5I paHrOBOfO cpaKTopa. B STOM CJIyqae )G = { (8, t) I
I 8 t} H
G(R) = I I I ( aij- ) = I I I aij-
s t iERs iE=Rt 2 s t IER, iER,
1 m
-4 IN:. (4.8)
8=1
nOCJIeAHee paBeHCTBO OO'bHCH5IeTC5I reM, qTO N s Nt =
s t
1 :
= - N: . 9TO y:me OTMeqaBIIleeCH B n. 1.3.3 CBOHCTBO:
2 s
KatleCTBeHHa51 JtHcnepCH5I npH3HaKa YMeHbIllaeTc5I BABoe, eCJIH
u
yqeCTb ynopHAOqeHHOCTb efO 3HaqeHHH.
AHaJIOfHqHble BbIpa)KeHH5I M01KHO nOJIYtlHTb ,l{JISI ,npyrHx
THnOB CTPYKTypHblX npH3HaKOB.
113 $OPMYJIbI (4.5) CJIeAyeT TaK>Ke, qTO SaAaqa 0 npeAcTaB-
JIHIOIneM $aKTope B KJIaCCe (I,O)-MaTpHl.( C OAHHaKOBbIM
tlHCJIOM e,l{HHHl.( (T. e. B KJIaCCe oTHolIleHHH p C oAHHaKoBoH
MO HOCTblO I pi) CBOAHTC5I K 3a,l{aqe MaKCHMH3al.(HH nOKa3a-
TMB
N
F(r)= aijriJ= aiJ' (4.9)
l.i= 1 (l,/) EP
B KOTOpOM HeT HHKaKOfO nopora cy ecTBeHHOCTH CB5I3eH.
B qaCTHOCTH, KpHTepHH (4.9) ,l{JIH KJIaCCa pa30HeHHH R C <pHK-
cHpoBaHHbIM pacnpe,l{eJIeHHeM (N 1 , ..., N m ) 6Y,l{eT HMeTb BHA
m
F(R)= at}, (4.10)
s= 1 I./ER,
a AJIH KJIaCCa CTpOfHX paH)KHpOBaHHH P = (ii' ..., iN) MHO-
>KeCTBa OO'beKTOB, onpeAeJI5IeMbIX OTHOIlleHH5IMH :rt = {(is, it) I
l.s t}, 6YAeT BbIfJI5IAeTb CJIeAYIO HM oopa30M:
F(i 1 , ..., iN) = ai ,. (4.11)
<t 8 t
176
<I>opMYJIa (4.10) xapaKTepH3yeT cYMMapHYIO BHYTpeHHlO1O
CB5I3b B pa30HeHHH R, a (4.11) - CYMMY HaMHarOHaJlbHbIX
9JIeMeHTOB B MaTpHne, nOJIyqeHHOH H3 a B pe3YRbTaTe O HO-
BpeMeHHoH nepecTaHoBKH ee CTpOK H CTOJIOnOB B nopSl Ke
P = (ii' ..., iN). KpHTepHH (4.10) H3BeCTeH B RHTepaType KaK
KpHTepHH 3a aqH 00 OnTHMaJIbHOM pa3pe3aHHH rpa a (c y-
raM" aij), a KpHTepHH (4.11) - KaK KpHTepHH TpHaHrYJISI-
UHH MaTpHnbI a (noTeHnHaR nepecTaHoBKH R [15]) (CM. c. 272).
PaccMoTpHM RR npHMepa perneHHe 3a aqH 00 annpoKcH-
MaLtHH aHHblX TaOJl. 3.1 B KJIaCCe pa30HeHHH (MaTpHn OTHO-
IIleHHH 3KBHBaReHTHocTH). B npocTeHrneM cJlyqae Bce HCXO -
HbIe npH3HaKH 3a aIOTC5I (I,O)-MaTpHnaMH, TaK qTO Cpe H5IH
1
MaTpHna a = - rk nOJIyqaeTC5I H3 MaTpHn bI XXT = rk e-
n
1
JIeHHeM ee Ha - npH n = 5:
n
a=
2/5 2/5 2/5 1/5 1/5
2/5 3/5 2/5 1/5 °
2/5 3/5 3/5 1/5 2/5
2/5 2/5 3/5 1/5 2/5
1/5 1/5 1/5 1/5 - 4/5
1/5 ° 2/5 2/5 4/5 -
.
( 4.12)
3 eCb H aJIee HarOHaJIbH e 3ReMeHTW He paCCMaTpHBa-
IOTCH.
. MaTpHna CBH3eH Me>K.L{Y OO'beKTaMH C BbIqTeHHbIM noporoM
1 u
a = 2" HMeeT CJIe.L{YlOmHH BH :
-0 1 -0 1 -0 1
, , ,
+ 0, 1 - 0,1
0,1
-0,1
a _ ..!.-- = - 0, 1 0, 1
2 -0,1 -0,1 0,1
\ -03 -0,3 -0,3 -0,3
- 0' 3 - ° 5 - ° 1 -- 0 1 ° 3
, , , , ,
nOJIO>KHTeRbHbIMH s eCb 51BR5IIOTC5I RHIllb 3ReMeHTbI (2,3),
(3,4), (5,6) H CHMMeTpHqHhIe K HHM. O HaKo OTHOIlleHHe,
COCT05l ee TORbKO H3 3THX nap (a TaK}Ke neTeJIb (i, i», He
51BJIHeTCH TpaH3HTHBHhIM H He xapaKTepH3yeT pa30HeHHe.
n03TOMY HeOOXO HMO RHOO ,lI.OOaBHTb HOBbIe 3ReMeHThI, JIHOO
y aJIHTb qaCTb 3THX, TaK qTOObI 3TO ObIRO COnp5lJKeHO C MHHH-
MaJIbH MH nOTep5lMH B <PYHKnHOHaJIe (4.7). OqeBH,lI.HO, qTO
MHHHMaJIbHhIe nOTepH CYMMbI BHYTpeHHHx CB5I3eH, paBHbIe
0,1, nOJIyqalOTCH JIIOObIM HS Tpex cnocoooB: a) YAaReHHeM
3JIeMeHTa (2,3); 0) YAaJIeHHeM 9ReMeHTa (3,4); B) )(OOaBJIeHHeM
-03 - 0 3
, ,
-03 -0 5
, ,
-0 3 -0 1
, ,
- 0,3 -0,1
0,3
171
........
3J1eMeHTa (2,4). B COOTBeTCTBHH C 3THM nOJlyqaeM TPH onTH-
MaJIbHbIX pa30HeHH5I:
R 1 = {1,2,3 - 4, 5-6}; R 2 = {1,2-3,4,5-6};
R 3 = {1,2-3-4, 5-6}.
4.1.2. BeCOBbte KoacpcpUl{UeflTbt oTHomeHuu
B 3TOM pa3,l{eJle MbI paccMoTpHM MO,l{H<pHKa[(HH
3a,l{aq annpOKCHMa[(HH (4.1) - (4.4) ,l{J151 cJlyqa5l, KOr,l{a B Ka-
qeCTBe <paKTopOB BbIcTynalOT oYJleBCKHe MaTpH[(bI co B3Be-
IlleHHbIMH 3J1eMeHTaMH KaK ,l{JI5I CJlYlIa51 C <pHKcHpoBaHHbIM
HaqaJlOM OTCqeTa «(A, O)-MaTpH[(bI), TaK H ,l{J151 npOH3BOJIhHbIX
BeCOB (((X, )-MaTpH[(bI).
PaccMoTpHM CHallaJIa CJIyqaH (A, O)-MaTpH[(. Tor,l{a MaT-
PH[(bI KJlaCCa HMelOT BH,l{ Ar, r,l{e r - oYJIeBCKa51 (1,0)-
MaTpH[(a (KapKac MaTpH[(bI Ar). OqeBH,l{HO, (A'ij)2 = A 2 'ij
(r,l{e 'ij, eCTb J1HOO HYJlb, JIHOO e,l{HHH[(a) H I tvr 1 2 = A 2 I:r 1 2 =
= A 2 I pi, r,l{e p - OTHOIlleHHe -KapKaC MaTpH[(bI Ar. B COOT-
BeTCTBHH C (4.2) H (4.4) nOJlYllaeM
N ( A [ N A ]
G(A.r)=A. afJ-T)riJ=A. aijrii-Tlr,1 .
1,/==1 1.]=1
( 4.13)
1
TaKHM oopa30M, A G (Ar) B TOqHOCTH COBna,l{aeT C G (r),
onpe,l{eJIeHHbIM <POPMYJIOH (4.4), 3a HCKJllOqeHHeM SHaqeHHSI
A
nopora a = 2 ' B A pa3 H3MeHeHHoro no cpaBHeHHIO C (4.5).
9TO 03HaqaeT, qTO npH <pHKcHpoBaHHoM A MaKCHMH3a[(H5I
(4.13) 3KBHBaJIeHTHa MaKCHMH3a[(HH
1 W N
):G(A.r)= (aiJ-+)r tJ = (atJ-a)rib
1./=1 1./==1
1
qTO nOJlHOCTblO aHaJlOrHqHO (4.5) npH nopore, paBHOM He 2 ' a
A A
2 ' TaK qTO <POPMYJIbI (4.6) - (4.11) C cooTBeTcTBYIOInHM a = 2
OCTalOTCH CnpaBe,l{JIHBbIMH H B 3TOM CJIyqae.
O,l{HaKO 3,l{eCb n05lBJl5IeTC5I B03MOiKHOCTb ,l{OnOJIHHTeJlbHOH
MHHHMH3a[(HH oCTaTollHoH ,l{HcnepcHH D (Ar) 3a CqeT nO,l{Oopa
BeJIHqHHbI A. OqeBH,l{HO, <PYHK[(H5I G (Ar) CTporo BbInYKJIa no
178
I
.....:
A, TaK qTO MHHHMYM G (A,r) npH 3a,l{aHHOM r ,l{OCTHraeTC5I ,l{JI5I
ao
3HaqeHH5I A, npH KOTOpOM a'A = o. HMeeM
ao (W
-= ( a'.-A ) r. } o=O
a'A l) l ,
l,/= 1
OTKY,l{a
N
aij'ij
A= I,/=1
ri.
i, J }
9TO 03HaqaeT, qTO OnTHMaJIbHa51 BeRHqHHa A, npH 3a,l{aH-
u
HOM KapKace p paBHa Cpe,l{HeMY 3HaQeHHIO BHYTpeHHeH CB5I3H
aij (no (i, j) E p). B qaCTHOCTH, eCJIH r COOTBeTcTByeT pa30He-
HHIO R = {Rt, ..., Rm}, TO
aij
(l,j)EP
(4.14)
-
-
I P I
m
aij
A = s= 1',/ERs
m
N:
s== 1
( 4.15)
TaKHM oopa30M, npH 3a,l{aHHOM r OnTHMaJIbHOe 3HaqeHHe
'A
nopora onpe,l{eJI5IeTCH KaK a (r) = 2 H3 CPOPMYJIbI (4.14): a (r)
eCTb nORQBHHa cpe,l{Heif BHYTpeHHeii CB5I3H aij (no (i, j) E p).
nO,l{CTaBJI5I5I 3TO 3HaqeHHe A = A, (r) B (4.13), nOJIyqHM
N
G ( ( » 1 [ aij 'ij]2 'A (r)
I\, r r = - = a. . r. 0
2 , . . 2 l} l J"
I,} i,/ = t
OT6paCbIBa51 nocTOSIHHbIH: COMHmf{IfTe.JIb H H3BJIeKa51
u
KopeHb KBa,l{paTHbIH, npH,l{eM K CJIe,l{YIO eMY 3KBHBaJIeHTHoMY
KpHTepHIO:
(4.16)
( ) aij 'ij
gr= y _=
rij
aij
(l,f)EP
YfPi
( 4.17)
nOCTpoeHHe r* E , MaKcHMH3HpYIO ero g (r), 3KBHBa-
JIeHTHO MaKCHMH3anHH G (Ar), npHqeM OnTHMaJIbHOe 3HaqeHHe
A, onpe)I.eJIHeTC5I no r* Ha OCHOBe CPOPMYRbI (4.14). OTMeTHM,
179
liTO B KJIaCCe pa36HeHHH R= {R I , ..., Rm} KpMTepHH (4.17)
u
HMeeT CJIeAYIO HH BHA:
m
aij
g (R) =-= s= t 1./ERs
-V N:
(4.18)
CYMMa BHYTpeHHHx cBSl3eH B R, OTHeceHHaSl K KOpHIO KBaApaT-
u
HOMY H3 qHCJIa STHX CB5I3eH.
,L{JISI pemeHHH 3aAaqH annpoKcHMaUHH B KJIacce MaTpHU
Af MOLKHO HCnOJIb30BaTb SKBHBaJIeHTHbIH nOKa3aTeJlb (4.17)
(AJIH pa36HeHHR - (4.18»). B OTJIMlIMe OT paccMoTpeHHoro B
n.4.1.1. nOKa3aTeJISI (4.5) <PYHKnHSI (4.17) 3aBHCHT OT P HeJIH-
HeRHo. C YlIeToM CPOPMY JI (4.12) II (4.14) MO}KHO npeAJIOLKHTb
TaK1Ke APyroH, HTepaTHBHbIH cnocoo pemeHH5I 3aAallH an-
npoKcHMaUHII.
COrJIaCHO 3TOMY cnoc06y BbIlIHCJIeHHH HallHHalOTCSI C npo-
H3BOJIbHOrO 3HalleHH5I a = a o . npH STOH BeJIHlIHHe nopora OTbI-
CKHBaeTC5I KapKac p E fS, MaKcHMH3HPYIO HH <PYHKUHIO
(4.13) (coBnaAalO YIO C TOllHOCTblO AO nopora C (4.5». 060-
3HallHM nOJIyqeHHOe pemeHHe lIepe3 fo- B COOTBeTCTBHH C <pop-
MYJIOH (4.14) onpeAeJI5IeM AJISI fo OnTHMaJIbHOe 3HaqeHHe nopo-
ra al = '" o) = ("i.aijr o lj)/21 Pol. nOCJIe qero nOBTOp eM pe-
l. I
meHHe 3aAaqH MaKCHMH3anHH (4.13) ,Z(JISI a = al. nOJIYlIeH-
Hoe pemeHHe (f 1 , AI) cpaBHHBaeTCSI C (ro, Ao) (At = 2at)
no 3HalleHHSlM CPYHKnHOHaJIa G (Af). ECJIH G (A 1 f}) < G (Aofo),
BbIqHCJIeHHSI npeKpa aIOTC5I; fo BbIAaeTCH B KalleCTBe OKOHlIa-
TeJIbHOrO perneHHSI. ECJIH )Ke G (AI f I ) > G (Ao fo), BbIqHCJIe-
HH5I nOBTOp5lIOTC5I (npH 3aMeHe fo Ha f I , Ao Ha AI). 9TOT npo-
necc SlBJI5IeTCSI KOHeqHbIM B CMJIY KOHeqHocTH oo ero KOJIM-
lIeCTBa KapKacOB (OTHorneHHH) Ha KOHeqHOM MHO>KeCTBe A,
TaK KaK BeJIHlIHHa G (Af) Ha Ka)K.L{OM mare YBeJIHlIHBaeTCH.
PaccMoTpeHHaH HTepaTHBHa51 npone.L{ypa HMeeT TO npeHMY-
eCTBO nepe.L{ perneHHeM 3a.L{allH MaKCHMH3a[(HH. (4.17), liTO
HBJI5IeTCH npSlMbIM oooo eHHeM npOneAypbI annpOKCHMaIlJIH
B KJIaCCe (I,O)-MaTpHn C H3MeH5IIO HMC5I noporOM cy ecTBeH-
HOCTH CBSl3H. 9TO n03BOJI5IeT npoCTo HHTepnpeTHpoBaTb nOJIY-
lIaeMbIe pemeHHSI. HeAocTaToK HTepaTHBHoH npOne.L{YPbI B TOM,
qTO B YCJIOBHSlX HCnOJIb30BaHHSI JIOKaJIbHO-OnTHMaJIbHbIX aJI-
ropHTMOB pemeHH5I 3a.L{all BH.L{a (4.13), (4.5) OKOH1JaTeJIbHOe
perneHHe M01KeT OKa3aTbC5I He CJIHIlIKOM 6JIH3KMM K rJIOOaJIb-
HO-OnTHMaJIbHOMY peweHHIO.
180
!
06paTHMCSI K HarneMY npHMepy. nYCTb - MHO)KeCTBO
BCeX pa30HeHIUi Ha 91, MaTpM[(a CB5I3eH a == (aij) HMeeT BH,l{
(4.I2) H Ao = 1. Tor.L{a, KaK nOKa3aHO Ha c. 178, OnTHMaJIbHO
JIlOooe H3 Tpex pa30HeHHH {I,2,3-4, 5-6}, {1,2-3, 4,5-6}
H {1,2-3-4, 5-6}. BbIoepeM nepBoe H3 HHX B KaqeCTBe
R o. OqeBH,l{HO,
G(Ao ro)= (aai-+) +(a'B-+) +( a S6 -+) +
+( a 6S -+ ) = 0,8.
COrJIaCHO <popMYJIe (4.14) HOBoe A onpe,l{eJISleTCSI KaK cpe,l{-
Hee apM<pMeTHqeCKOe BHYTpeHHHx CB5I3eH B Ro:
_ aS4+ a 4S+ a 66 + a65 _ a34+ a fi8 - 0 7
I'vl- - -,.
4 2
A,
MaTpHn a (aij - 2 ) nOJIyqaeTC5I BbIqHTaHHeM H3 KaLK,lI.oro
aij noporOBOH BeJIHqHHbI a = A l /2 = 0,35:
1 2 3 4
0,05 0,05 0,05
-- 0,25 0,05
0,25 - 0,25
0,05 0,25 --
-0,15 -0,15 0,15
-0,35 0,05 0,05
1
2 0,05
3 0,05
4 0,05
5 -0,15
6 -0,15
5
-0,15
-0 15
,
-0 15
,
-0,15
6
--0 15
. ,
-0 35
,
0,05
0,05
0,45
0,45
OnTHMaJIbHbIM ,l{JISI 3TOH MaTpHUbI SlBJISleTCR pa30HeHMe
Rl == {1-2-3-4, 5-6}: OHO yqHTbIBaeT Bce nOJIO)KHTeJIb-
HbIe CBSl3M, KpOMe (3,6) H (4,6), KaK BHYTpeHHHe, T. e. BXO,l{SI-
InHe B CYMMY (4.13). npM 3TOM HCHO, tlTO yqeT CB5I3eH (3,6)
M (4,6) KaK BHYTpeHHHx (T. e. OO'be,l{MHeHMe OO'beKTOB 3 H 6
MJIM 4 H 6 B O.L{HH KJIacc) HeB03MO)KeH oe3 YMeHbIlleHMH 3Haqe-
HM5I CYMMbI BHYTpeHHMX CB5I3eH (4.13). HanpMMep, BKJIlOqe-
HMe 3 (HJIH 4) B KJIaCC {5,6} ,l{ooaBJI5IeT B (4.13) BMeCTe c £la6 =
= 0,05 (HJIH a46 = 0,05) H a36 = - 0,15 (HJIH a45 =
= - 0,15), qTO B CYMMe,l{aeT OTpHnaTeJIbHbIH npHpocT (-0,10),
v
He CqMTa51 HCKJIlOqeHHH nOJIO)KHTeJIbHbIX CB5I3eH a1 3 , a 23 , a43
(MJIM a 1 4, a 2 4, a34). BeJIHqHHa nOKa3aTeJIR G npH 3TOM paBHa:
G (AIr!) = 0,7 (1,4 + 0,9) = 1,61; 3,l{eCb 1,4 - cYMMa BHYT-
peHHHX BeJIHqHH aij - /2 B nepBoM, a 0,9 - BO BTOpOM
KJIaCCaX RI. 3T0 cy ecTBeHHo OOJIbIlle, qeM G (Aoro) = 0,8.
BbIqHCJleHHe cpe,nHeH CB5I3H aij BHYTPH RI ,l{aeT
18 /
A2 =5 7 = 0,514.
181
A
BbP.IHTaHHe a = 2 2 = 0,257 H3 aij He ,!{aeT HI-ft.lero cyr.u.eCT-
BeHHO HOBoro: pacnpe,!{eJIeHHe nOJIO:>KHTeJIbHbIX H OTpH[(aTeJIb-
HbIX 3JIeMeHTOB B MaTpH[(e (aij - 0,257), TO }Ke, qTO H B MaT-
pH[(e (aij - 0,35). n03TOMY OnTHMaJIbHOe pa30HeHHe R 2
COBna,!{aeT c R l . BeJIHqHHa G (A 2 r 1 ) == 1,851 > G (A l r t ) ==
= 1,61, TaK qTO B KaqeCTBe OKOHqaTeJIbHOrO pe3YJIbTaTa npH-
HHMaeM R == {1-2-3-4, 5-6}, A == 0,514.
3aMeTHM, qTO TOT }Ke pe3YJIbTaT nOJIyqHTC5I, eCJIH HaqaTb c
JIIOOoro H3 ,!{BYX .L{pyrHx pa30HeHHM, nOJIyqeHHbIX Ha c. 178.
B qaCTHOCTH, npH Ro == {1,2-3,4, 5-6} nOJIyqaeM Al = 0,7,
H ,!{aJIee BbIqHCJIeHH5I O,!{HHaKOBbI. npH R 0 == {1,2 - 3-4,
A,
5-6} nOJIyqaeM Al = 0,6, TaK qTO nopor a == --l == 0,3 cpa3Y
,!{aeT B MaTpH[(e (aij - a) pacnpe,!{eJIeHHe nOJIO)KHTeJIbHbIX H
oTpHu.aTeJIbHbIX 3JIeMeHTOB, npHBO,!{Slrnee K R == {1-2-3-4,
5-6} KaK OnTHMaJIbHOMY pa30HeHHIO. O,!{HaKo pa30HeHHe
{1-2-3-4, 5-6} He 51BJIHeTC5I r JIOOaJIbHO-OnTHMaJIbHbIM.
OnTHMaJIbHbIM 51BJIHeTC5I YHHBepCaJIbHOe pa30HeHHe U, OO'be,!{H-
H5IIO ee Bce OO'beKTbI B O,!{HH H TOT 1Ke KJIacc. .QJIH Hero cpe.L{-
H5I5I BHYTpeHH5IH CBH3b A, = 0,36, TaK qTO BbIqHTaHHe nopora
cy ecTBeHHOCTH a = 0,18 OCTaBJISleT Bce 3JIeMeHTbI MaTpH[(bI
CB5I3eH, 3a HCKJIlOqeHHeM a 26 , nOJIO}KHTeJIbHbIMH. BeJIHqHHa
KpHTepH5I G (AU) == 0,36.5,4 = 1,944 OOJIbIlle, qeM G (AR)
npH JIIOObIX R, A. AHaJIOrHqHbIH BbIB O,!{ nOJIyqa eTCSI H3 paccMoT-
peHH5I nOKa3aTeJI5I (4.17) g (r)= V G (A (r) r).
nOJIyqeHHbIH pe3YJIbTaT OJIH30K K TeOpeTHqeCKOMY <paK-
TY, YCTaHOBJIeHHOMY B n. 3.1.4: npH KaqeCTBeHHOM cpaKTopHoM
aHaJIH3e Heooxo,!{HMO npe,!{BapHTeJIbHO BMqeCTb H3 Bcex CBSI-
3eM cpe,!{HIOIO CB5I3b Mem.L{y OO'beKTaMH. B ,!{aHHOM cJIY1Jae no-
JIyqeH TpHBHaJIhHbIH cpaKTop U, ,!{eHcTBHe KOToporo HeOOXO.L{H-
MO BbIqeCTb H3 MaTpH[(bI cBSl3eH, qTOObI ,!{aJIee y)Ke HCKaTb
HeTpHBHaJIbHYIO KJIaCcHcpHKa[(HIO (CM. CJIe.L{YIOIII.HM pa3,!{eJI).
OopaTHMcSI Tenepb K npOH3BOJIbHbIM (a, )-MaTpH[(aM.
B 3TOM CJIyqae MaTpH[(bI KJIaCCa HMelOT BH,!{ Ar + fl.,
r,!{e r - oYJIeBCKa51 (I,O)-MaTpH[(a - KapKac. npH 3TOM
a = A, + fl., P == fl.. OqeBH,!{Ho,
N N N
IAr+f1r 2 =a 2 rij+ 2 (l-rij)=A,2 rij +
1,1=1 l,j=1 1,1=1
N
+ 2Af1 rij -}- 2 N2 = a 2 e (r)+ 2d (r),
i,j = 1
182
r,ne
d (r) = d (U, r) N2_ e (r).
B COOTBeTCTBHH C (4.2) H (4.4) nOJIYQHM
G{Ar+It)= A (au-[It + - ])r iJ +
It} = t
N
+ It (a ij - ) ·
i,/ = 1
npH 3a,l{aHHbIX Iv H J..t MaKCHMH3a[(H5I 3TOH <PYHKIIJIH CBO-
,l{HTC5I K OTbICKaHHIO r E is, MaKCHMH3HpYIOInero
N ( A
aij-It -2") rib
1.1= t
A
T. e. On5lTb <PYHK[(HIO (4.5), npH a = J..t + 2.
npH 3a,l{aHHOM r E (S OnTHMaJIbHble 3HaqeHH5I A, H J..t onpe-
.L{eJI51 IOTC5I npHpaBHHBaHHeM HYJIIO qaCTHbIX npOH3BO,l{HbIX
<PYHK[(HH G (lvr +J..t) no A, H J..t:
.!!!.. = a. J ,.. - II. r. .-A ,..= O.
aA J .. J J ,
i.1 i,l i,J
.!!!.. = -A rij + aij-It 1 = O.
aJt 1,J I, i i ./
(4.19)
( 4.20)
113 nepBoro ypaBHeHH5I CJIe,l{yeT, qTO
aiJ 'ij
i.1
a=A,+ =.. rtJ
nO,l{cTaBJI5I5I (4.21) BO BTopoe ypaBHeHHe,
aij (l-rij)
= !1 = i./
(l-Tti)
I1HbIMH CJIOBaMH, npH 3a,l{aHHOM KapKace r OnTHMaJIbHOe
3HaqeHHe ex onpe,l{eJI5IeTC5I KaK Cpe,l{H5I5I BHYTpeHHSlH CB5I3b
aij B p (no 'ij = 1), a OnTHMaJIbHOe 3HaqeHHe - KaK cpe,l(-
H5I5I BHeIlIH5I5I-CB5I3b aiJ (no 'ij = 0). o TCIO,l{a
Iv = aij rij _ aij (1 - r ij) . (4.23)
riJ (l- r iJ)
(4.21 )
nOJIyqHM
( 4.22)
I
I
L,.
183
B. A. Tpoqn1MoB npe,nJIO>KHJI Ha3blBaTb 3TY BeJIHtIHHY
Ko3l/Jt/JUl{UeflmOM pa3petuenUfl, nOCKOJIbKY OHa xapaKTepH3Y-
eT pa3HHUY Me)K)J.Y CBH3S1MH aij, yqTeHHbIMH H He YllTeHHbI-
MH B r. qeM 60JlblIIe A, TeM oOJlee TOllHO YlIHTblBaeT r HepaB-
HOMepHocTb cBSlseH Me>KAY OO'beKTaMH.
nOACTaBJISlH B (4.19) BMeCTO A H Bblpa>KeHHH (4.23) H
(4.22), nOJIYlIHM, qTO G (Ar + !-t) C TOllHOCTblO AO KOHCTaHT
c I , c 2 > 0 HMeeT BHA G (Ar + J..t) = c i + c 2 f (r), rAe
f(r)= ( afJrfJ)I + ( afJ(I-rfJ»t . (4.24)
rij (l- r ij)
CneAoBaTeJIbHO, MaKCHMH3 UH5I G (Ar+!-t) no A,J..t H r SK-
BHBaJIeHTHa MaKCHMH3aUHH f (r) TOJIbKO no r; OnTHMaJIbHble
SH8tIeHHH A H,.., onpeAeJISlIOTCH no OnTHMaJIbHoMY r C nOMO blO
<POPMYJI (4.22) H (4.23). AnropHTMbl MaKCHMH3a[(HH nOKa3aTeJI5I
(4.24) OYAYT paccM8TpHBaTbcH B rJl8Be 5.
C ,nPyroH CTOpOHbI, AJISI perneHHH 3aAatlH (4.19) MO}KHO BOC-
u u
nonb30BaTbCH HTepaTHBHoH npoue,nypoH nonepeMeHHoro on-
peAeJIeHHH A, f.t H r 8HaJIOrHlIHO TOMY, KaK STO CAeJIaHO npH
8H8JIH3e npeAblAy ero CJIyqaH.
9Ta HTepaTHBHa51 npoueAypa COCTOHT B CJIeAYIO eM. Ha-
tlHHaSl C npOH3BOJIbHbIX 3H8qeHHH A = Ao H f.t = o (HanpH-
Mep, Ao = 1, J..to = 0) pelIIaeM 38,natIY MaKCHMH3aUHH nOKa3a-
TeJISI (4.5) npH 3H81leHHH nopora cy ecTBeHHocTH CBH3H ao =
= (J.to + ): 113 <POPMYJIbI (4: 19) C OqeBH.lI.HOCTblO CJIe.lI.yeT,
tlTO npH 3TOM 6y,neT nOJIyqeHO pemeHHe 3aAaqH MaKCHMH3a.
UHH G (M + !-t) npH cpHKcHpoBaHHbIx A H !-t (Iv = Ao, f.t = o).
0003HatlHM nOJIYtleHHOe pemeHHe qepes r 1 H no cpopMYJIaM
(4.22), (4.23) BbIqHCJIHM 3HaqeHHH Al H !-tl' cooTBeTcTBYIOIII.He
r l . .QaJlee npoueAypa nOBTOp5leTcH: no 3a,naHHbIM At H J..tt Bbl-
QHCJlSleTC5I ft, MaKcHMH3HpYIO ee G (r) (CM. $OPMYJlY (4.5» npH
nopore a = f.tt + (t = 0, 1, ...), nOCJIe qero onpeAeJI5IIOT-
2
CH HOBbIe 3HalleHHH At+l H f.tt+l no <POPMYJl8M (4.22) H (4.23).
KpHTepHeM OCTaHOBKH HBJI5IeTCH npeKpa eHHe YBeJIHlIeHH5I
3HalleHHH G (Atrt + J..tt), BbItIHCJlHeMbIX no cpopMYJIe (4.19).
npeHMyrnecTBo onHcaHHOH npoueAYPhI B TOM, qTO OHa HC-
nOJIb3yeT TOT 1Ke CT8H,n8pTHbIH OJIOK (onTHMH38UHH (4.5»)
u
B K8qeCTBe OCHOBHOH tIaCTH, qTO H B npe,nbIA HX CJIyqa5lx.
CJleAyeT 3aMeTHTb TaK>Ke, qTO npH At < 0 B COOTBeTCTBHH C
(4.19) CJIeAyeT He MaKcHMH3HpoBaTb (4.5) npH nOHCKe ft+l' a
MHHHMH3HpoBaTb.
184
06paTHMCH K paCCMaTpHBaeMOMY npHMepy C MaTpHueA
(4.1). npHMeM ,nJIH npOCTOTbI Ao = 1, !-to = 0. Tor,na corJlac-
ao paccY>K,neHH5IM c.I78 OnTHMaJIbHbIMH Oy,nYT TPH pa30HeHHH,
B TOM qHCJle Ro = {1,2,3-4,5-6}. BbIqHCJlHM ,l{JIH Hero Be-
JIIItIHHbI Al H 1-11 no <popMYJIaM (4.22) H (4.23). B CHJIY (4.21)
rJ., = A + !-t paBHa cpe,nHeH BHYTpeHHeH CBH3H B Ro:A + !-t =
= 0,7. BeJIHqHHa !-t = P no (4.22) onpe,L{eJI5IeTCSI KaK cpe,l{-
H5I5I «BHeIIlH5I5I» CB5I3b:
aiJ-(aS4+ a 4S+ll&S+aS&)
J.t= 1=1=/ = = 0,308.
6.5-4 13
3,L{eCb B 3HaMeHaTeJle CTOHT pa3HocTb oo ero tlHCJla cBH3eH
(6e3 yqeTa ,L{HarOHaJIbHbIX) H qHCJIa BHYTpeHHHx CB5I3eH.
OTclO,l{a A = 0,7-0,308=0,392, H HOBa51 BeJIHqHHa nopora
1
B (4.20) paBHa: a = + 2" = 0,308+0,196 = 0,504, qTO
1
nOqTH COBna,L{aeT C nepBOHaqaJIbHhIM noporoM a = 2". Cne)(o-
BaTeJIbHO, OllTHMaJIbHbIM On5lTb OY,l{eT Ro = {1,2,3-4, 5-6}
c Iv = 0,392, 1-1 = 0,308, a TaK>Ke ,L{pyrHe ,L{Ba pa36HeHHH c. 178.
C nOMO blO nOKa3aTeJIH (4.24) HeTpY,l{HO npoBepHTb, qTO
YHHBepCaJIbHOe pa30HeHHe B 3TOM cJlyqae 3HaqHTeJIbHO xY>Ke
OnTHMaJIbHoro.
4.1.3. Paa6uenuJl C 8ecoBou CTPYK,TypOr1
nYCTb fS COCTOHT H3 M8TPH[( (He 00S138TeJIbHO
BceB03MO}KHbIX) CTPYKTYPHbIX npH3HaKOB (R, ), r,L{e R =
= {R 1,.. · , R m } , " {I,. .. , m} X {I. ,.. . , m} C BeCOBbIMH
K09qxpHnHeHTaMH Ast «s, t) E ).
B COOTBeTCTBHH C (4.2), (4.6)
G{R, x, A)= Ast ( aij- Ast ) . (4.25)
(S,t)EX IERs/ERt 2
llpH 3a,L{aHHoM (R, x) OnTHMaJIbHble 3HaqeHH5I Ast «s,t) E
E x) onpe,neJI5IIOTC5I H3 YCJIOBH5I paBeHCTBa HYJIIO qaCTHbIX
n pOH3BO,nHbIX:
= (alJ-Aat)=O,
0')..,8t IER, /ER,
TaK qTO
Ast = 1 alJ= a s t -
N 8 Nt IER8/ER,
( 4.26)
185
cpe,l{HSlH CBH3b Mem,l{Y 06'beKTaMH KJIaCCOB Rs H R t. nOt(-
CTaBJIHH (4.26) B (4.25), HeTpY,l{Ho yoet(HTbcSI, qTO " ri-:t1.tR"
1 -
G(R, x, 'A(R, »==- a NsNt.
2 (S,t)EX
TaKHM oopa30M, BMeCTO MaKCHMH3anHH (4.25) no R, x H
'Ast MO)KHO MaKcHMH3HpoBaTb (4.27) no (R, x), onpe,l{eJI5IH
3aTeM 3HaqeHH5I Ast CPOPMYJIOH (4.26). , .
nOJIyqeHHbIe BbIpa)KeHHH ,l{alOT TaK:>Ke H ,l{pyroH, HTepa-
THBHbIH cnoco6 perneHHH 3a,l{a1JH, aHaJIOrHqHbIH paHee pac-
cMoTpeHHbIM. COrJIaCHO 3TOMY cnocooy npH 3at(aHHOM (R, )
BbIqHCJI5IIOTC5I OnTHMaJIbHbIe 3HaqeHH5I Ast (R, ) no cpopMYJIe
(4.26), a npH 3a,l{aHHbIX Ast BbIqHCJI5IIOTC5I OnTHMaJIhHbIe (HJIH
JIOKaJIbHO-OnTHMaJIbHbIe) (R, x) H3 pemeHH5I 3a,l{atlH MaKCHMH-
3anHH CPYHKnHOHaJIa
(4.27)
g= A:tN,N h
(S,t)EX
(4.28 )
nOJIyqaeMOrO H3 (4.27).
CJIe,l{yeT ocooo OCTaHOBHThCH Ha cJIyqae YHu8epcallbHou
CTPYKTYPbI, KOr,l{a x COCTOHT H3 BceB03MO)KHbIX nap (5, t).
npH 3TOM
G(r)=G(R. A)= f Ast . (aiJ-+Ast ) ;
s,t= 1 1ERs JERt
m
g('A)= ' A NsNt.
5,t==1
KaK BH,l{HM, B paCCMaTpHBaeMOM CJIyqae nopor cy ecTBeH-
1
HOCTH ast =2 Ast 3aBHCHT OT Toro, Me)K,l{Y KaKHMH KJIaCCa-
MH paCCMaTpHBaeTC5I CBSl3h.
B TOM CJIyqae, KOrt(a Ast 3acpHKCHpOBaHhI Ha ypOBHe 'Ast =
1
V ' nOJIyqHM, paCKpbIBa51 B (4.25) CKOOKH:
N s Nt
G(r) = aij- Ixl. (4.29)
(5.t)EK V N sN tiERs fERt 2
B 3TOM nOKa3aTeJIe COBMe eHbI ,l{Ba npOTHBOpeqHBbIX Tpe-
OOBaHHH: MaKCHMH3anHH CB5I3eH, yqTeHHbIX B CTpYKType x
(nepBoe CJIaraeMoe), H MHHHMH3au.H5I CJIO)KHOCTH CTPYKTYPbI
x (BTopoe CJIaraeMoe).
B TOM cJIyqae, KOr,l{a paccMaTpHBaeTcSI KJIaCC CTPYKTYP O,l{H-
u
HaKOBOH MO HOCTH, BTopoe CJIaraeMOe He BJIH5IeT Ha 3HaqeHHe
186
.".
G (r) H MomeT He YlIHThIBaTbC5I npH onTHMH3a[(HH. B CJIYlIae
YHHBepCaJIbHOH CTPYKTYPbI 1)( I = m 2 H (4.29) HMeeT BH,l(
m 1
G (r) = V NsNt . . aiJ- 2 ,
s,t= 1 tER s 1ERt
TaK 1JTO BTopoe CJIaraeMOe HrpaeT pOJIb .L{06aBKH, perYJIHpylO-
meH lIHCJIO KJIaCCOB pa36HeHH5I R.
PaccMoTpHM Tenepb Cne[(HaJIbHO CJIyqaH TpHBHaJIbHOH
CTPYKTYPbI )( = {(s, t) I s = t}, T. e. CJIYtIaH HeynOp5l.L{OtIeH-
HbIX pa36HeHHH R = {RI'...' Rm} C BeCOBbIMH K03<P<PH[(HeHTa-
MH KJIaCCOB As == Ass (s = 1, ..., m). B CHJIY (4.25)
m 1 m
G(r)=G(R, 1..)= A.s aij-- A.: N:. (4.31)
s= t l,jER s 2 s= 1
OnTHMaJIbHbIe As onpe.L{eJI5IIOTCH no 3a.L{aHHoMY R CPOPMYJIOH
1 '"' -
As= N'1. aij=a s ,
s l,fERs
nOJIYlIaeMOH H3 (4.26). AHaJIOrHlIHO nOJIYQaIOTC5I H .L{PyrHe
KOHKpeTH3a[(HH 06InHX COOTHOIlleHHH.
B tIaCTHOCTH, MaKCHMH3a[(H5I (4.31) no R H A .L{OCTHraeTCH
u .
MaKCHMH3a[(HeH
(4.30)
(4.32)
m _
G (R, A, (R» = a s '1. N:
s= 1
( 4.33)
no R, nOCJIe tIero OnTHMaJIbHbIe As onpe.L{eJI5IIOTC5I <POPMYJIOH
(4.32) .
MO)KHO TaK)Ke npH6erHYTb K HTepaTHBHoH npo[(e,nype
nOCJIe.L{OBaTeJIbHOrO onpe.L{eJIeHH5I As H R : npH 3a.L{aHHOM R Be-
JIHlIHHbI As onpe.L{eJI5IIOTC5I CPOPMYJIOH (4.32), a npH 3a.L{aHHbIX
As pa36HeHHe R onpeJ(eJI5IeTC5I KaK perneHHe 3a.L{aQH MaKCHMH-
3a[(HH YHK[(HOHaJIa
m
" A 2 N 2
S s,
s= 1
( 4.34)
paBHoro (4.33) npH As = a;. B CHJIY (4.32) B CTa[(HOHapHOH
TOllKe 3Toro npo[(eCca (4.34) COBna.L{aeT C (4.33), TaK tlTO pe-
3YJIbTaT 51BJI5IeTC5I peIlleHHeM 3a.L{allH [10).
B paCCMaTpHBaeMOM npHMepe KpHTepHH (4.33), TOllHee,
KpHTepHH G = I, a: N s (N s - 1), nOJIYlIaeMbIH, eCJIH CYM-
s
MHpoBaHHe B (4.31) H.L{eT no i =1= j, BbI.L{eJI5IeT H3 Tpex nepBo-
187
H3tJ3JIhHbIX pa30HeHHH KaK HaHJIyqrnee {I ,2-3-4,5-6},
.L{JISI KOToporo G = 2,987. Q.L{H3KO 3TO pa30HeHHe He OnT MaJIb-
HO; H {1-2-3-4,5-6}, H U = {1-2-3-4-5-6} JIyq-
rue. ,[{JIH nepBoro H3 HHX G = 3,893, .L{JI5I BToporo - G =
= 3,888, T3K tlTO OnTHM3JIbHbIM B .L{aHHOM CJIYtl3e SlBJI5IeTC5I
pa30HeHHe R = {1-2-3-4, 5-6}.
1
ECJIH nOJIO}KHTb A, = N, ' TO M3Tp H lI.a p330neHHSI R =
= {Rl,...,Rm} C 3THMH BeC3MH eCTb He qTO HHoe, K3K N X N-
MaTpHna onepaTop3 npoeKTHpOB3HHSI H3 nO.L{npOCTp3HCTBO,
nOpO)K.L{eHHOe OYJIeBCKOH N X m-MaTpHneH pa30HeHHSI R. npH
3TOM <PYHKQHOHaJI (4.31) oY.L{eT B COOTBeTCTBHH C (4.29) HMeTb
BH,l{
m 1
a (r) = - af}-.!!!.. .
s== I N, i,/ER, 2
( 4.35)
4.1.4. YtteT ufltjJopMaTu8flocTU npU3flaK,08
nOCKOJIbKY MaTpHn3 CBSl3eH Me)K,l{Y OO'beKT3MH
n
a = 1. r k CKJIa.l(bIBaeTC5.I U3 MaTpUII. CB5.l3eH no OT)J.eJIbHbIM
n k ==l
npH3HaK3M, TO Bec k-ro npH3HflKa B 3a,l{3qe KOHCTpYHpOBaHH5I
onpe,l{eJISleTCH BKJl3,l{OM r B aij. 1JeM OOJlbIlle r (i, j E 91),
TeM OOJIbIlle Bec k-ro npH3H3Ka npH nOCTpoeHHH <l>3KTopa.
B npe,l{eJIbHOM CJIyq3e, Kor,n3 r aij, cp3KTOp nOJIHOCTblO on-
pe,neJISleTCSI k-M npH3H3KOM.
B npOCTeHIlleM CJIyqae, Kor,na Bce HCxo,nHbIe npH3H3KH X3-
p3KTepH3YIOTCSI OYJIeBCKHMH (1,-1 )-MaTpHn3MH r k , HMeeT
MeCTO paBHonpaBHe npH3HaKOB H HX 3HaqeHHiL BC5IKHH npH-
3HaK, .nnH KOToporo r / = 1 «i, j) E pk), T. e. OO'beKTbI i H
. 1
J CBSl3aHhI, ,l{Ou3BJISleT n B aij, 3 npH3H3K, ,l{JI5I KOToporo
r = - 1 «i,j) $ p), T. e. OO'beKTbI i H j He CBSl3aHbI, BbIqH-
1
T3eT it H3 aij.
,[{JIH OTpa>KeHH5I <l>aKTHqeCKOrO Hep3BHOnpaBH5I npH3H3KOB
B 33,l{3qe KOHCTpYHpOBaHH5I eCTeCTBeHHO HCnOJIb30B3Tb Beco-
BbIe K03qxpHnHeHTbI HX HH<pOpM8THBHOCTH. nYCTb An - Bec
n k
npH3H8K8 r k ( Ah = 1), TOr,l{3 ail = I, Anri/ H K3}K,l{bIH npH-
k=l
3HaK ,l{aeT BKJI3.n B aij, paBHbIH ::I::Ak (B 33BHCHMOCTH OT 3H3Ka).
188
YlHbIMH CJIOBaMH, Bec A .L{aHHOro npH3HaKa OTpa>KeH B BeJIH-
GHHe, KOTOpa5I npHnHcbIBaeTC5I HaJIHqHIO HJIH OTCYTCTBHIO CB5I-
3H Me>K.L{Y 06'beKTaMH no .L{aHHoMY npH3HaKY. Becy A COOTBeT-
cTByeT HCnOJIb30BaHHe 6YJIeBCKOH (A,- A)-MaTpHI(bI, TaK tITO
,LlBOHHOe YBeJIHtIeHHe Beca B.L{BOe }Ke YBeJIHqHT BKJIa.L{ .L{aHHOrO
npH3HaKa B MaTpHny CBH3eH Me>K.L{Y 06'beKTaMH.
Beca npH3HaKOB MorYT Ha3HatIaTbCH H3 anpuopflblX co-
06pa>KeHHH, He CBH3aHHbIX HenOCpe.L{CTBeHHO C KOHCTPYHPY-
eMbIM cpaKTopOM, HJIH >Ke anocmepuopHO, Ha OCHOBe YtIeTa no-
CTpoeHHoro <jJaKTopa.
MO>KHO BbI,l{eJIHTb CJIe,nYIOIII.Me H,l{eH nocTpOeHH5I anpHopHbIX
BeCOBbIX O[(eHOK npH3HaKoB: a) 3KcnepTHbIM cnoc060M; 6)
Ha OCHOBe Cpe.L{H'eH HH<popMaUMH, .L{aBael'vtOH npH3HaKoM 06
06'beKTaX; B) Ha OCHOBe Cpe.L{HeH HHcpopManHH, ,l{aBaeMOH npH-
3HaKOM 0 3HatIeHH5IX HeKOTopbIX ,l{PyrHx, BbIXO,l{HbIX, nOKa-
3aTeJIeii; r) Ha OCHOBe Cpe,l{HeH HHcpopManHH npH3HaKa 0 BceH
cOBoKynHocTH HCXO,l{HbIX nOKa3aTeJIeH.
<PopMaJIbHbIe nOCTpoeHHH ,l{JI5I OneHKH BeCOB npH3HaKOB
BH,l{OB 6)- r) 06bltIHO OCHOBaHbI Ha KaKOM-JIM6o nOKa3aTeJIe
u
CB5I3H npH3HaKoB THna 3HTponHH, K8qeCTBeHHOH ,l{McnepCHH
H T. n. HanpHMep, MHcpopMaTMBHocTb npH3HaKa Rk MO>KeT xa-
paKTepH30BaTbcH: B cJIyqae a) ero ,nHcnepCMeH d (U, Rk);
B cJIYtIae 6) ero paCCT05lHHeM !; d (Rk, Rl) MJIM cYMMapHoH
I
CB5I3blO !; W (Rk, Rl), r,ne R' - BhIxoJtHble npH3HaKH; B
l
cJIyqae r) TaKHM }Ke cYMMapHbIM paCCT05lHHeM, HO Y>Ke,l{o HCXO,l{-
HbIX npH3HaKoB.
npH OneHKe no ,l{HcnepcHH d (Rk) (CM., HanpHMep, [121])
npH3HaK CtIHTaeTC5I TeM 60JIee HHcpOpM8THBHbIM, qeM 60JIbllle
d (Rk), T. e. qeM 60JIee paBHoMepHbIM BJIHeTC5I pacnpe,l{eJIe-
HHe (Rk). I1H<popMaTHBHocTb B 3TOM cJIyqae onpe,l{eJI5IeTC5I
Cpe,l{HeH OIllH6KOH nporH03a Rk. qeM 6JIH}Ke Rk K TpHBHaJIb-
HOMY pa36HeHHIO 0 MHO}KeCTBa m Ha N O,l{H03JIeMeHTHbIX KJIaC-
COB, TeM BbIllie Bec Rk. AHaJIOrHtIHO HHTepnpeTHpYIOTC5I Beco-
Bble K03cp<pMnHeHTbI B ,npyrHx CJIYtIa5lx.
06paTHMC5I K ,naHHbIM Ta6JI. 3.1 H BbItlHCJIMM BeJIHtIHHbI
d (Rk) =d (U, Rk): d (R 1 ) = d (R 2 ) = 6 2 - 3 2 - 3 2 = 18,
d (R 3 ) = d (R 5 ) = 36-1-4 = 31, d (R4) = 36-16-4=16.
nO,l{eJIHB 3TH 3HaqeHH5I Ha HX CYMMY 114, nOJIyqHM BeCOBbIe
K03<p<pHlI.HeHTbI npM3HaKoB:
Al = A 2 = 0,16, A3 = A5 = 0,27, A4 = 0,14.
C YtIeTOM BeCOB An BMeCTO 6YJIeBCKHX (I,O)-MaTpHlI. r k ,
1
HCnOJIb30BaHHbIx ,nJI5I nocTpOeHH5I MaTpH[(bI a = - rk, CJIe-
n
I !
I
I
189
'
.nyeT paCCMaTpHBaTb (Ak, O)-MaTpHQbI Akrk. MaTpHQa a = Akrk
u
HMeeT CJIe.nYIO HH BH.L{:
1 2 3 4 5 6
1 0,30 0,30 0,41 0,16 0,16
2 0,30 0,46 0,30 0,27 0
3 0,30 0,46 - 0,57 0,27 0,54
4 0,41 0,30 0,57 0,16 0,43
5 0,16 0,27 0,27 0,16 0,73
6 0,16 0 0,54 0,43 0,73
Pacnpe,l{eJIeHHe 3JIeMeHTOB B 3TOH MaTpHue HeCKOJIbKO OT-
JIHlIaeTC5I OT HX pacnpe,l{eJIeHH5I B MaTpHue (4.12). HanpHMep,
3.L{eCb BMeCTO a 2 3 3JIeMeHT a 36 npeBbIllIaeT nopor a = 0,5.
Haii,l{eM cpaKTop ,Z(JI5I ,l{aHHOH MaTpHl.(hI a B KJIaCCe MaTpHU
BH,l{a Ar + f.t, r,l{e r - (1 ,0)-MaTpHua, cooTBeTcTBYIO a51 HOMH-
HaJIbHOMY npH3HaKY (pa30HeHHIO). ,[{JI5I 3Toro BOCnOJib3yeM-
C5I HTepaTHBHbIM npOl.(eCCOM nOCJIe.L{OBaTeJIbHOrO nepeCqeTa
r H A, J..t, HallHHa51 C pa30HeHH5I Ro = {I, 2, 3-4, 5-6}, onTH-
MaJIbHOrO npHMeHHTeJIbHO K MaTpHue (4.12). Cpe.L{H5I5I BHYT-
peHH5I5I H Cpe,l{H5I5I BHellIH5I5I CB5I3H .L{JI5I R 0 paBHbI COOTBeTCT-
BeHHO: (X = 0,65 H = 0,29. Tor.L{a OnTHMaJIbHOe 3HaqeHHe
A, B CHJIY (4.23) paBHo A, = (X - = 0,36 H 3HaqeHHe nopora
A
paBHo a = J..t + 2 = 0,29 + 0,18 = 0,47. BOJIbIlle 3Toro
3HalleHH5I B MaTpHue a TOJIbKO TPH BeJIHlIHHbI: 4 = 0,57,
a 36 = 0,54 H a S6 = 0,73. OnTHMaJIbHbIM oY,l{eT On5lTb pa30He-
HHe {I, 2, 3-4, 5-6}, TaK KaK nonbITKa OO'be,l{HHeHH5I 00'b-
eKTOB 3 H 6 (.L{JI5I YlIeTa a 36 ) MomeT TOJIbKO YMeHblIIHTb CYM-
l\{Y BHYTpeHHHx CB5I3eH. I-IecKoJIhKO H3MeHHJIHCb, npaB,l{a,
3HalleHH5I A H J..t: npH paBHonpaBHhIx npH3HaKax HX 3HalleHH5I
ObIJIH 0,39 H 0,31, TOf,l{a KaK Tenepb 0,36 H 0,29, liTO OTpa}KaeT
OOJIbIllYIO paBHOMepHOCTb pacnpe,l{eJIeHH5I 3JIeMeHTOB ,l{aHHOH
MaTpHUbI no cpaBHeHHIO C MaTpHueH (4.12). nOJIyqeHHbIH B
u
,l{aHHOM npHMepe pe3YJIbTaT OTpa)KaeT cJIe.L{YIO HH XOpOlIIO
H3BeCTHbIH 3MnHpHQeCKHH <paKT: eCJIH ,l{aHHbIe xopolIIHe, T. e.
OOJIa,l{aIOT BHYTpeHHeH CTPYKTYPOH, TO HMeHHO 3Ta CTpYKTypa
H OY,l{eT BbI5IBJI5ITbC5I BCSlKHM pa3YMHbIM MeTO,l{OM. HMeHHO
3TOT <paKT H 51BJI5IeTC5I OCHOBOH nonWTOK nOCTpOeHH5I e,l{HHOH
TeopHH aHaJIH3a ,l{aHHbIX.
OueHKa BKJIa.L{a npH3HaKa B CYMMapHbIe CB5I3H Me}K,l{Y 000-
eKTaMH e,l{HHbIM BeCOBbIM nOKa3aTeJIeM HHOr,l{a CJIHllIKOM orpyo-
JI5IeT cHTyaUHIO. ,.[{OCTaTOQHO pa3YMHbIM KOMnpOMHCCOM MeJK,l{Y
HCXO,l{HOH CHCTeMOH qHCeJI r J (,l{JI5I Kam.L{OH napbI OO'beKTOB
CBoe QHCJIO) H e,l{HHbIM nOKa3aTeJIeM Ak 51BJI5IeTC5I yqeT B CB5I-
190
3HX aij BeCOB KOHKpeTHbIX 3HaqeHHH npH3HaKOB. ,[{JISI HOMH-
HaJIbHbIX npH3HaKOB npH 3a,!{aHHOH CHCTeMe BeCOB A , ..., A
Ka>K,!{OrO H3 m 3HaqeHHH k-ro npH3HaKa MaTpHna CBH3eH (rt)
XapaKTepH3yeTcSI 6JIOKaMH BHYTpeHHHX CBSl3eH A: Me:m,!{y OO'b-
eKTaMH s-ro KJIaCCa (s = 1, ..., m), npHqeM r = ° npH He-
9KBHBaJIeHTHbIX i H j (CM. c. 36). B qaCTHOCTH, npH 3a)\aHHH
npH3HaKoB HX npoeKnHOHH MH MaTpHnaMH Ph BeCa A paB-
Hbi , r,lle N - qaCTOCTb s-ro 3HaqeHHH k-ro npH3HaI a.
N.
3aMeTHM, qTO HMeHHO 9TH BeCOBbIe K09<p<pHUHeHTbI OhIJIH npe,!{-
JIO:>KeHbI He3aBHCHMO Ha OCHOBe npOCTOH CTaTHCTHqeCKOH Mo,!{e-
JIH B [92]. ToqHee rOBopH, B [92] npe,!{JIaraeTCH BCHKHH HC-
XO,!{HbIH npH3HaK 3aMeHHTb HaoopoM ,!{HXOTOMHqeCKHX, TaK
qTO B KaqeCTBe Xk y:me <PHrYPHPYIOT ,!{HXOTOMHqeCKHe npH3Ha-
KH, ,!{JISI KOTOpbIX H paCCMaTpHBalOTCSI BeJIHqHHbI, 06paTHbIe
qaCTOTaM.
MaTpHna CBH3eH no ,!{aHHbIM TaOJI. 3.1 c yqeTOM 3THX Beco-
BbIX 3HaqeHHH y:me nOCTpoeHa (CM. MaTpHny P Ha c. 121) ,!{JIH
HJIJIIOCTpanHH COOTBeTCTBYIO HX TeOpeTHqeCKHX nOJIO:>KeHHH.
,[{JISI y,!{o6cTBa npHBe,!{eM ee 3,!{eCb (6e3 ,!{HarOHaJIbHbIX 3JIe-
MeHToB):
1 2 3 4 5 6
1 0,58 0,58 0,75 0,33 0,33
2 0,58 0,92 0,58 0,5 0
3 0,58 0,92 - 0,92 0,33 0,67
4 0,75 0,58 0,92 0,33 0,67
5 0,33 0,5 0,33 0,33 - 1 ,5
6 0,33 0 0,67 0,67 1,5
nOCTpOHM <raKTop B KJIaCCe MaTpHn Ar + J..t, r,!{e r-
(1 ,0)-MaTpHII.a, OTBeqalO aH HOMHHaJIhHoMY npH3HaKY (pa3-
oHeHHIO). OnSlTb HaqHeM nponecc C pa36HeHHH Ro = {I, 2,
3 - 4,5-6}. Cpe,!{HSISI BHYTpeHHSISI CBSl3b B 3TOM pa30HeHHH
ex = (0,92 + 1,5)/2 = 1,21, cpe,!{HHSI BHeIllHSISI CBH3b p =
= 6,57/13 = 0,51, TaK liTO A = ex - = 0,7. BeJIHqHHa no-
A,
pora cymecTBeHHocTH a = J..t + 2 = 0,51 +0,35 = 0,86.
TIpH 3TOM nopore nOJIO}KHTeJIbHbI JIHlllb BeJIHqHHbI a23 - a =
= aM = a = 0,06 H a 56 -a = 0,64. CJIe,!{oBaTeJIbHO, OnTHMaJIb-
HbIMH ,!{JISI 3Toro cJIyqaSl 6Y,!{YT TOJIhKO pa30HeHHH {I, 2, 3-
4, 5-6} H {I, 2-3,4, 5-6} (o6'be,!{HHeHHe 2,3,4 B O,!{HH KJIaCC
npHBO,!{HT K CJIHIllKOM OOJIhlIlHM nOTepHM a S 4 - a = - 0,53
no cpaBHeHHIO C npHpOCTOM BHYTpeHHeH CYMMbI). TaK KaK
191
.............
a S 4 = a 2 3, 9TH pa30HeHHH COBepmeHHO O,!{HHaKOBbI C TOIIKH
3peHHH <PYHKUHOHaJla G (Ar + J..t) If, CJIe,!{OBaTeJlbHO, ooa ,nalOT
perneHHe 38,!{atlH. MbI e e pa3 yoe,!{HJIHCb B cnpaBe,!{JIHBOCTH
npHHUHna: eCJIH CTpYKTypa eCTb, OHa BbIHBJIHeTCH BC5IKHM pa-
3YMHbIM MeTo,nOM.
PaCCMOTpHM Tenepb BonpOCbI anOCTepHopHOH ou.eHKII Be-
COBbIX Ko9<fxPHnHeHTOB HH<popMaTHBHOCTH. AnOCTepHopHbIe
Beca npH3HaKOB ,!{OJILKHbI YtJHTbIBaTb HX OJIH30CTb K KOHCTPYH-
pyeMOMY <l>aKTOPY. nYCTb S = (Sij) - MaTpHna KOHCTPYH-
PyeMoro <paKTopa, a npH3HaKH 3a,!{aIOTCH MaTpHuaMH Anr lt ,
r,ne r lt - (1,0) -MaTpHna COOTBeTCTBYIO ero 6HHapHoro OT-
HomeHHH, a An - Bec, nO,!{JIejJ{a Hfi onpe,neJIeHHIO. KatJecTBo
annpoKcHMaUHH OneHHBaeTC5I nOKa3aTeJIeM (4.1), KOTOpbIH B
u
,!{aHHOM cJIYtJae HMeeT CJIe,!{YIO HH BH,!{:
n N
D = (Ah ':/-8tJ)2. (4.36)
k== t 1./=1
ECJIH MaTpHna S 3a,!{aHa, TO Beca An eCTeCTBeHHO onpe,!{eJIHTb
H3 YCJlOBHH MHHHMH3anHH (4.36): OnTHMaJlbHaH CHCTeMa BeCOB
TaKOBa, liTO S HaHJIYtJrnHM oopa30M annpOKcHMHpyeT MaTpHUY
. m k
CBH3efi atJ= Akrl/.
k=l
OnTHManbHbIe 3HalleHHH An onpe,!{eJI5IIOTCH npHpaBHHBa-
HHeM HYJIIO lIaCTHbIX npOH3BO,!{HbIX nOKa3aTeJIH (4.36):
iJD N
- = 2 (Ak '1/-80) r = 0,
OA", 1./ == 1
. I
I
TaK tJTO
,., k
r II Sij ( rk , S )
Ak = i,/ =
r (r k t r lt ) ·
1./
B Ba:>KHOM tJaCTHOM CJIYtJae S HMeeT BH,!{ S = Ar + J..t, r,!{e
r - oyneBcKa51 (I,O)-MaTpHna. BbIpa}KeHHe (4.37) npH STOM
,!{aeT
(4.37) !
A = A (r k t r) +
", ( k It ) ft,
r t r
N '
TaK KaK (r lt , rk) = rt ,!{JIH (I,O)-MaTpHUbI r lt . B lIaCTHOCTH,
I, / 1
eCJIH r k II r COOTBeTCTBYIOT HOMHHaJIbHbIM npH3HaKaM, TO
(r k , r) = (N t)'J., (r k , rk)::c (N:)2,
S,t s
(4.38) I
192
r){e N - QHCJIO OO'beKTOB, HMelO HX OAHOBpeMeHHO s-e 3Ha-
1JeHHe k-ro npH3HaKa H t-e 3HalleHHe cpaKTopa, aN;. = N:t-
t
tlHCJIO OO'beKTOB, HMelO HX s-e 3HatIeHHe k-ro npH3HaKa,
TaK liTO.
(N: t )2
An = A s ,t _ A-t- p, -A (RRk)_p,l> (Rk)
(N:.ra +J.t- 1-6(Rk) · (4.39)
s
rAe Rk H R - pa30HeHH5I, cooTBeTcTBYIO He MaTpH aM r k H
r.
TaKHM oopa30M, Beca An OTpa>KalOT KalleCTBO nporH03a Rk
npH H3BeCTHOM R no cpaBHeHHIO C KalleCTBOM nporH03a OAHoro
R.
H3JIO)KeHHOe n03BOJI5IeT npeAJIO>KHTb CJIeAYIO YIO HTepa-
THBHYIO npoueAYPY nocTpOeHH5I <t>aKTopa (B KJIaCCe MaTpHU
Ar + J..t) C o)tHoBpeMeHHoM o eHKoM BeCOB HCXOAHbIX npH3Ha-
KOB, HaHJIyqIIIHM oopa30M cooTBeTcTBYIO HX cpaKToPY.
Ha nepBoM IlIare paccMaTpHBalOTC5I oYJIeBCKHe (1,0)-
MaTpH bl r k HCXOAHblX KalleCTBeHHblX npH3HaKoB H CTpOHTC5I
MaTpH 8 cBH3eH a O = .!.. rk, paBHonpaBHo yqHTbIBalO aH
n k
Bce r k .
3aTeM AJIH 3TOM MaTpH bI pewaeTcH 3aAalla MaKCHMH3a-
[(HH nOKa3aTeJIH (4.19), ee perneHHe 0003HatIHM So = Aoro + J.to,
nOCJIe lIero BbIlIHCJI5IIOTC5I BeCOBbIe nOKa3aTeJIH At no CPOPMY-
JIaM (4.37) (HJIH, liTO TO JKe caMoe, (4.38) HJIH (4.39)) npH S =
= So, T. e. r = r o , A = A o ' J..t = J.to. Ha OCHOBe BeCOB A1 Bbl-
n
1JHCJI5IeTCH HOBaSl M8TPHI\8 CB5I3eH a 1 = A1r k .
k=l
Ha oo eM IIlare aJIrOpHTMa paCCMaTpHBaeTCH MaTpH a
CB5I3eH at (t = 0,1,...), )tJI5I KOTOpOH OTbICKHBalOTCH Ah J..tt, rt,
MaKcHMH3HpYIOI.I.(He (4.19), nOCJIe qero onpe.L{eJI5IIOTC5I Beca
A + 1 no <popMYJIaM (4.37) (HJIH (4.38), (4.39)) npH S = Sb T. e.
r = rt, A = At, J.t = J..tt. Ha OCHOBe 3THX BeCOB BbIQHCJI5IeTC5I
HOBa51 MaTpllll.a CB5I3eH a t + 1 == A +l r k .
k
npo ecc 3aKaH HBaeTC5I, KOf.L{a St + 1 = St, TaK liTO H
a t + 1 = at. rapaHTHeM 3Toro 51BJI5IeTC5I KOHetIHOCTb tIHCJIa
(I,O)-MaTpHlI. r B "'paccMaTpHBaeMoM KJIaCCe H YMeHblIleHHe
KpHTepH5I (4.36) Ha Ka}K.L{OM ware aJIfOpHTMa (CM. TaK}Ke
n. 4.2.1).
7 3aK. 1 743
193
npHMeHHM OnHCaHHYIO npO[(e,l{YPY K paCCMaTpHBaeMOMY
npHMepy (TaOJI. 3.1). TIepBbIH liar Y)Ke C,l{eJIaH Ha c. 185: ,l{JIH
MaTpH[(bI ao = 1. rk HaH,l{eHO OnTHMaJIbHOe pa30HeHHe R 0 =
n
= {I, 2,3-4, 5-6} C Ao = 0,392 H 1-10 = 0,308. YlIHTbIBa51
TaOJI. 3.1, nOJIYlIHM, liTO OTHOIIleHHe N:t (N: t - 1) K
N:. (N:. - 1) (3,l{eCb, KaK paHee, BMeCTO N2 OepeTC5I N (N-
- 1), TaK KaK He paCCMaTpHBaIOTC5I ,l{HarOHaJIbHble 3JIeMeHTbI
MaTpHIJ;bI cBsI3eii), paBHO 1; ()J.JISI k = 1, 3, 5) HJIH (.n.JISI k =
= 2,4). Il03ToMY B CHJIY (4.39) At = AA = A = 0,340,
A = A1 = 0,373. HopMHPyeM 3TH BeJIHlIHHbI TaK, 11 TOObI B
5
CYMMe nOJIYlIHTb e,l{HHHUY. Tor,l{a MaTpH[(a CB5I3eH a 1 = ALr k
k=l
HMeeT CJI e,l{YIO HH BH,l{:
1
2
0,404
3
0,404
0,615
4
0,404
0,422
0,615
5
0,211
o , 192
o , 192
0, 192
6
0,211
°
0,385
0,385
0,807
1
2
3
4
5
6
3,l{eCb H ,l{aJIee npHBO,l{HTC5I TOJIbKO BepXH5I5I lIaCTb CHMMeT-
pHlIHOH MaTpH[(M. 01leBH,l{HO, pacnpe,l{eJIeHHe ee 3JIeMeHTOB
npaKTHlIeCKH He OTJIHlIaeTC5I OT pacnpe,l{eJIeHH5I MaTpH[(bI
(4.12), TaK KaK Beca A ,l{OBOJIbHO 6JIH3KH ,l{pyr K ,l{pyry.. 1
,
Il03TOMY OnTHMaJIbHbIM On5lTb 51BJI5IeTCSI pa36HeHHe {I, 2, 3-
4, 5-6}, KOTOPOMY cooTBeTcTBYIOT Beca A, = 0,402 H 1-1 =
= 0,309. OTHOCHTeJIbHbIe 3HalleHH5I AI Te )Ke, liTO H A1. 3TO
YKa3bIBaeT Ha TO, qTO B ,l{aHHOM npHMepe Beca npH3HaKOB B
3a,l{alle <t>aKTopHoro aHaJIH3a MorYT 6bITb B351TbI O,l{HHaKOBbIMH.
Pa3BHBa51 onHcaHHbIH MeTO,l{, MO:>KHO npe,l{JIOJKHTb MeTO,l{
nOCTpOeHH5I ,l{BYX BeCOBbIX O[(eHOK npH3HaKOB: CHJI BHYTpeH- I
HHX H BHellIHHX CBSl3eH. ,aJIH 3Toro HY)KHO paCCMOTpeTb CJIe-
,l{YIO YIO KOHKpeTH3a[(HIO nOKa3aTeJISI (4.1):
n N
D= (Ahr7-+l-1k-Sij)2.
k::: 1 i,j= 1 1
3,l{eCb MaTpH[(bI r k 6epYTC5I B BH,l{e A;r k + I-1k, r,l{e ak ==
= Ak + I-1k H k = I-1k - Beca BHYTpeHHHx H BHelllHHX CB5I-
3eM no OTHOIlleHHIO pk. TIpH 3a,l{aHHOH MaTpH[(e cpaKTopa s
OnTHMaJIbHble 3HalleHH5I Ak H I-1k HaXO,l{51TCSI H3 YCJIOBHSI MHHH-
( 4.40)
194
L
aD aD
MH3al.UfM (4.40), T. e. paBeHcTBa HYJIIO aAk M aflk ' HeTPY):(Ho
nOKa3aTb, liTO 9TM 3HalleHH5I Ak M J..tk onpe):(eJI5IIOTC5I <POPMY-
JIaMM:
Sij r1 i
I\" h =
r .
1./
Sij (l-r: j )
(l-r )
(4.41)
-K03<p<pM MeHT pa3perneHH5I r k OTHOCMTeJIbHO s;
Sij (l-r )
J..t k =
(l-r )
( 4.42)
-Cpe,l{H5I5I CB5I3b «BHe» pk. HeTPY,l{HO npe,l{cTaBHTb 3TM Bbl-
pa*eHH5I no aHaJIOrHM C (4.39) B TepMHHax ,l{HCnepCHH npH3-
HaKOB Rk H <paKTopa R (fipH S == Ar + J..t).
HaJIHlIHe <POPMYJI (4.41), (4.42) n03BOJI5IeT ):(aTb HTepaTMB-
HbIH aJIrOpMTM o eHKM BeCOBbIX nOKa3aTeJIeH An, J..tk C O,AHO-
1
BpeMeHHbIM nocTpoeHMeM <paKTopa S: HallMHa51 C a = - rk
n k
nOCJIe):(oBaTeJIbHO BbIlIHCJI5IIOTC5I <paKTOpbl s, nOCJIe lIero
BbIlIHCJI5IIOTC5I no (4.41), (4.42) A,?-, J..tk M BbIlIMCJI5IeTC5I
HOBa51 MaTpH a CB5I3eH a = (An'; j + J..tn), ):(JI5I KOTOpOH
k
npo ecc nOBTOp5leTC5I BHOBb, TaK liTO npo e,l{ypa nOJIHOCTblO
aHaJIOrHlIHa BbllIleonHcaHHOH.
nO,l{OOHbIH npo ecc MO)KHO HCnOJIb30BaTb H ,l{JI5I nOCTpOeHH5I
BeCOB A: OT eJIbHbIX 3HalleHHH S (5 == 1, ..., mh) npH3HaKoB k
(k = 1, ..., n). TOJIbKO npM 3TOM paCC1\1aTpMBaeTC5I CJle,l{YIO a51
KOHKpeTH3a M5I nOKa3aTeJI5I (4.1):
n N I ) 2
D= . I A r:ij-Sij I
k = I l,J = 1 \ S I
(4.43)
f):(e ': === (r:ii) - (1 ,O)-MaTpM a ,l{JI5I MHO)KeCTBa OO'beKTOB,
Ha KOTOpbIX k-H npH3HaK npHHHMaeT s-e 3HalleHMe. IlpHpaBHM-
aD
BaHHe HYJIIO npOH3BO,l{HbIX s ):(aeT CJIe.L{YIOIJJ.He BbIpameHMH .L{JI5I
a1\k
7:1'
195
r
I
BeeOB, MHHHMH3HPYIOIlJ,HX paCXO>K.L{eHMe (4.43) Mem.ny MaTpH-
ueii-q,aKTopoM S H MaTpHueH CBSl3H a = A:r :
. k s
Sij
xk (l) = xk (j) = s
'A k = .
S N:(N:-J)
(4.44 )
4.1.5. AflaJl02 MerOaa 2JlaBflblX KOMnOfleHT
CPYHKlI.HOHaJI D (r) BH.L{a (4.1) xapaKTepH3yeT,
HaCKOJIbKO XOpOlllO MaTpHl.(a r E annpoKcMMHpyeT MCXO.L{-
HbIe MaTpMl.(bI r 1 ,..., r n . B TOM cJIyqae, Kor.L{a 3Ta BeJIHqHHa
.L{JI5I OnTHMaJIbHOrO r E OTHOCHTeJIbHO BeJIHKa, MomHO rOBO-
pHTb 0 TOM, qTO HMelO HeC5I .L{aHHbIe He MorYT 6bITb npe.L{CTaB-
JIeHbI B KJIaCCe (g C .L{OCTaTOqHOH TOqHOCTblO e.L{HHCTBeHHbIM <paK-
TOpOM. B03HHKaeT np06JIeMa nOCTpOeHH5I MHHHMaJIbHOH CHC-
TeMbI cpaKTopOB H3 (g (HJIH H3 f£ 1, 2, ..., q, eCJIH Ka}K.L{bui
H3 q KOHCTpYHpye1\1bIX cpaKTopOB .L{OJI}KeH npHHa.L{JIe}KaTb CBoe-
MY KJIaccy it), n pe.L{CTaBJI5IIO HX CHCTeMY r 1 , ..., r n C .L{OCTaTOq - J
HOH TOqHOCTblO. ,
npe.L{CTaBJI5IeTC5I BnOJIHe eCTeCTBeHHbIM HCnOJIb30BaTb ,Z(JISI
u u
3TOH l.(eJIH CJIe.L{YIO HH aHaJIOr MeTOAa rJIaBHbIX KOMnOHeHT.
0603HaqHM qepe3 SI perneHHe 3aAaqM MHHHMH3al.(HH D (r)
B KJIaCCe (8 H paCCMOTpHM MaTpHl.(bI r k1 = r k - SI OCTaTOq-
HbIX CB5I3eH Me:>KAY 06'beKTaMH, He yqTeHHbIX B MaTpMl.(e SI.
3TH OCTaTOqHbIe CB5I3H H 3aAalOT BeJIHqHHY norpernHOCTH
annpOKCHManHH D (r 1 ) = I r k1 12. nOCTpOHM cpaKTOp r = S2,
k
annpOKCHMHPYIO HH MaTpHl.(bI r k1 CB5I3eH, He YtlTeHHbIX nep-
BbIM <t>aKTopOM SI. ,[{JI5I 3Toro 6YAeM MHHHMH3HpOBaTb CYMMY
paCCT05lHHH
DI (s) = , rkl- s 1 2 (4.45)
k
i
no BceM sEtS (KJIaCC tS MO)KeT, Boo6III.e rOBopH, OTJIHqaTbCH ,
OT Toro, KOTOpbIH HCnOJIb30BaJIC5I,AJI5I OTbICKaHMSI SI).
PaCKpbIBa51 B (4.45) cK06KH, nOJlyqMM
k
D 1 (s) == (r k1 , rkl) + n (s, s)- 2 !, (rkl, s).
k k=l
196
-Ai.
nOCKOJIbKY r k1 <pHKCHpOBaHHO, MHHHMH3a[(HSI Dl (s)
1
CBO HTCSI K MaKCHMH3a[(HH - 2n D1 (s) + const, T. e.
Ql (s) = (at, s) -+ (s, s), (4.46)
r,ne
I n 1 n
a 1 =-;- r kt =--;; (r k -s 1 )=a-s 1 ,
k=l k=l
TaKHM oopa30M, G1 (s) nOJIHOCTblO COBna)(aeT C G (r) (CM.
<jJOPMYJIY (4.2)), BbIqHCJIeHHbIM JISI MaTpH[(bI OCTaTOqHbIX CBH-
3eH Me* y OOneKTaMH
a l = a-r 1 .
( 4.47)
H3JIO>KeHHOe npHBo HT K CJIe,l{YIOIl\eMY MeTO,l{Y cpaKTopHoro
aHaJIH3a, 3aKJIlOqalO eMycH B nOCJIeAOBaTeJIbHOM OT enJIeHHH
<t>aKTopoB sP, npe,l{cTaBJIHIO HX MaTpH[(bI a P - 1 OCTaTOqHbIX
CBSl3eH Me>K,l{Y OOneKTaMH.
Ha nepBOM mare nOJIaraeM a = aO H HaXO,l{HM cpaKTOp 8 1 E
E (g, MaKCHMH3HPYIO HH G (s) no s E , nOCJIe qero nepexo-
):lHM K MaTpH[(e OCTaTOqHbIX CBSl3eH a 1 = a O - Sl. Ha oo eM
(p + 1) -M lIIare (p=O, 1, ...) paccMaTpHBaeTcH MaTpH[(a oc-
TaTOqHbIX CBSl3eH MeJK,l{Y OOneKTaMH a P H OTbICKHBaeTCH MaT-
pH[(a SO + 1 E , MaKCHMH3HpYIO aH nOKa3aTeJIb
1
Qp (s) = (a P , s) - 2" (s, s) (4.48)
i
I \
, j
I
,
I '
no s E , nOCJIe qero BbIqHCJIHeTCH HOBaH MaTpH[(a OCTaTOq-
HbIX CBSl3eH a P + 1 = a P - sP + 1 H npOHCXO,l{HT nepeXO,l{ K CJIe-
,l{YlOrneMY, (p + 2)-MY mary.
P
OqeBHJJ.HO, a P = a - Sk H, CJIe.l(OBaTeJIbHO,
k=l
P
a = a P + Sk.
k= 1
nOCJIe,l{OBaTeJIbHOCTb MaTpH[( SI, ..., sP oopa3yeT CHCTeMY
<jJaKTopOB, TeM JIyqme npe,l{CTaBJIHIO YIO CBH3H Me*,l{Y OOneK-
TaMH, Onpe,l{eJIHeMbIe MaTpH[(eH a, qeM OJIH>Ke K HYJIIO 1\1aTpH-
u.a OCTaTOqHbIX CBH3eH a P .
HMeeM no onpe,l{eJIeHHIO
aP+l=aP-sp+l, (4.49)
197
I I
I
I
I
I
I
;
.
I '
1
TaK qTO
(a p + I, aP+ 1) = (ap, ap)-2 (aP, sP+ 1) +(SP+ 1, sP+ 1).
nOACTaBJI5I5I BbIpaLKeHHe GP (S)H3 (4.48), nOJIyqHM
I aP+112=(ap+l, a p + 1 )=(a p , a p )-1-0P(sP+l). (4.50)
2
TaKHM 06pa30M, nornarOBa51 MHHHMH3anH5I CYMMbI paCCT05l-
HHH BHAa (4.1), (4.45) OT <t>aKTopa AO CHCTeMbI npH3HaKoB
3KBHBaJIeHTHa nOlllarOBOH MHHHMH3anHH paCCT05lHH5I
laP-sI2=(aP-s, aP-s)
(4.51 )
OT S AO cYMMapHoH MaTpHnhI CB5I3eH MeLKAY 06'beKTaMH. STOT
pe3YJIbTaT e e pa3 nO.Lp.lepKHBaeT 3KBHBaJIeHTHOCTb ABOHCTBeH-
u
HbIX KOHCTpYKnHH B TepMHHax MaTpHn CB5I3H: annpOKCHManH H
CHCTeMbI npH3HaKOB H aHaJIH3a MaTpHnbI CB5I3eH Me:>KAY 06'beK-
TaMH.
npH 3TOM cpopMYJIa (4.51) nOKa3bIBaeT, qTO Ka:>KAbIH CJIe-
AYIO HH cpaKTop Sp+1 onpeAeJISleTC5I TaK, QT06bI MHHHMH3HpO-
BaTb ,lJ.JIHHY MaTpHnbI OCTaTOllHbIX CB5I3eH I a p + 1 1 2 (nOCKOJIbKY
Gp MaKCHMH3HpyeTC5I MaTpHneH Sp+1). n03TOMY npH 3Halle-
HH5IX Gp (sP-i 1), He 51BJI5IIO HXC5I CJIHIllKOM MaJIbIMH, BeJIH-
HHa I a p + 1 1 2 CTpeMHTCH K HYJIIO npH p -+00, H, CJIe)J.oBaTeJIbHO, I
SP B npeAeJIe paBHa a, TaK liTO MO:>KHO rOBopHTb 0 CXOAHMOCTH
p=l
MeTOAa, KOTOpa51 3aBHCHT OT KJIaCCa AonYCTHMbIX MaTpHn .
3aMeTHM, qTO 3TOT MeTOA 51BJI5IeTC5I aHaJIOrOM KOJIHlIeCTBeH-
Horo MeTOAa rJIaBHbIX KOMnOHeHT. Pa3HHna B TOM, qTO OMno-
HeHTbI sP BbI6HpaIOTC5I He npOH3BOJIbHbIM 06pa30M H3 nOAnpo-
CTpaHCTBa, opTOrOHaJIbHOrO Y:>Ke nOCTpoeHHbIM KOMnOHeHTaM,
a H3 3apaHee cpHKCHpOBaHHoro MHO)KeCTBa . n03TOMY B 06-
eM cJIYlIae KOMnOHeHTbI sP He opTOrOHaJIbHbI H, CJIeAOBaTeJIb-
q
HO, He AalOT peIlleHH5I 3aAallH MHHHMH3anHH I a - sP 1 2 no
p=1
BceM SP OAHOBpeMeHHO. nOCJIeAH5I5I 3aAalla (napaJIJIeJIbHbIH
<t>aKTopHbIH aHaJIH3) npaKTHlIeCKH He HCCJIeAOBaJIaCb B HHTepe-
CYIO HX Hac KJIaCCaX AonYCTHMbIX MaTpHn H He 6YAeT paCCMaT-
p HBa TbC5I AaJIee.
BonpOCbI CXOAHMOCTH H3JIO:>KeHHOrO aHaJIOra MeTOAa rJIaB-
H X KOMnOHeHT HaH60JIee nOJIHO HCCJIeAOBaHW AJISI CJIYlIa5l,
KorAa COCTOHT H3 (ex, p)-MaTpHn BHAa Ar+f.t, rAe r - (1,0)-
6YJIeBCKa51 MaTpHna. npH 3TOM oOHaPY)KHJIaCb AOCTaTOllHO
np03pallHa51 HHTepnpeTanH5I pellIeHHH. OCHoBHbIe pe3YJIbTaTbI
198
3,neCb nOJIyqeHbI B. A. TpOcpHMOBbIM [98, 99] (CM. TaK)Ke [51,
66]). nepeH,neM K paccMoTpeHHIO 3Toro CJIyqa5l.
fIpe1K,ne Bcero 3acpHKcHPyeM KJIaCC ,nonycTHMblX MaTpHU-
KapKacoB r. I1cKJIlOqHM H3 paccMOTpeHH5I ,nHarOHaJIbHbIe 3JIe-
MeHTbI MaTpHU (T. e. BCIO,ny oy,neT i =1= j) H B KaqeCTBe HX Kap-
KaCOB 6y,neM paccMaTpHBaTb (1,-1 )-npe,ncTaBJIeHHH OTHOlIle-
HHH (nepexo,n K (I,O)-npe,ncTaBJIeHHIO TpHBHaJIeH, CM. C. 36),
1JTO HeCKOJIbKO ynpocTHT <POPMYJIbI H HX HHTepnpeTaUHIO. By-
,neM CqHTaTb, qTO ,nJIH JIIOObIX i, j, (i =1= j) B co,nep1KHTCH MaT-
pHua r ij , TaKa5l, qTO ee e,nHHcTBeHHbIH: e,nHHHQHbIH: 3JIeMeHT
pacnOJI01KeH B (i, j)-if. KJIeTKe. B CJIyqae, Kor,na (8 COCTOHT H3
CHMMeTpHqHbIX MaTpH[(, r iJ co,nep>KHT pOBHO ,nBa e,nHHHqHbIX
3JIeMeHTa -B KJIeTKaX (i, j) H (j, i). I1HbIMH CJIOBaMH, Tpe-
6yeTc5I, qTOObI ,nJI5I JIIOObIX i, j (i =1= j) B BXO,nHJIO OTHOIlle-
HHe, CB5I3bIBaIOInee TOJIbKO i C j (noMHMo HepaccMaTpHBaeMbIx
,nHarOHaJIbHbIX cBH3eii, T. e. neTeJIb (i, i)). npHMepbl TaKHX
KJIaCCOB : KJIaCC (a, p)-MaTpHU, coo1;BeTcTBYIO HX BceB03-
MO>KHbIM pa36HeHH5IM (MaTpHua r lJ cooTBeTcTByeT pa30He-
HHIO 91, COCT05lIneMY H3 N - 1 KJIaCCOB, O,nHB H3 KOTOpbIX co..
CTOHT H3 06'beKTOB i, j, a OCTaJIbHbIe - o,nH03JIeMeHTbI); KJIaCC
(a, P) -MaTpHU, cooTBeTcTBYIO HX BceB03MO>KHbIM BbI,neJIeH'"
HbIM no,nMH01KeCTBaM (MaTpHua r lJ COOTBeTcTByeT BbI,neJIeH-
HOMY ,nByx3JIeMeHTHoMY no,nMHo:mecTBY); KJIaCC (a, )-MaTpHU,
cooTBeTcTBYIOInHX BceB03MOLKHbIM CTPYKTypHbIM npH3HaKaM
(noCKOJIbKY OH co,nep>KHT npe,nbI,nYInHe ,nBa KJIaCca) II T. n.
He y,noBJIeTBOp5leT CCPOPMYJIHpOBaHHoMY YCJIOBHIO KJIaCC MaT-
PH[(, COOTBeTCTBYIO HX BceB03MO>KHbIM paH)KHpOBaHH5IM HJIH
uenHbIM OTHOlIleHH5IM.
B C eJIaHHbIX npe,nnOJI01KeHH5IX MO)KHO ,nOKa3aTb CXO,nH-
MOCTb aHaJIOra MeTo,na rJIaBHbIX KOMnOHeHT ,naLKe B CJIyqae,
Kor,na 3a.L{aqa MaKCHMH3a[(HH (4.48) peIllaeTC5I JIHllIb JIOKaJIb-
HO-OnTHMaJIbHbIM aJlrOpHTMoM. npH 3TOM B COOTBeTCTBHH
C 3aMeqaHHeM Ha c. 147 MaTpHua a CqHTaeTC5I cHMMeTpHqHOif,
eCJIH KJlaCC [g COCTOHT 113 CHMMeTpHqHbIX MaTpllU (npH HeCHM-
u , a+a T )
MerpHQHOH a paCCMaTpHBaeTC5I a = 2 ·
YTOqHHM BH,n nOKa3aTeJI5I (4.51) ,nJI5I paccMaTpHBaeMoro
CJIyqa51 :
N
Ll (r, Iv, 11) = (aij-Arij-J1)2/N (N -1). (4.52)
I, j = 1
3,necb nO T05lHHbIH .L{eJIHTeJIb N (N - 1) BBe,neH ,nJIH npo-
CTOTbI HH)KeCJIe,nYIO Hx nocTpoeHHH. AHaJIOrHqHO cpopMYJIaM
(4.22) H (4.23) onpe,neJI5IIOTC5I OnTHMaJlbHble npH .L{aHHOM r 3Ha-
199
':II
qeHHSI A, H J..t, KOTOpbIe 3AeCb Y,!{06HO BbIpa3HTb CJIe,!{YIO HM
06pa30M:
A = COy (a, r)/cr 2 (r), J.t = M (a) - "A M (r), (4.53)
r,!{e M H 0'2 - cpeAHee 3HalleHHe H ,nHcnepCHH COOTBeTCTBYIO-
HX MaTpH[(, a
N
cov(a,r)= (aij-M(a»(rij-M(r»/N(N-l). (4.54)
't 1= 1
<1>0 p?vI YJIbI (4.53) npe,!{cTaBJISlIOT CODOn OOrneH3BecTHbIe Bbl-
pa)KeHHSI K03<P<PH[(HeHTOB JIHHeMHOM perpeccHH N (N - 1)-
MepHoro BeKTopa a no r [42].
nOACTaBJI5I5I B (4.52) J.t, onpeAeJI5IeMOe qepe3 A BbIpaJKeHHeM
(4.53), nOJIYlIHM nOCJIe HeCJIOJKHblX npeoopa30BaHHH
b. ::.: 0'2 (a) _')..,20'2 (r),
(4.55)
liTO TaK)Ke xapaKTepH3yeT H3BeCTHbIe CBOMCTBa npOeK[(HH TOll-
KH a Ha npSlMYIO "Ar + J..t. A HMeHHO BeJIHlIHHa N (N - I) 33-
,!{aeT KBaApaT eBKJIHAOBCKoro paCCT05lHH5I OT TOllKH a AO TOllKH
"Ar + J..t B N (N - 1)-MepHOM npocTpaHcTBe MaTpH[( 6e3 rJIaB-
HOM AHarOHaJIH. MHHHMH3aUHSI t1 no A H J..t BbIpa)KaeT onepa-
[(HIO npoeKTHpOBaHH5I a Ha nOAnpOCTpaHCTBO TOlleK BHAa
'A,r + J..t (npH 3aAaHHOM r), TaK 1ITO AO' (r) N (N - I) eCTb He
1ITO HHoe, KaK AJIHHa npOeK[(HH TOllKH a Ha 3TO nOAnpOCTpaH-
CTBO. TorAa paBeHCTBO (4.55) BbIpa)KaeT TeopeMY nH<paropa
(n.3): KBaApaT rHnOTeHY3bI 0'2 (a) paBeH CYMMe KBaApaTOB Ka-
TeTOB t1 + A, 2 O' 2 (r) (3AeCb 06rnHH MHO)KHTeJIb N (N -I) COKpa-
meH) .
npH 3TOM MHHHMH3a[(H5I 9KBHBaJIeHTHa MaKCHMH3a[(HH
KBa,npaTa eBKJIHAOBCKOH ,!{JIHHW npOeK[(HH 'A, 2 O' 2 ( . BeJIHlIHHa
"A2cr 2 (r) COrJIaCHO (4.55) nOKa3bIBaeT, KaKaSl qaCTb pa36poca
0'2 (a) BeJIHlIHH CB5I3eM aij YlITeHa B <t>aKTope r. ECJIH 3Ta lIaCTb
He CJIHlllKOM BeJIHKa, TO COrJIaCHO 8HaJIOry MeTOAa rJIaBHbIX
KOMnOHeHT CJIe,nyeT nOBTopHTb npO[(eAYPY annpOKCHMa[(HH AJI5I
lvlaTpHUbI OCTaTOllHbIX CB5I3eH a 1 = a - "Ar - J.t.
Ha 06rneM mare no MaTpH[(e OCTaTOllHbIX CB5I3eH a P - 1 OTbIC-
KHBaIOTC5I r P , "A p , J..tp, onTHMH3HpYIO He (4.42), ,nJI5I lIero MO-
u u
)KeT HCnOJIb30BaTbCSI JIOKaJIbHO-OnTHMaJIbHbIH HTepaTHBHbIH Me-
TOA (CM. c.184).!1aJIee paccMaTpHBaeTcSI ap=ap-1-Aprp-J.tp.
OlleBHAHo,
N(N-I)L\=(a P , ap)-(ap-l-"APrp_ p, a P - 1 -
- Ap r P - J,tp) . (4.56)
200
...
1XJI5I ,l{OKa3aTeJIbCTBa CXO):(HMOCTH 3TOrO npOQecca paCCMOT-
pHM MaKCHMaJIbHbIH no MO.nYJIIO 3JIeMeHT M rpliQbI OCTc:TOqHblX
CB5I3eH a p - 1 , KOTOpbIH 0603HallHM lIepe3 at}. ECJIH af} = 0,
3TO 03HallaeT, liTO a P - 1 = 0, TaK tITO r P = 0, Ap = J..tp = 0,
Ii, CJIe,l{OBaTeJIbHO, nponecc 3aKOHlIeH. nYCTb Tenepb ail =1= o.
PaCCMOTpHM Tor):(a MaTpHUY r il C e):(HHCTBeHHbIM e,l{HHHtIHbIM
06'beKTOM B KJIeTKe (i, j) (a npH CHMMeTpHlIHOCTH paCCMaTpHB£:-
eMhIX MaTpHII. H B (j, i». nOJIO)KHM 'A, = JL = I ail I H pac.
CMOTpHM MaTpHl.lY Ar il + J..t. OqeBH,l{Ho, Bce ee 3JIeMeHTbI HYJIe-
"
Bble, KpoMe (i, j)-ro, KOTOpblH paBeH I aij I (npH cliMMeTpHqHO-
CTH TO )Ke 3HaqeHlie npHHHMaeT (j, i)-H 3JIeMeHT MaTpliI.I.bl
"Ar il + J..t).
llOCKOJIbKY Aprp + J.1p OnTHMaJIbHO, liMeeM
(a P , a p ) = (aP-1-"Ap rP-'fJlp, a P - 1 -'Ap rP-J..tp)
(a p - 1 -'Ar ij - f.t, a P - 1 - 'Ar;l- f.t) = (a P - 1 , a P - 1 )-
-2 (a p - 1 , 'Ar 1j + f.t) -t- ('Ar lj + , Ar 1j + !1).
Ho
(a P - 1 , Ar 1j + IJ') ('Aril + fJI, Ar il + ) = all
(npH CHMMeTpHlIHOCTH 3TO 3HalleHHe paBHO 2alj), TaK 1ITO
(a p , ap) (ap-l, ap-1)-ah.
"
B CHJIY MaKCHMaJIbHOCTH aij HMeeM
"2 1
al" ( a p - 1 , ap-l ),
1 N (N - 1)
TOr)(a
1
(aI', aP) (aP-l, a P - 1 )- (ap-1, a p - 1 ).
N (N -1)
I
nOJIara51 e = 1 - N(N-l) , nOJIYlIaeM
(a P , ap) (ap-l, ap-l)B (a, a)BP.
CJIe,l{OBaTeJIbHO, npH p 00 CKaJI5IpHbIH KBa,l{paT (a p , a p )
cTpeMHTCH K HYJIIO co CKOpOCTblO, He MeHbIlleii, lIeM CKOpOCTb
reOMeTpHlIeCKOH nporpeCCHH CO 3HaMeHaTeJIeM B < 1. A 3TO
H 03HatIaeT, liTO a P -+ 0 npH p 00, liTO H Tpe60BaJIOCh ,l{OKa-
3aTb.
KaK BH,l{HO, ,l{aHHOe ,l{OKa3aTeJIbCTBO CXO,l{HMOCTH nO,l{pa3Y-
MeBaeT, 1ITO nOJIYlIaeMbIe JIOKaJIbHO-OnTHMaJIbHbIe perneH H5I
201
I
I
I
I I
I
'if
Ap r P + f.tp He xy:me npel{CTaBJI5IIOT l{aHHbIe, qeM 'Ar lJ + !J..
BnpOqeM, eC.lIH 9TO YC.lIOBHe He BbInO.lIHeHO, HCnO.lIb3yeMbIM Me-
TOl{ .lIOKa.lIbHOM onTHMH3alIJIH 51BJIHeTC5I HaCTO.lIbKO n.lIOXHM,
qTO BMeCTO Hero JIyqI.lIe npHMeH5ITb npOCTeMllIHM MeTOl{-BbIOOP
'Ar iJ + f.t.
4.1.6. H Hre.pnpera'1ufl t/JaKTOpHbtx aucnepcuu
B c.lIytJ:ae, Korl{a MeTOl{ CXO,lJ.HTCH, H3 (4.55) C.lIe-
l{yeT paBeHcTBo
0'2 (a) = 'A 0 2 (r p ).
P
( 4.57)
BeJIHqHHa 'Vp = 'A 02 (r p ) COrJIaCHO (4.57) xapaKTepH3yeT
BK.lIa,lJ. p-ro cpaKTopa r P B l{HCnepCHIO 0 2 (a) HCXO,lJ.HOH MaTpHnbI
CB5I3H Me:m,lJ.y OO'beKTaMH. n09TOMY 'Vp MO}KeT TpaKTOBaTbC5I KaK
xapaKTepHCTHKa HHcpopMaTHBHOCTH cpaKTopa r P B 3a,lJ.aqe Ka-
qeCTBeHHoro cpaKTopHoro aHaJIH3a. B 9TOM CMbICJIe OHa nOJI-
HOCTblO aHa.lIOrHqHa <paKTOpHOH ,lJ.HCnepCHH KO.lIHqeCTBeHHO-
ro cpaKTopHoro aHa.lIH3a.
DpH 3TOM HeOOXO,lJ.HMoe KOJlHqeCTBO cpaKTopOB B CH.lIY (4.57)
Mo:meT onpe,lJ.eJI5ITbC5I H3 YC.lIOBH5I npe,lJ.CTaBJIeHH5I ,lJ.HCnepCHH
0 2 (a) CYMMOH cpaKTopHbIX AHcnepCHH 'Vp = 'A 02 (r p ) C 3apa-
Hee 3aAaHHOM TOqHOCTblO.
Booo e, <popMYJIa (4.57) n03BO.lI5IeT cpOPMY.lIHpOBaTb pa3-
JIHqHbIe KpHTepHH npeKpa eHH5I nponecca OTbICKaHH5I cpaKTo-
pOB. HanpHMep, eCJIH nepBbIe l{Ba cpaKTopa l{alOT 70% OT 0'2 (a),
a TpeTHH - Bcero 5 % , 9TO MOjf{eT YKa3bIBaTb Ha TO, qTO AaHHbIe
xapaKTepH3YIOTC5I ABYMH cy ecTBeHHbIMH cpaKTopaMH.
<PaKTopHaH AHcnepCH5I l' = A, 2 02 (r) cpaKTopa r COCTOHT
H3 ABYX COMHO)KHTe.lIeH. nepBbIH H3 HHX, A,2, B CH.lIY (4.53) xa a
u
paKTepH3yeT CB5I3b rea, T. e. KaqeCTBO annpoKcHManHH CB5I3eH
Me)Kl{Y OO'beKTal'vIH, H, C.lIeAOBaTe.lIbHO, MO:>KeT ObITb HHTepnpe-
THpoBaH KaK Bec cpaKTopa r OTHOCHTe.lIbHO MaTpHnbI a. HaH- t
oO.lIee HarJISlAHOe BbIpa)KeHHe A, B 9TOM CMbIC.lIe AaeTC5I <POPMY-
.lIOM (4.23), TaK qTO 'A 2 npe,ZJ.CTaBJI5IeT COOOM KBaApaT K09qxpH-
nHeHTa pa3peI.lIeHH5I OTHOI.lIeHHH p A.lI5I a. BTOpOM COMHO}KHTeJIb,
0'2 (r), 5IB.lI5IeTC5I BHYTpeHHeH xapaKTepHCTHKOH r, TaK KaK OH I
AaeT HeKOTOpylO OneHKY C.lIO)KHOCTH COOTBeTCTBYIO ero OTHO- j
meHH5I p. KO.lIHqeCTBeHHa51 AHcnepCH5I 0'2 (r) npOCTO BbIpa1Ka- II
f
eTC5I qepe3 KaqeCTBeHHYIO l{HcnepCHIO COOTBeTCTBYIO ero OT-
1iOI.lIeHH5I :
0 2 (r) = 4i) (p) (1 - cS (p)).
( 4.58)
202
HanpHMep, ,l{JISI HOMHHaJIbHOrO (MJIH paHrOBoro) <t>aKTO-
pa R ,Z(HcnepCH5I {) (p) = I - p xapaKTepM3yeT ero Heon-
s
pe.L{eJIeHHOCTb, a I - () (p) = I pI, Hao60pOT, onpe,l{eJIeHHOCTb.
TaKMM 06pa30M, cpaKTOpHa51 .L{McnepCH5I l' = A 2 0'2 (r) pac-
na.L{aeTC5I Ha COMHO)KHTeJIH: ').}, KOTOpbIH xapaKTepH3yeT pa3-
pelIIalOInYIO cnoco6HoCTb (4.23) p OTHOCHTeJIbHO a; () (p), KO-
TOpbIH CJIe,l{MT, QTo6bI pacnpe,l{eJIeHHe R ObIJIO He CJIHIIlKOM paB-
HOMepHbIM, H I - () (p) = I pi, KOTOpbIH CJIe,l{MT, lIT06bI pac-
npe,l{eJIeHHe R 6bIJIO He CJIHIIlKOM onpe,l{eJIeHHbIM.
OCTaeTC5I HepelIleHHbIM Bonpoc 06 MHTepnpeTanHH BToporo
lIHCJIOBOrO nOKa3aTeJI5I- J..t. B CB5I3M C 3TMM 3aMeTHM, liTO npH
3a,l{aHHbIX Iv H J..t onTHMH3anHSI G (Ar + J..t) CBO.L{MTCSI K MaK-
CHMH3anHH nOKa3aTeJI5I (4.20), KOTOpbIH npH (I, - 1)-KapKa-
ce r MOJKeT 6bITb 3aMeHeH nOKa3aTeJIeM
N
( a.. - 11. ) ,. .
7,J r 7,J.
i, /= 1
,[{eHCTBHTeJIbHO,
(aij-!!) 'ij = (ai j- ) --t- (aij - ) =
(i, j)€P (i, i)fEp
=.2 (aij-!!)- (aij-f1), (4.59)
(i,/}€p ;,j
TaK liTO MaKcHMH3anH5I (4.59) 3KBHBaJIeHTHa M8KCHMH3anHH
(aij- ) (p E ),
(i,j)€p
(4.60)
1JTO COBna.L{aeT C (4.20) npH nopore J..t. TIpH (ex, )-npe.L{CTaBJIe"
HHH HCKOMOH MaTpMnbI J..t B (4.59) B TOqHOCTH COOTBeTcTByeT
'Jv
J..t + '2 B (4.20), TaK KaK 06a BbIpa)KalOT O,l{HO H TO )Ke 3Halle-
HMe (ex + P)/2.
TaKMM 06pa30M, B CHJIY (4.60) 3HatIeHMe J..t xapaKTepH3yeT
nopor cy ecTBeHHocTH MH,l{HBH,l{yaJIbHbIX CB5I3eH Mem,l{y o6'b-
eKTaMM, Tor,l{a KeM{ Iv 51BJISleTCSI rJI06aJIbHOH xapaKTepHcTHKOH
annpoKcHManHoHHoH CMJIbI cpaKTopa. 3TO 03HallaeT, tITO npH
(I, -1)-npe.L{CTaBJIeHHM OTHOlIleHMH OnTHMaJIbHbIe xapaK-
TepHCTHKH ex H BXom,l{eHH5I HJIH OTCYTCTBHSI napbI B p, paB-
u v u u
HbIe COOTBeTCTBeHHO Cpe.L{HeH BHYTpeHHeH H Cpe.L{HeH BHelIlHeH
CB5I3H (4.21) M (4.22) paC enJISlIOTC5I Ha Iv H f.t (ex == A - f.A.,
=== f.t - Iv), xapaKTepH3YlOInHe COOTBeTCTBeHHO fJIOOaJIbHbIe
Ii JIQKaJIbHbIe acneKTbI cpaKTopHoro npe,l{cTaBJIeHH5I.
203
't'
,[{JI5I MaTpHUbI (4.12) nepBbIM cpaKTop R 1 = {I, 2, 3-4,
5-6} C BecaMH A, = 0,392 M !1 = 0,308 nocTpoeH paHee (c. 185).
MaTpHua OCTaTOllHbIX CB5I3eH a 1 = a - A,r -,..., HMeeT CJIe-
u
.nYIO HH BH,l{:
1
p
1
2
3
4
5
6
2
0,092
3
4
0,092
0,092
-01
,
5
-0, 108
-0, 108
-0, 108
-0, 108
6
-0 , 108
-0,308
0,092
0,092
0,1
0,092
0,292
B KalleCTBe HCXO,l{HOro ,l{JI5I 3TOM MaTpHUbI pa36MeHH5I eCTe-
CTBeHHO BbI6paTb R = {1-2-3-4, 5-6}, TaK KaK 1, 2, 3
06'be,l{HH5IIOTC5I B O,l{MH KJIaCC M3-3a nOJIO}KHTeJIbHbIX CB5I3eM
Me}K,l{Y co6oii, npHCOe,l{HHeHHe K HHM 4 YBeJIHlIHBaeT CYMMY
CB5I3eH Ha 0,092+0,092-0,1=0,084. Cpe,l{H5I5I BHYTpeHH5I5I
CB5I3b B 3TOM pa36HeHHH ex = 0,660/7 = 0,094, Cpe,l{H5I5I BHern-
H5I5I CB5I3b = - 0,664/8 = - 0,083, TaK QTO OnTHMaJIbHbIe
SHaQeHH5I A, H !1 CYTb A, = ex - p = 0,177, = -0,083.
Tenepb CJIe,l{yeT HCKaTb pa36HeHHe, onTHMH3HpYIO ee (4.20),
T. e. CYMMY BHYTpeHHHx CB5I3eM npH nopore (ex + P)/2 =
= 0,005, T. e. ,l{JI5I MaTpHUbI
1 2 3 4 5 6
1 0,087 0,087 0,087 -0,113 -0, 113
2 0,287 0,087 -0, 113 -0,313
3 -0,105 -0, 113 0,087
4 -0,113 0,087
5 0,095
6
HeTpy,nHO y6e,l{HTbC5I, lITO H ,l{JI5I 3TOH MaTpHUbI OnTMMaJIb- ,
HbIM OCTaeTC5I R = {1-2-3-4", 5-6}. TaKHM 06pa30M, BTO-
pOM <l>aKTop eCTb pa36MeHMe R 2 = {1-2-3-4, 5-6} C Be-
caMH A,2 = 0,177, ""'2 = -0,083.
CJIe,l{YIO a51 MaTpHua OCTaTOllHbIX CB5I3eM HMeeT BH,l{ a 2 =
= a 1 - A,2 r2 - ""'2:
1 2 3 4 5 6
,
1 -0,002 -0,002 -0 , 002 -0,025 -0, 025
2 0, 198 -0 , 002 -0,025 -0,225
3 -0, 194 -0,025 o , 1 75
4 -0 ,025 0, 1 75
5 0,006
6
204
nO-BH)IJIMOMY, ,!{aJIbHeHIIHfX BbP.IHCJIeHHM MO)KHO H30e:>KaTb,
eCJIH paCCMOTpeTb ,!{HH3MHKY H3MeHeHH5I ,!{JIHHbI (a p , a p ) =
=1 a P 1 2 nOJIyqeHHOM nOCJIe,!{OBaTeJIbHOCTH MaTpH[(: 1 a 1 2 = 2,52,
I a l l 2 = 0,298, I a 2 12 = 0,193. cI>3KTHqeCKH, nepBbIH <paKTop
CH5IJI 88% Bcero pa30poca M3TPH[(bI, BTopoH-e e 5%. nOH5IT-
HO, qTO CJIe,!{YIO He <t>3KTOPbI 0Y,!{YT HeCTH BCe MeHblliHe ,!{OJIH
u
CYMMapHbIX CB5I3eH.
4.2. 3A AqA OnHCAHliSi Ii OIJ.EHKH CB5I3H
ME)I(AY KAQECTBEHHbIMH nPH3HAKAMH
4.2.1. JI uneunaR Mo8eAb
0003HaQHM M3TPH[(bI CB5I3eH Me:>K,!{Y OO'beKTaMH,
cooTBeTcTBYIOI..I.(He BXO,!{HbIM npH3H3K3M Xl,. . . , X P , Qepe3
r l , ...r p , 3 MaTpH[(bJ CB5I3eH, COOTBeTCTBYIOI..I.(He BbIXO,!{HbIM npH-
3H3KaM yl, ..., yq, - qepe3 Sl, ..., sq. BYAeM CQHT3Tb, QTO
MaTpHUbI r k , 51 npe,!{CTaBJIeHbI B HYJIb-e,!{HHHQHOM BH,!{e, a Be-
JIHQHHbI An, l-tz xap3KTepH3YIOT OTHOCHTeJIbHbIe Beca paCCMaT-
pHBaeMbIx npH3H3KOB Xk, yl (k = 1, ..., p; 1 = 1, ..., q).
Tor,!{a CYMM3pHaH MaTpH[(3 CB5I3eH Me:>K,!{y OO'beKTaMH no
p
BXO,!{HbIM npH3HaK3M eCTb a = Anrk, a CYMM3PH351 MaTpH[(3
k=1
q
CB5I3eM no BbIXO,!{HbIM npH3H3K3M - b = l-t ZS'. P33HHua
1=1
Me:>K,!{Y 3THMH MaTpH[(3MH
I a - b 1 2 = I A.I! r k - Itz SI I -
k= 1 1= 1
- ( A.I! rt- ItZS i ) 2
1,/=1 k=1 1=1
OTpa)K3eT pa3JIHQHH B OnHC3HM5IX MHOjJ{eCTBa OO'beKTOB, ,!{a-
B3eMbIX ,!{BYMSI CHCTeMaMH npH3HaKOB. npH 3TOM K3)K,!{aSl H3
M3TPH[( a, b npe,!{CT3BJl5IeT COOOH onHcaHHe B33HMOCBSl3eH Me>K-
,!{Y OO'beKT3MH no COOTBeTCTBYIOI..I.(eH CHCTeMe npH3HaKOB. 1JeM
MeHbIIle OTJIHQ31OTCH 3TH M3TPH[(bI, TeM OJIH)Ke onHCaHH5I B3a-
HMOCBSl3eH, TeM OOJIee COOTBeTCTBYIOT ,!{Pyr ,!{Pyry CHCTeMbI
npH3H3KOB Xl, ..., xP H yl, ..., yq.
HCXO,!{H H3 3Toro, CJIe,!{yeT onpe,!{eJI5ITb Beca npH3HaKoB.
KapKaCHbIe MaTpH[(bI r k , S' nOpO>K,!{aIOTcSI 6HHapHbIMH OTHO-
meHHSlMH, cooTBeTcTBYIO HMH npH3H3K3M, 3 Beca Ak, J..tl Ha-
3H3QalOTCH H3 YCJIOBH5I MHHHMH3a[(HH BeJIHQHHbI P3CXO)K,!{eHHH
(4.61 )
205
(4.61) Me)K.L{Y MaTpM[(aMH a M b. IlpM 3TOM .L{OJIlKHO BbInOJIH5ITb-
CSI YCJIOBMe
lal=)bl=l,
( 4 . 62)
KOTopoe <pMKcMpyeT O.L{MHaKOBbIH MacllITa6 M3MepeHM5I CBSl3eH
MeJK.L{Y OO'beKTaMM M, B qaCTHOCTH, npe.L{OXpaHSleT OT TpHBMaJIb-
Horo perneHMSI An == f.t l == O.
3a.L{aqa MMHMMH3a[(HM (4.61) npM YCJIOBMSIX (4.62) npe,L{-
CTaBJISleT COOOR He tlTO MHoe, KaK 3a.L{aqy KaHOHHqeCKOrO aHa-
JIM3a (CM. pa3.L{eJI n.B).
,aeHCTBMTeJIbHO, MaTpM[(bI r k , Sl SlBJISlIOTCSI BeKTopaMM
N X N-MepHoro npOCTpaHCTBa. 0003HaqMM (N 2 X p)-MaTpM-
[(Y co CTOJIO[(aMM r k (k = I, ..., p) qepe3 X, a (N2 X q)-MaTpM-
uY co CTOJIOQaMM Sl (l = 1, ..., q) - qepe3 Y. Tor Aa cor JIac-
HO (4.61)
I a- b 1 2 = I XA- Yt-t1 2 , (4.63)
r.L{e A == (AI' ..., A p ), f.t == (f.tl, ..., f.tq), TaK qTO X", M Yf.t-
BeKTopHbIe npe.L{CTaBJIeHH5I l'vlaTpMU a M b COOTBeTCTBeHHO. Y C-
JlOBMSI (4.62) B 3TMX TepMMHax MMelOT BMA
ATXT XA= T yr Y = 1,
( 4.64)
QTO AOKa3bIBaeT 3KBMBaJIeHTHOCTb 3aAaqM MMHHMH3a[(MM (4.61)
npM YCJIOBM5IX (4.62) 3aAaqe (Il.31), (Il.32) npMJIO)KeHM5I 0
KaHOHMqeCKHX KOppeJI5I[(M5Ix.
TaKHM 00pa30M, BeKTopbI K03c}xPMQMeHTOB A, M J..t SlBJI5IIOTCH
coocTBeHHbIMM BeKTopaMM MaTpH[( (n.36) pa3MepHocTM p X p,
q X q:
PXy = (XT X)-1 XT Y (YT Y)-1 YT X;
pYX = (YT Y)-1 YT X (XT X)-1 XT Y ,
( 4.65)
( 4. 66)
COOTBeTCTBYIO HMM MX MaKCMMaJIbHoMY coocTBeHHoMY qHCJIY,
cOBna.nalO eMY C MaKCHMaJIbHbIM coocTBeHHbIM qMCJIOM MaT-
PM[( PXPy M PyPx. 3aMeTMM, qTO npM 3TOM Bce napbI OO'beKTOB
aBTOMaTMqeCKM nOJIyqalOT Beca B CHCTeMax npM3HaKOB Xk M yl,
paBHbIe KOMnOHeHTaM COOCTBeHHbIX BeKTopOB 3TMX nOCJIeA-
HMX MaTpMQ: 3TH Beca 00pa3YIOT MaTpM[(hI CBSl3eH a M b, HaMOO-
JIee COOTBeTCTBYIO He Apyr APyry B CMbICJIe 3aAaqM (4.61),
(4.62).
reOMeTpMqeCKM perneHMe 3a.L{aqM MMHMMH3a[(MM (4.61) npM
YCJIOBMM (4.62) 03HaqaeT HaXO)Kj{eHHe TaKMX BeCOB An, npM
KOTOpbIX paccToSlHHe OT BeKTopa XIv.L{O ero npOeK[(HH Ha L (Y)
ObIJIO MMHMMaJIbHO (CM. TaK:>Ke n. 3.2.5).
206
Beca Ah Ii llz 51BJIHIOTC5I XapaKTepHCTHKaMM OTHOCHTeJIbHOH
Ba)KHOCTM npM3HaKOB Xk, y' J(JI5I OllHCaHli51 OJ(HOH CMCTeMhI
npM3HaKOB C nOMO blO J(pyroH Ii B 3TOM CMbICJIe BeCbMa YJ(06-
HbI ,!{JI5I MHTepnpeTa[(liM. _
MaTpM[(bI (4.65), (4.66) OCHOBaHbI Ha CKaJI5IpHbIX npoM3Be-
,!{eHM5IX 6YJIeBCKMX MaTpH[( r k , 51 KaK 3JIe1\1eHTOB (N X N)-
MepHoro npOCTpaHCTBa. B TOM CJIYlIae, Kor,!{a Xk, yl - CTPYK-
TypHbIe npH3HaKM Tlina HOMMHaJIbHbIX, paHrOBbIX M np., 3TH
CKaJlSlpHbIe npOli3Be,!{eHH5I MorYT 6bITb BbIpa>KeHbI lIepe3 lIac-
u
TOTbI COBMeCTHbIX pacnpe,!{eJIeHHM nap npli3HaKOB, TaK liTO ,!{JI5I
HX BbIlIMCJIeHli51 BOBce He 06513aTeJIbHO liCnOJIb30BaTb caMIi
MaTpM[(bI r k , 51. DOJIee Toro, npM (1, -1)-npe,!{cTaBJIeHMM
3TMX MaTpli[( HX CKaJl5lpHbIe KBa,!{paTbI (HOpMbI) paBHbI KOHC-
1 1
TaHTe N2, TaK liTO MaTpli[(bI N2 XTX, N2 yry MorYT paCCMaT-
pMBaTbCSI KaK KOppeJI5I[(MOHHbIe: Bce 3JIeMeHTbI rJIaBHOH ,!{Ma-
rOHaJIM paBHbI e,!{MHH[(e, OCTaJIbHbIe - CKaJI5IpHbIe npOH3Be-
,!{eHM5I HOpMMpOBaHHbIX BeKTopOB. TaKoe npOM3Be,!{eHHe JIM-
HeRHO onpe,!{eJI5IeTC5I paCCTOSlHMeM d M BbIpa>KaeT ,!{OJIIO nap
06'beKTOB, KOTopbIe CBSl3aHbI MJIM He CB5I3aHbI ,!{Pyr C ,!{pyroM B
060MX OTHOIlleHM5IX o,!{HoBpeMeHHO. ECJIM 06a npM3HaKa HOMM-
HaJIbHbIe, ,!{OJI5I paBHa Bep05lTHOCTM Toro, liTO ,!{Ba 06'beKTa
MMelOT O,!{HHaKOBbIe MJIM pa3HbIe 3HalleHM5I no 060HM npM3Ha-
KaM O,!{HOBpeMeHHO (CM. 1.3).
3aMeTMM TaK}Ke, liTO ,!{JI5I CJIYlIaH KOJIMlIeCTBeHHbIX npH-
3HaKOB Xk, yl, Kor,!{a 3JIeMeHTbI MaTpH[( r k , S' paBHbI COOTBeT-
CTBeHHO rlj = Xk (i) - Xk (j), S j == y' (i) - yl (j), CKaJISlp-
HbIe npoM3Be,!{eHMSI 3TMX MaTpM[( npOnOp[(MOHaJIbHbI K03cp-
<pM[(HeHTaM KOBapMa[(MM cooTBeTcTBYIO MX npM3HaKoB [118,
44]. n03TOMY B ,!{aHHOM cJIYlIae MaTpM[(bI (4.65), (4.66) C TOll-
HOCTblO ,!{O nOCT05lHHOrO MHO)KMTeJI5I COBna,!{aIOT C COOTBeT-
CTBYIO MMM MaTpM[(aMH (n.36) ,!{JI5I MCXO,!{HbIX npM3HaKOB, TaK
liTO 3a,!{alla (4.61), (4.62) npeBpa aeTC5I B 3a,!{allY 0 KaHOHMlIec-
KOH KOppeJISI[(MM npM3HaKoB Xk M yl.
TaKMM o6pa30M, 3a,!{alla MMHMMM3a[(MM (4.61) npM orpaHH-
lIeHM5IX (4.62) MO}KeT paccMaTpMBaTbcSI KaK 0606 eHMe 3a,!{a-
u u
lIH 0 KaHOHMlIeCKOM KOppeJI5I[(liM Ha CJIYlIaM npM3HaKoB npo-
H3BOJIbHOH npMpo,!{bI (B TOM QMCJIe npOM3BOJIbHbIX MaTpM[( CBSI-
3M r k , SI). PaCCMOTpeHMe npM3HaKOBOR MH<popMaUHM B TepMM-
Hax MaTpMU CB5I3eH Me)K,!{Y 06'beKTaMM M MCCJIe,!{OBaHMe COOT-
BeTCTBH5I Me>K,!{Y 3TMMH MaTpli[(aMM HenpMBbIlIHbI B aHaJIM3e
J(aHHhIX. Q,!{HaKO, Ha HaIll B3r JI5I,!{, aHaJlM3 MaTpM[( a, b CB5I3eH
Me>K,!{y 06'beKTaMM B pa3HbIX CHCTeMax npM3HaKOB H BbI5IBJIe-
HMe MX pa3JIMlIHH He MeHbIlle COOTBeTCTBYIOT TOMY MHTYM-
2 0 7
....-
THBHOMY CMbICJIY, KOTOpbIM BKJIa.L{bIBaeTC5I B npOOJIeMY HaHJIyq-
lllerO OnHCaHH5I, lJeM B KJIaCCHlJeCKHX TepMHHaX O.L{HOBpeMeHHO-
fa (CTPYKTYPHoro) perpeCCHOHHora aHaJIH38 [62], aHaJIOrOM
KOToporo SlBJISleTCH 3a.L{alJa (4.61), (4.62).
4.2.2. H CnOJlb308aflue JlUfleUflOU .M,08e AU
nOKa3aTeJIb (4.61) MO>KeT OblTb HCnOJIb30BaH TaK-
)l{e J{JISI pemeHH5I .L{PyrHx 3a,l{alJ onHCaHH5I H KOHCTpYHpoBaHH5I.
PaccMoTpHM HeKoTopbIe KOHKpeTHbIe 3a,l{alJH TaKoro po.L{a H Me-
TO.L{bI HX perneHHSI.
1. P erpeccHoHHbIH aHaJIH3 MaTpHLI-
C B 51 3 H. B cJIYlJae Kor,l{a.Ko9qxpHUHeIiTbI J..tz npH CHCTeMe MaT-
pHU Sl cpHKcHpoBaHbI 3apaHee, npOOJIeMa CBO,l{HTC5I K MH.HHMH-
3aUHH norpellIHOCTH (4.61) annpOKCHMaUHH 3a.L{aHHOH MaTpHUbI
b HCKOMOH MaTpHueH a = Ah,rk C nOMOrnblO nO,l{oopa BeCOBblX
k
K09cpcpHUHeHToB Ah" Ha KOTopble He HaJIaraeTCSI HHKaKI:fX an-
pHOpH X orpaHHlIeHHH.
0003HallHB, KaK H paHee, lIepe3 X MaTpHUY pa3MepHOCTH
N2 X n co CTOJIOuaMH r k , HeTpY,l{HO nOJIYlIHTb CJIe,l{YlOrnee pe-
weHHe nOCTaBJIeHHOH npOOJIeMW:
a-==P x b=P x Y!1, A,= (XTX)-lX T b=(XTX)-lX r YJ1, (4.67)
TaK liTO MaTpHua CB5I3eH a onpe,l{eJISleTC5I Qepe3 b CTaH,l{apTHbIM
oopa30M - C nOMOrnblO npoeKTHpOBaHHSI Ha npOCTpaHCTBO
u u
MaTpHU CB5I3eH, nOJIYlIaeMbIX BceB03MO>KHbIMH JIHHeHHbIMH
KOMOHHaUH5IMH MaTpHU r k .
3aMeTHM, liTO XTb - 9TO p-MepHbIH BeKTOp, k-51 KOMnOHeH-
Ta KOToporo Bblpa>KaeT CYMMY BHYTpeHHHX CBSl3eH b ij no k-MY
npH3HaKY, TaK liTO A OCHOBaHa Ha 3TOM BeKTope C yqeTOM MaT-
PHUbI KOBapHaUHH XTX.
06paTHM BHHMaHHe, liTO ,l{aHHOe perneHHe HMeeT CMbICJI H
llPH MaTpHue CB5I3eH b, 3a,l{aHHOH npOH3BOJIbHbIM oopa30M,
B TOM lIHCJIe H 3MnHpHQeCKH, a He 005l3aTeJlbHO npH3HaKaMH
yl.
DOJIee Toro, 3TO peweHHe HaHOOJIee YMecTHo HMeHHO ,l{JISI I
CJIYlIa5l, KOr,l{a npH3HaKH Xk HCnOJIb3YIOTCSI ,l{JISI annpoKcHMa-
UHH e,l{HHCTBeHHoH BbIXO,l{HOH MaTpHUbI b, B TOM IIHCJIe H Kor,l{a
3Ta MaTpHua 3a,l{aeT OTHorneHHe, nopO)K,l{aeMoe e,l{HHCTBeHHbIM
BbIXO,l{HbIM npH3HaKOM y. ,aelfCTBHTeJIbHO, npH npOH3BOJIbHOM f
Bece J..t MaTpHUbI b norpewHOCTb annpoKcHMaUHH (4.61) HMeeT
BH,l{ I Yh,rk-!.t b J2, TaK liTO 3a,l{alla MHHHMH3aUHH 3TOH BeJIH-
k
208
,
qHHbI no "111. (npH OTCYTCTBHH OrpaHHlJeHHH Ha Yk) 9KBHBaJIeHT-
Ha 3a.n.atIe MHHHMH3au.HH l,f Ak rk - b 1 2 , f.n.e Ak = I'k/IA-, pe.
weHHe KOTOpOH onpeAeJI5IeTC5I no <popMYJIe (4.67).
npH MaTpHne b, onpeAeJI5IeMOH MaTpHnaMH Sl, TaK qTO
b = YJ..t, BO BBeAeHHbIX 0603HaqeHHSlX <POPMYJIbI (4.67), AalOT
A=(XTX)-lXrYf1=A , (4.68)
rAe A=(X r X)-lX r y.
AHanOrHqHO onpeAeJISleTC5I npoeKnHH 3a aHHoH MaTpHU
a Ha nOAnpocTpaHcTBo L (Y) MaTpH BHAa J.tlS l = YJ..t:
. I
b = Pya, J.t = (YT Y)-l yr a = BA, (4.69)
rAe a = XA, B - MaTpHna (yrY)-1 yrx.
cl>OPMYJIbI (4.68), (4.69) n03BOJI5IIOT npOaHaJIH3HpOBaTb CJIe-
,nYIO HH nponecc HTepaTHBHoro nocTpoeHHH BeCOB A11., J.tz Ha
OCHOBe 3aAaHHbIX r k , S', 0600 alO Hii MeTOA n. 4.1.4. Ha nep-
I 1
BOM mare nOJIaraeM J.t1 = ... = J-l' = -, TaK qTO b = - Sl.
q q ,
nOcJIe SToro BbIqHCJI5IeM 3HaqeHHH A11. H MaTpHUY a no cpopMYJIaM
(4.67). 9TH 3HaqeHH5I A11. HCnOJIb3yeM AJIH BbIqHCJIeHH5I HOBbIX
u
J..tz, YTOqH5IIO HX HCxoAHhIe C ueJIblO AOnOJIHHTeJIbHOH MHHH-
MH3anHH (4.61), no <popMYJIaM (4.69). nYCTb J.t , A1- Beca
npH3HaKoB, nOJIyqeHHbIe Ha mare t. TorAa B COOTBeTCTBHH C
(4.68), (4.69) Ha (t + 1)-M mare BbIIIHCJI5IIOTCH Beca:
",t+l=Af1t, t+l =B",t+l. (4.70)
CXOAHMOCTb npouecca rapaHTHpyeTC5I TeM, tlTO Ha Ka1K,l{OM
mare BeJIHqHHa norpemHocTH (4.68) YMeHbmaeTC5I, TaK KaK
CPOPMYJIbI (4.70) ,lJ.alOT Beca, MHHHMH3HpYlOInHe (4.61).
Ha nepBbIH B3rJI5IA Ka:>KeTC5I, qTO STOT MeTOA npHBO,lJ.HT K
cy ecTBeHHO HHbIM pe3Y.lIbTaTaM, qeM MeTO,lJ. KaHOHHqeCKOrO
u
aHaJIH3a: 3AeCb HeT HHKaKHX OrpaHHqeHHH Ha HOPMY MaTpHn
a H b; CJIeAOBaTeJIbHO, OOJIbme CB060AbI B BhI60pe BeCOBbIX KO-
3<p<pHnHeHTOB. GAHaKO STO He TaK. B CHJIY (4.70) At+1 =
= ABAt, J.tt+1 = BAJ..tt, TaK qTO At H J.tt OJIH3KH K C06CT-
BeHHbIM BeKTopaM MaTpHU AB H BA, OTBeqalO HM HX MaKCH-
MaJIbHbIM c06cTBeHHbIM qHCJIaM (eCJIH 3TH c06CTBeHHbIe tlHCJIa
OTAeJIeHbI OT APyrHX). Ho AB H BA CYTb He liTO HHoe, KaK
MaTpHnbI (4.65), (4.66), TaK qTO AaHHbIH MeTOA aeT pe3YJIbTa-
TbY, 6JIH3KHe K pe3YJIbTaTaM KaHOHHQeCKOrO aHaJIH3a (npH 3TOM
CJIeAyeT HMeTb B BHAY, qTO At, J.tt -+ 0 (t -+ 00), TaK KaK MaK-
CHMaJIbHOe coocTBeHHoe qHCJIO a < 1).
209
2. <I> a K TOp H hI if a H a JI 11 3 MaT p II II. C B 51 3 H.
DOKa3aTeJIb (4.61) MOJKeT HCnOJIb30BaTbC5I H Kor,l{a npH3HaKH
yl (MaTpH[(bI Sl) He 3a,l{aHbI 3apaHee, 3 HX HYjJ{HO nOCTpOHTb
B 3aj(aHHbIX KJIaCCaX MaTpH[( l.
3TO MomHO OCYIneCTBHTb C nOMOInblO HTepaTHBHOH npon e -
,l{ypbI, BKJIlOqalOIneH ,l{Be OCHOBHbIe onepaUHH: a) onpe,neJIeHHe
no 3a,l{aHHbIM Sl, l-t l OnTHM3JIbHbIX 3HaqeHHH BeCOBbIX K09qxpH-
[(HeHTOB Ak C nOMOInhlO CPOPMYJI (4.67); 6) nocTpoeHHe 51,
III
l-tl no 3a,l{aHHOH MaTpH[(e CB5I3eH a = Akrk. HaqHHaTb npo-
k=l
IJ.e.n:ypy eCTeCTBeHHO C MaTplfIJ.bI Cpe.n:HIfX CBH3ea a = ..! rk.
P k
3. K 0 H C T P Y H P 0 B a H H e C 0 ,l{ HOB P e M e H-
H bI M 0 n H C a H HeM. B cJIyqae Kor,l{a MHHHMaJIbHOe 3Ha-
qeHHe (4.61) npH 3a,l{aHHbIX r k , 51 OTHOCHTeJIbHO BeJIHKO, MO}K-
HO annpOKCHMHpOBaTb HeyqTeHHbIe CB5I3H MeLK,l{Y 06'beKTaMH
B KJIaCCe (g cpaKTopOB TOH }Ke npHpO,l{bI, qTO H r k . A HMeHHO
npH 3a,l{aHHbIX MaTpH[(aX b H a KOHCTpYHpyeTC5I <paKTop r C
BeCOM A ,l{JI5I MaTpH[(bI OCTaTOqHbIX CB5I3eH a - b, nOCJIe qero
YTOqH5IIOTC5I 3HaqeHH5I K09qxpH[(HeHTOB !.tz (npH 3a,l{aHHbIX
Akrk), H npH 3a,l{aHHbIX l-t l - 3HaqeHH5I K09qxpH[(MeHTOB
AI' ..., Ap, Iv (no CPOPMY JIaM THna (4.70)) ,l{JI5I I Ivk rk +
+ Ar - l-tlSI12. npH OKOHqaHHH 9TOH npO[(e,l{ypbI ,nJI5I
YTOqHeHHbIx BeJIHqHH AI' ..., Ap H !-tI,..., l-tq BbIQHCJI5IIOTC5I
p q I
YTOqHeHHble MaTpH[(bI a = Akrk H b = l-t lSI H onRTb
k= 1 ,= 1
on pe,neJI 51 eTCH <paKTOp r C BeCOM A, MMHHMH3MPYlOInHH
I b - a - Ar 1 2 . 3TOT npo[(eCC nOBTOp5leTC5I ,l{0 Tex nop, TIOKa
YMeHbI.IIaeTC5I BeJIMqHHa I Akrk + Ar - !.tlSI12. CXO,l{HMOCTb
npouecca rapaHTHpyeTC5I TeM, qTO Ha Ka:>K,nOM ero Illare npo-
HCXO,l{HT YMeHblIleHHe HOpMbI MaTpH[(bI HeyqreHHbIX CB5I3eH.
MOLKHO npO,l{OJI}KHTb nOCTpoeHMe HOBbIX cpaKTopOB, eCJlH r
He,nOCTaTOqHO YMeHblIlaeT norpelliHOCTb annpOKCHMaU HH .
TaKoe KOHCTpYHpOBaHHe C BKJIlOqeHHeM B onHcaHMe MO>KHO
npOBO,l{HTb He TOJILKO B cxeMe perpeCCHOHHoro aHaJIH3a, HO H
HenOCpe,l{CTBeHHO B cxeMe KaHOHHqeCKOrO aHaJIH3a, nOCKOJIb-
KY <POPMYJIhI (4.70), KaK paHee ,l{OKa3aHO, npHBo,n5lT K perneHHIO
no MeTO,l{Y KaHOHHqeCKOrO aHaJlH3a.
3aMeTHM TaKLKe, qTO HOBbIe cpaKTopbI MOLKHO CTpOMTb ,l{JI5I
nOnOJIHeHH5I He TOJIbKO CHCTeMbI BXO,AHbIX npH3HaKOB Xk, HO H
CHCTeMbI BbIXO,l{HbIX npH3HaKOB yl. Pa3HH[(a B TOM, qTO OO'b5lC-
H5IeMbIH (T. e. BbIXO,l{HOH) <t>aKTOp ,l{OJILKeH BbIqHCJI5ITbC5I He no
MaTpH[(e a - b, a no npOTHBOnOJlOLKHOH eH MaTpH[(e b - a.
210
YMeCTHO TaKme OTMeTHTb, liTO BCe OnMCaHHbIe MO,l{eJIM M
MeTO,l{bI MOrYT ObITb npHMeHeHbI He TOJIbKO B KJIaCCe OYJIeBCKHX
(ex, O)-MaTpHQ (KaKOBbIMH 5IBJI5IIOTC5I Akrk, f.tlS1), HO H B OOJIee
IllHpOKOM KJIaCCe (a, )-MaTpHQ BH,l{a Ar + f.t, r):(e r - OYJIeB-
CKa51 (I,O)-MaTpHQa. Y CJIOmHeHHe nOJIYlIaeTC5I Hecy ecTBeH-
Hoe, TaK KaK Ar + f.t = Ar + f.tU, r,l{e u - MaTpHQa YHHBep-
caJIbHOrO OTHOIlleHH5I, COCT05lIII.a51 H3 Bcex e,l{HHH[(. IlpH 3TOM
B cpopMYJIe (4.61) Bce aMHTHBHbIe ,l{ooaBKH f.tu CYMMHPYIOTC5I,
TaK liTO OHa npHoopeTaeT BH,l{
Akrk_ f1lsl_vI2 ( 4.71 )
k I '
r):Le v - HeKOTOpa51 nOCT05lHHa5l. B 3a,l{alle KaHOHHlIeCKoro
aHaJIH3a B CHJIY orpaHHlIeHHH (4.62) lIJIeHbI f.tu CYMMHPYIOTC5I
no OT,l{eJIbHOCTH ,l{JI5I BXO,l{HbIX H BbIXO,l{HbIX npH3HaKoB.
TaKHM oopa30M, (a, )-MaTpHQbI aBTOMaTHlIeCKH YlIHTbI-
BaIOTC5I, eCJIH B lIHCJIO npH3HaKoB yl (a B 3a):(alle KaHOHHlIeCKO-
ro aHaJIH3a H Xk) ,l{OOaBHTb YHHBepCaJIbHbIH npH3HaK U, npH-
HHMalOlI1.HH O,l{HO H TO :>Ke 3HalleHHe Ha Bcex OO'beKTax.
4.2.3. H JlAIOCrpaTUBflbtu npuMep
npOHJIJIIOCTpHPyeM onHcaHHbIe MeTO,l{bI Ha npH-
Mepe ):(aHHbIX TaOJI. 3.1. nYCTb nepBbIe ,l{Ba npH3HaKa - BXO,l{-
HbIe, (1, O)-MaTpH[(bI cooTBeTcTBYIOIII.HX OTHOIlleHHH HMelOT BH,l{:
1 1 0 0 0
1 000
000
1 1
1
r 1 ===
- 0 0 0 l' 1
-- I 1 0 0
r 2 = - 1 0 0
0 0
]
3,l{eCb, KaK H paHee, ,l{HarOHaJIbHbIe 3JIeMeHTbI He paCCMaT-
pMBaIOTC5I. B CHJIY CHMMeTpHlIHOCTH 3):(eCb H ,l{aJIee npHBO,l{51TC5I
TOJIbKO HaMHarOHaJIbHWe 3JIeMeHTW. PaCCMaTpHBaeMbIe KaK
CTOJI6II.bI (15 X 2)-MaTpHII.bI X (r.n:e 15 = N ( -1) npH
N = 6 - KOJIHlIeCTBO 3JIeMeHTOB, onpe):(eJI5IIOIII.HX CB5I3H Me:>K-
):(Y OO'beKTaMH), 3TH MaTpH[(bI ,l{alOT CJIe):(ylOIII.YIO MaTpHQY:
XT X= (: :).
3TH :>Ke 3HalleHH5I nOJIYlIaIOTC5I K8K CYMMbI BeJIHlIHH
N s (N s - 1) (paccMaTpHBaeMbIx BMeCTO N: B CHJIY ocooeH...
HQCTeH npe,l{cTaBJIeHHSI HH<popMa[(MH ,l{JI5I ,l{aHHbIX pa30HeHHH).
211
HeTPY,l{HO BH,l{eTb, qTO
(XTX)-l = ( 3 -! ) .
16 _ 1
t,,1
B KaqeCTBe BbIXO,l{HbIX paCCMaTpHBaIOTC5I OCTaJIbHbIe TpH
npH3HaKa. MaTpHlI.bI COOTBeTCTBYIO HX pa30HeHHH HMelOT BH,l{:
- 0 0 1 0 0 - 1 1 1 0 0
- 0 0 0 0 - 1 1 0 0
51 = - 0 1 1 S2 = - 1 0 0
- 0 0 - 0 0
1 1
0 0 0 0 0
0 0 1 0
8 3 = 1 0 1
0 1
0
PaCCMaTpHBa51 HX KaK CTOJIOnbI (15 X 3)-MaTpHnbI Y, no-
JIYQHM:
( 4 2 1 )
YT Y. = 2 7 1 ; (yr Y)-l =
1 1 4
( 0,30 -0,08 -0'06 ) ( 1
= -0,08 0,17 -0,02 ; XTY = 1
-0,06 -0,02 0,27
: ).
B COOTBeTCTBHH C 3THM HaH,l{eM MaTpHnbI:
A = (XT X)-l XT Y = ( 0,12 0,5 0,12 ) ;
0,12 0,5 0,12
( -0 08 -0 08 )
B = (yr Y)-l yr X = 0:58 0:58,
0,13 0,13
npOH3Be,l{eHH5I KOTOpbIX onpe,l{eJI5IIOT BeKTopbI BeCOBbIX K03cp-
cpH HeHToB Ak H !1z KaK coocTBeHHble BeKTopbI, cooTBeTcTBYIO-
212
...
mile MaKCHMaJIbHOMY COOCTBeHHOMY IIMCJIY. 9TH npOH3BeAeHH5I
paBHbI:
( 1 1 ) ( -0,02 -0,08 -0,02 )
AB=O,296 1 1 ; BA = 0,14 0,56 0,14.
0,03 0,12 0,03
OlIeBHAHO, coocTBeHHbIH BeKTop MaTpHlI.bI AB HMeeT O.L{H-
HaKOBbIe KOMnOHeHTbI Al = A 2 , KOTopbIe BbIoHpaIOTC5I H3 yc-
JIOBHH (4.62) paBeHcTBa e.L{HHHlI.e HOpMbI MaTpHlI.bI CB5I3eH
a = A l r l + A 2 r 2 . 3TO YCJIOBHe 03HalIaeT, lITO Af (r\ rl) +
+ 2A 1 A 2 (r 1 , r 2 ) + A (r 2 , r 2 ) = 1. nOCKOJIhKY CKaJI5Ip-
HbIe npoH3BeAeHH5I CYTb 3JIeMeHTbI MaTpHlI.bI XTX, nOJIY-
lIaeM 6Ar + 4AIA2 + 6A = 1, npHlJeM Al = A 2 . n03TOMY
16AI = 1, TaK lITO Al == A 2 = 1/4.
CoocTBeHHbIH BeKTop J..t MaTpH[(bI BA C TOIIHOCTblO AO no-
CTOHHHoro MHO:>KHTeJI5I ex paBeH: J..t = BA, lITO B AaHHOM CJIYlIae
AaeT J..t = (-0,04ex; 0,29ex; 0,06ex). PaBeHcTBo (4.62) J..tTYTYJ.t ==
= 1 o3HalIaeT, lITO npOH3Be.L{eHH5I J..tiJ..tJ H8 (i, j)-e 3JIeMeHTbI
M8TPHlI.bI yr Y B CYMMe AalOT eAHHHlI.Y. OTCIOAa a = 1, 3 H
J..t = (-0,052; 0,376; 0,078).
3aMeTHM, QTO Bce BbIIIHCJIeHH5I AeJIaJIHCb He C «OOJIbIllHMH»
MaTpHlI.aMH CBH3eH Me*AY OO'beKTaMH, a C KOMnaKTHbIMH MaT-
pHlI.aMH «KoppeJI5IUHH» Me*AY npH3HaKaMH. B TO *e BpeM5I Bce
nOCTpOeHH5I OCMbICJIeHbI H B TepMHHax CB5I3eH Me*AY OO'beKTa-
MH.
Ou.eHHM norpeIllHOCTb
I a-b 1 2 = (a-b, a-b) = (a, a) +(b, b)-
-2(a, b)=2-2ATXTYJ.t.
.aJI'H 3Toro BbIIIHCJIHM ATXTYJ..t = 0,765, TaK qTO I a-b 1 2 =
= 2 - 1,53 = 0,47. 3TO 03HalJaeT, lITO HeCOOTBeTCTBHe CTPYK-
TYP CB5I3eH Me:>K.L{Y OO'beKTaMH no npH3HaKaM Xl, x 2 H r, x 4 , x 5
COCTaBJI5IeT 23,5% OT MaKCHMaJIbHOH BeJIHIIIIHbI 2.
MaTpHlI.bI a, b H b - a HMelOT CJIeAYIO HH BHA:
a=
0,25 0,25 0 0,25 0,25
0,5 0,25 0 0
0,25 0 0
0,25 0,25
0,5
213
b=
0,38 0,38 0,32 0 0
0,38 0,38 0,08 0
0,45 - 0,05 0,03
o 0,08
0,32
0,13 0,13 0,32 -0,25 -0,25
-012 0,13 0,08 0
,
b-a= 0,2 -0,05 0,03
-0,25 -0 17
,
-0,18
llpOBeJ\eM perpeccHIO nOCJIe,l{HMX Tpex npH3HaKoB no nep-
BbIM ,l{BYM.
B COOTBeTCTBHH C <popMYJIaMH (4.68), (4.69) npH perpeccHH
A = AJ..t, J..t = BA, r,l{e A H B - MaTpHQbI, BbIqHCJIeHHble Ha
c. 212. nYCTb Ha nepBoM mare ""1 =-= J..t2 = J..t 3 == 1/3. Tor,l{a B
CMJIY (4.68)
A = A = + C : ) G ) = c;: ) ·
T. e. O,l{HHaKOBbIM BecaM BbIXO,l{HbIX npH3HaKoB cooTBeTcTBYIOT
O,l{HHaKOBbIe Beca BXO,l{HbIX. BeJIHqHHa norpelliHocTH B 3TOM
cJIyqae paBHa:
la-bI 2 = ATXTXA,+ TYTYfJ;-2A,TXTYJ..t= 1 +
7 4
+"3- 2 ="3'
tITO COCTaBJHleT 57% OT HCXOJJ.HOH HOpMhI I b 1 2 = ; .
BhItIHCJIeHHe HOBhIX JLI no cpopMYJIe (4.69) npH A = C ' )
J\aeT
= BA = (-0,04; 0,29; 0,06).
MbI npHlIlJIH K TOM me CTpYKType BecoB, qTO H MeTO,l{OM Ka-
HOHHqeCKOrO aHaJIH3a, HO K MeHbllIHM HX 3HaqeHH5IM. !laJIb-
214
-k..
HeHlllee npO,nOJIJKeHlie liTepaluIii (4.70) He H3MeHHT CTPYKTY-
pbI, HO oY.LleM YMeHbIllaTb 3HalleHH5I BeCOB ,l{0 HYJI5I.
PaCCMOTpHM Tenepb B03MomHOCTb KOHCTpYHpOBaHli51 HOBO-
fO BXO,!{Horo pa30lieHH5I OO'beKTOB C [(eJIblO YMeHbIlleHH5I no-
rpelliHOCTH annpOKCHMa[(HH CB5I3eH Mem,!{y OO'beKTaMH no Bbl-
XO,!{HbIM npH3HaKaM CB5I351MIi Me*.LlY OO'beKTaMH no npli3HaKaM
BXO.LlHbIM. ,aJI5I 3Toro npOaHaJIH3lipyeM CTPYKTYPY MaTpH[(bI
b - a. OlleBH,!{HO, MaKCHMaJIbHa51 CYMMa BHYTpeHHHX CB5I3eH
,!{JI5I 3TOR MaTpH[(bl nOJIyqaeTC5I npH pa30HeHHH R =
== {1-2-3-4, 5-6}, TaK KaK npH 3TOM Tep5lIOTC5I TOJIb-
KO MaJIbIe nOJIOJKHTeJIbHhIe CB5I3H (2,5) Ii (3,6), a BKJIlOqaeTC5I
B CYMMY TOJIbKO O,!{Ha OTpH[(aTeJIbHa51 CB5I3b (2,3). BC5IKOe ,nPY-
roe pa30HeHHe npHBo,!{HT K OOJIblliliM nOTep5lM B <PYHK[(HO-
H aJI e.
B COOTBeTCTBHH C aJIrOpHTMaMH (c. 183) BbIqliCJIHM BeJIH-
qHHY cpe,!{HeH BHYTpeHHeH CB5I3H B R: (0,13 + 0,13 +
+ 0,32 - 0,12 + 0,13 + 0,20)/6 = 0,79/6 = 0,13, nOJIOBHHa
3Toro qHCJla ,!{aeT 3HaqeHHe nopora CYIlteCTBeHHOCTH HH,!{H-
BH,!{yaJIbHbIX CB5I3efi Me:>K,!{Y OO'beKTaMH (a = 0,06), KOTopoe
CJIe,!{yeT BblqeCTb H3 3JIeMeHTOB aHaJlH3HpyeMoH MaTpH[(bl. npH
3TOM cYMMapHa51 CB5I3b BToporo OO'beKTa C OO'beKTaMH 1, 3 H
4 CTaHOBHTC5I OTpH[(aTeJIbHOH, H npHxo,!{HTC5I HCKJIlOqHTb
BTOpOH OO'beKT H3 KJIaCCa {I, 2, 3, 4} , OO'be,!{HHHB ero C OODeK-
TOM 5 H3-3a HaJIHqH5I MaJIOH nOJIO:>KHTeJIbHOH CB5I3H Me:>K,!{Y
HHMH. Cpe,!{H5I5I BHYTpeHH5I5I CB5I3b B HOBOM pa30HeHHH R =
= {1-3-4, 2-5, 6} paBHa (0,13 + 0,32 + 0,20 +
+ 0,08)/4 = 0,18, TaK qTO npH HOBOM nopore cy ecTBeHHocTH
a = 0,09 CB5I3b (2,5) CTaHOBHTC5I OTpH[(aTeJIbHOH H OnTHMaJIb"
Hoe B CMblCJIe (4.20) pa30HeHHe npHHHMaeT BH,!{ R = {1-3-4,
2, 5, 6} co cpe,!{HeH BHYTpeHHeH CB5I3blO A, = (0,13 + 0,32 +
+ 0,20)/3 = 0,22. 3TO *e pa30HeHHe OnTHMaJIbHO H npH
HOBOM nopore a = 0,11, TaK qTO OHO npHHHMaeTC5I 3a peIlle-
HHe 3a,!{aQH annpOKCHMa[(HH MaTpH[(bl b-a OCTaTOQHbIX CB5I-
3eH. MaTpH[(a CB5I3eH no 3TOMY pa30HeHHIO HMeeT BH,!{
r 3 =
-0 1 1 00
-0000
- 1 0 0
- 0 0
-0
.
(4.72)
Tenepb MaTpH[(a X HMeeT TPH CTOJIO[(a: r 1 , r 2 , r 3 , TaK qTO
CJIe,!{yeT nepeClIHTaTb OCHOBaHHbIe Ha HeH MaTpH[(bI:
215
( 6 2 1 ) ( 1 4 1 )
XT X = 2 6 1 ; XT Y == 1 4 1 ;
1 1 3 1 3 1
( 0,193 -0,057 -0,045'
(X T X)-l= -0,057 0,193 -0'045 ) .
-0,045 -0,045 0,364
3TO ):(aeT HOBbIe Bblpa)l{eHH5I ,l{JI5I A, B H AB:
( 0,09 0,41 0,09 ) ( 0,033 0,033 0,054 )
A== 0,09 0,41 0,09 ; B = 0,565 0,565 0,391 ;
0,27 0,73 0,27 0,109 0,109 0,152
( 0,244 0,244 0,179 )
AD = 0,244 0,244 0,179 ·
0,451 0,451 0,341
HOpMHpOBaHHbIe no (4.62) coocTBeHHbIe BeKTopbI, COOT-
BeTcTBYIO He MaKCHMaJIbHoMY coocTBeHHoMY tlHCJIY, paBHbI:
Iv = (0,17; 0,17; 0,32), !.t = (0,031; 0,354; 0,096). HOBbIe MaT-
pHnbI CB5I3eH a = Xiv H b = YJ..t paBHbI:
0,17 0,49 0,32 0,17 0,17
0,34 0,17 0 0
a= 0,49 0 0 .
,
0,17 0,17
0,34 I
0,35 0,35 0,38 0 0
0,35 0,35 ° 1 1 0
,
b= 0,45 0,03 0,13 .
0 0,10
0,38 ,
Pa3HocTb 3THX MaTpHn
0,18 -0 14 0,06 -0,17 -0,17
,
0,01 0,18 0,10 0
b-a= -004 0,03 0,13
,
-0 17 -0 07
, ,
0,04
216
:J.
------
HMeeT ,Z(JIHHY I b - al 2 = 0,21, T.e. 10,5% OT MaKCHMaJIbHOrO
3HaqeHHSI. CJIe,l{OBaTeJIbHO, CKOHCTpYHpoBaHHoe pa30HeHHe
{1-3-4, 5,6, 2} n03BOJIHJIO ,nonOJIHHTeJIbHO YtIeCTb 23,5-
- 10,5 = 13% CB5I3eH Me)K.L{Y OO'beKTaMH, nopO)K.L{aeMbIX
npH3HaKaMH x 3 , x 4 , x5.
4.2.4. Koal/Jl/Jul1ueflTbl KoppeAfll1uu Marpul1
oJ
U OTflOUleHUU
PaCCMOTpHM npocTeHwylO MO.L{eJIb onHCaHH5I 3a-
.L{aHHOH MaTpHnbI CB5I3eH b C nOMO blO MaTpHnbl CB5I3eH a.
KpHTepHH (4.61) ,l{JI5I SToro CJIYQa51 npHHHMaeT BH,l{
A = I b - Aa 1 2 . ( 4. 73 )
aA
npHpaBHHBa51 HYJIIO npOH3BO,l{HYIO a", ' nOJIyqHM OnTHMaJIb-
Hoe 3HaQeHHe
A== Ca, b) (4.74)
(a, a) ·
llpH STOM 3HaQeHHH A, nOrpeI.I1HOCTb annpoKcHManHH paBHa:
L\=(b-Aa, b-Aa)=(b, b)-2A(a, b)+
+ ').,2 (a, a) = (b, b) _ \a, b); .
a, a
TaK llTO
L\ = 1 b 1 2 ( 1 - (a, b)2 ) ==1 b 1 2 (1 _ p2),
(a, a) (b, b)
(4.75)
r.u-e
(a, b)
p=
y(a, a) (b, b)
(4.76)
BeJIHtIHHa p nOKa3bIBaeT, KaKaH .L{OJI5I .L{JIHHbI MaTpHnhI b
YQHTbIBaeTc5I MaTpHneH M. qeM JIyqrue KaqeCTBO onHCaHH5I,
T. e. qeM MeHbI.I1e L\, TeM OJIH:>Ke p K ::f:: 1. T aKHM oopa30M, p
xapaKTepH3yeT KatIeCTBO onHCaHH5I b tIepe3 a H B STOM CMbICJIe
51BJI5IeTC5I MepoH CB5I3H Me)K,l{Y a H b, a TaK)Ke MepOH: CB5I3H
npH3HaKoB, 3a.L{aBaeMbIX STUMH l\faTpHlI.aMH.
113 (4.75) CJI, e,l{yeT QTO
I b 1 2 - A f1
p2= = 1 - - (4.77)
I b 1 2 I b 1 2 '
217
T. e. BeJIHqHHa p2 Xap3KTepH3yeT OTHOCHTeJIbHYIO BeJIHqHHY
B3PH3UHH b, OOYCJIOBJIeHHYIO M3TpHII.eH a, qTO COOTBeTcTByeT
MaTepHaJIY c. 135.
BeJIHQHHY p oY,l{eM Ha3bIBaTb o606U1renHblM KoaifJifJUl{uen-
mOM KoppellRl{UU MampUl{ a, b, a p2 - KoaifJifJUl{uenmOM oemep-
Munal{UU 8 MOaellu (4.73).
33MeTHM, qTO BeJIHqHHa 1 (4.74) T3K)Ke MO)KeT paccM3TpH-
B3TbC5I K3K Mep3. CB5I3H, x3p3KTepH3YIO 351 CTeneHb onHC3-
HH5I b qepe3 a. HeTpY,l{Ho BH,l{eTb, qTO
p = VA (a, b) Iv (b, a) ,
o606 eHH H KO HII.HeHT KOppeJIHII.HH p3BeH Cpe,l{HeH reo-
MeTpHqeCKOH 3cHMMeTpHqHbIX x3p3KTepHcTHK 1 CTeneHeH onH-
caHHH b qepe3 a H a qepe3 b.
,aJIH OTHOllIeHHH p H pi, 3a,l{aB3eMbIX (1,0)-M3Tp H u 3MH , Be-
JIHqHHa K03qxpHII.WeHTa (4.76), OqeBH,l{HO, paBH3:
p = I p n p' I (4.78)
-V I p I I pi I
ECJIH me ,l{JIH 3a,l{aHH5I OTHOIlleHHH HCnOJIb3YIOTC5I (1, -1)-
MaTpHII.bI, TO
_N2-2d(p,p')_ 1 _ 2 ( , )
p - - u p, p ,
N2
(4. 79)
TaK qTO P JIHHeHHbIM OOp330M npe,l{CTaBJI5IeTC5I qepe3 P3CCT05l-
HHe Me)K,l{Y OTHOIlleHH5IMH.
C yqeTOM B03MO)KHOrO C,l{BHr3 IllK3JI H3MepeHHH CB5I3eH
onHcaHHe MaTpHII.bI CB5I3eH b qepe3 a CJIe,l{yeT HCK3Tb B BH,l{e
!va + J..tU, r,l{e U - N X N MaTpHII.3 H3 e,l{HHHLJHbIX 3JIeMeHTOB.
B 3TOM CJIyqae norpeIllHOCTb onHCaHH5I H3Mep5leTCH BeJIHqHHOH
6. = I b- Aa-!112,
( 4.80)
nOJIHOCTblO 3H3J10rHqHOH (4.57) (c 33MeHOH a H3 b, a HCKOMoro
r - Ha 33,l{3HHOe a), npHqeM OnTHMaJIbHbIe 3H3qeHH5I 1, A BhI-
pa*3IOTC5I (4.74), (4.75) (cpaBHH C (4.58», C TeM H3MeHeHHeM,
qTO BMeCTO a H b paccMaTpHBalOTC5I MaTpHII.bI a - M (a) H
b - M (b) (3,l{eCb M - Cpe,l{Hee 3H3qeHHe 3JIeMeHTOB COOTBeT-
cTBYIO eH M3TpHII.bI). BJIH5IHHe J..t np05lBJI5IeTCH B C,l{BHre 3JIe-
MeHTOB ooeHX M3TPHLI- K HX cpe,l{HHM 3HaLJeHH5IM. TaKHM oopa-
30M, B CHJIY (4.57), OnTHMaJIbHbIe A, H HMelOT BH,l{:
A = (a, 6) .
(a, a) ,
218
--
= , b I (1 - 1'2) == N2 a 2 (b) ( 1 - 1'2),
(a, 6) cov (a, b)
1'= -
V(a, a)(6, b) a(a)a(b)
H a=a - M(a), b==b-M(b), (J2(a)==
= I aj2/N2- AH cnepcH5I MaTpH[(bI a.
BeJIHqHHa (4.81) Ha3bIBaeTC5I KoacjJifJUl{uenmo.M KOppeAJl-
l{UU MaTpM[( a H b. TaK me KaK H (4.76), OHa M3Mep5leT KaqeCT-
BO onHCaHH5I b qepe3 a, HO B OTJIHqHe OT (4.76) He 3aBHCHT OT
Bbloopa HaqaJIa OTCqeTa B H3MepeHMM CB5I3eH Me)K):(Y OO'beKTa-
MH.
BOJIee TOqHO, Mepa (4.76), KaK HeTpY):(Ho BH):(eTb, HHBapHaHT-
Ha OTHOCHTeJIbHO H3MeHeHH5I MacIllTaooB H3MepeHH5I cBSl3eH
aij H b ij , TaK qTO p (a, b) = p (/va, /vb) (ufleapuaflmflocmb B
IllKaJIe oTHoIlleHHii). Mepa )Ke (4.79) HHBapHaHTHa He TOJIbKO
OTHOCHTeJIbHO H3MeHeHH5I MaCIllTaOOB, HO H TOqeK OTCqeTa B
H3MepeHHH CB5I3eH, T. e. HHBapHaHTHa B IllKaJIe HHTepBaJIOB
H3MepeHH5I CB5I3eH.
CBOHCTBa K03cpcpH[(HeHTa KOppeJI5I[(HH (4.81) oocY)K):(aIOTC5I
B [52, 79]. BnepBbIe K03<p<pH[(HeHT KOppeJI5I[(HH MaTpH[( KaK
KOscpcpH[(HeHT KOppeJI5I[(HH 3a):(aBaeMbIX HMH npH3HaKoB pac-
CMaTpHJ3aJIC5I r. ,[{sHHeJICOM [118]. Ho B ero paooTe, KaK H B
AaJIbHeHllIHx [162-164], paccMaTpHBaJIHCb TOJIbKO paHro-
Bble H KOJIHqeCTBeHHbIe npH3HaKH.
3aMeTMM TaK)Ke, qTO B CHJIY (4.81)
A, = T a (b) .
a (a)
CMbICJI STOB BeJIHqHHbl npH (1 ,O)-MaTpH[(e a KaK xapaKTepHC-
THKM ee pa3peIllalOIneH cnocooHoCTH OTHOCHTeJIbHO b oocYJK-
AaJIC5I B n. 4.1.2.
IlpoaHaJIH3HPyeM KOscpcpH[(HeHT KOppeJI5I[(HH (4.81) AJI5I
(a, )-MaTpH[(. B COOTBeTCTBHH C BbIIlleCKa3aHHbIM ero 3Haqe-
HHe HHBapHaHTHO OTHOCHTeJIbHO H3MeHeHH5I ex H , nOSTOMY
AOCTaTOqHO OrpaHHqHTbC5I aHaJIH30M KapKacoB, (I,O)-MaTpH[(,
npeACTaBJI5IIOInHX cooTBeTcTBYIO He OTHOllIeHH5I. IlycTb p H
pi _ ABa OTHOIlleHH5I C COOTBeTCTBYIO HMH MaTpH[(aMH r H r'.
OqeBHAHO,
r):(e
(4.81 )
( 4.82)
(r, r ') = I p n p' I, M (r) = I p 1/ N2 , (r , r I ) =
= (r, r ') - N2 M (r) M (r ') = I p n p I I-I p lip' 1/ N2 .
219
A .....
llpH r = r' nOJIyqaeM
(r, r)=lpl-IPI2/ N 2=IPI(1- 21 ) =c5(P)(1-c5(p»N2.
T8KHM OOpa30M,
't (p, p') = I pnp' I-I p II p' I/N? . (4.83)
V I p I (1 - I p 1/ N2) I p' I (1 - J p' 1/ N2)
c:l>OPMYJI8 (4.83) cpa3Y n03BOJISleT onpe,l{eJIHTb, Kor,na Bbl-
nOJIHeHO COOTHOIlleHHe 't (p, p') = 0, T. e. MaTpHna A.r' + J..t
u
HHKaK He CB5I3aHa C MaTpHneH r, He ,l].8eT HHKaKoro BKJIa a B ee
onHcaHHe. A HMeHHO K03c}xPHnHeHT KOppeJISlnHH OTHOIlleHHH
p, p' npHHHM8eT HYJIeBOe 3HalleHHe TOr,l{8 H TOJIbKO Tor,l].8,
KorAa
I p n p' I = I p I r p' 1/ N2 ,
(4.84)
T. e. BepOSlTHOCTb Toro, qTO npOH3BOJIbH8S1 napa (i, j) E m X m
npHH8,l{JIe:>KHT p n p', paBHa npOH3Be,l{eHHIO BepoSlTHocTeii ee
npHH8,l{JIe:>KHOCTH p H p' no OT,l{eJIbHOCTH. HHbIMH CJIOB8MH,
q:opMYJIa (4.84) 03H8l1aeT CT8THcTHlIecKYlO He38BHCHMOCTb co-
ObITHH, COCTOSl HX B TOM, liTO (i,j) E P HJIH (i, j) E p'.
B TOM CJIYlI8e, KOr,l{8 OTHOIlleHHe p xapaKTepH3yeT pa30He..
HHe R = {Rt, ..., Rm} CO CTPYKTYPOH x, 8 p' - pa36HeHHe
R' = {Rl, ..., R:n,} CO CTPYKTYPOH ', <popMYJIa (4.83) MOJKeT
6bITb nepenHC8Ha CJIe,l{YIO HM 06pa30M:
Pss' ( Ptt' ) - ( Ps. Pt. ) ( P.s,P.t' )
s, s' tex(s) t'ex'( ') (5, t)ex (5',t')&,,'
T= ,
{( Ps. Pt. ) ( 1- Ps. Pt. ) ( P.s' P.t' ) X
(5, t)e" (s,t)ex (5'. t')ex'
X ( l- P.s' P.t' )
(5', t')ex' (4.85)
r,l{e HH,l{eKC S npooeraeT 3H8QeHHSI 1,..., m, s' - 3H8l1eHH5I
1, ..., m' , a Pss', Ps. ,Pt. - 3JIeMeHTbI Ta6J.{HnbI COnp5lJKeHHOCTH
pa36HeHHH R H R'.
<1>OPMYJI8 (4.85) n03BOJI5IeT ,l{OK83aTb CJIe,l{YIO ee YTBepJK,l{e-
HHe [107,52], nYCTb p830HeHHSI R H R' CT8THCTHlIeCKH He3a-
BHCHMhI, T. e. Pss' = Ps. p. s' .L{JI5I JII06bIX S, s'. Tor.L{8
T ( (R, )(), (R', x')) = 0 npH JIIOObIX CTPYKTYP8X x H X'.
220
- "
4.2.5. Mepbl CBR3U ynopflaOtleHHbtx npU3flaKOB
C KaLK,IlbIM OTHOllleHHeM JIHHeHHOro KBa3HnOp5l -
Ka p A x A, xapaKTepH3yeMblM ynop5l,l{OqeHHbIl\f pa30HeHH-
eM R = (R l , ..., Rm) MHO)KeCTBa _OO'beKTOB, MO)KHO CB5I3aTb
CJIe YIO He MaTpHnbI CBSl3eH a J a J b J c:
a. . = { I, (i, j) E p,
l} 0, (i, j)$p,
a..= { 1,(i,j)EP } =2a.._l
l,) _ 1 ( . . ) rr I,)'
, t, J CJ=p
I 1, (i, j) E p - p - ,
bi}' = 0, (i.. j E p p-=p,
-1, (t, J)Ep -P,
Cij=S-t npH iERs, iERt. (4.89)
. nepBbIe ,l{Be MaTpHUbI - CTaH,l{apTHbIe (1, 0)- H (1, -1)-
KapKacbI paccMaTpHBaeMoro OTHOlIleHH5I. TpeTb51 xapaKTepH-
u .
3yeT paH}KHpOBaHHe TpeM5I QHCJIaMH: e,l{HHH[(eH, eCJIH t CTpO-
. · . U
fO JIY1JlIle J, HYJIeM, eCJIH t H J paBHoneHHbI, H MHHYC e,l{HHHneH,
eCJIH i cTporo xY}Ke j. TaKOH cnocoo npe,l{CTaBJIeHH5I paH>KH-
pOBaHH5I lIIHpOKO pacnpocTpaHeH B JIHTepaType [44, 41]. B
u
QeTBepToH MaTpHne paH}KHpOBaHHe xapaKTepH3yeTC5I paHra-
MH (HoMepaMH S KJIaCCOB Rs, CO,l{ep)Ka HX OO'beKTbI), H B Ka-
QeCTBe CB5I3H Me}K,l{Y i H j BbloHpaeTC5I B COOTBeTCTBHH C (4'.89)
pa3HocTb paHroB 3THX OO'beKToB.
K03<p<pHnHeHThI OOOOrneHHoH KOppeJI5IUHH MaTpHU, xapaK-
TepH3YIO HX paH)KHpOBaHH5I, ,l{alOT pa3JIHQHbIe MepbI CB5I3H
paHroBbIx npH3HaKoB. BY,l{eM 0003HaQaTb 3TH MepbI OYKBOH 't
H HH,l{eKCaMH, YKa3bIBalO HMH Ha BH,l{ HCnOJIb3yeMbIx MaTpHn.
nYCTb paHrOBbIe npH3HaKH 3a,l{aIOTC5I ynop5l,l{OQeHHbIMH pa3-
OHeHH5IMH R = (R l , ..., R m) H R' = (Rl, ..., R:n,) MHO}Ke-
CTBa OO'beKTOB. Tor,l{a
( 4. 86)
( 4. 87)
(4.88)
fpnp'l= NstNs,t"
s s' t t'
(4.90)
r.u-e
Nst=IRsnR/ I; Ipl=N2_ d (p)=+ ( N:. +N2);
Ip'I=N2_ d (p')=+ ( N t+N2} (4.91)
221
a d (p), KaK OOhItIHO 0003HatIaeT KaqeCTBeHHYIO .L{HcnepClilO co-
OTBeTCTBYIO ero npli3HaKa. Tor a B CliJIY (4.78)
T -2
a-
N st N s ' t'
s s' t t'
( 4. 92)
V(N2+ N:.)(N2+ N t)
<l>OPMY JIa (4.79) .L{aeT
Ta = 1 -26 (R, R'),
(4.93)
r.L{e cS (R, R') - OTHOCHTeJIbHOe paCCT05lHHe MeLK.L{Y paHJKHpO-
BaHH5IMIi R H R' (CM. CPOPMYJIY (1.50)).
CKaJI5IpHOe npOH3Be.L{eHHe MaTpH[( BH.L{a (4.88) paBHo:
2 (N st N s ' t' -N st ' N s ' t) =
5>5' t>t'
= d (R-, R') -d (R, R').
(4.94)
nOCJIe.L{Hee paBeHcTBo CJIe.L{yeT H3 CPOPMYJIbI (1.50) .L{JI5I pac-
CT05lHH5I MejJ{.L{Y paH)KHpOBaHH5IMH.
HeTpY.L{HO BH.L{eTb TaK)Ke, liTO ,Z(JIHHa MaTpH[(bI BH,l{a (4.88)
paBHa .L{HcnepcHH cooTBeTcTBYIO ero HOMHHaJIbHOrO npH3Ha-
Ka. CJIe.L{OBaTeJIbHO,
(N st N 5 , t' -N st ' N s ' t)
IT' _ d (R-, R') - d (R, R') ._ 2 5>S' t>t'
b- - ·
V d (R) d (R ') 11 (N2 - N:. )( N2- N t)
( 4.95)
K03cpcpH[(HeHT Tb XOpOI.IlO H3BeCTeH KaK Koatjxput{uenm pan-
208011 Koppe.//Jlt{uU Kenaa./llla. B OTJIHtIHe OT 'fa, HenOCpe.L{CT-
BeHHO 3aBHC5I erO OT paCCT05lHHSI d (R, R'), K03cpcpH[(HeHT
Tb onpe.L{eJI5IeTC5I TOJIbKO pa3HOCThlO d (R-, R') - d (R, R') =
= d (R- * R', R' * R-) - d (R * R', R' * R) H, CJIe.L{Q-
u u
BaTeJIbHO, He 3aBHCliT OT HOMHHaJIbHOH COCTaBJI5IIOIII.eH pac-
CT05lHH5I, KOTOpa51 nopO)K.L{aeTC5I He ynop5l OtIeHHOCTblO KJIaC-
COB, a pa3HOCTblO B HX HanOJIHeHHH.
B CHJIY (4.95) Tb eCTb OTHOllIeHHe pa3HocTH BepoSlTHocTeH
Toro, liTO npOH3BOJIbHa51 nepeCTaHOBKa OO'beKTOB COrJIaCOBaHa
Ii He COrJIaCOBaHa B CTporOM CMbICJIe no R H R', K cpe,l{Heii reo- I
222
MeTpHlIeCKOH OillMOKe npOnOp HOHaJIbHOrO npOrH03a 3HalleHHH
npH3HaKOB.
TaK KaK Cpe,l{Hee 3HalleHHe MaTpHlI.bI BH,l{a (4.88) paBHo HY-
JIIO, KosqxpHlI.HeHT KeH,l{aJIJIa SlBJISleTC5I KOscp<pHlI.HeHTOM KOp-
peJI5I HH MaTpHlI. BH,l{a (4.88) H KaK TaKOBOH MeH5IeTC5I oT-l
,l{O + 1, npHlIeM 't'b = 1 Tor,l{a H TOJIbKO Tor,na, KOr,l{a R H R'
COBna,l{aIOT, H 't'b = -1 Tor,l{a H TOJIbKO TOr,l{a, KOr.L{a R Ii R'
B3aHMHo-oopaTHbI: R-= R'. npH CTaTHCTHlIeCKH He3aBHCHMbIX
-'
R H R' KOSqxpHlI.HeHT Tb paBeH HYJIIO.
B paMKax cTaTHcTHlIeCKOH: MO,l{eJIH H3BeCTHO, liTO B npe.L{-
nOJIO:>KeHHH CTaTHCTHlIeCKOH He3aBHCHMOCTH npH3HaKOB pacn-
pe,l{eJIeHHe 't'b KaK BWOOpOqHOrO KOScpcpHlI.HeHTa KOppeJI5IlI.HH
aCHMnTOTHlIeCKH HOpMaJIbHO, tITO MO:>KeT HCnOJIb30BaTbCSI .L{JI5I
npOBepKH rHnOTe3bI 0 He3aBHCHMOCTH paHrOBbIX npH3Ha-
KOB [44].
PaccMoTpHM Tenepb KOSqxpHlI.HeHTbI, KOTopbIe nOJIYtIaIOTC5I,
Kor,l{a npH3HaKH 3a aIOTC5I MaTpHuaMH pa3HbIx BH,l{OB. TaK
KaK rlj = 2r/j - 1, TO (r,?) = ri jrh = 2 (r, r') - rij.
. l,j i,f
CJIe,l{oBaTeJIbHO, B CliJIY (4.92)
2 NstN s ' t' -: N s . N s '
s>s' t>t' s>s'
't' aa =
(4.96)
N V N2_+ N:.
AHaJIOrHtIHO nOJIYtIaeM ,l{JISI MaTpH BH.L{a (4.86) H (4.88)
(a, b)= (NstNs,t,-Nst,Ns,t)==
s s' t>t'
= (NstNs,t,-Nst,Ns,t)+
s>s' l>t'
+ (NstN s ' t' -N st ' N s ' t),
s=s' t>t'
npHtIeM nepBoe CJIaraeMOe paBHO nOJIOBHHe CKaJI5IpHOrO npOH3-
Be,l{eHHSI (4.94) MaTpHlI. BH,l{a (4.88), a BTopoe - HYJIIO, TaK
KaK NstNs't, = N st ' Ns,t npH S = s'. CJIe,l{oBaTeJIbHO, COOT-
BeTcTBYIO HH K03cpcpHlI.lieHT paBeH:
(NstN s ' t,-Nst,Ns,t)
s>s' t>t'
Tab = V (N2 -+ N:.) (N2_}':N t) ·
( 4. 97)
223
nO.L{06HbIM 1Ke OOpa30M .L{JIH a BH.L{a (4.87) H b BH.L{a (4.88)
nOJIyqHM (B CHJIY TOfO, tlTO ( a, b) = 2 (a, b»:
(N st N s ' I,-N st ' Ns,t)
't- - 2 s>s' t>t'
ab-
NVN9_ N.t2
(4.98)
9TH nOKa3aTeJIH OTJIHqalOTCH OT K03qxpHUHeHTa KeH,ltaJIJIa
CBoeH aCHMMeTpHqHOCTblO (B 3H8MeHaTeJIe).
ACHMMeTpHqHbIH K03qxpHUHeHT (4.74) OTJIHqaeTC5I OT KO-
3qxpHUHeHTa 06oo eHHofi KOppeJI5IUHH TOJIbKO 3HaMeHaTeJIeM.
,lXJIH Ma:rPHU BH.L{a (4.88) OH paBeH no aHaJIOfHH C 't b :
d(R-, R')-d(R. R')
A.b = d ('R') , (4.99)
npHqeM, KaK H B oOIneM cJIyqae, K03cpcpHUHeHT KeH.L{aJIJIa eCTb
Cpe.L{Hee reOMeTpHqeCKOe 3THX aCHMMeTpHqHbIX xapaKTepH-
CTHK CTeneHH npe.L{CTaBJIeHH5I R qepe3 R' H R' qepe3 R:
1:b (R, R') = VAb (R, R') Ab (R', R).
( 4.1 00)
B 3aKJIIOQeHHe paCCMOTpHM tlaCTHbIH cJIyqafi, KOf.L{a pac- I
CMaTpHBaeMble npH3HaKH .L{HXOTOMHqeCKHe. npH 3TOM TaOJIHua
conp5DKeHHOCTH npH3HaKOB R H R' HMeeT cJIe.L{YIO HH BH.L{:
c
b
d
b+d
a+b
e+d
1
a
a+c
TOf.L{8 B(R,R')=2(a+d)(b+c), TaK QTO
B(R, R')=(a+d)(b+c)+2bc, B(R-, R')=
=(a +d)(b+c)+2ad.
CJI e.L{ OBa TeJI bHO,
(b,b')=cS(R-,R')-B(R,R')=2(ad-bc). (4.101) i
N2 l
j
3r8 BeJIHQHHa paBHa HYJIIO TOf.L{a H TOJIbKO TOf.L{a, KOf.L{a
a c R R '
r;= d ' T. e. npH3HaKH H He38BHCHMbI..
B cpopMYJIe (4.101) ocooeHHo tleTKO np05lBJlHeTC5I TO CBOH-
CTBO K03qxpHUHeHTOB, OCHOBaHHhIX Ha 3a.L{aHHH npH3HaKOB MaT-
224
pHnaMH BH.L{a (4.88), liTO HX lIHCJIHTeJIH He 3aBHC5IT OT HOMH-
HaJIbHOH COCTaBJI5IIOIneH paCCT05lHH5I, T. e. <PHKCHPYIOT He Bce
COrJIaCOBaHHbIe napbI, a TOJIbKO Te, KOTopbIe COrJIaCOBaHbI (HJIH
He cOrJIaCOBaHbI) B CTporOM CMhICJIe.
113 .L{HXOTOMH3HpOBaHHbIX K03$PHlI.HeHTOB HaH60JIee HHTe-
peCHbI K03<t$HnHeHTbI (4.95) H (4.99):
ad - be
't'b =
y(a+b) (c+d) (a+ c) (b+d)
ad-be a b
A b = -
(a+e)(b+d) a+c b+d ·
( 4. 1 02)
( 4. 1 03)
nOKa3aTeJIb (4.103) lIlHpOKO H3BeCTeH KaK nOKa3aTeJIb npo-
QeHTHOH pa3HOCTH .L{OneH nepBoro 3HaqeHH5I R npH nepBOM H
BTOpOM 3HalleHH5IX R'.
K03<P<PHlI.HeHT (4.102) paccMaTpHBaJICSI B n. 2.2.1; HHTe-
peCHO OTMeTHTb, qTO .L{JI5I .L{HXOTOMHlIeCKHX npH3HaKOR 'tX =
= X 2 / N. 3TH nOKa3aTeJIH He 3aBHCSlT OT nOp5l.L{Ka 3HalleHHH
npH3HaKOB. TaKOB LKe H K03<P<PHnHeHT (4.93):
Ta = a 2 + b 2 + e 2 + d 2 - 2 (ad - be).
( 4.1 04)
4.2.6. CBfl3b eTpozux ynopR8otleHUU
AHaJIH3 cTporHx ynopSl.L{OlleHHH ynpo aeTCSI TeM,
liTO B 3TOM CJIYlIae HOpMbI MaTpHQ 3a.L{aHHOrO BH.L{a (4.86)-
(4.88) nOCT05lHHbI H He 3aBHC5IT OT KOHKpeTHoro ynopSl.L{olle-
-' -'
HM5I. ECJIH R H R' - CTporHe ynopSl.L{OlleHH5I, TO R = R' =
= RR' - TpHBHaJIbHOe pa36MeHHe, TaK qTO <S (R, R') =
=: 1 - <S (R-, R') == <S (R* R', R' * R). CJIe.L{OBaTeJIbHO,
1-2<S(R, R')=<S(R-, R')-<S(R, R')=
=Tb(R, R')=Ab(R, R')=Ab(R', R),
T. e. nOKa3aTeJIH (4.93), (4.95), (4.99) COBna,naIOT If onpe,L{eJISI-
IOTC5I BeJIHQlfHOH: 6 (R, R') YABoeHHoH Bep05lTHOCTH pa3JI HlIH5I
nopSl,nKa ABYX 06'beKTOB no R H R' (noCKOJIbKY ynopSl,l{OlleHH5I
cTporHe, Hepa3JIHQHMbIX 06'beKTOB HeT). 3TOT CMbICJI BeJIHlIH-
HbI <S (R, R') 03HallaeT TaKme, liTO d (R, R') paBHO y.L{BOeH-
HOMY KOJIHlIeCTBY nepeCTaHOBOK nap 06'beKTOB, KOTopbIe Hym-
HO npOH3BeCTH, lIT06hI nonYlIMTb H3 O,AHOrO ynop5l.L{OlleHH5I
,L{p yroe.
8 3aK. 1743
225
MaTpHua BH,l{a (4.89) ,l{JIH 3a,l{aHHH CTpOfHX ynopH,D-otJeHHA
onpeAenHeTCH qoPMYJIOH
CiJ = sJ - Si,
(4.105)
r,ne Si - paHf i-fO OO'beKTa B ynopH,l{otJeHHH R. PaHfH qaCTQ
u
npHMeHHIOTcH B aHaJIH3e CTpOfHX ynopSl,l{otJeHHH.
B CJIYllae KOf,l{a OO'beKTbI ynop5l,l{otJeHbI B COOTBeTCTBHII co
3HatJeHH5IMH KOJIHtJeCTBeHHOfO npH3HaKa x, YMeCTHO paccMaT-
pHBaTb TaK}Ke KOJIHlIeCTBeHHYIO MaTpHUY (XiJ) , r,ll.e
xii = Xi - Xj, (4.106)
npHtJeM Xi =1= Xj (i =1= j).
npHMeM ,l{JI5I y,l{OoCTBa, tJTO OO'beKTbI HYMepOB8HbI B COOT-
BeTCTBHH C R, T. e. R = (1,2, ..., N) - ynopH,l{otJeHHe oOneK-
TOB B nOp5l,l{Ke HX HOMepOB.
PaccMoTpHM npe)K,l{e BcefO CKaJIHpHOe npOH3Be,ll.eHHe MaTpHU
(Xii) H (Yii) BH,l{a (4.106):
(Xi-Xj)(Yi-YJ) = 2N2 ( XiYi-X Y ) =
I, I I
= 2N 2 cov (x, y), (4.107)
f,l{e COy (x, y) - oObIlIHa51 KOBapHaUH5I N-MepHbIx BeKTopoB
X = (Xi) II Y = (Yi). 113 (4.107) CJIe,l{yeT, tJTO
(Xi -X J)" = 2N2 0'2 (X),
I, I
f,l{e 0 2 (X) = 1.. xf - (X) 2 - OObltJHa51 ,l{HCnepCHH BeKTopa
N I
x. TOf,l{a npH Xi = i (paHfH ynop5l,l{OtleHHH R)
(i - j) (Yi -Yj) = N ( 2 iYt -(N + 1) Yi ) , (4.108)
1,/ I I
OTKY,l{a, B llaCTHOCTH, CJIeAyeT, liTO npH Yi, 3aAalO HX paHfH i
B ynopHAolleHHH R',
(i-j)(Yi-Yj) =2N ( iYi- (N+l)I ). (4.109)
i, I I 4 I '
nOJIYtJeHHbIe cpOPMYJIbI nOKa3bIBaIOT, liTO K09qxpHUHeHT
oooo eHHOH KOppeJI5IUHH Me}K):(Y KOJIHtJeCTBeHHbIMH MaTpHua-
MH Xii H Yii paBeH
2N2 cov ex, y) _ cov (x, y) (4.110)
Y 2N2 a 2 (x) 2N2 0 2 (y) a (X) (J (y)
226
i
,..
06bPIHOMY K09<P<lnn.{HeHTY KOppeJI5IQHH Mem.L{Y BeKTOpaMH x
H y.
AHaJIOfHqHbIH K03<fxPH[(HeHT KOppeJIH[(HH Me}K.L{Y BeKTOpa-
MH paHrOB npe.L{CTaBHM B BH.L{e
TC=[ !.iYi- (N I)2 ]/+(N-I)(N+I), (4.111)
1
f.L{e 12 (N - 1) (N + 1) - .L{HCnepCH5I BeKTopa nepBbIx N
HaTypaJIbHbIX qHCeJI. 9TOT Ko3<jxpHUHeHT CB5I3H Me>K.L{Y cTporH-
MH paH)KHpOBaHHHMH Ha3bIBaeTC5I Koat/JcjJulisueflmoM pafl2080u
KoppeJlfllluu CnupMaHa.
PaCqeT I, iYi MO}KHO 3aMeHHTb BbIQHCJIeHHeM paCCT05lHHH
I
If Me:>K.L{Y BeKTopaMH (1, ..., N) H (Yl' ..., YN), TaK KaK
N
d = (i-Yi)2 = Y12 + i2_2 iYi
1= 1 i I i
H, CJle.L{OBaTeJIbHO,
iYt = N (N + 1)(2N + I)- d.
i 6 2
,[(eHcTBHTeJIbHO, i2 = yl eCTb cYMMa KBa.L{paTOB nepBbIx
N HaTypaJIbHbIX qHCeJI, KOTopaH, KaK H3BecTHo, paBHa
N (N + I) (2N + I).
nO.L{CTaBJIHH nOCJIe.L{Hee paBeHcTBo B (4.111), nOJIyqaeM
N
'rc= 1- 6 d=l- 6 (i-Yi)2. (4.112)
N(N2-I) N(N2-1) l 1
<1>opMYJIa (4.112) BbIpa)KaeT K09<P<PH[(HeHT CnHpM9Ha qe-
pe3 CYMMY KBa.L{paTOB pa3HocTeH paHfoB 06'beKTOB B .L{BYX
ynopH.L{OqeHH5IX; 3TO npe.L{CTaBJIeHHe qaCTO HCnOJIb3yeTcH npH
pyqHhIX BbIqHCJIeHHHX.
MOLKHO BhIpa3HTb TaK)Ke'r c qepe3 KOJIHqeCTBO nepecTaHoBoK
nap oOneKToB, He06xo.L{HMbIX JIH nepeBO.L{a O.L{HOfO ynopH.L{O-
qeHHH B ,npYfoe. A HMeHHO d eCTb CYMMa BeJIHqHH
I riJ - rlJ I, HO He paBHonpaBHbIX, KaK d (R, R'), 8 B3BerneH-
HbIX BeC8MH Ii - j I, xapaKTepH3YIOInHMH Y.L{aJIeHHOCTb ne-
peCT8BJIHeMhIX 06'beKToB.
Ko3<fxPHUHeHT CnHpM9Ha HBJI5IeTCH KJIaCCHqeCKHM II npH-
MeHHeTCH T8K>Ke npH HecTpofHX paH)KHpOBaHHHx. npH STOM
8.
227
OOblqHO HCnOJIb3YIOTCH paHfH, BhIqHCJI5IeMhIe cJIe,LlYIOmHM 06-
pa30M. llYCTb R == (R 1 , ..., Rm) - 3a,LlaHHoe paHLKHpoBaHHe,
npHqeM tlHCJIeHHOCTb KJIaCCa Rs paBHa N s (s == 1, ..., m).
Tor,l{a Kam,l{OMY OO'beKTY i E Rs npHnHcbIBaeTC5I oaJIJI
N + +N + Ns(Ns+l) u
1 ... s - 1 2 ' paBHbIH Cpe HeMY 3Ha-
qeHHIO paHrOB OO'beKTOB Rs npH npOH3BOJIbHOM CTpOrOM yno-
pSl,l{OqeHHII (BHYTPII KJIaCCOB Rs). K03<pqninHeHT CnHpM3Ha
eCTb KOJIHqeCTBeHHbIH K03cpqHfnHeHT KOppeJI5InHH MeJK,l{Y Ta-
KHMH BeKTOpaMH OaJIJIOB.
4.2.7. Mepbl CBR3U HOMUHallbflblX npu3HaKoB
HOMHHaJIbHbUi npH3HaK R == {R 1 , ..., R m} MO-
)KeT ObITb oxapaKTepH30BaH MaTpHnaMH CJIe,LlYIOInHX BH,l{OB:
a . . __ { I, i R b
lJ - .-
0, tRjt
a . . =- 2 a. . - 1
lJ - 'lJ ,
( ' i, j E Rs, S == 1, ..., m,
b . . .- N s
'lJ-
0, iRj.
HOpM 3THX MaTpHn paBH :
(a, a) = N: = N2- d (R), ( a , a) = N2, (b, b) = m,
s
1
a cpe,l{HHe 3HatleHH5I- 1 - () (R), 1 - 2{) (R) H N COOTBeT-
CTBeHHO. CKaJI5IpHbIe npOH3Be,l{eHHH MaTpHn, xapaKTepH3YIO-
InHX R = {R 1 , ..., Rm} II R' = {Rl, ..., R:n}, paBHbI:
(a, a') = N t,
s. t
(a, a')=2 N:t- N:.,
s, t s
(a, a ') =N2-2d (R, R'), (8, b') = 2 ;:t - N,
s, t . t
, N: t
(b, b ) =
N s . N. t
s, t
(a, b')=
s, t
N: t
N .t
228
HaHOOJIblIlHH HHTepeC npe.L{CTaBJI5IIOT CJIe.L{YIOIIIJie K03cp-
cpHl.lHeHTbI KOppeJI5Il.lHH:
N t - N:. "£N t/N2
T = s, t
a
V N:. (N2- N:.) N t (N2 - N t)
(4.113)
N: t
.... -1
s, t N s . N. t
Tb=
r(m - l)(m' - 1)
_, "I}
.. N rem-I) (m' -1)
(4.114)
N2
st -N N2
N .... s.
s, t · t S
Tab = ·
rem' - l) N:. (N2_?JN:.)
(4.115)
nOKa3aTeJIb Tb B TOqHOCTH COBna.L{aeT C KoatjJtjJu""ueflmOM
cOnpJl'JICeflflOCmu lfynpoea (2.59), a Tab OTJIHlIaeTC5I JIHIlIb 3Ha-
MeHaTeJIeM OT nOKa3aTeJI5I BaJIJIHCa (2.68) OTHOCHTeJIbHOrO
YMeHblIleHH5I nponopnHOHaJIbHOrO nporH03a.
MbI e e pa3 nOJIyqHJIH nO.L{TBep1K.L{eHHe, tlro 3TH .L{Ba nOKa-
3aTeJI5I 51BJI5IIOTC5I OCHOBHbIMH B aHaJIH3e HOMHHaJIbHbIX npH-
3HaKOB.
r AaBa 5
AHAJIH3 CTPYKTYPbl CBSl3ER
ME)I(AY 061>EKTAMH
5.1. PA36HEHHE CHCTEMbl
B3AHMOCBSl3AH HblX 061>EKTOB
5.1.1. floKa3aTeJlU KatleCTBa pa36uenufl
B npe,l{bI,l{Y HX rJIaBaX MbI YCTaHOBHJIM, liTO 3a-
allH KOHCTpYHpOBaHMH HOMHHaJIbHWX npH3HaKOB CBO,l{HTCH K
3a,l{allaM 0 nOCTpOeHHH B TOM HJIH HHOM CMbICJIe OnTHMaJIbHOrO
pa30HeHHH no MaTpHne CBH3eH Me)K Y OO'beKTaMH.
llYCTb a = (ail) (i, j E m) - 3a,l{aHHaH MaTpHna CBH3eH
Me)K Y OO'beKTaMH H R = {R l' ..., R m} - HeKOTopoe pa3-
6HeHHe MHO)KeCTBa m Ha m KJIaCCOB. TeopHH KOHCTpYHpOBa-
HMH HOMHHaJIbHbIX cpaKTopOB-npoeKTopOB B TepMHHax TaOJIMU
OO'beKT-npH3HaK HJIH MaTpMU npoeKTHpOBaHHH KaK MaTpHU
CBH3H OO'beKT-06'beKT npMBO MT K CJIe YIO eMY KpHTepMIO Ka-
qeCTBa pa30HeHHSl R:
m 1
F (R, a)= - ail-am, (5.1)
N 8 l .
s == 1 , J €R 8
KOTopoe BbIpa)KaeT Cpe,l{HIOIO CYMMapHYIO CBH3b B KJIaCCaX R
tot
3a BbIlIeTOM BeJIHlIMHbI am, perYJIHpM3YIO eH lIHCJIO KJIaCCOB
pa30HeHHH R. MaKcHMH3anMH (5.1) 3KBHBaJIeHTHa MHHHMH3a-
u tot
UHH cpe HeH BHYTpHKJIaCCOBOH ,l{HCnepCHH npH3HaKOB.
TeopHH KOHCTpYHpoBaHHH HOMHHaJIbHbIX <t>aKTopoB-npo-
eKnHH HJIB oYJIeBCKHX MaTpHn CBH3H OO'beKT-OO'beKT npHBO-
JtHT K nOKa3aTeJIIO
m
G (R, a) = (aiJ- a),
s==li,/€Rs
(5. 2)
BbIpa1KalO eMY CYMMapHYIO BHYTpeHHlO1O CBH3b B pa30HeHHH
R C YlIeTOM nopora cy ecTBeHHOCTH CBSl3eH a (CM. 4.1), on-
pe,l{eJIHeMOrO nOJIycYMMOH 3JIeMeHTOB (ex + P)/2 MaTpH bI KOH-
230
CTpYHpyeMoro pa30HeHHH. npH OnTMMaJIbHOM C TOqKH 3peHHH
annpOKCHM8UHH 8HH X B OOpe 3HaqeHHH a, p nOKa38Tenb
(5.2) npeBpa aeTcH B cy ecTBeHHO oOJIee CJIO)KHYIO KOHCTPYK-
UfIlO:
g(R)= [G(R,O)]2 +
N:
s
r QiJ-O(R.O) ]
1,1==1
N2 - N:
8
(5.3)
KOTopaH nOJIyq8eTCH 113 CPOPMYJIbI (4.24) npH 'ij, COOTBeTCTBYIO-
InHX pa30HeHHIO R.
M8KcHMH3aUH5I (5:3) no R 3KBMBaJIeHTH8, M8KCMMH3aUHH
G (R, a) npM 3HaqeHHH nopora a, p8BHOM nOJIYCYMMe cpe HeH
BHYTpeHHeH H cpe HeH BHeIllHeH CB5I3eH .L{JI5I R (CM. (4.21) H
(4.22)).
nOKa3aTeJIH (5.1) H (5.3) HeJIHHeMHO 3aBHC5IT OT pa30HeHHH
R H 3H8qMTeJIbHO CJIO)KHee JI5I M8TeMaTHqeCKOrO 8HaJIH3a, qeM
KpHTepMM (5.2). ,[{aJIee MbI oY.L{eM npOBO.L{HTb 8H8JIH3 rJI8BHbIM
OOp830M JI5I KpHTepH5I (5.2); KpHTepHH (5.1) H (5.3) 0Y.L{YT
p8CCM8TpHB8TbC5I npH CPOPMYJIHpOBKe aJIrOpMTMOB nOCTpoeHHH
pa30HeHHH. ';
3aMeTHM,1ITO 38.L{a1la 0 pa30MeHMH MHO)KeCTBa OO'beKTOB Ha
OTHOCHTeJIbHO .L{8JIeKHe rpynnbI TeCHO CB5I3aHHbIX MeLK.L{Y co-
oOH OD'beKTOB IllHpOKO p8CCM8TpHBaeTC5I B JIHTepaType nO.L{Q8C
no.L{ P83JIHQHbIMH H8HMeHOBaHM5IMH. HCnOJIh3YIOTC5I TepMHHbI:
8BTOMaTHQeCK851 MJIM OO'beKTHBHa51 KJIaCcH<pHKaUH5I, TaKCO-
HOMHH, KJI8CTepHbIH aHaJIH3, pacno3HaB8HMe oe3 YQHTeJI5I, KOM-
nOHOBK8, .L{HarOH8JIH3aUHH M8TpMU CB5I3H, pa3pe38HHe rpacpoB
H T. n. B JIHTepaType npe.L{JIOJKeHbI .L{eCHTKH H COTHH P83JIMqHbIX,
3aQacTYIO M8JIOOTJIMQaIOInHXC5I MeTO.L{OB H aJIrOpHTMoB KJI8C-
cM<pHKaUHH,ocHOB8HHbIX Ha C8MbIX pa3JIHQHbIX 3BpHCTHqeCKHX
M.L{e5IX H nO.L{XO.L{8X.
B QaCTHOCTH, MO)KHO BbI.L{eJIMTb H.L{eM KJI8CcMcpHKaUHH Ha
OCHOBe 51BHoro onpe.L{eJIeHMH KJIaCCa (KJI8CTepa, T8KCOH8, DJIO-
Ka), Kor.L{a K8m,l{bIH KJI8CC MCKOMoro p830MeHH5I .L{OJI)KeH 51B-
JIHTbC5I KJI8CTepOM B CMWCJIe HeKOToporo TOQHOrO onpe.L{eJIe-
HH5I. Llpyroe Hanp8BJIeHMe He TpeoyeT YTOQHeHM5I nOH5ITM5I
OT.L{eJIbHOrO KJIaCCa, a HCnOJIb3yeT cpQpM8JIM30BaHHoe npe.L{CT8B-
JIeHHe 0 xopOIlleH KJI8CCH<pHK8UHM B UeJIOM, OObIQHO B BH.L{e
KpHTepH5I KaQeCTB8 p830HeHM5I. H8KOnJIeHHbIH onbIT nOK83bI-
BaeT, QTO B TaKOM KpHTepHM B TOM MJIH MHOH Mepe .L{OJILKHbI ObITb
oTpameHhI CJIeAYlOmHe HecpopMaJIhHhIe Tpe6oBaHH5I K KJIaCCH-
<pHKaQMH: 8) BHYTPH rpynn OO'beKThI .L{OJI)KHbI ObITb TeCHO CB5I-
231
saHbI Me)K,l{Y CODOR; 6) OD'beKTbl pa3HbIx rpynn ,L{OJI)K HbI ObITb
,naJIeKH ,L{pyr OT ,l{pyra; B) npH npOllHX paBHbIX YCJIOBH5IX pac-
npe,l{eJIeHHe 06'beKTOB no rpynnaM ,L{OJI)KHO DbITb paBHOMepHO.
TpeOOBaHH5I a) H 6) BbIpa)KalOT CTaH,l{apTHYIO KOHnennHIO KOM-
naKTHOCTH KJIaCCOB pa30HeHH5I [5]; TpeooBaHHe B) COCTOHT
B TOM, qTOObl KpHTepHM He HaB5I3bIBaJI 06'be,l{HHeHH5I OT,l{eJIbHbIX
rpynn 06'beKTOB.
HMeIOTC5I TaK)Ke nO,L{XO,l{bI K KJIaCCH<pHKanHH, OCHOBaHHbIe
Ha 51BHOM BbI,l{eJIeHHH Ii OTopaCbIBaHHH MaJIbIX CB5I3eH (HanpH-
Mep, CHallaJIa CTpOHTC5I TpHBHaJIbHOe pa30HeHHe CO CTPYKTY-
pOM, Ha OCHOBe KOTOpOM Y)Ke npOH3BO,L{HTCH KJIaCcHcpHKanH5I).
npH 3TOM Bce CB5I3H, MeHblllHe HeKoTol?bIX noporoBbIx 3Haqe-
HHH, OO'b5lBJIHIOTC5I HecyrnecTBeHHbIMH H HCKJIlOlIaIOTC5I H3
,l{aJIbHeHI.I1ero aHaJIH3a. B ,l{PyrHx MeTO,L{aX HecyrnecTBeHHbIe
CB5I3H BbI5IBJI5IIOTC5I anocTepHopHo, TaK liTO HecyrneCTBeHHOH
MO)KeT OKa3aTbC5I Ii OTHOCHTeJIbHO OOJIbI.I1a51 CB5I3b.
HMelOTCH TaK)Ke aJIrOpHTMHlIeCKHe KOHnennHH KJIaCCH-
Q>HKanHH, KOr,L{a <popMHpyeTc5I aJIrOpHTM, OOJIa,l{alOrnHM p5l,L{OM
u u
XOpOI.I1HX C 3BpHCTHqeCKOH TOllKH 3peHH5I CBOHCTB; pe3YJIbTaT
npHMeHeHH5I 3Toro aJIrOpHTMa H OO'bHBJI5IeTC5I HCKOMOH KJIaC-
CHcpHKanHeM. .QpyrHe nO,L{XO,L{bI BbIpa)KalOT KOHellHYIO neJIb
HenOCpe,L{CTBeHHO B TepMHHax HCKOMOH KJIaCCH<pHKanHH.
B KpHTepHH (5.2) COBMerneHbI lIepTbI Bcex ynoM5IHYTbIX no,L{-
XO,L{OB.
npO,L{eMoHcTpHpyeM npe)K,l{e Bcero, KaK peaJIH3YIOTC5I H
YTOQH5IIOTC5I B HeM TpeOOBaHH5I a), 6), B) K xopoI.I1eii KJIaCCH-
<pHKanHH.
0603HatIHM tIepe3 G (R , a) = I, (aiJ - a) CYMMY Bcex !
s=l=t iE Rsl E Rt
BHeIllHHX CB5I3eii (T .e. CB5I3eH Me)K,L{y OO'beKTaMH pa3HbIx KJIaC- t
COB) 3a BbItleTOM nopor.a a. OtleBH,L{HO,
_ N
. G(R,a)+G(R,a)= (aij-a),
i,/ = 1
npHlIeM npH 3a,L{aHHOM a npaBa51 lIaCTb 3Toro COOTHOI.I1eHH5I no-
CT05lHHa. CJIe,L{OBaTeJIbHO, pa30HeHHe R, MaKCHMH3HpYlOrnee
G (R, a), O,l{HOBpeMeHHO MHHHMH3HpyeT CYMMY BHellIHHX CB5I-
3eM 7i (R, a). 9TO 03HallaeT, liTO KpHTepHM (5.2) HapH,l{y C Tpe-
60BaHHeM a) aBTOMaTHlIeCKH YllHTbIBaeT TpeooBaHHe 0).
PacKpbIBa51 B (5.2) CKOOKH, nOJIYllHM
m m
G(R,a)= aiJ-a N:, (5.4)
s== 1 /,/ER, s= 1
232
T. e., KaK Y)Ke OTMellaJIOCb, MaKCHMH3aUH5I G (R, a) ,nOCTH-
raeTC5I, C O,l{HOH CTOpOHbI, 3a ClIeT MaKCliMH3aUHH CYMMbI BHYT-
u u
peHHl1X CBSl3eH ail H, C ,l{pyroH CTOpOHbI, 3a ClIeT MHHHMH3a-
II.liH BeJIHqHHbI onpe,l{eJIeHHOCTH pacnpe,l{eJIeHH5I e (R) = N:
8
(npH YCJIOBHH Ns = N). 3TOT BTOpOH tlJIeH a N: ,l{OCTHraeT
8 S
MliHHMYMa npH paBHOMepHOM pacnpe,l{eJIeHHH (N1=...=N m),
TaK liTO B HeM YllHTbIBaeTC5I TpeOOBaHHe B), nopor a HrpaeT
pOJIb nOK33aTeJI5I KOMnpOMHcca MejK,l{Y TpeOOBaHH5IMH 3), 0)
Ii B).
nOKa*eM Tenepb, liTO eCJIH pa30HeHHe R OnTHM3JIbHO B
CMbICJIe (5.2), TO H caMO OHO, H Ka:>K,l{bIH ero KJIaCC 51BJI5IIOTC5I
XOpOI.I1HMIi oe30THOCHTeJIbHO K KaKOMY-JIHOO nOKa3aTeJIIO onTH-
MaJIbHOCTH.
BY,l{eM rOBopHTb, liTO pa30HeHHe R KOMnaKmHO, eCJIH
cpe,nHSl5I BHYTpeHH5ISI CB5I3b Me1K,l{Y OO'beKTaMH B Ka>K,l{OM KJI3C-
ce npeBbII.I1aeT cpe,l{HIOIO BHelIIHlO1O CBSl3b Me}K,l{Y pa3HbIMH
KJIaCCaMH:
1 1
a. a. ·
N ( N -1 J N N JJ'
8 8 ) I, fERa q , IERq JER t
f,l{e N s - KOJIHlIeCTBO OO'beKTOB B Rs (s, t, q = 1, ..., m).
3,necb II ,l{aJIee ClIHTaeTCSI, qTO M3Tp H lI.a (aij) cHMMeTpHlIHa,
npHlIeM ,l{HarOHaJIbHbIe ee 3JIeMeHTbI He 3a,l{aHbI. B COOTBeTCTBHII
CO CKa3aHHbIM Ha c. 174 peaJIbHO 3TO HHKaKHX OrpaHHqeHHH
Ha ,l{aHHbIe He HaKJIa,l{bIBaeT.
MHO)KeCTBO S m OY,l{eM Ha3bIBaTb KllacmepoM, eCJIH
Cpe,l{HSISI CB5I3b Me*,l{Y OO'beKTaMH B S npeBbII.I1aeT Cpe,l{HIOIO
CB5I3b OO'beKTOB S C OCTaJIbHbIMH OO'beKTaMH:
(5.5)
1 1
a. - a ·
ISI081-1). J '81(N-ISI). il.
I./ES tES JfES
(5.6)
OKa3bIBaeTCSI, OnTliMaJIbHOe no (5.2) pa30HeHHe 005138-
TeJIbHO KOMnaKTHO, 8 KaJK,l{bIH ero KJIaCC S1BJI5IeTCSI KJIaCTepOM.
,[{JI5I ,l{OKa3aTeJIbCTBa 3Toro YTBep}K,l{eHH5I 0003HallHM npH
,naHHOM pa30HeHHH R = {R 1 , ..., Rm} H nopore a qepe3
Ast= (a'j-a)
IER, JER,
CYMMY 3JIeMeHTOB KJIeTKH (ail - a) (i E Rs, j E Rt) MaTpHlI. bI
CB5I3eM (3a BbIlIeTOM nopora), COOTBeTCTBYIO eH KJIaCCaM RBt
R. t pa30HeHHH R. DOCKOJIbKY CBH3H OO'beKTOB C caMHMH COOOM
..
(5.7)
233
......-.....
He paCCMaTpHB8IOTC5I, TO eCTeCTBeHHO npHHHTb .L{JI5I 0.L{H03JIe-
MeHTHorO KJIaCCa R8 == {S}, liTO Ass = O.
Cnp8Be.L{JIHBO CJIe,l{YIO ee HeOOXO,nHMOe YCJIOBHe OnTHMaJIb-
JiOCTH pa30HeHH5I.
ECJIH R OnTHM8JIbHO no (5.2) B KJIacce- Bcex B03MO:>KHblX
pasoHeHHH MHO)KeCTBa al, TO ...
8) Ast 0 npH Bcex S, t = 1, ..., m (8 =1= t);
6) Ass 0 npH Bcex s = 1, ..., m.
,l{eHcTBHTeJIbHO, nycTb R OnTHM8JIbHO, npH1JeM Ass < 0
,nJI5I HeKOToporo s. B STOM CJIYtIae paCCbIneM 06'beKTbI i E R.
B O,l{HOSJleMeHTHbIe KJI8CCbI {i}. Tor,l{8 Ait = 0 > Ass, TaK
IER,
liTO JtJI5I HOBoro pa36HeHHSI R' G (R', a) > G (R, a) = Att,
t=l
tJ:TO npOTHBOpe1JHT OnTHM8JIbHOCTH R.
ECJIH, C ,l{pyroH CTOpOHbI, Ast > 0 npH S =1= i, TO YBeJIHlIe-
HHS! KpHTepHH (5.2) MO)KHO ,nOCTHlIb, 06'be,l{HH5IS! KJI8CCbI Rs
II Rt. B STOM CJIYQ8e B CYMMY (5.2) ,n06aBJISleTC5I Ast + Ats =
= 2A,t > 0, qTO onSlTb npOTHBOpeqHT OnTHM8JIbHOCTII.
nOJIyqeHHbIe npoTHBOpeqH5I ,nOKa3hIBaIOT YTBep>K,l{eHHe.
Tenepb JIerKO nOJIyqHTb Tpe6yeMbIH pe3YJlbTaT. llYCTb R
OnTHMaJIbHO, TOr,l{a H3 ,l{OKa3aHHoro CJIe.nyeT, liTO
O As,= afj-aN,(N.-l) (s=I,.. ,In);
I. J ER s
O Atq= aiJ-aNtN g (t,q=l,...,m; t=l=q).
lERt JER q
CYMMHPYH nOCJIe,l{HHe HepaBeHCTB8 no q, nOJIyqaeM TaK>Ke
o Atq = aij-aNt (N -Nt) (t= I, ..., m).
... q+' iERt fER,
Ho TOr,l{a
1
N r(N -1) aiJ a;
8.' I./ER,
1 alJ a;
NtNq iERtlERq
I . ..-
N (N-N) aiJ a.
, t fER, IER,
OTKy,n8 CJIe,l{YIOT Hep8BeHCTB8 (5.5) H (5..6) ,l{JI5I JIIOObIX KJIaCCOB
pa36HeHHSI R, liTO Ii TpeOOBaJIOCb ,l].oKasaTb.
234
TaKHM OOpa30M, paCCMaTpMBaeM8H KOHUennM5I COBM rnaeT
no,nXO,!{bI, OCHOBaHHble Ha 51BH01v1 onpe,neJIeHMH nOH5ITHH. KJIaC-
Tepa, H nO.L{XO,!{bI B TepMHHax CHCTeMbI KJI8CCOB. npH 3TOM B
KpHTepHH (5.2) B 51BHOH <popMe BXO,!{HT nopor cyrneCTBeHHOCTII
u
HH,!{HBH,!{yaJIbHbIX CB5I3eH, HO MaJIble CB5I3H He 3JIHMHHHPYIOT-
C5I, a YllMThIBaIOTC5I C OTpHuaTeJIbHblM 3HaKOM, liTO ,!{aeT B03-
MO>KHOCTb He>KeCTKOro HX YlleTa, T8K liTO B OKOHlI8TeJlbHOM pa3-
OHeHMH MaJIa51 CB5I3b MO>KeT nonaCTb BHYTPb, 8 60JIblIIaH-
BO BHe pa36HeHHH R, JIHlIlb ObI BblnOJIH5IJIMCb B Cpe.L{HeM COOT-
HOIlIeHH5I (5.5) M (5.6). _
B TOM lI8CTHOM CJIYll8e, KOr.L{8 rp8cp a-CXO.L{CTB8 r a =
= { (i, j) I aij> a} C8M 51 BJI51eTC5I OTHOIlleHHeM 3KBHBa-
JIeHTHOCTH, OH onpe,!{eJI5IeT 86COJIIOTHO OnTHMaJIbHOe pa30He-
HHe R, TaK KaK npH 3TOM B G (R, a) CO.L{ep)K8TCSI Bce HeOTpH-
u8TeJIbHbIe CJI8r8eMble H TOJIbKO OHM; BC5IKOe ,!{pyroe pa30He-
HHe MO>KeT TOJIbKO YMeHbllIHTb 3H8l1eHHe G (R, a). KJIaCCbI
3Toro p830HeHH5I 51BJI5IIOTC5I He TOJIbKO KJIaCTep8MH, HO H cry-
rneHH5IMH. CZYUIf!flUeM H83bIBaeTC5I TaKoe MHO>KeCTBO 06'beK-
TOB S, B KOTOpOM JIIOOa51 ero BHYTpeHH5I5I CB5I3b ail (i, j E S)
npeBbIIll8eT BHeIllHHe akl (k E S, 1 S).
,[{JI5I BC5IKOrO Rs B paccMaTpHB8eMoH CHTY8n M H ail a,
akl a npH it j, k E S, 1 $. S, liTO II .L{OK83bIB8eT YTBep>K,lte-
HMe.
HHTepecHbIM 51BJI5IeTC5I TaK>Ke nOHSlTMe cIla6ozo CZYUIf!flUR.
CJI86blM cryrneHHeM H83bIB8eTC5I T8Koe MHomeCTBO S, B KOTO-
pOM JIIOOble .L{Ba ero 3JIeMeHT8 MorYT 6blTb COe.L{HHeHbI nOCJIe)(o-
BaTeJIbHOCTblO OO'beKTOB S, K8:>K.L{hIH H3 KOTOpblX CB5I3aH C co-
Ce.L{HMH (no nOCJIe.L{OBaTeJIbHOCTH) CHJIbHee, lIeM C JIIOObIM OO'b-
eKTOM, He BXO.L{51rnMM B S. OlleBH.L{HO, KOMnOHeHTbI rpacpa
a-CXO.L{CTB8 51BJI5IIOTC5I CJI80blMH cryrneHMSlMH (CM. c. 263).
HCCJIe.L{yeM Tenepb Bonpoc 0 BJIHSlHHH nopora a H8 onTO-
M8JIbHOe pa36MeHMe. npe)K.L{e Bcero OTMeTHM, liTO eCJIH nopor
HeBeJIHK H aij a .L{JI5I Bcex i, j E m, TO Bce BeJIHlIHHbI
ail - a HeOTpHU8TCJIhHhl H M8KCHMYM (5.2) .L{OCTMr8eTC5I H8
pa30HeHHH, COCT05lrneM H3 e.L{HHCTBeHHoro KJI8CCa, BKJIlOqalO-
rnero Bce OO'beKTbI. HaooopoT, eCJIH ail a .L{J151 Bcex i, j E 91,
TO OnTHM8JIbHOe HYJIeBOe 3HalleHMe G (R, a) .L{OCTHraeTC5I Ha
TpMBHaJIbHOM pa30HeHHH m Ha N O,!{H03JIeMeHTHbIX KJI8CCOB.
To :>Ke cnp8Be,!{JIHBO H .L{JI5I nOKa38TeJIH (5.1), TOJIbKO aij B HeM
cpaBHHBalOTCH He c a, a c :8 npH i, j E R8'
npH YMeHhIlIeHHH a OT CHTY8UHH a ail .L{O cHTyal(HH
a ail (.L{JI5I Bcex i, j E ) QHCJIO KJI8CCOB OnTHM8JIbHOrO
p836MeHH5I T8KJKe YMeH llIaeTC5I OT N o 1. 3HalleHHe nopor8
235
I
I
, i
a, KaK H qHCJIO KJIaCCOB m, BblpaLKaeT CTeneHb rPYOOCTH npe,l{-
CTaBJIeHHH HCXO,l{HOH HH opManHH B pa30HeHHH.
.
MO}KHO JIH YTBep}K,l{aTb, lITO KOJIHlIeCTBO KJIaCCOB OnTHMaJIb-
Horo pa30HeHHH MOHOTOHHO 3aBHCHT OT a? ,L{a, eCJIH HCnOJIb-
3yeTcSI KpHTepHH (5.1). ECJIH }Ke HCnOJIb3yeTcH (5.2), TO HeT:
u
MOLKHO npHBeCTH npHMepbI HeMOHOTOHHOH 3aBHCHMOCTH IIHCJIa
KJIaCCOB OnTHMaJIbHOrO pa30HeHHH OT a. DOJIee Toro, ,l{JIH
HeKOTopbIX MaTpHn CBSl3H He Bce 3HalIeHHSI m OT 1 ,l{0 N MorYT
ObITb nOJIYlIeHbI MaKcHMH3anHeH (5.2) npH pa3JIHIIHbIX a [48].
Q,l{HaKO HMeeTCSI ,l{PyraH xapaKTepHcTHKa OnTHMaJIbHOrO
pa30HeHHSI - ero ,l{HCnepCHH d (R) = N 2 - N:, KOTopaH
s
MOHOTOHHO 3aBHCHT OT a. A HMeHHO HMeeT MeCTO CJIe,l{YIOInHH
aKT: ,l{HCnepCHSI cS (R) OnTHMaJIbHOrO no G (R, a) pa30HeHHSI
R MOHOTOHHO He YObIBaeT C YBeJIHlIeHHeM a [48].
TT ' ", "
AJISI ,l{OKa3aTeJIbCTBa paccMoTpHM 3HalIeHHH a , a ,a > a .
nYCTb R' H R" - OnTHMaJIbHbIe pa30HeHHH no G (R, a') H
G (R, a") B KaKOM-JIHOO KJIaCCe ,l{OnycTHMbIX pa30HeHHH .
B CHJIY (5.4) HMeeM
G (R, a) = G (R, 0) + ad (R) - aN (N + 1).
no onpe,l{eJIeHHIO R' H R"
G (R', a') G (R", a'); G (R", a") G (R', a").
nO,l{CTaBHB B 3TH HepaBeHCTBa paCCMOTpeHHoe BbIpa:>KeHHe
G (R, a), nOJIYIIHM
a' (d (R') - d (R"» G (R", 0) - G (R', 0) a" (d (R') -
- d (R"»,
TaK lITO
a' (d (R') - d (R"» a" (d (R') - d (R"»,
T. e.
(a' - a") (d (R') - d (R"» o.
nOCJIe,l{Hee HepaBeHCTBO 03HalIaeT, qTO COMHO)KHTeJIH
a' - a" H d (R') - d (R") HMelOT O,l{HHaKOBbIH 3HaK, H TaK
KaK a' - a" > 0, TO d (R') - d (R") 0, lITO H TpeOOBaJIOCb
,l{OKa3aTb.
,[{OKa3aTeJIbCTBO Toro, lITO IIHCJIO KJIaCCOB m OnTHMaJIbHOrO
no F (R, a) pa30HeHHH MOHOTOHHO 3aBHCHT OT a, nOJIHOCTblO
nOBTopHeT npHBe,l{eHHOe paccYLK,l{eHHe.
236
I
--
5.1.2. AZAoMepaTuBflbte aAzopuTMbl
KAaccuifJuKat{uu
PaCCMOTpHM 3a,!{aqy onTHMH3alIJiH nOKa3aTeJISI
(5.2) B KJIaCCe Bcex B03MOJKHhIX pa30HeHHH MHO)KeCTBa ffi.
3Ta 3a,!{aqa HMeeT KOMoHHaTopHYIO npHpo,!{y H, Booo e rOBO-
p5I, He MOJKeT ObITb pelIleHa npOCTbIMH MeTOAaMH. ,[{OCTaTOqHO
u
CKa3aTb, qTO pa3JIHqHbIe ee qaCTHbIe CJIyqaH CBO,!{SlTC5I K TaKOH
KJIaCCHQeCKOH nepeoopHOH 3 a,!{aqe, KaK 3a,!{aqa 0 paCKpalIlH-
BaHHH rpacpa [37]. Hey,!{oBJIeTBOpHTeJIbHbIM 51BJI5IeTC5I TaK)Ke
pacnpocTpaHeHHbIH B ,!{HcKpeTHoH MaTeMaTHKe cnocoo pac-
cMOTpeHH5I TaKHX npocTbIX aJIrOpHTMoB, KOTopbIe ,!{alOT TOqHOe
perneHHe He Bcer,!{a, HO ,!{JI5I no,!{aBJI5IIO erO OOJIblIlHHCTBa CH-
Tya HH (CM., HanpHMep, [106], [70, c. 151-153]).
n09TOMY OCHOBHoe 3HaqeHHe 3,!{eCb npHoopeTalOT 1l0Kallb-
flo-onmUMallbflble aJIrOpHTMbI (HX Ha3bIBalOT TaK}Ke cy6onmu-
MallbflbtMU), KOTopbIe paooTalOT CJIe,!{YIO HM oopa30M. Ha Ka}K-
OM lIlare paccMaTpHBaeTC5I HeKOTopoe pa30HeHHe R MHO}KeCT-
Ba OO'beKTOB H HeKOTopaSl cOBoKynHocTb pa30HeHHH 0 (R),
COCTaBJI5IIO HX oKpecmHocmb R. B 9TOH oKpecTHocTH BbIoHpa-
eTCSI OnTHMaJIbHOe no ,!{aHHOMY KpHTepHIO pa30HeHHe, KOTopoe
Ii paCCMaTpHBaeTC5I Ha CJIe,!{YIO eM mare. nponecc 3aK8HqH-
B8eTC5I, eCJIH HOBoe pa30HeHHe COBna,!{aeT C npe,!{bI,!{y HM.
C BbIqHCJIHTeJIbHOH TOqKH 3peHH5I KaqeCTBO 8JIrOpHTMa on-
pe,!{eJI5IeTC5I ,!{BYM5I OCHOBHbIMH xapaKTepHCTHKaMH: OO'beMOM
BbIqHCJIeHHH, onpe,!{eJI5IeMbIM OO'beMOM OKpeCTHOCTeH 0 (R) H
B03MO)KHOCTSlMH ObICTporo onpe,!{eJIeHHSI onTHMYMa B 0 (R),
a TaK}Ke OO'beMOM 38nOMHHaeMOH npH BbIqliCJIeHH5IX HH<popMa-
nHH (onpe,!{eJl5leMOH CJIO}KHOCTblO CHCTeMbI 0 (R) OKpeCTHOCTeH
H BbIqHCJI5IeMOrO KpHTepHSI). nOHSlTHO, qTO YJIyqlIleHHe 9THX
X8paKTepHCTHK CB5I38HO C YMeHbrneHHeM OO'beMa nepeoopa H
TeM C8MbIM YXY,!{IlleHHeM K8QeCTBa cyoonTHMaJIbHbIX pemeHHH.
CJIe,!{yeT 3aMeTHTb TaK)Ke, qTO OO'beM BbIQHCJIeHHH H OO'beM
naM5ITH BO MHorOM B38HM03aMeHSleMbI: MO)KHO OOOHTliCb MeHb-
u u
eH naM5ITblO 3a CqeT YCJIOJKHeHH5I BbIqHCJIeHHH HJIH YMeHb-
lIlHTb OO'beM BbIQHCJIeHHH 3a CqeT YBeJIHqeHH5I naM5ITH. 3TH
u
B03MO)KHOCTH cy ecTBeHHO 3aBliC5IT OT ypOBH5I HCnOJIb3yeMoH
BWQHCnHTeJIbHOH TeXHHKH.
OObIqHO BbIOOp CHCTeMbI OKpeCTHOCTeH onpe,!{eJI5IeTC5I H8
OCHOBe 9BpHCTHqeCKHX COOOp8)KeHHH nYTeM pa3YMHoro KOM-
npOMHcca MeJK,!{Y KaqeCTBOM BhIqHCJIHTeJIbHOrO nponeCca H Ka-
qeCTBOM pelIleHH5I. O,!{HaKO ,!{JI5I KpHTepH5I (5.2) He06xo,!{HMbIe
YCJIOBHSI OnTHMaJIbHOCTH (c. 234) 51BJISIIOTC5I TeOpeTHqeCKOH
6830H cJIe.AYIOInero aJIrOpliTM8.
237
A JI r 0 p H T M «n P 0» (n p 0 B e p K a, pac C bI-
n a Hue, 0 0 'b e ,l{ H H e H He). Ha oow.eM mare aJIrOpHTMa
paCCM8TpHBaeTC5I pa30HeHHe MHOLKeCTBa in H8 P KJI8CCOB
R (P) = {Rl' ..., R p } H pXp M8TpHua BeJlHllHH Ast =
IER.
(ai - a) BH,na (5.7) - cYMMapHbIx CB5I3eH: MeLK.L{Y KJI8CC8MH
iERt
(38 BbIlleTOM noporOBoro 3HaqeHH5I). npOH3BO,nHTC5I npOBepKa
BbInOJIHeHH5I HeOOXO,l{HMbIX YCJIOBHH: OnTHMaJIbHOCTH. ECJIU
Ass< 0 ,l{JlSI HeKOToporos, TO HeOOXO,l{HMO paCCbInaTb KJlaCC Rs
Ha N s O,l{H03J1eMeHTHbIX KJlaCCOB H ,l{aJlee paCCMaTpHBaTb nOJIY-
qeHHOe TaKHM oopa30M pa30HeHHe R (p + N s - 1). ECJlH
Ass 0 ,l{JlSI Bcex s = 1, ..., p, TO npOH3BO,l{HTC5I BbIOOp MaK-
CHMaJIbHOrO 3J1eMeHTa Ast(s=l=t). npH Ast>O KJlaCCbI Rs H
R t OO'be.L{HH5IIOTC5I, a MaTpHna CYMMapHhIX CB5I3eH: .L{Jl5I TaK no-
JlyqeHHOrO pa30HeHH5I R (p - 1) BbIqHCJI5IeTC5I npocTbIM no-
KOMnOHeHTHbIM CYMMHpoBaHHeM CTpOK (8 TaKLKe CTOJIOUOB)
e HOMepaMH s H t.
BhIQHCJleHH5I OKaHqHBaIOTCSI, eCJlH .L{Jl5I paCCMaTpHB8eMoro
pa30HeHH5I Ass 0, Ast 0 npH s, t = 1, ..., p; s =1= t.
Y,l{OOHO HaqHHaTb BbIqHCJleHH5I C TpHBHaJIbHOrO pa30HeHHSI
R (N) = { {I}, ..., {N} } MHO)KeCTBa in Ha N 0,l{H03J1eMeHT-
HbIX KJlaCCOB; MaTpHna (Ast) COBna.L{aeT C (ail - a). B 3TOM
CJIyqae HeT HeOOXO,l{HMOCTH B npOBepKe HeOTpHnaTeJIbHOCTH
JtHarOHaJII HbIX BeJIHqHH Ass (H, CJIe,l{OBaTeJIbHO, npOBe,ll.eHI.B
npou.e,l{ypbI paCCbInaHH5I), TaK KaK OHH Bcer.L{a 0Y.L{YT HeOTpH-
naTeJIbHbI no nOCTpoeHHIO: OO'be,l{HHeHHe Rs H Rt npH Ast =
= Ats > 0 TOJIbKO YBeJIHqHBaeT BeJIHqHHbI ,nHarOHaJIbHbIX
3JIeMeHTOB. n03TOMY MO)KHO HCnOJIb30BaTb CJIe,l{YIO HH ynpo-
v
rneHHbIH aJIrOpHTM.
A JI r 0 p H T M «0 0 'b e ,l{ H H e H He». nYCTb R (P) =
= {Rl' ..., R p } - npOH3BOJIbHOe pa30HeHHe MHO)KeCTBa m
Ha p KJIaCCOB. 0003HaLJHM qepe3 Rst pa30HeHHe, nOJIyqeHHOe
H3 R (P) OO'be,nHHeHHeM Rs H Rt. COBoKynHocTb pa30HeHHH:
R.st (s, t = 1, ..., m) oopa3yeT OKpeCTHOCTb 0 (R (p». OnTH-
MaJIbHbIH 3JIeMeHT 0 (R (P», eCJIH OH He COBna,l{aeT C R (P),
0003HaQHM qepe3 R (p - 1).
AJIrOpHTM HaqHHaeT c R (N) H CTpOHT nOCJIe,l{OBaTeJIb-
H.OCTb pa30HeHHH R (N - 1), R (N - 2), ..., R (m), r,l{e m
TaKOBO, qTO R (m) - OnTHMaJIeH B 0 (R (m».
DpH HCnOJIb30BaHHH nOKa3aTeJI5I (5.2) BbIqHCJIeHH5I MOLKHO
npOBeCTH TOJIbKO Ha OCHOBe MaTpHQ CB5I3eH. Ha HaLJaJIbHOM
llIare (Ast) = (aij - a).
238
I
I
-
DpM 3a,l{8HHOH MaTpMlI.e (Ast) BbIOMpaeTCH MaKCHMaJlbHOe
Ast (5 -+ i). ECJlH Ast > 0, TO CYMMHPYIOTC5I CTpOKH 5 H i, a
T8KLKe CTOJIOnbI S H t. BbIqHCJIeHHSI 3aKaHqHBaIOTC5I, eCJIH Bce
Ast (8:::/== t) OTp HuaTeJI bHbI. MaTpHn a (Ast) ,l{aeT CYMMapHbIe
CBH3H Me>K.ny KJIaCCaMH, a caMH KJlaCCbI onpe,l{eJI5IIOTCH nOCJIe-
u
,nOBaTeJIbHOCTblO onepaIlHH CYMMHpOBaHHH.
OO'beM BbIqHCJleHHH no aJlrOpHTMY «OO'be,l{HHeHHe» MOLKHO
oneHHTb BeJlHQMHOH nOp5l,l{Ka N3, TaK KaK Ha KaLK,l{Olvf Illare
Cp8BHHB81OTCH Bce SJleMeHTbI MaTpMnbI (Ast) (MX QHCJlO p8BHO
P (p-l) npH Cpe,l{HeM p = N ,qTO ,l{aeT BeJIHqHHY nOpH,l{ 8
2 2
N 2 ), 8 qHCJIO IllarOB He npeBbIIllaeT N. OO'beM HeOOXO,l{HMOH
naMSlTH HMeeT nOp5l,l{OK N2 - ,l{J1H xpaHeHH5I HCXO.L{HOH MaTpH-
UbI (CYMMHpOBaHHe MOLKHO BbInOJIHHTb Ha STOM 1Ke MeCTe).
npHMeHHM aJIrOpHTM «OO'be,l{HHeHHe» K ,l{aHHbIM TaOJI. 3.1
.nJIH M8TPHI{bI a = P (c. 121) npH a = 0,6. MaTpHua (atJ - a)
u
HMeeT CJIe.nYIOIl\HH BH)(:
1 2 3 4 5 6
1 0 -002 -002 0,15 -0,27 -027
, , ,
2 -002 0 0,32 -002 -0 I -06
, , , ,
3 -002 0,32 0 0,32 -027 0,07
, ,
4 0,15 -002 0,32 0 -0,21 0,07
,
5 -0,27 -0,1 -027 -021 0 0,9
, ,
6 -027 -06 0,07 0,07 0,9 0
, , MaKCHMaJIeH 3J1eMeHT (5, 6). nocJle CJlOLKeHM5I CTpOK (H-CTOJlO-
nOB) 5, 6 nOJIYQaeM
2
-002
,
o
0,32
-002
,
5,6 -0,54 -0,7 -0,2 -0,14 1,8
Tenepb M8KCHMaJIbHbI SJleMeHTbI (2, 3) H (3, 4). CYMMHPY"
eM, H8npHMep, CTpOKH H CTOJIOUbI 2, 3:
I
2
3
4
1
o
-002
,
-002
,
0,15
3
-002
,
0,32
o
0,32
4
0,15
-002
,
0,32
o
5,6
- 0,54 _
-07
,
-02
,
- 0,14
1 2,3 4 5,6
1 0 -004 0,15 -054
, ,
2,3 -004 0,64 0,3 -0,9
,
4 0,15 0,3 0 -014
,
5,6 -054 -0,9 -014 1,8
, ,
239
B 3TOM MaTpHLle MaKCHMaJIbHbIM SlBJISleTCSI 3JIeMeHT (2, 3,
4), qTO .L{aeT
1
2,3,4
5,6
1
o
0,11
-054
,
2,3,4
0,11
1,24
-023
,
5,6
-- 0 54
,
-023
,
1,8
OCTaeTC5I e,l{HHCTBeHHbIH nOJIO)KHTeJIbHbIH 3JIeMeHT; npo-
Be,l{51 OO'be,l{HHeHHe, nOJIyqHM OKOHqaTeJIbHOe pa30HeHHe
{I-2-3-4, 5-6}, KOTopoe 51BJI5IeTCSI OnTHMaJIbHbIM.
XOT5I onHcaHHbIH aJIrOpHTM «oO'be.L{HHeHHe» nOJIyqeH HaMH
H3 TeOpeTHqeCKOrO YCJIOBHSI OnTHMaJIbHOCTH, COOTBeTCTBYIO a51
CHCTeMa oKpecTHocTeH H opraHH3a HSI BWqHCJIeHHH HCnOJIb30-
BaJIaCb pa3HwMH aBTopaMH 3a.L{OJIrO ,l{0 n05lBJIeHHSI KpHTepH5I
(5.2) H CBSl3aHHOH C HHM TeopHH.
B tIaCTHOCTH, cxeMa cyoonTHMaJIbHbIX aJIrOpHTMOB no
npOH3BOJIbHOMY KpHTepHIO KaqeCTBa pa30HeHHSI Ha OCHOBe
CHCTeMbI OKpeCTHOCTeH 0 (R) = {Rst} onHcaHa B CTaTbe
Y op,l{a [170). Ho H 3a,l{OJIrO ,l{0 3TOM CTaTbH BbIqHCJIeHH5I Ha
v v
OCHOBe 3TOH CHCTeMbI OKpeCTHOCTeH HCnOJIb30BaJIHCb, HanpH-
Mep, B HCCJIe,l{OBaHHSlX no OHOJIOrHqeCKOH CHCTeMaTHKe oe3-
OTHOCHTenbHO K onTHMH3anHH KaKHX-JIHOO KpHTepHeB, a KaK
COOCTBeHHO 3BpHCTHqeCKHe MeTO,l{W, OCHOBaHHwe Ha aJIrOpHT-
MHqeCKOM npe,l{CTaBJIeHHH 0 xoporneH KJIaCCH<pHKanHH. npH
3TOM HCnOJIb30BaJIHCb JIOKaJIbHbIe nepeCqeTbI Ha Ka:>K.L{OM mare,
OCHOBaHHbIe TOJIbKO Ha OneHKe CB5I3eH Me:>K,l{Y 06'be,l{HH5IeMbI-
MH 06'beKTaMH, qTO MOJKHO ,l{eJIaTb, He HMe51 B BH.L{Y HHKaKoro
KpHTepH5I.
B HaCT05l ee BpeM5I H3BeCTHO 3HaqHTeJIbHOe KOJIHqeCTBO
aJIrOpHTMOB, peaJIH3YIO HX cxeMY 06'be,l{HHeHHSI KaK Ha OCHO-
Be 3a,l{aHHOrO KpHTepHSI OnTHMaJIbHOCTH, TaK H Ha OCHOBe 3B-
pHCTHqeCKHX C006pa:>KeHHH (.L{JISI KOTOpbIX, BnpOqeM, KaK .L{a-
JIee 6Y,l{eT nOKa3aHO, COOTBeTCTBYIO He KpHTepHH qaCTO MO-
rYT 6bITb nO,l{06paHbI). 3TH aJIrOpHTMbI npHHSlTO Ha3bIBaTb
a2/l0MepamUBflblMU.
nYCTb I (R) - HeKOTopbIH nOKa3aTeJIb KaqeCTBa pa36He-
HHSI. 0603HaqHM 1st = I (Rst) - I (R), r,l{e Rst E 0 (R) -
pa36HeHHe, nOJIyqeHHOe 06'be.L{HHeHHeM 8-ro H i-ro KJIaCCOB R.
3aMeTHM, qTO ,l{JISI KpHTepHSI (5.2) 1st = 2Ast. ECJIH npHpa e-
HH5I 1st MorYT 6bITb OneHeHbI TOJIbKO Ha OCHOBe 3HaqeHHM 1st
,l{JISI npe.L{bI,l{Y HX pa36HeHHH, TO BbIqHcJIeHH5I MorYT 6bITb op-
raHH30BaHbI HenOcpe,l{CTBeHHo no MaTpHne (fst), KaK 3TO
ObIJIO onHcaHO ,l{JISI aJIrOpHTMa «oO'be.L{HHeHHe»: Ha Ka*,l{OM Illa-
re OO'be,l{HH5IIOTC5I Te 8 H t, ,l{JI5I KOTOpbIX 1st MaKCHMaJIbHO.
240
'<
EC H OrpaHHqeHHH no 06'beMY xpaHHMOH HH opManHH He
SlB HIOTCH :>KeCTKHMH, BbIQHCJIeHHSI opraHH3YlOTCH Ha OCHOBe
,nBYX MaTpHn. ,a H .L{aHHOro pa36HeHHSI R == {R 1, ..., R m}
BbIIlHC SlIOTCH MaTpHna Be HIlHH B st (S, t = 1, ..., m), xa paK-
TepH3YIO HX CBSl3H Mem.L{Y KJIaCTepaMH Rs H Rt, H MaTpHna
BeJIHIlHH npHpa eHHH fst, xapaKTepH3YIO HX KaIleCTBO 06'be-
,nHHeHHSI Rs H Rt no <PYHKII,HOHa y f, npHIleM 3HaIleHHSI fst on-
pe.L{e HIOTCH 3HaIleHHHMH Bst. ,aJIH nOKa3aTe SI (5.2) Ha.L{06HO-
CTH B 9TOM He 6bIJIO, TaK KaK .L{JIH Hero HMeeT MeCTO nponopnHO-
HaJIbH3S1 CBH3b 1st = 2Ast = 2Bst.
PaCCMOTpHM COOTBeTCTBYIO YIO ar OMepaTHBHYIO npo[(e-
.L{YPY .L{JISI KpHTepHH (5.1). nYCTb R = {R 1 , ..., R p } - He-
KOTopoe pa36HeHHe. Onpe.L{eJIHM .L{JISI Hero BeJIHqHHbI
Bst= s ail.
IERs /ERt
(5.8)
Bblpa:>KalOrnHe Cpe.L{HIOIO cYMMapHYIO CBH3b OT Rs K Rt. OIle-
BH.L{HO,
p
F(R,a)= Bss-ap.
s= 1
Bhlpa3HM npHparneHHe F st nOKa3aTeJIH F npH 06'be.L{HHeHHH
R s H R t IJepe3 BeJIHqHHbI (5.8). OIleBH.L{HO,
Fst = F (RS t , a) -F(R, a) = [ 1 aij-a (P-l) ] -
Ns+N t l,jERsUR t
- [ aiJ+ aiJ-ap ] .
s i,/ERs t i,/ERt
npH 9TOM BBH.L{Y (5.8)
1 1
a J (N s (Bss+B.t>+Nt(Bts+Btt».
Ns+Nt 1/ ..= Ns+N t
t ERsURt
(5.9)
nO.L{CTaBJISlH Bblpa:>KeHHe c araeMbIX Fst Ilepe3 Bst, nOJIY-
IJHM <POPMY Y .L{JlSl BbIIlHc eHHSI MaTpHnbI (Fst) IJepe3 (Bst):
Fst= Ns Nt (Ns(Bst-Btt) + Nt (Bts-Bss))+a. (5.10)
241
----
OoeJIe 001>e,LlHHeHH5I KJIaCCOB Ra H Ra, ,!{JI5I KOTOpbIX Fst
MaKCHMaJ1bHO, nOJIYlIHM pa36HeHHe R (p - 1) = RaP. 060-
SH811HM Qepe3 Raup = R U Rf}, TaK 11 T06bI HOMepa KJI8CCOB,
He yqaCTBYIO HX B 061>e,!{HHeHHH, OCTaJIHCb npe)KHHMH, H YKa-
)KeM cpOPMYJIbI ,lJ.JI5I nepeCqeTa BeJIHlIHH Bst ,lJ.JIH HOBoro pa30He-
HH5I. Bo H36e)KaHHe nYTaHHnbI Cpe,lJ.HHe CYMMapHbIe CBSl3H
Me:>K,!{Y KJIaCCaMH pa36HeHH5I R (p - I) 6Y,lJ.eM 0603HaqaTb
IllTpHXOM - B;,: '
B: t = Bst (s, t =F a, p);
B u/I.t = N 1 N (N a Bat + N /I B/lt);
a+ P
(5.11 )
B;.aUii = Bsa + Bsp;
I
B U/I. aUI' = N +N (N a (Baa + Ba/l) + Nt\ (BIIr:x.+Bt\/I»'
a p
9TH paBeHCTBa JIerKO npOBepHTb, HCnOJIb3YH (5.8); nOCJIe.n-
Hee paBeHCTBO COBna,lJ.aeT C (5.9).
<I> OPMYJIbI (5.10) H (5.11) HBJI5IIOTCH OCHOBOH peaJIH3a[(HH
arJIOMepaTHBHoro 8JIrOpHTM8, onHcaHHe KOToporo COBnaAaeT
C onHeaHHeM aJIrOpHTMa «06'be,!{HHeHHe» C TOllHOCTblO AO H3Me-
HeHHoro nOKa3aTeJI5I KaqeCTBa. Ha K81KAOM Illare H8 OCHOBe
M8TpHl.(bI (Bst) BbltlHCJI5IIOTCH BeJIHqHHbI Fst (5.10), aInoCJIe
001>eAHHeHH5I Tex KJIaCCOB Ra, H Rf3, ,!{JI5I KOTOpblX Fa,B =
= max Fst > 0, M8TpHua (Bst) nepeClIHTbIBaeTCH no cpOPMY-
JIaM (5.11).
OnHilleM Tenepb arJIOMepaTHBHbIH 8JIrOpHTM AJIH nOKa-
3aTeJISI (5.3). POJIb BeJIHlIHH Bst HrpalOT 3AeCb cYMMapHbIe CBH-
3H Me}K,lty AKJIaCCaMH pasoHeHHH R = {R 1 , ..., R p }:
B.t= ' a'l.
'ERa /eR,
(5.12)
nOK83aTeJIb (5.3) HMeeT CJIe,!{YIO ee npe.l{CTaBJIeHHe B Tep-
MHHax (5.12):
g(R)= ( Bssr + ( tBstr
N . 2
N2_ N
S
242
.j»
-----c
OTCIO,!{a BbIBO,!{HTC5I BbIpa>KeHHe ,!{JI5I npHpa eHMH gst =
= g (RBt) - g (R):
4EN2 B:t+4EBst [ N2 Bkk-Est ail ] +
k i,l = t
+2N s Nt [Est g {R)-N2 (. Bkk r]
EEst (N2_ E sd
E= NZ, Eat=E+2N s N t .
k
,[(JI5I YCKopeHMH paclIeToB no 3TOM cpopMYJIe He06xo,!{MMO
HapH,!{Y C Bst BbIlIMCJI5ITb 3HaqeHH5I N8 E, Est, G (R, 0) =
= Bkk H g (R).
k
HOBaH MaTpHna BeJIHlIHH Bst nOJIYlIaeTC5I CYMMHpoBaHHeM
CTpOK II CTOJIOnOB, cooTBeTcTBYIO HX 06'beAHHHBIllHMC5I KJIac-
CaM.
<1>opMYJIa (5.13) HarJI5IAHO nOKa3bIBaeT, .HaCKOJIbKO YCJIO>K-
HBeTCH paClIeT npHpa eHH5I <PYHKnHOHaJ a [0 xapf KTepI:CTM-
u
KaM JIOKaJIbHbIX CB5I3ell Me>KAY KJIaCCaMH npH YCJIO:>l:eHHM 3TO-
ro YHKUHOHaJIa. .
ArJIOMepaTHBHbIe aJIrOpHTMbI, pa3pa60TaHHbIe H3 3BpHC-
THlIeCKHX co06pa:>KeHMH, B 3TOM CMbICJIe 3H qMTeJIbHO npo e:
OHM MCXO,!{51T H3 OneHKH KalleCTBa 06'be,!{HHeHH5I KJIaCCOB TOJIb-
KO Ha OCROBe JIOKaJIbHbIX CB5I3eM Mem y 060eKTaMH 3THX KJIaC-
COB. npHBeAeM HeKOTopbIe cymeCTBYIO Me KpMTepHH OO'be,!{M-
HeHHH, oe3 KOTOpbIX JIIOOoe M3JIOmeHHe AaHHOM TeMaTHKH ObIJIO
6bI HenOJIHO.
npH 3TOM B COOTBeTCTBHH co CJIOmHBIIleMC5I TepMMHOJIOrH-
eM 6y,!{eM B KalleCTBe nOKa3aTeJIeH CB5I3H OO'beKTOB HCnOJIb30-
BaTb paCCT05lHMe d ij Mem,!{y HMMH (neoTpHQaTeJIbHOCTb H Hepa-
BeHCTBO TpeyrOJIbHMKa He npe):(nOJIarafOTC5I BbInOJIHeHHbIMM),
TaK qTO YMeHbIIleHHe d ij 03HaQaeT B03paCTaHHe OJIH30CTH 000-
eKTOB i H j.
HTaK, nYCTb HMeIOTC5I MaTpHna paCCT05lHHH (d ij ) H pa30He-
HMe R = {R 1 , ..., R p } Ha MHomeCTBe OO'beKTOB. PaCCMOTpMM
CJIe,nYIO He xa paKTepHCTHKM OJIH30CTM KJIaCCOB R s M R t (8, t =
= 1, H., p).
gst=
, (5.13)
r.u-e
b st = min d ij ;
iERa' fERt
b st = max d ij ;
lER s , iERt
(5.14 )
( 5.15 )
243
1
b st = NsN t d ij ;
iERs JER t
b st = dl* i*,
( 5.16)
(5.17)
r,l{e i* H j* - neHTpaJIbHbIe 3JIeMeHTbI Rs H Rt, HanpHMep,
onpe.L{eJISlMbIe YCJIOBHSlMH
d i * i = min d ij , dij* = min d ij .
iERs iERs i ER s iERt ic.R t IERt
BeJIHlIHHa (5.14) xapaKTepH3yeT MHHHMaJIbHOe, (5.15)-
MaKCHMaJIbHOe, (5.16) - Cpe,l{Hee paCCTOSlHHSI MeJK.L{Y KJIaC-
caMH, a (5.17) - paCCTOSlHHe MeJK):(Y Cpe,l{HHMH 06'beKTaMH,
«neHTpaMH T5I:>KeCTH» paccMaTpHBaeMbIx KJIaCCOB.
ArJIOMepaTHBHbIe aJIrOpHTMhI, B KOTOpbIX 06'be,l{IIHeHHe npo-
II3BO,l{HTC5I .L{JI5I KJIaCCOB Rs H R t, MHHIIMH3IIPYIOIIIJIX paCCT05l-
HHe b st , Ha3bIBaIOTCSI: aJIrOpHTM 6.1UJlCfle20 coceaa TIPH b st (5.14),
aJIrOpHTM aa.llbHe20 coceoa npH b st (5.15), aJIfOpIITM cpeofleu
C8J13U TIPH b st (5.16) H aJIrOpHTM 2pynno8blX l{eHmpo8 npH b st
(5.17) .
3TH aJIfOpIITMbI 6bIJIH TIpe,l{JIOmeHbI H3 3BpHCTHlJeCKHX CO-
06paJKeHHH. ,[{JI5I HeKOTopbIX H3 HHX MomHO YKa3aTb KpHTe-
pHH KallecTBa, ,l{JI5I KOTOpbIX MepbI 6JIH30CTH b st xapaKTepH3Y-
lOT npHpa eHHe KpHTepH5I 1st npH 06ne,l{HHeHHH Rs II Rt. TaK,
,l{JI5I Mep (5.14) H (5.15) 3TH KpHTepHH HMelOT CJle,l{YIOIl\HH BH,l{:
f (R) = max min d ij ;
s i,iERs
f (R) = max max d ij .
s l,iER s
,aeHCTBHTeJIhHO, npHpameHHe KpHTepH5I (5.18)
XO.L{e OT R H Rst paBHO 0, eCJIH
min d ij <f (R),
lERs' iERt
(5. 18)
(5.19)
npH nepe-
(5.20)
H paBHO pa3HOCTH 9THX BeJIHlIHH - B npOTHBHOM CJIYlJae. no-
u
CKOJIbKY arJIOMepaTHBHWH aJIrOpHTM HallHHalOT C TpHBHaJIb-
Horo pa36HeHH5I, TO CHallaJIa oT6HpaIOTC5I MeHblIlHe 3HalleHH5I
paCCT05lHH5I, TaK liTO HepaBeHcTBo (5.20) npH 9TOM HeB03MOJK-
HO. TaKHM 06pa30M, min d ij onpe,l{eJI5IeT KalleCTBO 06'be,l{II-
IERs,iERt
HeHH5I Rs H Rt C T01JKH 3peHH5I KpHTepHH (5.18), lITO II Tpe60-
BaJIOCb ,l{OKa3aTb.
244
--
AHa.lIOrI-ft.IHO .L{OKa3bIBaeTCH, liTO (5.19) COOTBeTcTByeT Mepe
(5.15). B TO )Ke BpeMSI .L{.lISI MepbI (5.16), no- BH.L{HMOMY, He CY-
ecTByeT KpHTepH5I, npHpaIneHHe KOToporo ObI.lIO ObI npHMO
CB5I3aHO C (5.16).
AJIrOpHTM O.lIHmHerO coce.L{a OO'be.L{HH5IeT KJIaCCbI, B KOTOpbIX
paCCT05lHHe Me)K.L{Y OJIHmaHllIHMH 3.lIeMeHTaMH MHHHMa.lIbHO no
cpaBHeHHIO C .L{pyrHMH K.lIaCCaMH. CJIe,l{OBaTeJIbHO, .L{Ba OO'beKTa
nona.L{alOT B O.L{MH KJIaCC, eC.lIH cy ecTByeT COe.L{HH5IIO a51 HX
nenOllKa o.lIHmaihllHX .L{pyr K ,l{pyry OO'beKToB. n03TOMY TaKOH
a.lIrOpHTM lIaCTO Ha3bIBalOT MemOOOM OaflOU ce.,fl3U. 5ICHO TaK-
me, qTO nO.lIYtIaeMbIe K.lIaCCbI 51B.lI5IIOTCH C.lIaObIMH cry eHH5IMH.
HanpoTHB, aJIrOpHTM ,l{a.lIbHerO COce,l{a OO'be.L{HH5IeT K.lIaCCbI,
B KOTOpbIX MHHHMa.lIbHO paCCT05lHHe Mem.L{y caMbIMH ,l{aJIeKHMH
,l{pyr OT .L{pyra OO'beKTaMH. 9TO 03HatIaeT, liTO Bce OCTaJIbHbIe
OO'beKTW B nOJIYtIeHHOM nOC.lIe OO'be.L{HHeHH5I K.lIaCCe CBH3aHW
,l{pyr C .L{pyroM e e TeCHee, tIeM «COCe,l{H». n03TOMY a.lIrOpHTM
.L{a.lIbHerO COce,l{a Ha3bIBalOT TaK)Ke MeTO.L{OM nO.llflOU ce.,fl3U.
IlO.lIYllaeMbIe B ar.lIOMepaTHBHOH npOne,l{ype K.lIaCCbI 51B.lI5IIOTCH
crYIII.eHH5IMH.
3a,l{allH K.lIaCcHqHfKan HH C KpHTepHHMH (5.18), (5.19)
npe,l{cTaB.lI5IIOT caMOCT05lTeJIbHbIH HHTepec, nOCKOJIbKY 3TH
KpHTepHH npHMeHHMhI H B C.lIYllae, KOr,l{a CBH3H Me:>K,l{y OO'b-
eKTaMH H3MepeHbI He B KO.lIHlIeCTBeHHOH, a B paHrOBOH IllKa-
.lIe. IlpH 3TOM OllTHMH3anH5I (5.18) CBO,l{HTCH K 3a,l{aqe 0 pac-
KpaCKe rpa<poB a-CXO,l{CTBa (nO.L{pOOHbIH aHaJIH3 CM. B [66,
c. 131-136]). IloKa3aTeJIb )Ke (5.19) COBna,l{aeT C (5.18) ,l{.lIH
CB5I3eH, B351TbIX C npOTHBOnOJIO)KHbIM 3HaKOM (-d ij BMeCTO d ij ).
ArJIOMepaTHBHbIe aJIrOpHTMbI npHMeH5IIOTCH TaK:>Ke ,l{.lIH
.L{aHHbIX, onHCbIBaeMbIX KOJIHQeCTBeHHbIMH npH3HaKaMH. 3TO
u
npHBO.L{HT K cy ecTBeHHoMY cOKpa eHHIO BbItIHCJIeHHH BC.lIe,l{-
CTBHe HCnOJIb30BaHH5I BHYTpHKJIaCCOBWx Cpe,l{HHX KaK npe,l{-
cTaBHTeJIeH K.lIaCCOB. TaK, npH HCnO.lIb30BaHHH B KalleCTBe Me-
pbI CBH3H Me)K.L{y ,l{BYMSI OO'beKTaMH CKaJI5IpHOrO npOH3Be,l{e-
HH5I COOTBeTCTBYIO HX CTpOtIeK Cpe,l{HHH CBSl3b (5.16) eCTb He
liTO HHoe, KaK CKa.lI5IpHOe npOH3Be,l{eHHe neHTpOB T5ImeCTH
(cpe.L{HHX apH<pMeTHlIeCKHX) K.lIaCCOBR s H Rt, T. e. CBH3b Me)K,l{y
cpe,l{HHMH. IlpH HCnO.lIb30BaHHH B KalleCTBe MepbI OJIH30CTH
KBa.L{paTa eBK.lIH.L{OBCKOro paccToHHHH Cpe.L{Hee paccToHHHe
(5.16) Mem.L{Y R s H R t eCTb
1 d 1j = cr: + crr + dt* J*,
N s Nt I I .ER
s t
r.L{e a:, a1 - BHYTpeHHHe .L{HCnepCHH B Rs H Rt, a i* II j* -
neHTpbI TH)KeCTH 3THX K.lIaCCOB [28]. AHaJIOrHtIHO eC.lIH B Ka-
245
qeCTBe KpHTepHSI pa36HeHH5I no KOJIHIIeCTBeHHhIM npH3HaKaM
B3S1Tb CYMMY KBa.L{paTOB eBKJIH.L{OBCKHX paCCT05lHHH Me:>K.L{Y
06'beKTaMH H eHTpaMH COOTBeTCTBYIO HX KJIaCCOB,TO npHpOCT
3Toro KpHTepH5I BbIpa:>KaeTC5I BeJIHIIHHOH [28]
f Ns Nt d
st = - ,. ,*,
Ns+N t
r,ne i*, j* - [(eHTpbI TSlmeCTH (BeKTopbI Cpe.L{HHX) B KJIaCCaX
Rs, R t.
nepeCIIeT pacCTOSlHH5I Mem.L{Y 06'be.L{HHeHHbIM KJIaCCOM
H OCTaJIbHbIMH npOH3BO.L{HTC5I no CJIe.L{YIOlI1.eH <popMYJIe [28]:
bsut, h = as b sk + at b th + bst + l' J b sh -b tk I, (5.22)
KOHKpeTHbIe 3Ha l leHH5I a, H l' B KOTOpOH cooTBeTcTBYIOT pa3-
JIHIIHbIM KOHKpeTHbIM aJIrOpHTMaM: 6JIHJKHerO COce.L{a (as =
1
== at == -"I == "2' == 0); .L{aJIbHerO coce.L{a (as == at =
= '\' = ' = 0); cpe.ll.Heii CBSI3H «(Xs = N s / (N s + Nt),
at = N tf (N 8 + Nt), == "I = 0); [(eHTpOB T5ImeCTH (.L{JIH
KOJIHIIeCTBeHHbIX npH3HaKoB, KOr.L{a i* H j* B (5.17) - BeKTopbI
N s Nt A
apH\fIMeTHlleCKHX Cpe.L{HHX, as == N s + Nt ' at = N s + Nt ' t"=
= -asat, l' == 0); .Memoo Yopoa MHHHMH3a[(HH OTKJIOHeHHH
OT Cpe.ll.HHX, oCHoBaHHbIH Ha (5.21) «(Xs = N: t SNt ' (Xt =
Nk+Nt A -Nk ) B
- Nk+Nt+N s ' P = Nk+Ns+Nt ' "1==0. 3TY JKe cxeMY YKJIa-
.L{bIBaeTcSI H aJIrOpHTM «o6'be HHeHHe» (as == at == 1, ==
="1 == 0), H arJIOMepaTHBHbIH aJIrOpHTM .L{JI5I KpHTepH5I (5.1)
( CM. <t 0 P MY JIbI (5. 11».
CJIe.L{yeT OTMeTHTh, lITO npH KaqeCTBCHHbIX npH3HaKax, 33-
.L{aHHbIX B KOJIHIIeCTBeHHOM BH.L{e 6YJIeBCKHX MaTpH[( 06'beKT-
npH3HaK, TaKme MorYT 6bITb BbIIIHCJIeHbI eHTpbI T5IJKeCTH
KJIaCCOB 06'beKTOB, KOTopble 6Y YT xapaKTepH30BaTbC5I IIaCTO-
TaMH COOTBeTCTBYIO HX 3HaqeHHH npH3HaKOB B .L{aHHOM KJIaCce.
HCnOJIb30BaHHe BeKTopOB IIaCTOT H paCCT05lHHH Mem y HHMH
lIlHpOKO pacnpOCTpaHeHO B aHaJIH3e ,l{aHHbIX.
(5.21 )
5.1.3. H epapXUtteCKue 2pynnupoBKu
U y.abTpaMerpuKa
Pe3YJIbTaTbI npH\1eHeHH5I arJIOMepaTHBHbIX aJI-
ropHTMOB MorYT 6bITb npe CTaBJIeHbI B BH e ,l{eH,LlpOrpaMMbI -
epeBa, oTpalKalOIII.erO HepapxHIO 06'be HHeHHH. Ha P-M ypOB-
246
He 3TOM Hepa p Xliii (p = 1, ..., N) 1i leeTC51 p BepIIIHH, COOTBeT-
CTBYIO HX KJIaCCaM Rs pa36ueHH5I R (p), npHQeM 3TH BepllIli-
HbI coe,l{HHeHbI pe6paMIi C TeMH BepIllHHaMH (p - I)-ro ypOB-
H5I, KOTopble COOTBeTCTBYIOT KJIaCCaM R (p -1), B KOTopble
nepexo,Z(SlT Rs. OqeBH,l{Ho, npH 3TOM, qTO Ka*,l{a51 BepllIHHa
(p - I)-ro ypOBH5I CB5I3aHa JIHllIb C O,l{HOH BepllIHHOH p-ro
u u
ypOBH5I, 3a HCKJIlOqeHHeM O,l{HOM BepllIHHbI, CB5I3aHHOH C TeMH
BYM5I BepI.I1HHaMH p-ro 1 #
ypOoeHb
YPOBH5I, KOTopbIe COOT- _ _ _ _ _ - - - - - -
BeTcTBYIOT 06'be)J.H H 51 eMbIM
KJIaCcaM. TaK, Ha pHC. 5.1
li306paiKeHa ,l{eH,l{porpaM-
Ma, OTBeQalOlI.(a51 pe3YJIb-
TaTaM npHMeHeHH5I aJIro-
pHTM8 «06oe,l{HHeHHe» K
I.I1eCTH 06'beKTaM (CM.
c. 239).
Crre,l{yeT OTMeTHTb npH
3TOM, liTO KpHTepHH TH-
na (5.1) - (5.3) He n03-
BOJI5IIOT ,Z(eH,l{porpaMMe
,nOXO,nHTb ,l{0 KOpH5I (T. e.
u
e,l{HHCTBeHHOH BepllIliHbI
I-ro YPOBHSI) nYTeM no-
CTeneHHbIX 06'be,l{HHeHHH.
HaqHH8S1 C HeKOToporo p
npOBO,l{HTb ,l{aJIbHeHI.I1He
06'be,l{HHeHH5I CTaHOBHTCSI
HeBbIrO,l{HbIM H nOJIyqeH-
HbIe p BepI.I1HH cpa3Y COe,l{HH5IIOTCH B KopeHb. B TO LKe BpeMH
60JIbllIHHCTBO ,l{PyrHx npol.(e,l{YP, B qaCTHOCTH, OTMeqeHHbIe
BeJIHqHHaMH (5.14)-(5.17), (5.21) H T. n., He HMeeT TaKHX
eCTeCTBeHHbIX KpHTepHeB OCTaHOBKH, H B HHX 06'be,l{HHeHH5I
no ,Z(Ba MorYT npOBO,l{HTbCSI nOCTeneHHO ,l{O nonHoro CJIHHHH5I
06'beKTOB B e,l{HHbIH KJIacc.
r'''' HHOr,l{a (HanpHMep, B HCCJle,l{OBaHHHX no 6HOJIOrHlIeCKOH
CHCTeMaTHKe) nOJIyqaeMOe ,l{epeBO caMO npe,l{CTaBnHeT HHTepec
,l{JI5I CO)(ep*aTeJIbHOrO aHaJIH3a H MOJKeT paCCMaTpHBaTbCH
K8K pe3YJIbTaT npHMeHeHH5I aJIrOpHTMa.
TaKHe HepapXHqeCKHe rpynnHpoBKH OTRHqalOTCH OT pac-
CMOTpeHHbIX B n. 3.2.8 TeM, liTO, Bo-nepBbIX, BhInOJIHHIOTCSI CHH-
3Y, nYTeM 06'be,l{HHeHHH, a He.CBepxy, nYTeM pa3,l{eJIeHHH, H, BO-
BTOpbIX, KJIaCCbI <POpMHPYIOTCH He B TepMHHax npH3HaKOB H
HX 3HaQeHHH, a B TepMHHax OO'beKTOB H CBSl3eH MeLK,l{Y HHMH.
-
2 ypa!ellb
6 ypoBeHb
1
2
3 "5 6
PHC. 5.1
247
Co BC5IKOH HepapXHtIeCKOH rpynnHpoBKOH OO'beKTOB MO}l{HO
CB5I3aTb CJIe.L{YIO YIO ynpo eHHYIO Mepy CB5I3H Me)K.L{Y 000-
eKTaMH: a (i, j) paBHO HOMepy Toro YPOBHSI HepapXHH, Ha KO-
TOpOM 06'beKTbI i H j BnepBbIe nona,l{alOT B O,l{HH KJIacc. TaK,
,nJIH Hepa PXHH pHC. 5.1 MaTpHna CB5I3eH a (i, j) HMeeT CJIe,l{YIO-
u
HH BH,l{:
1
1
2 2
3 2
4 2
5 1
6 1
2
2
4
3
1
1
3
2
4
3
1
1
5 6
1 1
1 1
] 1
1 I
5
4
2
3
3
1
1
5
TaK, BBe,l{eHHaSl Mepa a (i, j) 06JIa.L{aeT CJIe,l{YIO HM Y,l{H-
BHTeJIbHbIM CBOHCTBOM. .QJI5I JII06bIX i, j, k E in BbInOJIHeHO He-
paBeHcTBo
a (i, j) min [a (i, k), a (j, k)J.
(5.23)
9TO HepaBeHcTBo BbIpa)KaeT olleBH,l{HbIH <paKT: eCJIH i Ha-
XO,l{HTC5I B O,l{HOM KJIaCCe C k H j HaXO,l{HTC5I B O,l{HOM KJIaCCe c
k, TO i H j npHHa,l{JIeJKaT O,l{HOMY KJIaccy.
Mepa, Y,l{OBJIeTBOp5l1O a51 (5.23), Ha3bIBaeTCSI YllbmpaJvtem-
pUKOt1.
Y JIbTpaMeTpHKa OCTaeTC5I YJIbTpaMeTpHKoH npH JII060M ee
MOHOTOHHOM npe06pa30BaHHH, TaK KaK npH 3TOM He MeH5IeTCSI
ynop5l,l{OtIeHHe 3HalleHHH, a 3HallHT, COXpaH5IeTC5I (5.23).
BC5IKaSl YJIbTpaMeTpHKa YCTpoeHa OlleHb npOCTO. A HMeHHO
B CHJIY (5.23) OlleBH,l{HO, liTO ,l{JI5I JII06bIX i, j, k Bce TPH lIHCJIa
a (i, j), a (i, k), a (j, k) He MorYT 6bITb pa3JIHlIHbIMH: MHHHMaJIb-
Hoe H3 HHX Bce-TaKH He MeHbIlle KaKorO-TO H3 OCTaBIllHXC5I, Tal{
tITO ,l{Ba 06513aTeJIbHO COBna,l{aIOT, a TpeTbe He MeHbIlle HX 06-
ero 3HalleHHSI.
OTclO,l{a CJIe,l{yeT, liTO cOBoKynHocTb nap (rpacp CXO,l{CTBa)
r = { ( i, j) I a (i, j) = max a (i, j) }
I,f
51BJI5IeTC5I OTHOIlleHHeM 3KBHBaJIeHTHOCTH. .QeHcTBHTeJlbHO,
nycTb a (i, j), a (j, k) MaKCHMaJIbHbI. Tor,l{a a (i, j) = a (j, k)
H a (i, k) He MO)KeT 6bITb MeHbIlle 3Toro 06 ero 3HalleHH5I, TaK
KaK Tor,l{a 6bIJIO 6bI a (i, k) < min (a (i, j), a (j, k», tlTO npo-
THBopellHT (5.23). TaKHM 06pa30M, rpacp r nopOm,l{aeT pa3-
oHeHHe MHOJKeCTBa m Ha KJIaCCbI 06oeKToB, CB5I3aHHbIX Mem,l{y
COO OR MaKCHMaJIbHbIM 3HatIeHHeM CBSl3H al = max a (i, j).
I, f
248
-.r.tI.CI
Yoe,!{HMC5I Tenepb, liTO CBH3H Me:>K,!{Y pa3HbIMH 06'beKTaMH
,!{BYX KJIaCCOB 3TOrO pa30HeHH5I O,!{HHaKOBbI. nYCTb i, j E Rs;
k, i E Rt, nOKameM,. liTO a (i, k) = a (j, i).
PaccMoTpHM CHallaJIa CJIYlIaM, Kor,!{a k = i, HMeeM a (i, k),
a (j, k) < a (i, i), TaK KaK (i, k) $ r, (j, k) $ r, HO Tor,!{a
a (i, k) = a (j, k), TaK KaK HepaBeHCTBO Bcex BeJIHIIHH HeB03-
MO:>KHO. nYCTb Tenepb k =1= i. no ,!{OKa3aHHOMY a (i, k) =
= a (i, k) H, CHMMeTpHlIHO, a (k, j) = a (i, j), TaK lITO
a (i, k) = a (j, k) = a (i, j), liTO H TpeOOBaJIOCb ,!{OKa3aTb.
TaKHM 06pa30M, MaKCHMaJIbHOe paCCTOHHHe al H pa30He-
HHe R = {R l , ..., Rm} no rpaCPY al-cxO,!{CTBa C MaTpHneH
CB5I3eH (A (s, t)) Me:>K,!{Y KJIaCCaMH, r,!{e A (s, t) paBHO oo eMY
3HalleHHIO CB5I3eH aij ,!{JI5I i E Rs, j E R t , nOJIHOCTblO onpe,!{eJI5I-
lOT HCXO,!{HYIO YJIbTpaMeTpHKY. npH 3TOM CHCTeMa BeJIHlIHH
A (s, t) = aij (i E Rs, i E Rt) Y,!{OBJIeTBOp5leT (5.23) H, CJIe,!{o-
BaTeJIbHO, HBJIHeTCH YJIbTpaMeTpHKoH. n03TOMY K HeM npHMeHH-
MO npe,!{bI,!{y ee nocTpoeHHe.
ECJIH 0003HaIIHTb pa3JIHIIHbIe 3HalleHHH, npHHHMaeMbIe
aij (i, j E m) lIepe3 al > a 2 > ... > a p , TO rpacpbl ak-cxo,!{-
CTBa (k = 1, ..., p) xapaKTepH3YIOT BJIO)KeHHbIe ,!{Pyr B ,!{Pyra
pa30HeHHH R k : R l R 2 c= ... c= Rp. npH aTOM, OlJeBH,!{HO,
rpa<t> a p CXO,!{CTBa BKJIIOIIaeT Bce napbI H, CJIe,!{OBaTeJIbHO, RP -
YHHBepCaJIbHOe pa36HeHHe. TaKHM oopa30M, ,!{aHHaH YJIbTpa-
MeTpHKa 0,!{H03HallHO onpe,!{eJIHeT HepapXHlIeCKYIO KJIaCCH<pH-
KanHIO, r,!{e pa30HeHHe Rk COOTBeTcTByeT (p - k)-MY YPOBHIO
HepapXHH.
ECJIH a (i, j) = ak, TO OO'beKTbI i H j nona,!{alOT B O,!{HH KJIaCC
BnepBbIe Ha (p - k)-M ypOBHe HepapXHH, TaK liTO nOJIYlIeHHoe
,!{epeBO ,!{aeT Mepy CBH3H a' (i, j) = P - k. COOTBeTCTBHe
ak -+ (p - k) 51BJI5IeTCH MOHOTOHHbIM B CHJIY ynOp5l,!{OlleHHOCTH
a l > ... > ape 3TO 03HaIIaeT, liTO ,!{OKa3aHO CJIe,!{YIO ee YT-
Bep:>K,!{eHHe.
BC5IKaH YJIbTpaMeTpHKa 0,!{H03HaIIHO 3a,!{aeT HepapXHIIeC-
KYIO KJIaCCHcpHKanHIO, KOTOpaH XapaKTepH3yeT 3TY YJIbTpaMeT-
PHKY C TOllHOCTblO ,!{O MOHOTOHHOrO npeOOpa30BaHH5I.
npHBe,!{eM e e HeKOTOpbIe nepecpOpMYJIHpOBKH pe3YJIbTa-
TOB npOBe,!{eHHOrO aHaJIH3a YJIbTpaMeTpHKH.
,[{JI5I BCHKOrO nopora a MaKCHMYM nOKa3aTeJI5I G (R, a)
(5.2) ,!{OCTHraeTCH Ha pa30HeHHH, nopO:>K,!{aeMoM rpacpoM a-
CXO,!{CTBa {(i, j) I aij a}, paBHoM Rk ,!{JI5I MaKCHMaJIbHOrO k,
npH KOTOpOM ah a. npH 3TOM BC5IKHH KJIaCC Rk 51BJI5IeTC5I
crYIneHHeM. BOJIee Toro, cnpaBe,!{JIHBO H 06paTHoe YTBepm,!{e-
HHe: eCJIH npH JIIOOOM a KJIaCCbI OnTHMaJIbHOH no G (R, a) KJIaC-
cHcpHKanHH 51BJI5IIOTC5I crYIneHHHMH, TO Mepa CB5I3H aij-
249
YJlbTpaMeTpHKa. Mepa CB5I3H aij 51BJI5IeTC5I YJIbTpaMeTpHKOH TOr)(a
II TOJIbKO TOr,l{a, KOr,l{a npH JIIOOOM a rpa<p a-CXO,l{CTBa ,l{aeT
OTHOllIeHHe 3KBHBaJIeHTHOCTH (pa30HeHHe).
HeTPY,l{HO YBH,l{eTb TaK:>Ke, qTO onHcaHHble HaMH arJIOMe-
paTHBHble aJIrOpHTMbI npHMeHHTeJIbHO K YJIbTpaMeTpHKe npHBe-
,l{YT K HepapXHH pa30HeHHH, BKJIlOqalO eH HepapXHIO pa30He-
HHH Rk (k = 1, ..., p).
Bce CKa3aHHoe J1pHBO,l{HT K BbIBO,l{Y, qTO MaTpHua CB5I3efi,
Y,l{OBJIeTBOp5l1O a51 (5.23), 51BJI5IeTC5I cpOpMOH npe,l{CTaBJIeHHH
cooTBeTcTBYIOIneH HepapXHqeCKOH rpynnHpOBKH. n03TOMY B03-
HHKaeT H,l{e51 0 TOM, tJTO npOOJIeMY nOCTpOeHH5I HepapXHqeCKOH
u u u
rpynnHpoBKH no 3a,l{aHHOH npOH3BOJIbHOH MaTpHue CB5I3eH MO:>K-
u
HO pernaTb KaK 3a,l{aqy annpOKCHl\1aUHH 3TOH MaTpHUbI CB5I-
3eH C nOMOInblO YJIbTpal\leTpHKH [135, 142, 143].
9Ta H,l{e51 ObIJIa peaJIH30BaHa B [142] CJIe,l{YIOInHM oopa30M.
llYCTb (aij) - MaTpHua CB5I3eH Me:>K,l{Y OO'beKTaMH. PaccMoT-
u .
pHM 3a,l{aqy HaXO:>K,l{eHH5I TaKOH YJIbTpaMeTpHKH ail, KOTopaH
MHHHMH3HpyeT d (a, a*) npH YCJIOBHH, 1JTO aij all )(JI5I Bcex
i, j E ffi. -
OKa3bIBaeTC5I, ee perneHHeM 51BJI5IeTC5I YJIbTpaMeTpHKa, co-
OTBeTCTBYIOIna51 HepapXHqeCKOH rpynnHpoBKe no MeTO,l{Y OJIH:>K-
Hero coce,l{a [142]. ECJIH:me 3HaK HepaBeHCTBa B 3a,l{aqe annpoK-
cHMaUHH H3MeHHTb Ha oopaTHbIH: HCKaTb a* a, MHHHMH3H-
PYIOIII.YIO I a - a* I , TO perneHHe COOTBeTcTByeT HepapXHqeCKOH
rpynnHpoBKe no MeTO,l{Y ,l{aJIbHerO COce,l{a [111] (TaK KaK KPH-
TepHH (5.18) npH 3TOM nepeXO,l{HT B (5.19)).
5.1.4. EbtCTpbte aA20puT Mbl 2pynnupoBKu
ArJIOMepaTHBHbIe aJIrOpHTMbI TpeoYIOT ,l{OBOJIb-
HO OOJIbrnOrO qHCJIa rnarOB (nOp5l,l{Ka N3) no nepeCqeTY pac-
u u
CT05lHHH H cpaBHeHHIO BeJIHqHH npHpaIlJ,eHHH.
Kor,l{a qHCJIO OO'beKTOB HMeeT nOp5l,l{OK TbIC5IqH, BblqHCJIe-
HH5I cy ecTBeHHo YCJIO:>KH5IIOTC5I H3-3a OTCYTCTBH5I B03MO)KHO-
CTH xpaHeHH5I O,l{HOBpeMeHHO Bcero OrpOl\fHOrO KOJIHqeCTBa
K03<P<PHUHeHToB CB5I3H: npHXO,l{HTCH nOJIh30BaTbC5I HH<popl\1a-
lI.HeH B HCXO,l{HOM BH,l{e OO'beKT-npH3HaK, a K03<P<PHUHeHTbI CB5I-
3H MeLK,l{Y OO'beKTaMH paCCqHT BaTb no Mepe HeOOXO,l{HMOCTH.
3TO YMHO:>KaeT OO'heM BbIqHCJIeHHH Ha BpeM5I, HeOOXO,l{HMOe
,l{JI5I paCqeTa K03<P<PHUHeHToB CB5I3H.
B CB5I3H C 3THM B03HHKaeT HeOOXO,l{HMOCTb B TaKHX aJIro-
pHTMax KJIaCcH<pHKaUHH, B KOTOpbIX 6bIJIO ObI YMeHbrneHO KO-
JIHqeCTBO marOB H KOJIHqeCTBO OOpaIII.eHHH K OT,l{eJI bHhIM K03<p-
<pHUHeHTaM CB5I3H. Y,l{OBJIeTBOp5leT 3THM TpeOOBaHH5IM KJIaCC
250
4
aJIrOpHTMOB, B KOTOphIX Kam,LlbIH OO'beKT paCCMaTpHBaeTC5I
JIHWb O,LlHH pa3 Hero CB5I3H OIJ.eHHBaIOTC5I ,l{JI5I MaJIOrO IBICJIa
06'beKTOB.
PaCCMOTpHM CJIe.L{YIO HH aJIrOpHTM TaKoro pO,l{a.
A JI r 0 p H T M «p a 3 M e e H He». no 3TOMY aJI-
ropHTMY CTpOHTC5I pa30HeHHe R = {R 1 , ..., Rm} Ha 3a.L{aH-
Hoe 1JHCJIO m KJIaCCOB (eCJIH HeOOXO.L{HMO, TO H C 3a.L{aHHbIM pac-
npe,l{eJIeHHeI\.I (R)) C nOMO blO pa3Me eHH5I OO'beKTOB no m
51l1eHKaM.
Ha npe.L{BapHTeJlbHOM HYJIeBOM mare B Ka:>K.L{YIO H3 m
51l1eeK nOMe aeTC5I no O.L{HOMY OO'beKTY, T. e. oopa3yeTc5I TPH-
BHaJIbHOe pa30HeHHe RO Ha MHomeCTBe m OO'beKTOB.
CnOCOObI Bbloopa 9THX m OO'beKTOB OOCY.L{HM HHme.
Ha Kam.L{OM H3 nOCJIe.L{YIO HX N - m lllarOB paCCMaTpH-
BaeTC5I O,l{HH H3 elI.(e He pa3Me eHHbIX OO'beKTOB H nOMe aeT-
C5I B TY 51l1eHKY, KOTOpa51 ooeCneqUBaeT MaKCHMaJIbHWH npH-
pOCT nOKa3aTeJI5I 1 (R).
TOllHee, Ha llIare C HOMepOM i = 1, ..., N - m paccMaTpH-
BaeTC5I pa30HeHHe RI- 1 y*e pa3Me eHHbIX m + i-I OO'b-
eKTOB Hero OKpeCTHOCTb, COCT05lma51 113 pa30HeHHH R'-l (s),
nOJIYlIaeMbIX npHCOe.L{HHeHHeM HOBoro OO'beKTa K S-MY KJIaccy
RI-l (s = 1, ..., m). OO'beKT nOMe aeTC5I B TY 51l1eHKY S,
.L{JI5I KOTOpOH MaKCHMaJIbHO npHpa eHHe 1118 = f (RI-l (s)) -
- 1 (RI-l). nOJIYlIeHHOe pa30HeHHe RI-l (s) npHHHMaeTCSI
3a RI, nOCJIe qero HaIlHHaeTC5I CJIe.L{YIO HH mar aJIrOpHTMa.
0IleBH,l{HO, ,l{JI5I KpHTepH5I (5.2) 18 C TOllHOCTblO ,l{O KOHCTaH-
TbI paBHo B cYMMe CB5I3eH (c YlIeTOM nopora cy ecTBeHHocTH
a) HOBoro OO'beKTa C OO'beKTaMH s-ro KJIaCCa pa30HeHH5I RI-I.
,[{JIH KpHTepH5I (5.1) paCtleT Is HeCKOJIbKO CJIO>KHee. nYCTb
1 - HOMep npHcoe,l{HH5IeMOrO K S-MY KJIaccy OO'beKTa, Tor,l{3
f = 1 ( 2 a l e - a. . ) .
s N 8 + 1 N s J
iERs l,iER 8
3aMeTHM, liTO Cpe.L{H5I5I CYMMapHa51 CB5I3b B S-M KJIaCCe
RI-l (S) paBHa CYMMe Cpe,l{HeH CYMMapHOH CB5I3H B S-M KJIaC-
ce RI-l H 18' TaK lITO 3TY BeJIHlIHHY MO:>KHO nepeCqHTbIBaTb
OT lIIara K mary Ha OCHOBe I1f8 H xpaHHTb B naM5ITH. Tor,l{a no-
JIYlIeHHa51 cpopMYJIa .L{JI5I f 8 nOKa3bIBaeT, lITO ,nJI5I ero O[(eHKH
,nOCTaTOllHO BbIIIHCJIHTb CYMMapHYlO CB5I3b HOBoro OO'beKTa c
OO'beKTaMH R8 (KaK H npH HCnOJIb30BaHHH (5.2)), TaK KaK cpe,n-
1
H5I5I cYMMapHa51 CB5I3b N s aij xpaHHTC5I B naM5ITH.
TaKa51 B03MomHOCTb HCnOJIb30BaHHH TOJIbKO JIOKaJIbHOH
251
"
HH<popMalI.HH 0 CB5I35IX 001>eKTa C .L{aHHbIM KJIaCCOM MOmeT HC-
nOJIb30BaTbCSI .L{JISI nOpO)K.L{eHH5I pa3JIHQHbIX 3BpHCTHQeCKHX
aJIrOpHTMOB, paOOTalO HX no cxeMe «pa3MeIneHHe» Ha OCHOBe
KaKOM-JIHOO KOHKpeTHOH MepbI CB5I3H 061>eKTa C KJIaCCOM 000-
eKTOB.
3TO ocooeHHO Ba:>KHO, KOr.L{a BblqHCJIHTb CYMMapHbIe CB5I3H
.L{JI5I Ka:>K.L{orO 001>eKTa (T. e. OneHKa CB5I3eH Bcex 061>eKTOB
Mem.L{Y COOOH) 3aTpY.L{HHTeJIbHO H3-3a OOJIbIllOrO 001>eMa BblqHC-
JIeHHH. B TaKHX CJIyqa5lX CB5I3b 001>eKTa C KJIaCCOM MomHO o[(e-
HHBaTb He CYMMapHbIMH CB5I351MH co BceMH 001>eKTaMH KJIaCCa,
a JIHIllb O.L{HOH CB5I3blO C KaKHM-JIHOO npe.L{CTaBHTeJIeM, [(eHTpOM
KJIaCca. npH KOJIHqeCTBeHHbIX npH3HaKax 3a neHTp KJIaCCa
06bI1JHO npHHHMaeTcH ero n.eHTp TH)KeCTH N 1 x (i). TIpH HH-
slER
8
cpOpManHH TOJIbKO 0 CB5I3S1X Me)K)lY OO'beKTaMH B KaqeCTBe neH-
Tpa KJIaCCa Rs MO)KHO paCCMaTpHBaTb HCXO.L{HbIH 001>eKT, no-
u u
Me eHHbIH Ha npe.L{BapHTeJIbHOM Illare B S-IO SlqeHKY, HJIH :>Ke
Me.L{HaHY-001>eKT i+, MaKcHMH3HPYIO HH CYMMapHYIO CB5I3b
C OCTaJIbHbIMH 001>eKTaMH KJIaCCa ai = aij. npH HCnOJIb30Ba-
jERs
HHH Me.L{HaHbI Ha.L{O xpaHHTb B naM5ITH HaKOnJIeHHYIO CYMMap-
HYIO CBSl3b ai .L{JISI Kam.L{oro 001>eKTa H nepeCQHTbIBaTb ee npH
BKJIlOqeHHH B KJIaCC HOBoro 001>eKTa. nOCKOJIbKY <paKT npHcoe-
.L{HHeHHSI HOBoro 001>eKTa K KJIaccy onpe.L{eJISleTCSI Ha OCHOBe
ou.eHKH ero CB5I3H C Me.L{HaHOH, 3TO 03HaqaeT, qTO CYMMapHbIe
CB5I3H OO'beKTa Tenepb BbIqHCJISlIOTCSI He co BceMH Y:>Ke pa3Me-
eHHbIMH OCTaJIbHbIMH 001>eKTaMH, KaK paHbIlle, a JIHIllb C
001>eKTaMH CBoero KJIaCca. 3TO CYIneCTBeHHO (B Cpe.L{HeM B
Nm ) If!:. u u
2 pa3 CHH)KaeT Ou1>eM BbIqHCJIeHHH no OneHKe CBSl3eH.
EIne oOJIee pe3Ko CHH:>KaeTC5I 001>eM BblqHCJIeHHH npH HC-
nOJIb30BaHHH B KaqeCTBe neHTpa HCXO.L{HorO npe.L{CTaBHTeJI5I
S-H SlQeHKH.
BJIaro.L{apSl TOMY, qTO 3TOT neHTp He MeH5IeTC5I, MO)KHO Bce
001>eKTbI pa3Me aTb no SlqeHKaM O.L{HOBpeMeHHO (a He nOCJIe.L{O-
BaTeJIbHO). .QJISI 3Toro HeOOXO.L{HMO .L{JI5I Ka:>K.L{OrO 001>eKTa i E
OneHHTb ero CB5I3b aij C Ka:>K.L{bIM H3 m neHTpOB j H nOMeCTHTb
ero B TY 51qeHKY S, .L{JI5I KOTOpOH ais = max aij (j = 1, ..., m)
j
(B cJIyqae HeO.L{H03HaqHOCTH MaKcHMYl\tla BbIoHpaeTC5I, HanpHMep,
SlqeHKa C MeHbIIlHM HOMepoM).
3aMeTHM, qTO B OTJIHqHe OT OCHOBHOH npOne.L{ypbI aJIrOpHT-
Ma pe3YJIbTaT STOH MO.L{H<pHKalI.HH He 3aBHCHT OT nOp5l.L{Ka npe.L{1>-
SlBJIeHHSI 001>eKTOB.
252
-
PaCCMOTpeHHbIe 9BpHCTHQeCKHe npHeMbI MOJKHO ClIHTaTb
npHOJIH)KeHH5IMH K pa3Me eHHIO no KpHTepHIO (5.2) B YCJIOBH-
51X, KOr,l{a HeT B03MO)KHOCTH BbIlIHCJIHTb CB5I3H HOBbIX OO'beKTOB
co BceMH Y)Ke pa3MeIneHHbIMH OO'beKTaMH.
Ilpone,l{ypa aJIrOpHTMa «pa3Me eHHe» MomeT nOBTOp5lTbC5I
MHoroKpaTHo, eCJIH Ha OCHOBe pe3YJIbTaTOB pa3MeIneHH5I Bbl-
OHpaeTC5I HOBa51 CHCTeMa HCXOAHbIX m OO'beKTOB. HOBbIe HC-
XO,l{HbIe OO'beKTbl BbIOHpaIOTC5I KaK neHTpbl KJIaCCOB, nOJIYlIeH-
HblX B pe3YJIbTaTe npe,l{bI,l{y ero npHMeHeHH5I aJIrOpHTMa. 3TO
BO MHorHX CJIYlIa5lX n03BOJI5IeT YCTpaHHTb nOCJIe,l{CTBH5I He-
y,nallHoro BbIoopa HCXO,l{HbIX m OO'beKTOB. H3JIO)KeHHYIO npo-
u.e,l{YPY YAOOHO Ha3bIBaTb .MemOOOM l{eHmpoe 1 . MeTO,l{ neHTpOB
u
B eBKJIH,l{OBCKOM npOCTpaHCTBe, HCnOJIb3YIOlI.(HH npH pa3MemeHHH
B KaqeCTBe l'leHTpOB HCXO.L{HbIe OO'beKTbI, a npH nOBTopeHHH-
neHTpbl T5I)KeCTH KJIaCCOB, HCCJIe.L{OBaJIC5I MaK KHHOM B [153].
OH ,l{OKa3aJI, B lIaCTHOCTH, liTO npH CJIYlIaMHOM nOpOm.L{eHHH
OO'beKTOB B pe3YJIbTaTe npHMeHeHH5I 3TOH npoue,l{Ypw aCHMn-
TOTHlIeCKH nOllTH Bcer,l{a nOJIYlIaeTC5I pa30HeHHe, MHHHMH3H-
PYIO ee Cpe,l{HIOIO BHYTpHKJIaCCOBYIO ,l{HCnepCHIO (T. e. MaK-
CHMH3HpYIO ee KpHTepHH (5.1)).
BbIOOp HCXO,l{HbIX m OO'beKTOB B 3HallHTeJIbHOM Mepe onpe-
.L{eJI5IeT pe3YJIbTaTbI pa3MeIneHH5I. 3TOT BblOOp MO)KeT npoHcxo-
u
,l{HTb CJIYlIaHHO - B Ha,l{e:>K,l{e, qTO nOCJIe,l{YIO He HTepan HH
HCnpaB5IT OIllHOKH, OOYCJIOBJIeHHbIe B03MO)KHOCTblO OTHeCeHH5I
K pa3HbIM KJIaCCaM TeCHO CB5I3aHHblX OO'beKToB. Cyw.ecTBeH-
HbIM npeHMy ecTBoM CJIYlIaMHoro Bbloopa npH OOJIbIllHX KO-
JIHlIeCTBax OO'beKTOB 51BJI5IeTC5I npocToTa.
HeCJIYlIaMHbIH BbIOOp HCXO,l{HbIX OO'beKTOB ,l{OJI)KeH OCHO-
BbIBaTbC5I Ha npe,l{CTaBJIeHHH 0 CTpYKType KJIaCTepOB B ,l{aHHOM
MHO)KeCTBe. nO-BH,l{HMOMY, B oo eM cJIyqae B KalleCTBe HC-
XO,l{HbIX CJIe.L{yeT BbIoHpaTb caMbIe OT,l{aJIeHHbIe .L{Pyr OT .L{pyra
OO'beKTbI. 3TO ,l{aeT HeKoToPYIO rapaHTHIO, liTO OHH npHHa,l{JIe-
>KaT K pa3HbIM KJIaCTepaM.
npHBe,l{eM O,nHH npOCTOH 3BpHCTHlIeCKHH aJIrOpHTM, n03-
BOJI5IIO HH BbIOHpaTb He TOJIbKO OO'beKTbl, HO H HX lIHCJIO.
COrJIaCHO 3TOMY aJIrOpHTMY CHallaJIa BbIOHpaIOTC5I TaKHe
i, j E m, tlTO HX CB5I3b aij MHHHMaJIbHa no BceM napaM OO'beKTOB.
ECJIH 3TO HeB03MO:>KHO no KaKHM-JIHOO npHlIHHaM, TO i BblOHpaeT-
CH CJIyqaHHO, a j MHHHMH3HpyeT CB5I3b atj. TpeTHH OO'beKT Bbl-
OHpaeTCH TaK, qTOObl ero CYMMapHa51 CB5I3b C y)Ke OToopaHHbIMH
1 B JIHTepaType HMelOTCH TaK}Ke Ha3BaHHH: MeTO MaK KHHa H
MeTO k-cpe,nHHX (CM., HanpHMep, [2]).
253
06'beKTaMH l H j 6bIJIa MHHHMaJIbHa. 11 TaK ):(aJIee: eCJIH OTOOpaHO
1 06'beKTOB, TO B KaqeCTBe (I + I)-ro 6epercSI 06'beKT, MHHM-
MM3HPYIOI..I.(MH CYMMapHYIO CB5I3b C HMelOI..I.(HMHC5I 1 OO'beKTaMH.
BbI60p lIHCJIa STHX OO'beKTOB OCYI..I.(eCTBJISleTC5I Ha OCHOBe
CJIe):(YIOI..I.(HX COOOpa)KeHHH. nYCTb B peaJIbHOH CHCTeMe HMeeT-
C5I HeKOTOpOe lIHCJIO, HanpHMep m, KOMnaKTHbIX KJIaCCOB. Tor-
):(a nepBbIe m OO'beKTOB 0Y,!{YT npHHa,!{JIejJ{aTb pa3HbIM KJIaCCaM,
TaK tITO BeJIHlIHHa CYMMapHOH CBSl3H XOT5I H oy,!{eT paCTH C YBe-
JIMlIeHHeM tIHCJIa OO'beKTOB, HO He CJIHIIlKOM ObICTpO. CJle,llYIO-
I..I.(HH, (m + I )-H OO'beKT 005l3aTeJIbHO oy,!{eT HaXO,!{HTbCH B O,!{-
HOM KJIaCCe C KaKHM-JIHOO H3 Y)Ke oToopaHHbIx OO'beKTOB H
oy,!{eT HMeTb C HMM OOJIhllIYIO CBSl3b, TaK liTO npOH30H,!{eT pe3-
Koe H3MeHeHHe CYMMapHOH CBSl3H. aTO pe3Koe H3MeHeHHe H
SlBHTC5I CHrHaJIOM, liTO (m + I)-if OO'beKT y:me JIHIIlHHH H HY*-
HO OCTaHOBHTbC5I Ha nepBbIX m OO'beKTax. Pe3KOCTb H3MeHe-
HH5I MO)KHO OneHHBaTb 3apaHee BbIopaHHOH noporOBOH BMH-
u
tIHBOH.
npHMeHHM paCCMOTpeHHbIe H,!{eH K MaTpH[(e CB5I3eH lIleCTH
OO'beKTOB (C. 121 ) :
1 2 3 4 5 6
1 0,58 0,58 0,75 0,33 0,33
2 0,92 0,58 0,5 0
3 0,92 0,33 0,67
4 0,33 0,67
5 1,5
6
MHHHMaJIbHbIM 51BJI5IeTC5I 3JIeMeHT 6 = 0; oJlH)KafIIIIHH
K {2, 6} 9JIeMeHT I HMeeT cYMMapHYIO CBSl3b C 2 H 6, paBHYIO
0,58 + 0,33 = 0,91. TaKoe H3MeHeHHe (no cpaBHeHHIO C HY-
JIeM) CJIe,!{yeT ClIHTaTb CJIHIIlKOM OOJIbIlIHM, TaK KaK OHO cyw.e-
CTBeHHO npeBbIlIlaeT cpe,!{HIOIO CB5I3b Me>K,!{y OO'beKTaMH 0,6.
TaKHM oopa30M, HCXO,!{HbIMH 51BJI5IIOTC5I OO'beKTbI2 H 6. OO'beK-
TbI I, 3, 4 OJIH)Ke K 2, tIeM K 6, a 5 OJIH)Ke K 6. DOJIYtIaeM pa3-
oHeHHe {1-2-3-4, 5-6}. Me,!{HaHa KJIaCCa {I, 2, 3, 4} -
3TO OO'beKT 3, a B {5, 6} JIIOOOH OO'beKT Me,!{HaHHbIH. HallHHa51
C OO'beKTOB 3, 5 KaK HCXO,!{HbIX, On5lTb nOJIYtIHM pa30HeHHe
{1-2-3-4, 5-6}, KOTopoe 51BJI5IeTC5I OKOHtIaTMbHbIM.
PaccMoTpHM Tenepb npone,!{yPY npH CJIYlIaHHoM BhIoope
OO'beKTOB I, 2 KaK HCXO,!{HbIX. OO'beKTbI 4,6 OJlHjJ{e K I, 001>eK-
TbI 3,5 - K 2, tITO ,!{aeT pa30HeHHe {1-4-6, 2-3-5}.
Me,!{HaHOH nepBoro KJIaCCa 51BJI5IeTCH 4, BToporo - 2. 001>-
eKTbI 1,3,6 OJIH>Ke K 4, OO'beKT 5 - K 2, TaK liTO Ha STOR HTe-
panHH nOJIYtIaeM {1-3-4-6, 2-5}.
254
Me,nHaHOH nepBOrO KJIaCCa 51BJI5IeTC5I 4, BToporo - JII060R
H3 06'beKTOB 2, 5. TaK KaK 4, 2 y}Ke paccMaTpHBaJIHCb, 6epeM
B KaqeCTBe HCXO,nHbIX 4, 5, qTO npHBO HT K YCTOHqHBOMY pa3-
6HeHHIO {1-2-3-4, 5-6}, MeAHaHaMH KJIaCCOB KOTOpOrO
HBJIRIOTCR CHOBa 06'beKTbl 4, 5.
5.1.5. AAZOPUT Mbt nepeMeU{eliUR o6'OeKToB
PaccMoTpHM aJIrOpHTMbI rpynnHpoBKH, OCHOBaHHble Ha
HAee nepeMe eHH5I 06'beKTOB H3 KJIaCCa B KJIacc. CHCTeMa OK-
pecTHocTeH AJI5I pa36HeHH5I R B 3TOM CJIyqae COCTOHT H3 pa3-
oHeHHR R', nOJIyqeHHblX H3 R ,nonYCTHMhIMH nepeMerneHH5I-
MH OO'beKToB.
nYCTb R = {Rt, ..., Rm} 3aAaHHoe pa30HeHHe MHO}KeCT-
Ba m H 8 - HeKoTopoe nO.L{MHOmeCTBO OO'beKTOB. AcconHH-
pyeM co BCHKHM t = 1, ..., m HOBoe pa30HeHHe R , KOTopoe
nOJIyqeHO H3 R CJIeAYIO HM o6pa30M: Bce OO'beKTbI i E 8 ne-'
peMe aIOTC5I B KJIacc R t; COCT05lHHe OCTaJIbHblX OO'beKTOB He
MeHHeTCH. <l>HKcHpyeM HeKOTopoe ceMeHCTBO MHO}KeCTB
= {SI' ..., 8M}, B KaqeCTBe oKpecTHocTH O (R) npHHH-
MaeTC5I 01:(R) = {R klt= l,...,m, k= 1,...,M}.
OOWqHO B KaqeCTBe Sk paccMaTpHBaeTC5I o.L{H03JIeMeHTHoe
MHO}KeCTBO {k} (k E in), TaK qTO M = N. B 3TOM CJIyqae
O}:t (R) COCTOHT H3 BceB03MO}KHbIX pa36HeHHii, nOJIyqaeMbIX
H3IR nepeMe eHHeM JIIOOoro O.L{HOrO OO'beKTa H3 KJIaCCa B
KJlacc. ,[{JI5I YMeHbIlleHH5I 06'beMa nepeoopa HCnOJIb3YIOTC5I
TaK}Ke OKpeCTHOCTH Of (R) = (R It = 1, ..., m}, r.L{e R:-
pasoHeHHe, nOJIyqeHHOe nepeMe eHHeM .L{aHHOrO OO'beKTa
i E m B KJIaCC Rt (OqeBHAHO, npH 3TOM = { {I}, ..., {N}}.
.- t B 3THX TepMHHax MO}KeT ObITb C<POPMYJIHpOBaHa cxeMa
u
aJIrOpHTMoB JIOKaJIbHOH onTHMH3anHH.
A JI r 0 p H T M «n e p e M e rn. e H He». HCXOA5I H3 3a-
AaHHoro pa30HeHH5I R H CHCTeMbI MHO}KeCTB B O (R) Bbl-
oHpaeM TaKoe pa30HeHHe R', KOTopoe .L{aeT MaKCHMaJIbHblH
u
nOJIO}KHTeJIbHbIH npHpocT onTHMH8HpyeMoMY nOKa3aTeJIIO
f (R). 9TO pa30HeHHe R' npHHHMaeTC5I 3a R H nponecc
nOBTOp5leTC5I .L{O Tex nop, nOKa npHpocT f (R) He CTaHeT Heno-
JIO*HTeJIbHbIM .L{JI5I Bcex R' E O!, (R).
,QJI5I YMeHbIlleHH5I 06'beMa nepe60pa Ha Ka*.L{OM lIlare MO-
>KeT' paCCMaTpHBaTbCH TOJIbKO Ta qaCTb OKpeCTHOCTH 01:(R), KO-
TopaH nopO:>K.L{aeTC5I O.L{HHM KOHKpeTHbIM MHO:>KeCTBOM S E ,
T. e. Os (R) = {R I t = 1, ..., m} - cOBoKynHocTb pa36He-
HHii: nOJIyqaeMbIX 113 R nepe lern.eHHeM 8 i (npH 3aAaHHOM i)
255
"
B i-H KJIaCC (no BceM t = 1, ..., m). CcpOPMYJIHPyeM TOqHee
B03HHKalO YIO cxeMY aJIrOpHTMOB.
A JI r 0 p H T M «n 0 C JI e ,l{ 0 BaT e JI b Hoe n. e p e-
M e In e H H e». O HH mar SToro aJIrOpHTMa COCTOHT H3 M STa-
nOB. Ha k-M STane (k == 1, ..., M) 0603peBaeTc5I pa36HeHHe
R, nOJIyqeHHOe B pe3YJIbTaTe npe.L{bI.L{y ero STana, H OTbIC-
KHBaeTCH TaKoe pa36HeHHe B oKpecTHocTH OSk (R) =
= {R It = I, ..., m}, KOTopoe ,l{aeT MaKCHMaJIbHOe nOJIO-
11.
)KHTeJIbHOe npHpa eHHe nOKa3aTeJIIO f (R). 3TO pa36HeHHe
51BJI5IeTC5I pe3YJIbTaTOM k-ro STana H 0603peBaeTc5I Ha CJIe-
,l{YIO eM (k+ l)-M STane. fIpH k=M mar aJIrOpHTMa 3aKaHqH-
BaeTC5I, nOJIyqeHHOe pa36HeHHe 51BJI5IeTC5I HCXO,l{HbIM ,l{JI5I CJIe-
,l{YIO ero mara.
Ilponecc HaqHHaeTC5I C HeKoToporo 3a,l{aHHOrO pa36HeHH5I
R H OKaHQHBaeTC5I, eCJIH Ha ,l{aHHOM mare pa30HeHHe He H3Me-
HHJIOCb. HHOr,l{a ,l{JI5I YMeHblIleHHSI 06'beMa BbIqHCJIeHHH qHCJIO
marOB cpHKcHpyeTc5I 3a paHee.
AJIrOpHTM nOCJIe,l{oBaTeJIbHOrO nepeMe eHH5I npH S 11. =
= {k} H3BeCTeH TaK)Ke KaK aJIrOpHTM «nepeHH,l{eKCan H 5I»;
B HeM nOCJIe,l{OBaTeJIbHO 06'beKTbI i E m nepeMe aIOTC5I B HaH-
JIyqmHe (Ha ,l{aHHOM STane) KJIaCCbI l [5, 25].
nOH5ITHO, qTO pelIIeHHe no aJIrOpHTMaM nepeMe eHH5I 3a-
BHCHT OT HaqaJIbHOrO pa36HeHH5I H CHCTeMbI , a npH nOCJIe,l{o-
BaTeJIbHbIX npoue,l{ypax - H OT nOp5l,l{Ka HYMepau.HH MHO-
meCTB S 11. (B aJIrOpHTMe «nepeHH,l{eKcanH5I» - OT HaqaJIbHOrO
pa36HeHH5I H HYMepanHH 06'beKTOB).
PaCCMOTpHM npHpa eHHe nOKa3aTeJI5I (5.2) npM nepexo,l{e
OT R K R npH ,l{aHHOM S c= m.
0603HaqHM R g - S = Sq (q = 1, ..., m), TOr.L{a
hG=G(R )-G(R)=2 [ (aij-a)-
IES iERt
- (au-a) ] . (5.24)
iES q= 1 iES q
OTCIO.L{a CJIeJI.yeT, qTO npHpaIneHHe G nOJIOmHTeJIbHO Tor-
,l{a H TOJIbKO Tor,l{a, KorJ{a CYMMapHa51 CB5I3b (3a BbIqeTOM no-
pora) 06'beKTOB H3 S C 06'beKTaMH KJIaCCa R t 60JIblIle, qeM
C OCTaBlIlMMHC5I 06'beKTaMH Tex KJIaCCOB, H3 KOTOpbIX OHH
B351TbI.
I
t
'"
I
1 3TOT aJirOpHTM HCnOJlb30BaJlCH, B tJaCTHOCTH, C. PeHbe npHMe-
HHTeJlbHO K CPYHKII.HOHaJIY (4.7) [159].
256
ECJIH R OnTHM3JIbHO, TO G HenOJIOJKHTeJlbHO npH JII060M
S m. TaKHM oopa30M, cnpaBe,nJIHBO CJIe,l{YIOll1.ee Heooxo..
,l{HMOe YCJIOBHe OnTHMaJIbHOCTH. ECJIH R - OnTHMaJIbHOe pas..
6HeHHe, TO ,l{JI5I JIlOooro MHomeCTBa S CYMMapHa51 CB5I3b (33
BblqeTOM nopora) ero OO'beKTOB C OCTaJIbHbIMH OO'beKTaMH co..
,l{ep>Kall1.HX HX KJIaCCOB npeBbIlIlaeT CYMMapHYIO CB5I3b S C JIIO.
6bIM H3 KJIaCCOB R.
OTCIO,na, B qaCTHOCTH, CJIe,l{yeT, 1JTO JIIOOoe nO,l{MH01KeCTBO
OO'beKTOB, CO,l{ep)Kall1.eec5I B KaKOM-JIH60 KJIaCCe OnTHMaJIbHOro
pa30HeHHH, HMeeT CYMMapHYIO CB5I3b (c yqeTOM nopora) C OC-
TaJIbHbIMH OO'beKTaMH 3Toro KJIaCCa, He MeHblIlYIO, qeM C OO'b"
eKTaMH JIIOOoro ,l{Pyroro KJIaCca. nepeXO,l{51 K ell1.e oOJIee qaCT-
HOMY cJIyqalO, nOJIY1JaeM, qTO B OnTHMaJIbHOM pa30HeHHH Ka}K..
,l{blH OO'beKT CB5I3aH (c yqeTOM nopora) C OO'beKTaMH CBoero
KJIaCca. TeCHee, qeM C OO'beKTaMH JIIOOoro ,l{pyroro KJIaCca. .a-lIH
OO'beKTa i, TaKoro, 1JTO S = {i}, <popMYJIa (5.24) ynpocTHTCSI
cJIe,l{YIOll1.HM oopa30M:
L\a=G(R ).-G(R)=2 (aij--a)-2 (aij-a), (5.25)
iER, iER,
I=/=i
r.ne Rs - 3TO TOT KJIaCC R, KOTOpbIH CO,l{ep:>KHT i.
<l>opMYJIa (5.25) HCnOJIb3yeTc5I B aJIrOpHTMe «nepeHH,l{eK..
CaIl.HSI» ,l{JISI BbIqHCJIeHHSI npHpall1.eHHH <PYHKII.HOHaJIa (5.2) npH
nOCJIe,l{OBaTeJIbHblX nepeMell1.eHH5IX OO'beKToB.
AHaJIOrHqHYIO <POPMYJIY HeTPY,l{Ho nOJIyqHTb H ,nJIH nOKa.
SaTeJIH (5.1):
2 aiJ - ail
IERt 1,/ER t
L\F=F(R )-F(R)= -
Nt(Nt+l)
2 atJ- ail
IERs 1,/ERs
N s (N s +l)
CJIe,l{yeT OTMeTHTb, qTO nOJIyqeHHbIe 3,l{eCb Heooxo,nHMbIe
YCJIOBH5I OnTHMaJIbHOCTH pa30HeHHSI no nOKa3aTeJIIO (5.2)
n03BOJIHIOT YCHJIHTb aJIrOpHTM npo, oCHoBaHHbIM TOJIbKO Ha
YCJIOBH5IX n. 5.1.1, CJIe.nYIOIII.HM oopa30M.
A JI r 0 p H T M «n P ° n» (n p 0 B e p K a, pac C bI-
n a H H e, 0 0 'b e ,n H H e H He, n e p e M e III. e H H e If JI H
n e p e II H ,l{ e K C a II. H 51). Ha OOIII.eM mare aJIrOpHTMa BbI-
nOJIHHeTCH OOll1.HH lIlar aJIrOpHTMa npo, T. e. ,nJI5I paCCMaT
(5.26)
K. \ 74
57
pMBaeMoro pa36MeHMSI R == {Rt, ..., Rm} npOH3BO.L{liTCSI pac-
CblnaHHe KJIaCCa R s C A 8S < 0 MJIH 06'be.L{MHeHMe KJIaCCOB R s
Ii R t C A 8t > O. O.L{HaRo nOCJIe BC5IKOrO paccblnaHM5I MJIM 06'be-
,l(liHeHMSI 3.L{eCb npliMeH5IeTC5I e e aJIrOpMTM «nepeMH.L{eKCc.p.{H5I»
MJIM «nOCJIe.L{OBaTeJIbHOe nepeMeIII.eHMe».
BbP.IMCJIeHMSI OKaHlIliBaIOTC5I, Kor,l{a ,l{JI5I paccMaTpMBaeMo"
ro pa36MeHMR BbI nOJI HeHbI He TOJIbKO YCJIOBM5I Ass 0,
Ast 0, HO M G (5.25) (HJIM (5.24)) HenOJIO)KMTeJIbHO .L{JI5I Bcex
i M t.
Oc06eHHo 3qxpeKTMBHo npMMeHeHMe «nepeMH.L{eKCaUMM»
nOCJIe Ka:>K.L{OrO IIlara aJIrOpMTMa «06'be,l{MHeHMe».
3aMeTMM, lITO npM BbInOJIHeHMM nepeMeIneHMH OCHOBHa51
CJIO:>KHOCTb COCTOMT B BbIIIHCJIeHMM npHpa eHMH I1f paCCMaT-
pMBaeMoro nOKa3aTeJI5I npM nepeMe eHM5IX. ,[{JI5I YCKopeHM5I
M ynpOIneHli51 3Toro nponecca MO:>KeT npMMeH5ITbC5I MeTO,l{ BTO-
pbIX pa3HocTeH [8], oCHoBaHHbIH Ha npe.L{CTaBJIeHMH nepBoH
pa3HocTM f B BM.L{e KOM6HHanMM pa3HocTeH npHpaIneHHH I1f
(npM nOBTopHbIX nepeMe eHH5Ix), KOTopbIe qaCTO 6bIBalOT 60-
JIee npocTbIMM li3-3a npocToH CTPYKTYPbI <PYHKQMOHaJIa f. MbI
npo,l{eMoHcTpHpyeM 3TOT MeTO.L{ ,l{JISI nOKa3aTeJISI (5.3) B n. 5.2.6.
PaccMoTpMM aJIrOpMTM «nepeMH,l{eKcaUM5I» ,l{JI5I MaTpliQbI
CB5I3eH IlleCTH 06'beKTOB npM nopore a == 0,6 (CM. c. 239).
nYCTb MCXO,l{HOe pa36MeHMe eCTh R == {1-2-6, 3-5, 4}.
npMpa eHMe G npM nepeHoce 1 BO BTOpOH KJIaCC paBHo B
CMJIY (5.25) 0, a B TpeTMH KJIaCC - 0,88. nOJIO:>KMTeJIbHOCTb
npHpa eHM5I ,l{HKTyeT He06xo,l{MMOCTb nepeMeIneHMSI 1 B Tpe-
-THH KJIaCC, TaK lITO nOJIYlIaeM HOBoe pa36MeHMe {2-6, 3-5,
1-4}. ,aJI5I3TOrO HOBoro pa36MeHM5I paccMaTpMBaeM BapHaHTbI
nepeHoca 06'beKTa 2: npM nepeHoce 2 B KJIaCC {3, 5} nOJIYlIa-
eM B CMJIY (5.25)
I1G = 2 [ (0,32 - 0,1) - (-0,6)] == 1,64,
npH nepeHoce 2 B {1,4}
I1G == 2 [ (-0,02 - 0,02) - (-0,6)] == 1,12;
TaK, HOBoe pa36HeHMe R == {6, 2-3-5, 1-4}. PaccMoTpMM
Tenepb BapMaHTbI nepeHoca 3 B nepBbIH KJIaCC ( G ==
== 2 [0,07 - (0,32 - 0,1)] < 0) H B TpeTMH KJIaCC ( G =
== 2 [ (-0,02 + 0,32) - (0,32 - 0,1)] > 0), lITO .L{aeT R =
= {6, 2-5, 1-3-4}. CB5I3M 4 C {I, 3} 60JIbIlle, lIeM C 6
MJIM C {2, 5}, TaK lITO 06'beKT 4 He HY:>K.L{aeTC5I B nepeMe eHMM.
B TO :>Ke BpeM5I 06'beKT 5 liMeeT CBSl3b C 6, npeBocXO.L{51mylO Bce
-ero OCTaJIbHbIe CB5I3H (KaK oTpMnaTeJIhHbIe), M .L{OJI)KeH 6bITb
258
t
nepeMerneH B nepBbIH KJIaCC, liTO .L{aeT R = {5-6, 2, 1-3-4}.
OO'beKT 6 HMeeT C 5 60JIee TeCHble CBH3H, qeM C 2 HJIH {I, 3, 4},
II OCTaeTCH Ha MeCTe.
TaKHM o6pa30M, B pe3YJIbTaTe nepBoro IIlara «nepeHH-
.L{eKCanHH» pa36HeHHe {1-2-6,3-5,4} nepelIlJIO B R =
= {1-3-4, 2, 5-6}. HaqHHaeM onHTb npoQecc nepeMerne-
HHH 06neKTOB B nopH.L{Ke HX HOMepOB.
06neKT 1 HMeeT cYMMapHble CBH3H C {3, 4} 0,13, KOTopble
OOJIbIlle, qeM ero CBH3H C {5, 6} HJIH C {2} , n09ToMY OCTaeTC5I
Ha MeCTe. 06'beKT 2 OJImeH ObITb nepeMerneH B KJIaCC {I, 3, 4 },
TaK KaK ero cYMMapHaH CBH3b C 3THM KJIaCCOl\f nOJIO}KHTeJIbHa
(0,28). nOJIYlIeHHoe pa30HeHHe {1-2-3-4, 5-6} Y.L{OBJIeT-
BOp5leT HeOOXO.L{HMOMY YCJIOBHIO OnTHMaJIbHOCTH: B CHJIY (5.25)
G 0 npH JIIOObIX nepeMerneHHHX OOneKToB, qTO OKaHqH-
BaeT nponecc. JIoKaJIbHbIH (OH me H rJIOOaJIbHbIH) onTHMYM
eCTb R = {1-2-3-4, 5-6}.
5.2. 3AAAIJH BbISlBJlEHHSI CTPYKTYPbI
5.2.1. lloKaaaTeAu KatteCTBa paa6uefluu
oJ
co CTPYKTYpOU
HanoMHHM, QTO pa36HeHHe co CTPYKTYPOH
(CTPYKTYPHbIH npH3HaK) (R, x) 3a.L{aeTCSI pa30HeHHeM R =
= {R 1 , ..., Rm} C npoHYMepoBaHHbIMH KJIaCCaMH H oHHap-
HbIM OTHOIlleHHeM % c= {I, ..., m} X {I, ..., m}, xapaKTe-
pH3YIOrnHM CTPYKTYPY CBH3eH Me)K Y KJIaCCaMH. 3Ta napa
cooTBeTcTByeT 6HHa PHOMY OTHOlIleHHIO p c= m X 91, .L{JIH KOToporo
(i, j) E p TOr.L{a H TOJIbKO TOr.L{a, KOr.L{a (8, t) E x, r.L{e 8 - HOMep
KJIaCCa R, CO.L{epJKarnerO i, a t - HOMep KJIaCCa R, cO.L{ep-
)Karnero j. HHbIMH CJIOBaMH, p = U Rs X Rt. CTpYKTypa x
( S , t) E)'.
MomeT 6bITb 3a.L{aHa m X m 6YJIeBOH MaTpHneH, (8, t)-H 3JIe-
MeHT KOTOpOH paBeH 1 TOr.L{a H TOJIbKO Tor.L{a, Kor,na (5, t) E x.
Oo06rneHHeM 3TOH KOHcTpYKnHH 51BJI5IeTCSI BeCOBaH CTPYKTY-
pa, 3a,l{aBaeMaH m X m MaTpHneH A = (Ast) CBH3H Me)K,l{Y
KJIaCCal\1H, r,l{e Ast He OOSl3aTeJIbHO paBHO HYJIIO HJIH e,l{HHHne,
a MO)KeT npHHHMaTb JIIOObIe 3HaqeH H5I. B qaCTHOCTH, npe,l{CTaB-
JISleT HHTepec 3a,l{aHHe Ast = I/VNsN t .
llYCTb MaTpHna a = (aij) CBSl3H Me)K,l{Y OOneKTaMH 3a,l{aHa.
TpeoyeTcH HaHTH pa30HeHHe CO CTPYKTYPOH (H3 3a,l{aHHOro
KJIaCCa E), HaHJIyqIIIHM oopa30M XapaKTepH3YIOrnee 3TY MaTpH-
ny.
9*
259
--
...
t1>6pMYJIa (4.25) npHBoAMT K cJte.nYlOrneMY KpHTepHIO ka-
tIeCTBa pa30MeHHH C npOH3BOJIbHOH 3a,l{aHHoH BecoBoH CTPYK-
TYPOR A:
m
G (R, A) = Ast (aij-A st ).
s,t = 1 iERs iERt
(5.27)
I1pH J.st = V ,lJ;JHI (s, t) E x H J.st = 0 )],.1151 (s, t) $. x
NsNt
<PoPMYJIa (5.27) ,l{aeT B COOTBeTCTBHH C (4.29)
F (R, x, a) = y aij -a I x I. (5.28)
(s,t)EX s t lERs IERt
B tJaCTHOCTH, npH ASh npHHHMalOInHX O,l{HO H3 ,l{BYX <p HK -
cHpOBaHHhIX 3HalleHHH a, p, KpHTepHeM KalleCTBa 5IBJI5IeTC5I
G (R; 'X, a) = (aiJ-a), (5.29)
(s,t)EX IERs/ERt
r,l{e a - <pHKcHpoBaHHbIH nopor cy ecTBeHHOCTH CB5I3H, a
x - CTpYKTypa, OTBeqalO a51 O.llHOMY H3 3HatIeHHH ex, p.,
npH nOHCKe OnTHMaJIbHOrO OTHOllIeHH5I B KJIaCCe npOH3-
BOJIbHbIX (ex, p)-MaTpHU B03HHKaeT oOJIee CJIO}KHbIH nOKa3a-
TeJIb BH,l{a (4.24):
g(R,x)= [G ' 60»tS + [G( ;C,O)]S ,(5.30)
N2- N s Nt
(s,t)Ex (s,t)fx
rAe 'X-)(OnOJIHeHHe 'X.
nOKa3aTeJIH (5.28)-(5.30) oooOInalOT KpHTepHH (5.1)-
(5.3), KOTopbIe nOJIYllaIOTC5I npH 'X = { (s, t) I s = t}. Pa30He-
.HHe co CTPYKTYPOR (R, 'X), MaKCHMH3HpYIOInee (5.30), O,l{HO-
BpeMeHHO MaKcHMH3HpyeT (5.29) npM nopore a = (ex + P)/2
(rAe a H p onpeAeJIeHbI cpopMYJIaMH (4.21) H (4.22», xapaKTe-
v
_pH3YIO eM cpeAHee cpeAHHx BHYTpeHHHx H BHeIllHHX CB5I3eH
)(JI5I (R, 'X).
AHaJIOrHtIHbIH nOKa3aTeJIb npM OnTHMaJIbHbIX Ast nOJIYlIa-
eTC5I H3 (5.27) (CM. (4.27»:
a (R, x) = N IN [ }:}: a iJ ] 2 · (5.31)
(s,t)Ex s t IfRs IERt
OnTHMaJIbHa51 B CMbICJIe (5.5) napa (R, 'X) o,nHoBpeMeHHo
OnTHMaJIbHa M ,l{JI5I (5.1) npM Asf = as t.
nOKa3aTeJIH (5.28)-(5.31) HCqepnbIBalOT KJIaCC nOKa3aTe-
JIeH KatIeCTBa pa30HeHHH co CTpYKTypal\'IH, KOTopbIe B03HHKalOT
,
260
B paCCMOTpeHH X 3aAaqaX CTPYKTypHOrO aKTOpHOrO aHanH-
38. Cne,l{yeT OTMeTHTb, qTO, BOOO e rOBOp5l, HHOr,l{a (5.28)
TpeoyeT He MaKCHMH3aUHH, a MHHHMH3aUHH (TIpH OTpHUa-
TeJIbHOM A B (4.19», OAHaKO cpopMaJIbHO MHHHMH3aUH5I (5.28)
3KBHBaReHTHa MaKCHMH3aUHH 3Toro TIOKa3aTenH OTHOCHTeRb-
HO HCXO,l{HOH: MaTpHUbI CB5I3eH: (-aij) TIPH TIopore -a. n09TO-
MY ,l{aRee OrpaHHQHMC5I TOJIbKO 3a,l{aQaMH MaKCHMH3aUHH pac-
u
CMaTpHBaeMbIX TIOKa3aTeJIeH.
MHO)KeCTBO Bcex B03MOLKHbIX pa30HeHHH: co CTPYKTYPOH
(R, x) 0003HaQHM qepe3 P, a MHO)KeCTBO Bcex (R, x), AR5I
KOTOphIX R CO,l{ep)KHT pOBHO m KJIaCCOB, -qepe3 Pm. ,[(PyrHe
CTIeUHaRbHbIe MHO>KeCTBa MaKpOCTpYKTYP OYAeM BBOAHTb npH
paCCMOTpeHHH.
HeTPYJtHO BH,l{eTb, qTO OTITHMaRbHOe 3HaqeHHe TIOKa3aTe-
JIeM (5.29) (npH 3a,l{aHHOM a) H (5.30) B KRacce Pm He YObIBa-
eT c YBenHqeHHeM m H, CJIeAOBaTeJIbHO, ,l{OCTHraeT MaKCHMYMa
npH m = N, T. e. B KRacce CTPYKTYP C TpHBHaJIbHhIMH pa3-
OHeHHHMH. ,[(eH:CTBHTeJIbHO, eCJIH (R, x) E Pm - onTHMaRb-
Hoe pa30HeHHe co CTPYKTYPOM TIPH m < N, TO TIOcTynHM CJIe-
,l{YIOIII.HM oopa30M. Pa300beM O,l{HH H3 KJIaCCOB Rs Ha ,l].Be qac-
TH Rs = R; U R; H onpe,l{eRHM x' TaKHM oopa30M, qTOObI
R If R: ObIJlH CB5I3aHbI B ooe CTOpOHbI ,l{Pyr C ,l{pyroM, a HX
CB5I3H C OCTaRbHbIMH KRaccaMH ObIJIH OAHHaKOBbI co CB5I351MH
Rs B x. OCTaRbHbIe CBSl3H OCTaBHM HeH3MeHHbIMH. OqeBHAHO,
qTO OTHOIlleHHe p HOBoro pa30HeHH5I co CTPYKTYPOH TO 1Ke, QTO
H ,lJ.RH (R, x), TaK qTO 3HaqeHH5I (5.29) H (5.30), BbIpa>KaeMbIe
qepe3 p, He H3MeH5ITC5I. B TO LKe BpeM5I HOBoe pa30HeHHe co
CTPYKTYPOH npHHa,l{Re}KHT Pm + 1 H He 005l3aTeJIhHO onTH-
MaJIbHO, qTQ H ,l{OKa3bIBaeT YTBep)KAeHHe.
B TO }Ke BpeM5I ODTHMaJIbHbIe 3HaqeHH5I TIOKa3aTeJIeR (5.28)
H (5.31) B Pm He 3aBHC5IT MOHOTOHHO OT m, KaK HeTpY,lJ.HO
u
YBH,l{eTb Ha npOCTeHIllHX npHMepax.
5.2.2. TpuBuaAbflble pa36ueflUJl co CTpgKTYPOU:
2patjJbl a-cXOaCTBa
CRyqaH: TpHBHaRbHbIx pa30HeHHH co CTPYKTY"
pOR (R, x), rAe pa30HeHHe R TpHBHaJIbHO, T. e. COCTOHT H3
N O,l{H03ReMeHTHbIX KRaCCOB, HHTepeceH He TOJIbKO H He CTORb-
KO nOTOMY, qTO, KaK MbI YCTaHOBHJIH B n. 5.2.1, 3TH pa30He-
HH5I CO CTPYKTYPOH onTHMH3HPYIOT (5.29) H (5.30). ,[LeJIo B TOM t
qTO npe,l{CTaBReHHe CB5I3eH M:eLK,ny OO'beKTaMH B BHAe CTPYK-
TypbI AOCTaTOQHO npocTOro BHAa caMO no ceoe, oe3 o,l{HOBpe-
MeHHoro arperHpOBaHH5I OO'beKTOB B KRaCCM, H3,l{aBHa 51BJIH-
261
P-r-
--------.
eTC5I MOI..I.(HbIM Cpe CTBOM aHaJIH3a aHHbIX, 3alJaCTYlO He MeHee
3<p<peKTHBHbIM, lJeM rpynnHpoBKa OO'beKTOB. HarJI5I,l{HOCTb BH-
u u u
,l{a CTPYKTYPbI, HanpHMep JIHHeHHbIH HJIH HepapXHqeCKHH no-
pSl OK OO'beKTOB, OOeCneqHBaeT Y OOCTBO co epLKaTeJIbHOrO
aHaJIH3a H HHTepnpeTanHH 3a CqeT paCCMOTpeHH5I MaJIOrO H
npOCTO YCTpoeHHoro MHO:>KeCTBa CB5I3eH.
ECJIH R TpHBHaJIbHO, npHqeM Rs == {s} (5 Em), TO
x £; m X m, OqeBH,l{HO, COBna):(aeT C OTHOllIeHHeM p, COOT-
BeTCTBYlOI..I.(MM .nape (R, x). OqeBH,l{HO TaKme, 1JTO au =
i ER s j ER t
== ast, TaK tlTO nOKa3aTeJIb (5.28) HMeeT B 3TOM CJIYlJae £H):(:
F (x, a) = ast-a I x J = (ast-a), (5.32)
(s,t)EX 5,t)EK
H, CJIe,l{OBaTeJIhHO, COBna,l{aeT C (5.29) npM Rs == {s} (s Em).
B CHJIY (5.30) g (R, x) HMeeT BH,n:
g(x)=( a st )2/1 x[+( ast) !(N2_1 xf), (5.33)
(s,t)EX (s, t)(fX
a (5.31) ,l{aeT
G (R, ) -== a:t. (5.34)
(s, t)Ex
cI>opMYJIa (5.34) nOKaSbIBaeT, qTO G (R, x) (5.31) ,l{OCTH-
raeT MaKCHMYMa B P N npH x == m X m, T. e. npH YHHBep-
CaJIbHOH CTpYKType, npHqeM OnTHMaJIbHhIe Beca Ast COBna,l{a-
lOT C TOqHOCTblO ,l{O 3HaKa C BeJIHqHHaMH CB5I3eH ast.
OnTHMaJIbHa51 CTpYKTypa B CMbICJIe (5.33) onpe,l{eJI5IeTC5I
KpHTepHeM (5.32) C COOTBeTCTBYlOI..I.(HM noporoM a == (ex + )/2.
1JTO KacaeTCSI CTPYKTypbI, OnTHMaJIbHOH B CMbICJIe (5.32), TO
OHa onpe,l{eJI5IeTC5I CJIe YlO HM OqeBH HbIM oopa30M. B x CJIe-
yeT BKJIlOqaTb Te (s, t), JI5I KOTOpbIX ast > a, H CJIe,l{yeT HC-
KJIlOqaTb Te (s, t), JI5I KOTOpbIX ast < a. ,aeHcTBHTeJIbHO,
(5.32) npH 3TOM oy eT paBHO CYMMe Bcex nOJIO}l{HTeJIbHbIX
BeJIHqHH aij - a, H OOJIbillee 3HaqeHHe F (x, a) HeB03MO:>KHO
(npH ,l{aHHOM a), TaK KaK ,l{JI5I BC5IKOH ,l{PyroH CTPYKTypbI x'
JIHOO He 0Y,l{YT CYMMHpOBaTbC5I HeKOTopbIe H3 nOJIO)KHTeJIbHbIX,
JIHOO, HaooopOT, OY YT ,l{OoaBJIeHbI HeKOTopbIe H3 OTpHna-
TeJIbHbIX BeJIHqHH aij - a, a TO H ,l{Pyroe YMeHblIlHT 3HaqeHHe
F. TaKHM oopa30M, OnTHMaJIbHa51 B CMbICJIe (5.32) CTpYKTypa
onpe,l{eJI5IeTC5I noporOM a: x COCTOHT H3 Tex H TOJIbKO Tex nap
(5, t), ,l{JI5I KOTOpbIX ast > a (HeKoTOpa5I HeOnpe eJIeHHOCTb
,l{OnycKaeTC5I npH 'ast == a). TaKa51 CTpYKTypa onpe,l{eJI5IeT
rpa<p Ha MHO)KeCTBe 91, KOTOpbIH B JIHTepaType npHH5ITO Ha-
3bIBaTb zpaifJoM a-cxoacmea. KaK BH,l{HM, Ha caMOM ,l{eJIe rpacp
a-CXO,l{CTBa ,l{aeT OnTHMaJIbHOe pellIeHHe 3a,l{aqH MaKCHMH3a-
262
lIJIH nOKa3aTeJI5I (5.29), B KJIaCCe Bcex pa30HeHHM co CTPYKTY-
pOM P. .
OnTHMH3anHIO (5.33) MO>KHO npoH3BecTH C HCnOJIb30Ba-
HHeM rpa oB a-CXO.L{CTBa Ha OCHOBe HTepaTHBHoro aJIrOpHTMa
onpe.L{eJIeHH5I nopora, H3JIO)KeHHOrO B n. 4.1.3. npH 3TOM Ha
OCHOBe HaM.L{eHHOro rpa<jJa a-CXO.L{CTBa BblqHCJISleTCH HOBoe
3HalleHHe nopora cy ecTBeHHOCTH a, paBHoe nOJIYCYMMe .L{BYX
Cpe.L{HHX -BeJIHlIHH ast: O.L{HOM - no (5, t) E x, BTOpOM - no
(5, t) $. x. npone.L{ypa 3aKaHlIHBaeTC5I, eCJIH HOBbIH rpa<p
a-CXO.L{CTBa COBna.L{aeT C npe.L{bI.L{YI..I.(HM. nO-BH.L{HMOMY, 3TOT Me-
TO.L{ nOKOOp.L{HHaTHOro cnYCKa .L{aeT rJIOOaJIbHbIM onTHMYM
(5.33), O HaKO cTpororo .L{OKa3aTeJIbCTBa 3Toro Y Hac He
HMeeTC5I.
OObIlIHO rpacp a-CXO.L{CTBa CJIHIliKOM CJIO:>KeH .L{JI5I HenOcpe.L{-
CTBeHHoro CO.L{ep:>KaTeJIbHOrO aHaJIH3a H HCnOJIb3yeTC5I J<aK
OCHOBa ,l{JISI .L{aJIbHeMlllero npHMeHeHH5I <t>opMaJIbHbIX MeTO.L{OB.
HanoMHHM, [6], liTO nOCJIe.L{OBaTeJIbHOCTb HeCOBna.L{alOI..I.(HX
nap OO'beKTOB (iI' i 2 ), (i 2 , i 3)' ..., (i p _], ip) Ha3bIBaeTC5I ny-
meM rpa a (91, r), r.L{e r s;; X m - 3a.L{aHHOe MHO:>KeCT-
BO .L{yr (i, j) Em x , eCJIH (i j , ij + 1) E r .L{JI5I BC5IKOrO
j = 1, ..., p - I, a Cpe.L{H i j HeT o,nHHaKOBbIX. BepI.lIHHa i l
Ha3bIBaeTC5I HattalloM, a ip - KOHl{OM 3Toro nYTH, npo KOTO-
pbIH rOBOp5lT TaK:>Ke, liTO OH Be.L{eT H3 i 1 B ip. nOCJIe,l{OBaTeJIb-
HOCTb HeCOBna.L{alOI..I.(HX nap (iI' i 2 ), (i 2 , i 3 ), ..., (i p -1, ip)
Ha3bIBaeTCSI l{enblO rpa<t>a (in, f), COe,l{HHSlIO eH i l Hip, eCJIH
(i j, i j + 1) E r HJIH (i j + 1, i j) E r .L{JI5I JIIOOoro j =
= 1, ..., p-I. B OTJIHlIHe OT nYTH B nenH HeKoTophIe napbI
MO:>KHO npOXO.L{HTb npOTHB HX HanpaBJIeHH5I B r.
ClIHTaeTC5I, liTO Ka)K,l{a51 BeplIlHHa COe.L{HHeHa nYTeM H nenblO
(HYJIeBOH .L{JIHHbI) caMa C COOOM. BepI.lIHHbI iI' i 2 Ha3bIBaIOTC5I
C8Jl3HblMU B rpa<jJe on, r), eCJIH cy ecTByeT nenb, COe.L{HHSlIO-
a51 i l H i 2 , H 6UC8J13flblMU, eCJIH cYI..I.(ecTBYIOT nYTH, Be.L{Y He
H3 i 1 B i 2 H oopaTHO. 113 onpe.L{eJIeHH5I HenOCpe.L{CTBeHHO CJIe-
yeT, liTO BepIIlHHbI OHCBSl3HbI TOr.L{a H TOJIbKO TOr.L{a, Kor.L{a
OHH npHHaJiJIe:>KaT KOflmypy rpa<pa (T. e. TaKOMY nYTH, Y KO-
Toporo HallaJIO COBna.L{aeT C KOHnOM).
HeTPY.L{HO npOBepHTb TaK:>Ke, liTO OHHapHble OTHOIIleHH5I
CB5I3HOCTH H OHCBSl3HOCTH BeplIlHH pe<pJIeKCHBHbI, CHMMeTpHlI-
HbI H TpaH3HTHBHbI, T. e. 51BJISIIOTC5I OTHOI.lIeHH5IMH 3KBHBa-
JIeHTHOCTH. A 3TO 03HallaeT, liTO MHO:>KeCTBO 91 pa30HBaeTC5I
Ha KJIaCCbI, KOTopble Ha3bIBaIOTC5I KOMnonenmaMU (.L{JI5I OTHO-
IIleHH5I CBSl3HOCTH) H 6UKOMnOHeflmaMU (.L{JI5I OTHOllIeHH5I OH-
CBH3HOCTH). I1HbIMH CJIOBaMH, KOMnOHeHTa - 3TO MaKCR-
MaJIbHOe MHO>KeCTBO OO'beKTOB, KOTopwe nonapHO CB5I3aHW.
263
------.
..-- -.- -- ------- -- -"III
Cne,nOBaTeJIbHO, BepIIlMHbI, npHHa,nJIeLKa He pa3HblM KOMnO-
HeHTaM, He CB5I3aHhI ,npyr c ,npyrOM.
BHI{OMnOHeHTa - 3TO MaKCHMaJIbHOe MHOLKeCTBO nonap-
HO OHCB5I3HbIX OO'beKTOB. OTCIO,na CJIe,nyeT, qro, XOTH CBH3H
Me}K)J.Y OHKOMnOHeHTaMH MorYT cy ecTBOBaTb, OHH KaK OhI
o,nHOHanpaBJIeHHbl. ,aJI5I YTOqHeHH5I Onpe.L{eJIHM ,nJIH r 2pat/J
Fept{a. BeplIlHHbI rpacpa repua - 3TO OHKOMnOHeHThI rpacpa
(m, f) (MO>KHO paccMaTpHBaTb HOMepa OMKOMnOHeHT), a ,nyra
_ Be,l{eT H3 O,l{HOM TaKoH BeplIlHHbI B ,l{pyrylO TOr,l{a H TOJIbKO
Tor,na, Kor,na B HCXO,l{HOM rpa<t>e HMeeTC5I ,l{yra H3 KaKOH-JIHOO
BepIllHHbI nepBoH oHKoMnoHeHTbI B KaKYIO-JIHOO BeplIlHHY
BTOpOM. rpacp repna He CO,l{epLKHT KOHTYPOB (3TO H eCTb O,l{HO-
HanpaBJIeHHOCTb), TaK KaK npOTHBHoe 03HaqaJIO ObI, qTO B r
u
cy ecTByeT KOHTYP, npOXO,l{51 HH qepe3 BeplIlHHbI pa3HbIX
OHKOMnOHeHT, a 3TO npOTHBopellHT onpe,l{eJIeHHIO OHKOMnOHeH-
TbI KaK MaKCHMaJIbHOrO OHCBSl3Horo nO,l{MHOLKeCTBa.
TaKHM 00pa30M, OHKOl\-lnOHeHTbI rpacpa r MorYT OblTb pa3-
HeceHbI no C/lO.flM rpa<t>a repua. nepBhIH CJIOM 00pa3YlOT Te
6HKOMnOHeHTbI, B KOTopble He 3aXO,l{HT HH o,nHa ,nyra. TaKHe
OMKOMnOHeHTW, B CHJIY KOHellHOCTH HX QHCJIa, OOH3aTeJIbHO
CYIneCTBYIOT, TaK KaK B rpa<t>e repna HeT KOHTypOB. Cne,l{YIO-
InHH CJIOH onpe,l{eJISleTCH KaK MHO>KeCTBO OHKOMrIOHeHT, B
KOTopbIe Be,l{YT ,nyrH TOJIbKO nepBoro CJIOH. ECJIH k cnoeB Y>Ke
onpe,l{eJIeHbI, TO (k + I)-A CJIOM Onpe,l{eJIHeTCH KaK MH01KeCTBO
oHKoMnoHeHT, B KOTopbIe Be,l{YT ,l{yrH TOJIbKO H3 npe,l{w,l{Y HX
CJIOeB. nOCJIe,l{HHM CJIOH oopa30BaH oHKoMnoHeHTaMH oe3 Bbl-
XO,l{HInHX ,l{yr. Onpe,l{eJIeHHhIe TaKHM oopa30M CJIOH nopo}K)la-
lOT ynopSl,l{OlleHHOe pa30HeHHe MHO}KeCTBa m Ha KJIaCChI OOD-
eKTOB, COOTBeTCTBYIO He OT,l{eJIbHbIM CJI05lM. XapaKTepHCTH-
qeCKOM ocooeHHOCTblO 3Toro pa30HeHHH HBJIHeTCH TO, qro B
rpa<t>e r HeT ,u.yr, HanpaBJIeHHbIX OT nOCJIe,nYIO HX KJIaCCOB
K npe,nbI,l{y HM. Bonee Toro, MeLK,l{Y OHKOMnOHeHTaMH O,l{HOrO
u
H Toro 1Ke cnoS! HHKaKHX CBH3eH HeT.
TaKMM oopa30M, B CHJIY rpa<t>a r MHO}KeCTBO OO'beKTOB -. m
pa30HBaeTCS! Ha HeCB5I3aHHble ,l{pyr C ,npyroM KOMnOHeHTbl,
u
CTpYKTypa CB5I3eH BHYTPH KOTOpbIX onpe,neJI5IeTCH COOTBeTCT-
BYIOInHMH rpacpaMH repua. ,[(n51 nOCTpoeHHSI KOMnOHeHT H OH-
KOMnOHeHT B TeopHH rpacpOB pa3paOOTaHbI 3qxpeKTHBHbIe aJIro-
pHTMbI (CM., HanpHMep, [37, 39]), H3JIOLKeHHe KOTOphIX YBeJIO
ObI Hac CJIHIIlKOM ,naJIeKO OT OCHOBHoro MaTepHana KHHrH.
Ka1K aH KOMnOHeHTa, OlleBH,nHO, HBJISleTCH- CJIaObIM cry-
Ir(eHHeM. ll03TOMY pa30HeHHH rpa<t>a a-CXO.L{CTBa Ha KOMnOHeH-
TbI qaCTO HCnOJIb3YlOTCH B nHTepaType. I1Hor,na 3TH KOMnOHeH-
TbI Ha3hIBalOT KOppe/lR1{UOHHblMU nlleJloaMu (5], B OCHOBHOM
264
npH aHaJIH3e MaTpHn CB5I3H MeLK,l{Y npH3HaKaMH. CTaHAapTHhlM
u
B aHaJIH3e ,l{aHHbIX 51BJI5IeTC5I paCCMOTpeHHe CeMeHCTBa BJlO"
>KeHHbIX ,l{pyr B ,l{pyra rpa<t>OB r 1 c= ... c= r p a k -CXO,l{CTB8
npH al > a2 > ... > ape Cy ecTBeHHYIO pOJIb npH STOM Hrpa-
eT COOTHOllIeHHe pa30HeHHM, ,l{aBaeMbIX f k H f li + 1
(k = 1, ..., p - 1). ECJIH pa30HeHHe R Ii + 1 nOJIyqaeTCH r JIaB-
HhIM oopa30M C nOMO blO OO'be,l{HHeHHH KJIaCCOB R k , TO HMeeT
MeCTO Ta >Ke TeH,l{eHUHH: B Rk + 1 YKpynHHIOTcH OCHOBHbIe 38-
KOHOMepHOCTH MaTepHaJIa. ECJIH LKe KnaCCbI Rk + 1 nOJIyqeHbI
B OCHOBHOM H3 qaCTeH pa3HbIX KJIaCCOB R k, TO npOHCXO,l{HT
v
nepeCTpOHKa MaTepHaJIa, ak + 1 - CJIHIliKOM MaJIOe 3H8qeHHe
nopora, Cne,l{yeT OCTaHOBHTbC5I Ha R'k' B HaHOOJIee YKpynHeH-
HOM BH,ne, npe,l(CTaBJI5IIO eM HCXO,l{HbIe CBH3H.
.1
1 " :5 2 1 I( 3 2 2 3 "
. . . . . . .
. . . .
5 6 5 6
aaf16
a=O,7
a;:: 8
PHC. 5.2
3TOT MeTO,l{ 8H8JIH38 rpacpOB a-CXO,l{CTBa HenpHrO,neH, eCJIH,
K8K 3TO qaCTO ObIBaeT, CBH3H MeLK,l{y OO'beKTaMH opraHH30Ba-
HhI B BH,ne KJIYOK8. B STOM CJIyqae rpacp a-CXO,l{CTB8 COCTOHT H3
O,l{HOH OOJIbWOH H MHorHX MaJIbIX KOMnOHeHT, npHtleM YBeJIH-
qeHHe a npHBO,l{HT K ,l{aJIbHeMlIleMY OT enJIeHHIO MaJIbIX (O,l{-
H8-,l{Be BepI.I1HHhI) KOMnOHeHT OT HCXO,l{HOH OOJIbI.I10H KOMno-
HeHThI. MO,l{eJIH 9 5.1 HecpaBHeHHo oOJIee TOHKO BCKpbIBalOT
CTPYKTYPY pa30HeHHH.
rpacpbI a-CXO,l{CTBa npH a = 0,6; 0,7; 0,8 ,l(JIH MaTpHUbI
CBH3H p (c. 121) npe,l{CT8BJIeHbI Ha pHC. 5.2. KaK BH,lJ.HO, npH
a = 0,7 KOMnOHeHTbI oopa3YlOT pa30HeHHe Ha ,l{Ba KJI8CC8,
8 npH a = 0,8 - H8 TPH KJIaCca. npH a = 0,6 Bce 001>eKTbI
BXO,lJ.HT B O,l{HY KOMnOHeHTY.
5.2.3. P.epeBo MaKCUMaAbflOU 8AUflbl
PaCCMQTpHM Tenepb 3a,l{aqH '0 TpHBHaJIbHOM P83-
oHeHHH 3apaHee 3a,l{aHHOro BH,l{a, HMeH B BH y nOKa3aTeJIb
(5.32) (TaK KaK (5.33) CBO,l{HTCH K (5.32) npH OnTHMaJIbHOM a,
a (5.34) - npH a = ° H MaTpHne CB5I3eH (alj)).
265
---..
TaKHe 3a aqM npe CTaBJI5IIOT cy ecTBeHHbIH MHTepeC ,nJISI
npOCTbIX BM.L{OB CTPYKTYP TMna ynop5l OqeHM5I MJIM MepapXMH.
O HHM M3 MHTepeCHbIX BM.L{OB npOCTbIX CTPYKTYP 5IBJI5IIOTCSI
.L{epeBbH. HanOMHHM, qTO OepeeOM Ha3bIBaeTC5I CBSl3HbIH rpa<p
6e3 KOHTYPOB. 3KBHBaJIeHTHble Onpe.L{eJIeHH5I: .L{epeBO - 3TO
a) CBSl3HbIH N-BepIllHHHbIH rpa<t> C N -1 .L{yroii, 6) rpa<p,
B KOTOpOM BC5IKHe ,l{Be BepIIlHHbI coe HHeHbI O.L{HOH H TOJIbKO
O.L{HOH nenblO, B) CB5I3HbIH rpa<t>, KOTOpbIH paCna,l{aeTC5I Ha KOM-
nOHeHTbI npM YJ(aJIeHMH JII060ro pe6pa. MHomeCTBO Bcex CMM-
MeTpHqHbIX ,l{epeBbeB 0603HaqHM qepe3 D. HanOl'vIHHM, qTO B
CHJIY CHMMeTpHqHOCTH annpOKCHMMPYIO MX OTHOIlleHMH
MCXO,l{HaSl MaTpHna CB5I3eH 3KBMBaJIeHTHbIM 06pa30M 3aMe-
1
HSleTCSI Ha CHMMeTpHqHYIO no <popMYJIe a' == 2 (a + aT).
n 0 C T P 0 e H H eon T H M a JI b H 0 r 0 ,l{ e p e B a.
3a,l{aqa onTHMH3anHH (5.32) B KJIaCCe D H3BeCTHa KaK 3a,l{a-
qa 0 ,l{epeBe MaKCHMaJIbHOH ,l{JIHHbI. J!.IlUflOU oepeea d E D Ha-
3bIBaeTC5I CYMMa f (d) == I,ast Bcex CB5I3eH no ,l{epeBY. OqeBH,l{-
( S , t) Ed
HO, TaK KaK qHCJIO ,l{yr ,l{epeBa nOCTOSlHHO, MaKCHMH3aUM5I
G (d, a) == I, ast - a I d I paBHOCHJIbHa MaKCHMH3anHI1 f (d).
(s, t ) Ed
AJIrOpHTMbI nOCTpOeHH5I ,l{epeBa MaKCHMaJIbHOH ,l{JIHHbI OC-
HOBaHbI Ha CJIe,l{YIO eM <paKTe. nYCTb it, ..., ip - nYTb Ka-
KOrO-JIH60 ,l{epeBa d, TaK qTO pe6po (it, ip) He npHHa,l{JIe)KHT
d (HHaqe it, ..., i p , it 51BJISleTC5I nHKJIOM). HeTPY,l{HO nOKa3aTb
[105], qTO ,l{epeBO d HMeeT MaKCHMaJIbHYIO ,l{JIHHY TOr,l{a H TOJIb-
KO TOr,l{a, KOr,l{a ,l{JISI JII060H ero nenH BbInOJIHeHO CJIe,l{YIO ee
YCJIOBHe:
ai i p min [ai i ,ai i ,..., ai l i p ]'
1 1 2 2 3 p-
KOTopoe 03HaqaeT, qTO BCSlKaSl napa (i, j) BHe ,l{epeBa HMeeT
CBSl3b aij, He npeBOCXO,l{51 YIO CB5I3eH nYTH ,l{epeBa, COe,l{HH5IIO-
ero i H j.
OTCIO,l{a BbITeKaeT KoppeKTHOCTb CJIe,l{YIO ero ,aJlrOpHTMa
[105]. YnopSl,l{OqHM Bce 3JIeMeHTbI aij MaTpHnbI CBSl3eH no y6bI-
BaHHIO H 6Y,l{eM no nOpSl,l{KY npocMaTpHBaTb 3JIeMeHTW 3TOH
nOCJIe,l{OBaTeJIbHOCTH, oT6Hpa51 COOTBeTCTBYIO He pe6pa B on-
THMaJIbHOe ,l{epeBo. Ha HaqaJIbHOM Illare MHO)KeCTBO oT06paH-
HbIX pe6ep nYCTO. ,[{JISI BHOBb paccMaTpHBaeMoro aij BKJIlOqa-
eM B ,nepeBO d pe6po (i, j), eCJIM OHO He 06pa3yeT UMKJIa C Ka-
KMMH-JIH60 H3 paHee BKJIlOqeHHbIX pe6ep. nponecc 3aKaHqH-
BaeTCH nOCJIe oT60pa N - 1 pe6pa..
COrJIaCHO ,l{PyroMY MeTO.L{Y [125] nepBOHaqaJIbHO CTp05lT-
CH OT,l{eJIbHbIe BeTBH ,l{epeBa. ,LlJI5I 3Toro, HaqMHa51 C npOH3BOJlb-
(5.35)
Horo (paHee He paCCMaTpHBaBlIlerOC5I) 06'beKTa i E 91, onpe-
,!{eJI5IIOT it, Ha KOTOpOM .L{OCTMraeTCH max aij, 3aTeM - i 2 , Ha
j
KOTOpOM ,!{OCTMraerC5I max ai J . M TaK .L{O Tex no p , nOKa oqe-
1. l'
pe.L{HOH i h He COBna.L{aeT C O.L{HMM Ji3 paHee nOJIyqeHHbIX. nOCJIe-
AOBaTeJIbHOCTb i, it, ..., i h - t 06pa3yeT HCKOMYIO BeTBb. noc-
JIe nOJIYQeHM5I Bcex BeTBeM npOHCXO.L{HT HX ClIlHBaHHe. L{JI5I
3Toro CTpOHTC5I MaTpMQa CB5I3eM Me:>K.L{Y nOJIyqeHHbIMH KOM-
nOHeHTaMH a, ; yqHTbIBaeTC5I, qTO aal3 = max aij, nocJle
iEa,iE(3
u u
qero onHcaHHWH nponecc nOCTpoeHHH BeTBeH npHMeHHeTCH K
nOJIyqeHHbIM a, ,... npH
3TOM ,!{yra (a, ) COe.L{HHHeT
Te 06'beKTbI i H j, ,!{JIH KOTO-
pbIX aafi = aij. TaK nOBTO-
p5leTC5I .L{O Tex nop, nOKa He
nOJlyqHTCH O,!{Ha KOMnOHeH-
Ta; BbI,!{eJleHHble B HeM CB5I-
3H Ii ,!{alOT ,!{epeBo MaKCH-
MaJIbHOM ,!{JIHHbI.
npHMeHHTeJIbHO K MaT-
pHne CB5I3eM HalIlero npHMe-
pa 3TOT nponecc, HaqHHaH
C 1, ,!{aeT BeTBb 1-4-3-2,
nocJle qero nOJIyqaeM BeTBb
5 - 6. COe.L{HH5IIOTCH 3TH
BeTBH JII06bIM H3 pe6ep
(3, 6), (4, 6), HMelO HX MaKCHMaJIbHYIO CBH3b 0,67. no-
JIyqeHHOe ,!{epeBo H306pa:>KeHO Ha pHC. 5.3.
TaKoe ,!{epeBO HCnOJIb3yeTcH He TOJIbKO ,!{JI5I Henocpe,!{CTBeH"
Horo aHaJIH3a, HO H KaK OCHOBa ,!{JI5I ,!{aJIbHeMlIlero npHMeHeHH5I
cpopMaJIbHbIX MeTO,!{OB. B qaCTHOCTH, cor JIaCHO MeTO,!{y B pOt{-
Ila8CKOU maKCOHOMUU [125] npOH3BO,!{HTCH pa3pe3aHHe m-l pe6.
pa HaHMeHbIlleH ,!{JIHHbI B nOJIyqeHHOM ,!{epeBe. npH 3TOM MHO"
LKeCTBO in pa36HBaeTC5I Ha m KJIaCCOB TaKHX, qTO CYMMa ,!{JIHH
MaKCHMaJlbHbIX ,!{epeBbeB B 3THX KJIaCCaX MaKCHMaJIbHa no
BceM B03MO:>KHbIM pa36HeHH5IM MHO:>KeCTBa R Ha m KJIaCCOB.
Pa3pe3aHHbIe pe6pa xapaKTepH3YlOT CTPYKTYPY CBH3eH Me)K-
y KJIaCCaMH pa36HeHH5I.
,[{JI5I Harnero npHMepa KJIa CCbI npH m = 2 Ha pMC. 5.3
H300pa:>KeHbI IllTpHXOBOH JIHHHeH. HeTPY,!{Ho BH,!{eTb, qTO
KJIaCCbI BpOnJIaBCKOH TaKCOHOMHH COBna,!{aIOT C KOMnOHeHTa-
MH rpa<pa a-CXO,!{CTBa npH a, paBHOM CBH3H aij nOCJIe,!{Hero
(MHHHM8JIbHOrO no CBH3M) pa3pe3aHHoro pe6pa (i, j).
---'-
-------.......
"", .", 3 "'" .....
// " "
/ '
/ \
1-2 \
, )
\ ,.,..-- ....... /
\ ,/ "./
/ ......--
" ,/ -
'-- ,/
/
I
I
,
\ 5 I
\ /
'.......__./
PHC. 5.3
267
n03TOMY KJIaCCbl BpOUJIaBCKOH TaKCOHOMHH MOrYT ObITb
nOJIyqeHbI npOCTO KaK KJIaCChI rpa<jJa a-CXO,l{CTBa npH no xo-
,nH eM a.
He06xo,l{HMO TaKJKe OTMeTHTb eIne O.L{HO CBOHCTBO KJIaCCOB
BpOUJIaBCKOH TaKCOHOMHH: OHH COBna,naIOT C KJIaCCaMH, no-
JIyqeHHbIMH MeTO,nOM 6JIHJKHerO coce,na (npH TOM >Ke 3a,naHHOM
qHCJIe KJIaCCOB). !{eHCTBHTeJIbHO, ,nepeBO MaKCHMaJIbHOH ,l{JIH-
HbI nOJIHOCTblO XapaKTepH3yeT COe,l{HHeHHe KJIaCCOB no MeTO-
,ny OJIH)KHerO coce,na.
Y CTaHOBJIeHHaH CBH3b Me)K,ny MeTo,nOM OJIHJKHerO coce,na,
KOMnOHeHTaMH rpa<t>a a-cxo,nCTBa H BponJlaBCKOH TaKCOHOMHH
v
xapaKTepH3yeT C pa3HbIX CTOpOH KaqeCTBO nOJlYlIeHHOH KJIaC-
CH<pHKanHH. .
K CO)KaJIeHHIO, B npaKTHlIeCKHX cHTyaUHHx pa3pe3aeMbIe
peopa 3aqaCTYlO HaXO,l{HTCH Ha KOHuax ,nepeBa (CM. 3aMeqaHHe
Ha c. 265), liTO CYIneCTBeHHO CHH}KaeT 3BpHCTHqeCKYIO ueH-
HOCTb nOJIYllaeMoro pa30HeHHH. B CBH3H C 3THM B. H. EJIKH-
Ha H H. r. 3aropYHKO npe,nnO)KHJlH aJlrOpHTM «Kpao», cornac-
HO KOTOPOMY pa3pesaeMbIe peopa ,nOJI)KHbI ,naBaTb MaKCHMYM
nOKa3aTeJIIO F, yqHTbIBalOIneMY cpe,nHIOIO CBH3b B nOMepeBb-
SIX BHYTPH KJIaCCOB p, cpe,nHee apH<pMeTHtleCKOe CBH3eH no
pa3pe3aeMbIM pe6paM ,l{epeBa d, CTeneHb paBHOMepHOCTH h
pacnpe,neJIeHH5I pa30HeHH5I, a TaK1Ke nOKa3aTeJlb O,l{HHa-
KOBOCTH CTPYKTYPbI ,l{epeBa A, OKOJIO rpaHHU KJIaCCOB: F --=
dh
= log (l+p)(J+A) (CM. [32]).
He,nocTaTKH Bcex aJIrOpHTMoB, OCHOBaHHbIX Ha ,nepeBe MaK-
v
CHMaJIbHOH ,nJIHH , nOMHMO HX 3BpHcTHlIecKoro xapaKTepa,
CBH3aHbl C TeM, qTO 3TO ,nepeBO onpe,neJ1HeTCH ,naJIeKO He e,nHH-
CTBeHHbIM 06pa30M npH HaJIHlIHH COBna,l{alO HX CBH3eH aij,
llTO THnHlIHO ,nJIH KaqeCTBeHHbIX npH3HaKOB. OnTHMaJIbHOe
v v
,nepeBO BeCbMa HeYCTOHlIHBO OTHOCHTeJIbHO MaJIWX H3MeHeHHH
CBH3eH. HeTPY,l{HO npHBeCTH npHMepbI cyrnecTBeHHoro H3Me-
HeHHH CTPYKTYPbI ,nepeBa npH ManbIX ,ne<PopMaUHHX MaTpHUbi
a: TaK, eCJIH ,nJIH BeTBeH a H p aa = max aiJ Hast = aaa- 8 ,
lEft, IEP
TO npH YMeHbIlleHHH aij 60JIbIlle, lIeM Ha 8, nOJlYtlHM aa = ast.
n03TOMY ,naHHaH TeXHHKa HCnOJIb3yeTcH KaK cpe,nCTBO npe,l{-
BapHTeJIbHOrO aHaJIH3a.
5.2.4. l!enHble 2pat/Jbl U ynOpJlaOfleHUJl
CB5I3HhIH rpacp, peopa KOToporo oopa3YIOT e,nHH-
cTBeHHYIO nenb, TalOKe 6y,neM Ha3bIBaTb uenblO. rpacp, Ka>K-
,!taH KOMnOHeHTa KOToporo HBJIHeTCH nenblO, oy,neM Ha3bIBaTb
268
qenflblM. OlleBH,nHO, Uenb 3a,naeT ynop5l,n01leHHe OO"beKTOB B
nOpH,nKOBOM IIlKaJIe (CM. n. 1.1.2); pa3JIHlIHbIX u.eneii pOBHO
NI/2 IIlTYK. KaK B 9 4.1, OlleBM,nHO 3a,nalla annpoKcHMaU HH
MaTpHII.bI cBSl3eH B KJIaCCe II.eneH 9KBHBaJIeHTHa 3a,na1Je 0 Ha-
u
xo*,neHHH uenH MaKCHMaJIbHOH ,nJIHH , KOTopaH M3BeCTHa B
,nHcKpeTHoM aHaJIH3e KaK 3a,natIa 0 raMMJIbTOHOBOM nYTH
(nYTb MaKCHMaJIbHOH 9qxpeKTHBHoCTH 1Iepe3 Bce BepIIlHHbl).
3a,nalla 0 raMHJIbTOHOBOM nYTH 5IBJIHeTCSI nepeoopHoH [145],
nOSTOMY JIOKaJIbHhIe aJIrOpHTMbI ee pelIleHM5I He 5IBJISlIOTCSI T01l.
HhIMH.
PaCCMOTpHM aJIrOpHTM «npHcoe,nHHeHHe», OCHOBaHHbIH
Ha H,nee nOCJIe,nOBaTeJIbHOrO npHcoe,nHHeHHH OO'beKTOB no CBSI-
3H C npe,nbI,nyrnHM. HCXO,nHbIM ,nJI5I k-ro IlIara 9Toro aJIrOpHT-
Ma SlBJISleTCSI ynopSl,n01leHHe k OO'beKTOB iI' ..., i k H CB5I3H
atl'., ..., alk_llk Me}K,ny coce,nHMM. !{aJIee cpe,nH OCTaB-
lIIHXCH N - k OO'beKTOB BbIOHpaeTCSI TOT OO'beKT i1&+l' ,nJIH
KOTOporo CBH3b C ill. MaKCHMaJIbHa: a l ,lk+1 = m x al,,/.
nOCJIe 1Iero Ha1lHHaeTCH BbInOJIHeHHe CJIe,nYlOrnero (k + I)-ro
lIIara. nepBbIM OO'beKT BblOHpaeTcn CJIY1laHHO.
KaK II B npe){bI,nyrnHX pa3,neJIaX, nOJIYlIeHHoe ynopSl,no-
lleHHe Mo*eT HCnOJIb30BaTbCSI ,nJI5I KJIaCCH<pHKau.HH. TaK, 03-
BeCTHbIR 9BpHCTHtIeCKHH aJIrOpHTM «CneKTp» [5] OCHOBaH Ha
CJIe,nYlOrnHx npe,nCTaBJIeHH5IX. OO'beKT, HaHOOJIee CBSl3aHHbIH
C ,npyrHMH OO'beKTaMH, ,nOJI*eH Haxo,nHTbCH B TOM }Ke KJlaCce.
Cne,nOBaTeJIbHO, eCJIH OO'beKTHBHO cymeCTBYlOrnHe KJIaCCbI KOM-
naKTHbI, TO CHatIaJIa 6y,nYT oT6HpaTbC5I 06'beKTbI H3 Toro }Ke
KJIaCCa, tlTO H i 1 , a 3aTeM YLKe n05lB5ITC5I 06'beKTbI H3 ,npyrHx
rpynn. npH nepexo,ne H3 O,nHOR rpynnbI B ,npyrylO ,nOJI>KHO
npOH30RTH pe3Koe YMeHblIleHHe CBSl3H. ECJIH a',ik+l MeHb..
lIIe, 1IeM coce,nHHe CB5I311 a'k'k_l'...' a',lt' TO 9TO rOBopHT
o TOM, tITO OO'beKT i1&+1 H3 ,npyroro KJIaCTepa. CKatIO:K CBSl3H
coce,nHHX OO'beKTOB HBJlHeTCH CHrHaJIOM ,nJlH OT,neneHHSI O,nHO-
ro KJIaCca OT .npyroro.
H300pa3HM Ha OCH a6CUHCC HOMepa OO'beKTOB B nopSl,nKe HX
Bbloopa, a no OCO op,nHHaT - tlHCJIeHHbIe 3HatleHHSI HX CBSl3H
C npe,nbI,nyrnHMH OO'beKTaMH. nOJIYllHTCSI KpHBa5l, no,nOOHaH
H300pa>KeHHOA Ha pHC. 5.4, r,L{e BocnpOH3Be,neHbI pe3YJIbTaTbI
ynopH,nOtIeHHH pa3JIHtIHbIX BapHaHTOB HanHcaHHH apaocKHx
UH p I, 2, 3, 4, 5 B O,nHOM Jl3 9KcnepHMeHTOB [5], -HHTepBaJlbl t
Ha KOTOphIX H306pa>KeHHaH KpHBaH COBeplIlaeT CKatIKH BHH3,
COOTBeTCTBYIOT rpaHHnaM MeJK,ny H30JIHpOBaHHbIMH rpynnaMH
o&beKTOB.
269
nO.L{1JepKHeM JII060nblTHYIO OCOOeHHOCTb onHCaHIIOro 8JI-
ropHTMa: eCJIM Mem.L{Y KJIaCCaMH MMeeTC5I eCTeCTBeHHoe yno-
p5l.L{OtIeHHe, OTpa)KeHHOe B CB5I351X aij, npHQeM nepBbIH OO'beKT
BbI6paH M3 KpaHHero KJIaCCa, TO nOJIYtIeHHOe pa30HeHHe TaKR<e
oy,neT ynop5l.L{OlJeHHbIM; KJIaCCbI n05IBJI5IIOTC5I no Mepe HX OT"
,naJIeHHOCTM OT Toro KpaHHero, H3 KOToporo B351T nepBhIH OOn-
eKT. PHC. 5.4 MJIJIIOCTpHpyeT STY ocooeHHOCTb: peaJIH3aUHH
l.(H<PP OOJIa.L{aIOT eCTeCTBeHHbIM ynop5l.L{OtIeHHeM 5, 3, 4, 2, 1
B CTOPOHY YMeHbUIeHH5I CJIO:>KHOCTH HX <PHrypbI.
5 3 4- 2 1
-"' :.:....t......
PIIC. 5.4
,
3aMeTHM, tITO no OCHOBHOMY BapHaHTY aJIrOpHTMa «CneKTp»
KpHTepHeM npHcoe,nHHeHH5I OO'beKTa j K ynop5l,nOtIeHHIO it,...,i k
1 k
BJI5IeTC5I He BeJIHtIHHa a l kf , a cpe,nH5I5I CB5I3b k aisi. OtIe-
5=1
BIf;llHO, BnpOtIeM, tITO BblnOJIlteHHe 3Toro BapHaHTa ynop5l,nOQe-
HRH He TpeoyeT HHKaKOM ,nonOJIHHTeJIbHOH HH<popMaUHH no
cpaBHeHHIO C aJIrOpHTMOM «npHcoe,nHHeHHe». ,[{eJIO B TOM,
-tITO BeJIHtIHHbI ai j ,nJI5I Bcex j =1= it, ..., i k, YtIaCTBYIO He B
s k
1
BbItIHCJIeHHH cpe,nHero - a i i' H3BeCTHbI, TaK KaK YtIaCTBYIOT
" k s=1 s
B cpaBHeHH5IX npH BbIoope is+l.
OOCJIe.L{Hee 03HalJaeT, tITO np" nOHCKe ynop5l.L{OlJeHHH TaK
}1JIH MHatIe npMXO,l{HTC5I onpe.L{eJI5ITb Bce BeJIMtIHHbI CB5I3H aij
(c yqeTOM HX CMMMeTpHH). I1MeHHo n03TOMY aJIrOpHTM «CneKTp»
He BXO,nHT B tIHCJIO ObICTpbIX aJIrOpHTMOB KJIaCCH<pHKaQHH.
B TOM CJIY1Jae, Kor,na Bce aij ,nocTynHbI 0003peHHIO O,nHO-
.Bp MeHHo (3TO He BCer.L{a B03MO)KHO, HanpHMep, H3-3a HeXBaT-
}<H MalIlHHHOH naM5ITM), MO:>KHO npe,nJIO)KHTb p5l,n J{pyrHx 3B-
pHCTMK ,nJI5I opraHH38QHH CHCTeMbI OKpeCTHOCTeH. HanpHMep,
MO)KHO OTOHpaTb HCKOMbIH rpa<l> napbI (i, j) B nOp5l.L{Ke YObJ-
BaHH5I aij, nponYCKa51 Te H3 HHX, KOTopbIe BbIBO,n5lT, H.3 KJIaC-
270
--""'"
ca [(enHbIX rpacpOB npH npHCOe HHeHHH K yraM, Y)Ke OTOOpaH-
HbIM.
B npHMeHeHHH K KOHKpeTHOMY MaTepHaJIY paCCMOTpeHHbIe
aJIrOpHTMbI OOJIaj{aIOT CJIe.L{YIO HM cy ecTBeHHbIM He.L{OCTaTKOM.
HOBbIe OO'beKTbI He MorYT ObITb BCTaBJIeHbI BHYTPb ynop5l oqe-
HH5I Y>Ke OToopaHHbIX OO'beKTOB. n03TOMY eCJIH nepBbIH OO'beKT
BwopaH HeY.L{aqHO, H3 Cepe.L{HHW OnTHMaJIbHOrO ynop5l.L{oqe-
HHH, TO ynop5l.L{OqeHHe, nOJIyqeHHOe «npHCOe.L{HHeHHeM» OY-
.L{eT CKOJIb yrO.L{HO CHJIbHO OTJIHqaTbC5I OT HCKOMoro. ,[{JIH CM5Ir..
qeHH5I YKa3aHHoro He.L{OCTaTKa MO>KHO HCnOJIb30BaTb CJIe.L{YIO-
He B03MO>KHOCTH. Bo-nepBbIX, MomHO Hapa HBaTb uenb He
C O.L{HOrO, a C OOOHX KOHnOB. BO-BTOpbIX, nOCJIe nOJIyqeHHH
nenH MO)KHO pa3pe3aTb ee H CKJIeHBaTb .L{pyrHMH KOHnaMH.
TOqHee, H3 nenH i l ... iN pa3pe3aHHeM k-H .L{yrH nOJIyqalOTC5I
.L{Ba KYCKa: i l ... i k H i k + 1 ... iN; B 3aBHCHMOCTH OT BeJIHqHH
CB5I3H aitik + l' ai 1 i N , aiki k + l' aikiN BbIOHpaeTC5I O.L{HH
H3 BapHaHTOB CKJIeHKH iN ... i h +lil ...i k , i k + 1 ...iNi l ... ik'
i l ...iki k + 1 ... iN, i 1 ... iki N ...i h +1 (H3MeHeHHe nOp5l.L{Ka Ha
oopaTHbIH He MeH5IeT .L{JIHHbI nenH) - TOT, KOTOpbIH COOTBeT--
cTByeT MaKCHMaJIbHOH H3 3THX BeJIHqHH (k == 1, ..., N - 1).
nOCJIe nOJIyqeHH5I nenH OnTHMaJIbHbIH BapHaHT (no (5.32))
ee pa3pe3aHH5I .L{JI5I nOJIyqeHH5I nenHoro rpa<jJa ,l{HKTyeTcH CJ1e-
.L{YIO HM OOCT05lTeJIbCTBoM. npH m KOMnOHeHTax qHCJIO peoep
B nenHOM rpacpe paBHo N - m, TaK qTO 3HaqeHHe (5.32) .L{JIH
m-KOMnOHeHTHoro nenHoro rpacpa paBHO CYMMe CB5I3eH Me>K.L{Y
COCe.L{51MH MHHYC a (N - m). OTCIO.L{a CJIe.L{yeT, tlTO 3HatleHHe
(5.32) npH pa3pe3aHHH nenH (T. e. HCKJIlOqeHHH pa3pe3aHHbIx
.L{yr), C O HOH CTOpOHbI, YMeHblIlaeTCH Ha CYMMY CBH3eH no
v
HCKJIlOqaeMbIM .L{yraM, C .L{PyroH CTOpOHbI, YBeJIHqHBaeTCH
Ha am, r.L{e m - qHCJIO pa3pe3aHHbIx .L{yr.
TaKH1\1 oopa30M, OnTHMaJIbHbIH nenHoH rpacp .L{JI5I .L{aHHOH
nenH nOJIyqaeTC5I B COOTBeTCTBHH C aJIrOpHTMOM «CneKTp»:
HCKJIlOqeHHeM Tex peoep nenH, CBH3H no KOTOpbIM MeHbllle no-
pora a. MO>KHO nO.L{OOpaTb H OnTHMaJIbHOe 3HaqeHHe a - no..
JIOBHHY Cpe.L{HeH CB5I3H Me>K.L{Y oooeKTaMH no peopaM, BXO.L{51-
HM B nOJIyqaeMbIH nenHoH rpacp, TeM >Ke HTepaTHBHbIM aJI-
ropHTMOM, QTO H3JIO)KeH Ha c. 184 .L{JIH oo eH 3a.L{aqH annpoK-
CHManHH.
PaccMoTpHM npHMeHeHHe onHcaHHbIX aJIrOpHTMoB K Ha-
IlleMY npHMepy.
IlycTb nepBbIM BbIopaH OO'beKT 1, TOr.L{a nenOqKa npHCOe.L{H-
HHeMbIX OO'beKTOB eCTb 4 C a14 = 0,75, 3 C a43 == 0,92, 2 c
a32 == 0,92, 5 c a26 == 0,5, 6 c a 56 = 1,5, qTO .L{aeT ynopH.L{O-
271
...
qeHHe 1, 4, 3, 2, 5, 6 (MJIM oopaTHoe 6) 5, 2, 3, 4, 1) C CYM-
MOH CBSl3eH 4,59. TIpM nopore a = 0,6 nOJlyqaeM OnTMMaJIbHOe
pa3pe3aHMe no pe6py (2, 5), lITO ,naeT ,nBa OT,neJIbHbIX ynopSl-
-nOqeHHSI 1, 4, 3, 2 H 5, 6. Cpe.L{H5ISI CB5I3b B cooTBeTcTBYIO eM
II.enHOM rpacpe paBHa: A, = 4,09/4 = 1,02, liTO .L{aeT HOBoe 3Ha-
tIeHHe a = 0,51, npMBO.L{51 ee K TOMY)Ke pa36HeHHIO - no pe6-
PY (2, 5).
PaccMoTpHM Tenepb cJIYtIaii, KOr.L{a B KatIeCTBe HCXO.L{HOrO
B3S1T 06'beKT 3. TIOJIYlIaeM CJIe.L{YIO YIO nenOqKY npHCOe.L{HHe-
HHif: 2 C a32 = 0,92, 1 C a21 = 0,58, 4 C a44 = 0,75, 6 C a46 =
= 0,67,5 c a 6 5 = 1,5, tITO .L{aeT ynopSl.L{OlleHHe 3, 2, 1, 4, 6, 5
C npSlMoH CBSl3blO 4,42. XOTSI 3TO 3HalleHHe CPYHKII.HOHaJIa OT-
JII-ItIaeTCSI OT npe.L{bI.L{ymero Ha 3-4 %, nOJIYlIeHHoe ynopSl,no-
tIeHHe cy ecTBeHHo HOBoe, npHlIeM npH a = 0,6 pa3pe3aTb
npHXO.L{HTCH pe6po (1, 2), a 2 1 < 0,6. aTO .L{aeT KJIaCCbI 3-2,
1--4--6-5, KOTop e OlleHb nJIOXO COrJIaCOBaHM co CTPYKTY-
POR Me)K.L{Y 06'beKTaMH.
TaKHM 06pa30M, OpHeHTanHSI JIHWb Ha CBSl3H Me>K.L{Y coce-
,nHMH MO)KeT npHBO.L{MTb K. Hea.L{eKBaTHbIM pe3YJIbTaTaM .L{a>Ke
npH HaJIHqHH SlBHOH CTPYKTypHpoBaHHocTH ,naHHbIX. BnpOtIeM,
B ,naHHOM CJIYtIae XOpOI.I1He pe3YJIbTaTbI nOJIYtIaIOTCSI, eCJIH pa3-
pellIHTb npHCOe.L{HHeHHe OO'beKTOB K 060HM KOHuaM uenH. Tor-
.Ita, HatIHHaSl C 3, CHallaJIa npHCOe.L{HHSleM 2 C a32 = 0,92. Te-
nepb BH.L{HM, liTO K 2 6JIH)KaHlIIHMH SlBJISlIOTCSI 1, 4 C a12 =
= a42 = 0,58, a K 3-4 C a43 = 0,92. 3HaqHT, CJIe,nyeT npH-
COe.L{HHHTb 4 K HMelO eHcSI u.enH 3-2 co CTOpOHbI 3, liTO .L{aeT
4, 3, 2. ,[{aJIee onSlTb BH.L{HM,tITO CJIe,nyeT npHcoe,nHHHTb 1 CJIe-
Ba: 1, 4, 3, 2; 3aTeM 5 cnpaBa: .}, 4, 3, 2, 5; 3aTeM CHOBa cnpa-
Ba 6. aTO .L{aeT paHee nOJIYlIeHHoe ynopSl.L{OlleHHe 1, 4, 3, 2, 5, 6.
5.2.5. llocTpoeflue C7'p02UX paHOICupo8aHUa
CTporoe paH.iKHpOBaHHe MO>KeT 6bITb 3a.L{aHO OTHO-
meHHeM JIHHeHHOro KBa3HnOpSl.L{Ka. OCTaBJIHeM llHTaTeJIIO ,no-
R83aTeJIbCTBO Toro, QTO 3a.L{alla OTWCKaHHSI CTpororo paH)KH-
poBaHHH iI' ..., iN, onTHMH3HpYIO ero nOKa3aTeJIH (5.28),
(5.32) H (5.33), 3KBHBaJIeHTHa 3a.L{atIe HaXO)K.L{eHHSI TaKoif ne-
peCTaHOBKH (iI' ..., iN) CTpOK H cToJI6nOB MaTpHUbI a o.L{HOBpe-
u
MeHHo, npH KOTOpOH npHHHMaeT MaKCHMaJIbHOe 3HatIeHHe CYM-
Ma Ha,LLnHarOHaJIbHbIX 3JIeMeHTOB:
N-l N
e (it, ..., iN) -= ai s it.
s.- t t..s+ 1
(5.36)
212
B OT JIHqMe OT npe,L{bI,L{y ero pa3J{eJIa B (5.36) YtITeHa CB5I3b
MeJK Y BceMH OO'beKTal\1U, Tor,l{a KaK TaM paCCMaTpHBaJ1HCb
JIHIllh CB5I3H Me*,l{Y coce,l{HHMH 06'beKTaMM.
CcpOPMYJIHpOBaHHa51 3a,l{atIa 0606 aeT HeKOTopbIe H3BeCT-
HbIe B JIHTepaType 3a,l{atIH ynopSl,l{OtIeHH5I. TaK, npH 3a,l{aHHOM
opHeHTHpOBaHHOM rpa<pe r H YT TaKYIO nepecTaHoBKY
(ii, ..., iN), QT06bI KOJIHtIeCTBO 06paTHbIX yr B rpa<t>e r (T. e.
TaKHX (is, it) E r, tITO S > t) ObIJIO MHHHMaJIhHO. OtIeBH,nHO,
CCPOPMYJIHpOBaHHbIM KpHTepHH 3KBHBaJIeHTeH (5.36) npH MaT-
pHne a, paBHoM (1, O)-MaTpHne OTHOIlleHH5I r. STY 3a,l{atIY MO>K-
HO paCCMaTpHBaTb TaK)Ke KaK 3a,l{atIY npeBpa eHH5I rpa<pa B
rpacp 6e3 KOHTYPOB C nOMOInblO Y,l{aJIeHHSI H3 r MHHHMaJIbHOro
tIHCJIa ,l{yr. I1Hor,l{a paCCMaTpHBalOT B03MO)KHOCTb npeBpa e-
HHSI r B rpa<t> 6e3 KOHTypOB C nOMOInblO He TOJIbKO y,naJIeHHH,
HO H ,l{06aBJIeHH5I ,l{yr. MHHHMM3anHSI tIHCJIa TaKMX onepaUHH
9KBHBaJIeHTHa 3a,l{atIe HaXO)K,l{eHHSI TaKoro CTpororo paHLKHpo-
BaHHSI, KOTopoe 6bI OTJIHtIaJIOCb OT r MHHHMaJIbHbIM qHCJIOM
nap (i, j), T. e. .aa,l{atIe annpOKCHMaUHH rpacpa r B KJIaCCe OT-
u u
HOIlleHHH JIHHeHHOro nopSl,nKa, KOTopaSl, KaK nOKa3aHO B
n. 4.1.1, TaK:>Ke 3KBHBaJIeHTHa MaKCHMH3anHH (5.36) ,l(JIH
(1, O)-MaTpHnbI a rpacpa r.
TOtIHOe peIlleHHe 3TOM 3a,l{atIH, Bo06 e rOBopH, TaK>Ke
npaKTHtIeCKH ,nOCTH)KHMO JIHIllb npH He60JIbIllOM tIHCJIe 06'beK-
TOB. Ho BO MHorHX npaKTHtIeCKHX cHTyanH5IX Tpe6yeTcSI HaM-
TH cTporoe ynop5l,l{OtIeHHe HMeHHO MaJIOrO KOJIHtIeCTBa 06'beK-
TOB. HanpHMep, eCJIH 06'beKTbI - npOMbIIllJIeHHbIe H3,l{eJIHH,
cpaBHHTeJIbHOe KatIeCTBO KOTOpbIX OneHHBaeT 9KcnepTHaH
KOMHCCHSI (Rk xapaKTepH3yeT npe,l{nOtITeHH5I k-ro 3KcnepTa,
aij - BepOSlTHOCTb npe,l{nOtITeHHH i nepe,l{ j), HJIH HeCKOJIbKO
.
KOHKYPHPYIOInHX TeXHHtIeCKHX MJIH xY,l{O)KeCTBeHHbIX npoeK-
TOB H T. ,no B TaKoro pO,l{a cHTyanH5IX npe,l{CTaBJI5IeT HHTepec
TOIIHbIM aJIrOpHTM peIlleHHH 3a,l{allH (5.36) [58, 15j.
npe)K,l{e Bcero saMeTHM, liTO OnTHMaJIbHOe ynopSl,nOtIeHHe
He H3MeHHTCSI, eCJIH ,l{JI5I npOH3BOJIbHbIX i H j M3 aij H aji OJtHO-
BpeMeHHO BbIlIeCTb O,l{HO H TO :>Ke 11 HCJIO. ,[{eHCTBHTeJIbHO, npH
JII060M nepeCTaHOBKe B CYMMY (5.36) BXO,l{HT O,l{HO H TOJIbKO
u
O,l{HO H3 lIHCeJI aij HJIH ajh TaK liTO BbIlIHTaHHe H3 HHX O,l{HOM H
TOM :me BeJIHtIHHbI npocTo H3MeHHT 3HalleHHe e (ii' ..., iN) Ha
3TY :>Ke KOHCTaHTY npH JII060M nepecTaHoBKe (ii' ..., iN). nos-
TOMY 6Y,l{eM ,l{aJIee paccMaTpHBaTb MaTpHny, nOJIYlleHHYIO BbIIIH-
TaHHeM min [aij, aji] H3 aij H aji ,l{JISI Ka)K,l{OM napbI i, j E 91.
0603HalleHHe aij ,l{JI5I Y,l{06CTBa OCTaBHM TO :me. 113 CKa3aHHoro
BH}1HO, liTO KpHTepHH (5.36) HMeeT CMbICJI npHMeH5ITb TOJlbKO
K HecHMMeTpHQHbIM MaTpHnaM.
273
HeTpY HO BM eTb ,!{aJIee, liTO BC5IKOe ynOp5l,!{OlleHHOe pa3-
OHeHHe R = (R I , ..., R p ), COOTBeTCTBYIO ee rpacpy r ==
== { (i, j) I ai j > O}, xa paKTep H3yeT OnTHMaJI bHoe ynop 51,!{0-
lIeHHe C TOllHOCTblO ,!{O nepeCTaHOBOK OO'beKTOB BHYTPH OMKOM-
nOHeHT Rs: B OnTHMaJIbHOM ynopSl,!{OlJeHHH Bce OO'beKTbI H3 R s
npe,!{llIeCTBYIOT OO'beKTaM OHKOMnOHeHT R s + I' ..., R t. .ap Y-
rHMH CJIOBaMH, ,!{yrH rpacpa rep a G HanpaBJIeHbI OT npe,!{lIleCT-
BYIOIII.MX B OnTHMaJIbHOl\f ynop5l,!{OlJeHHH OO'beKTOB K nOCJIe,nylO-
III.HM. ,[{eHCTBHTeJIbHO, BC5IKOe ,!{pyroe pacnOJIO)KeHHe OHKO 1no-
HeHT MO:>KeT TOJIbKO YBeJIHlIHTb CYMMY nOMHarOHaJIbHbIX 3Jle-
MeHTOB H, CJIe,!{OBaTeJIbHO, YMeHblIlHTb CYMMY HaMHarOHaJIb-
HbIX.
TaKHM oopa30M, ,!{JI5I nOCTpOeHH5I OnTHMaJIbHOrO ynop5l,!{0-
lIeHH5I OCTaeTC5I TOJIbKO npOpaH:>KHpOBaTb OO'beKTbI BHYTPH
OHKOMnOHeHT Rs, oopa a51 BHMMaHHe TOJIbKO H8 nO,!{MaTpH bI
(ail) (i, j E Rs), nOCKOJIbKY CYMMa H8MHarOHaJIbHbIX 3JIeMeH-
TOB (5.36) npH nepeCTaHOBKax BHYTPH Rs 3aBHCHT TOJIbKO OT
u
ynOp5l,!{OlleHH5I COOTBeTCTBYIOIII.eH nO,!{MaTpH bl.
HTaK, nYCTb S - HeKOTOpa51 oHKoMnoHeHT8, COCT05l a51
H3 M OO'beKToB. ,[{JI5I BC5IKOH nepeCTaH OBKH (iI' i 2 , ..., iM)
lIepe3 I It oo03HallHM ee HallaJIbHYIO qaCTb (iI' ..., ik) (k EM),
a lIepe3 I It - MHO)KeCTBO OCTaBlIlHXC5I M - k OO'beKTOB, HO
He ynopSl,!{OlleHHbIX. 5y,!{eM ,!{aJIee no,!{ I It nOHHMaTb TaK:>Ke MHO-
:meCTBO Bcex nepecTaHoBoK MHO)KeCTBa S C HallaJIbHOH lIaCTblO
(i l , ..., i k).
npone,!{ypa, onHCbIBaeMaSl ,!{aJIee, npe,!{JIOJKeHa B. H. 5yp-
KOBbIM H B. O. rponneHoM ,!{JI5I ynop5l,!{OQeHHSI npOH3BOJIbHOH
MaTpH bI CB5I3eH [15].
· 5y,!{eM nOCJIe,!{OBaTeJIbHO HaoHpaTb B OnTHMaJIbHYIO nepe-
CTaHOBKY no O,!{HOMY OO'beKTY, npOBep5l51 3aTeM, ,!{eHcTBHTeJIb-
HO JIH MomeT ObITb nOJIYlIeHO OnTHMaJIbHOe paHJKHpOBaHHe.
ECJIH k OO'beKTOB Y)Ke HaopaHbI (k == 1, ..., M - 2), T. e. no -
JIYlIeHa lIaCTb I It, TO ,!{aJIee paCCMaTpHBaIOTC5I MHOJKeCTBO I It
H CB5I3H BXO,!{51 HX B Hero OO'beKTOB. ECJIH no,!{rpacp rpacpa r
Ha Tn He 51BJI5IeTC5I OHCB5I3HbIM, TO I It pacna,!{aeTC5I Ha KJIaCCbI
COOTBeTCTBYIOIII.erO ynOp5l,!{OlleHHoro pa30HeHH5I (R , ..., R It).
0003HallHM lIepe3 Sk MHO:>KeCTBO OO'beKTOB nepBoro CJI05l.
B lIaCTHOCTH, Sk MomeT COBna,!{aTb C I k . OlleBH,!{Ho, TOJIbKO
OO'beKTbI Sk MorYT npeTeH,!{oBaTb Ha (k + 1)-e MeCTO B on-
THMaJIbHOH nepecTaHoBKe. 3HallHT, Heooxo,!{HMO HCCJIe,!{OBaTb
Ha OnTHMaJIbHOCTb Bce B03MO)KHbIe nepecTaHoBKH (k + 1)
OO'beKTOB BM,!{a (I It, j) (j E Sk). 3TO 03HallaeT, liTO Ha KaJK OM
274
-
k-M mare npo[(ecc BeTBHTC5I: HY}I{HO OT.L{eJIbHO HCCJIe,l{OBaTb
BC5IKYIO nepeCTaHOBKY BH,l{a (I k, j) (j E Sk). L{JI5I 3Toro O[(eHHM
CHH3Y CYMMY nOMHarOHaJIbHbIX 3JIeMeHTOB B nepecTaHoBKax
BH,l{a (Ik, -Ik). Ooo3HallHM ee V (I h). OtIeBH.L{HO, V (I k) eCTb
cYMMa CJIe,l{YIO Hx BeJIHlIHH: e (I h) - nO.lIJ1.HarOHaJIbHa51 CYM-
Ma Ha nepBbIx k OO'beKTaX; aij -cYMMa CB5I3eH OO'beKTOB
iEI k j(f.I k
H3 I k C OO'beKTaMH H3 I k ; e ( I Ii) - nOMHarOHaJIbHa51 cYMMa
CB5I3eH nOCJIeAHHX M - k OO'beKToB. nepBbIe .L{Be H3 3THX Be-
JIHlIHH onpe,l{eJIeHbI 3a.L{aHHeM I Ii, nOCJIe,l{H5I5I )Ke On pe JISleTCH
e e He npOH3Be,l{eHHbIM ynOp5l,llOlleHHeM OO'beKTOB 1 k. 5ICHO,
liTO MHHHMYM 3HaqeHHH V (I k) .L{OCTHraeTC5I npH MHHHMaJIb-
HOM 3HalleHHH e (I k). Ho e (I k) He npeBOCXO,l{HT BeJIHtIH-
- -
HbI ex (I h)' JIerKO BbIlIHCJISleMOH .L{JI5I I h C nOMO blO cJIe YIOIII.ero
npaBHJIa TI.
n p a B H JI 0 n. Onpe eJIHM npOH3BOJIbHbIH KOHTYP c
nO.L{rpa<pa r Ha 7 k H BbIlIHCJIHM ex (c) = min aij - MHHHMaJIb-
(l, j) Ec
HYIO CB5I3b Ha c. Tenepb BbIlITeM H3 Bcex CB5I3eH 3Toro KOHTypa
BeJIHtIHHYex (c); XOT5I ObI O,l{Ha H3 HHX CTaHeT HYJIeBOH, TaK liTO
KOHTYP OY,l{eT HCKJIIOQeH H3 cOBoKynHocTH Bcex KOHTypOB no,l{'"
rpa<pa r Ha I k. nOBTOp5l51 9TY npOne,l{YPY H CYMMHPY5I nOJIY-
lIaeMbIe BeJIHtIHHbI ex (c), nOJIYtIHM nO,l{rpa<p oe3 KOHTypOB,
,l{JI5I KOToporo MHHHMaJIbHa51 cYMMa nOMHarOHaJIbHbIX 3JIe-
MeHTOB paBHa O. OTCIO,l{a CJI, e,l{yeT liTO HaKOnJIeHHa51 CYMMa
ex (I Ii ) He npeBbIlIIaeT min e ( I h).
HTaK, BeJIHtIHHbI V(I k) .L{JI5I BCSlKoro I k MorYT ObITb OneHe-
HbI C nOMOIII.blO <PYHKnHH ex ( I k ).
Tenepb nponecc BWlIHCJIeHH5I OnTHMaJIbHOH nepecTaHOBKH
opraHH3yeTC5I CJIe,l{YIOIII.HM oopa30M. Ha k-M ma re AJI 5I BC5IKOH
nepecTaHOBKH (I k, j) C j E Sk BhIlIHCJI5IeTC5I ex (I k, j).. B Ka-
lIeCTBe I k + 1 BbIOH paeTC5I Ta nepeCTaHOBKa (I k, jo), ,l{JI5I KO-
TOpOH 3HaqeHHe ex (I k, jf MHHHMaJIbHO, nOCJIe lIero nponecc
BeTBJIeHH5I, T. e. nepeXO.L{a K (I k + l' j), j E Sk + 1, nOBTOp5l-
eTC5I ,l{O Tex nop, nOKa He OY,l{eT nOJIY tIeHO 11.1 . EcJIH 3HalleHHe
V (1M) He npeBOCXO.L{HT OneHOK min e (I k, j), j =1= jo, j E Sk,
TO nOJI YtIeHHa 51 nepeCTaHOBKa 1 M OnTHMaJIhHa. l{eHcTBHTeJlbHO,
min e (I k, j) 3TO OneHKa CHH3Y BeJIHlIHH V ,lJ.JI5I nepecTaHoBoK t
HallHHalO HXC5I (I k, j): eCJIH HH}KHHe oueHKH He MeHbIlle t lIeM
V (1 M ), TO peaJIbHbIe 3HalleHH5I MorYT ObITb eIne OOJlbIlle, TaK
qTO HX H npOCMaTpHBaTb He3alleM.
275
-.,-
ECJIH >Ke V (1 M ) 60JIbIlIe KaKOH-JIH60 H3 BeJIHlIHH V (I k, i1)
)VIH i1 E Sk, TO He06xo,l{HMO TaK>Ke npOBeCTH npOQeCC BeTBJIe-
HH5I ,l{JI5I I k + 1 = (I k, jl) nOKa He ,l{OH,l{eM .L{O HeKOTopOH .L{PY-
roB: nepecTaHoBKH 1 M , .L{JIH KOTOpOH CHOBa npOBep5leTC5I .L{OCTa-
TOtIHOe YCJIOBHe OnTHMaJIbHOCTH: 60JIbille HJIH HeT V (1 M ),
tIeM V (I k, j) (j E Sk). nponecc npo.L{OJI>KaeTCSI ,l{0 Tex nop,
nOKa He oy eT nOJIYlIeHa TaKa51 nepeCTaHOBKa, 3HalleHHe KO-
TOpOH He npeBbIIllaeT HHLKHHX OneHOK V ,l{JI5I e e He HCCJIe O-
BaHHhIX BeTBeH (I k, j) H, CJIe OBaTeJIbHO, OnTHMaJIbHO. Mo-
>KeT CJIYlIHTbC5I, liTO npH,l{eTC5I nOCJIe,l{OBaTeJIbHO npOaHaJIH-
3HpOBaTb Bce BeTBH: aJIrOpHTM,Bo06rne rOBOp5l, OT nOJIHOrO
nepe60pa He rapaHTHpyeT.
O HaKO, KaK npaBHJIO,TaKOH aJIrOpHTM ,l{OBOJIbHO CHJIbHO
COKpa aeT nepe60p 3a CtIeT OTCelleHH5I BeTBeH, HatIHHa51 C Ma-
JIbIX k. Crre,l{yeT OTMeTHTb TaK:>Ke, liTO onHcaHHbIH aJIrOpHTM
npHHa JIe}KHT TaK Ha3bIBaeMOMY MemoiJy 8em8eu U 2paflUl{.
BeTB5IMH HBJISIIOTCH B .L{aHHOM CJIYllae B03MO:>KHbIe nepeXO,l{bI
OT I k K (Ik' j), rpaHHnaMH-HH:>KHHe OneHKH V (Iii).
npH peaJIH3aQHH aJIrOpHTMa BOBce He 06513aTeJIbHO pac-
CMaTpHBaTb Bce BeTBH (I k, j), j E Sk. qacTo H B Sk MO:>KHO
3apaHee oTopaKoBaTb 06'beKTW, 3aBe,l{OMO HenpHrO,l{HbIe ,l{JI5I
(k + I)-ro MeCTa.
nOJIe3Ha, B tIaCTHOCTH, CJIe YIO a51 oueHKa. ,[{JIH BepIllHH
113 Sit 0603HallHM tlepe3 ' MHO:>KeCTBO TaKHX 06'beKTOB j, JIH
KOTOpbIX CYMMa BbIXO,l{H HX CB5I3eH ajl 60JIbIlle CYMMbI BXO-
lES k
AHIII.HX "'5'.a,J. ECJIH S' =1= 0., TO 06513aTeJIbHO cy ecTByeT on-
IE
THMaJIbHaSl nepeCTaHOBKa BepWHH Sk (a HMeHHO OHH ,l{OJI:>KHbI
OhITb pacnOJIO)KeHbI 3a I k)' TaKa5l, tITO Ha (k + 1)-M MeCTe
pacnOJIO:>KeH O.L{HH H3 06'beKTOB MHO)KeCTBa S' [15]. n03TOMY
u
KOJIHtIeCTBO BeTBeH MO>KHO orpaHHlIHTb, paccMaTpHBa51 TOJIb-
KO BeTBH (I k, i) JI5I i E S'.
5.2.6. AA20pUTMbl Bbt8eAeHUfl no8Mflo{)/CecTB
ECJIH B npe,l{bI,l{Y HX nYHKTax MbI paccMaTpH-
BaJID 3a,natIH 0 nOCTpoeHHH TpHBHaJIbHbIX pa30HeHHH co CTPYK-
u
TYPOH, TO 3a atIY 0 BbI eJIeHHH nO.L{MHO:>KeCTBa MO:>KHO TpaKTO-
BaTb KaK 3a aqy 0 nOCTpoeHHH pa36HeHH5I C TpHBHaJIbHOH
CTPYKTYPOH, COCT05lmeH H3 e HHCTBeHHOH neTJIH.
PaCCMOTpHM OCHOBHbIe nOKa3aTeJIH KalleCTBa nO,l{MH0-
1KeCTBa S 91, BbITeKalOIl{He H3 npe bI y ero MaTepHaJIa:
276
CYMMapHa BHYTpeHHHH CBH3b C yqeTOM nopora
G(S,a)= (aij-a)= afj-aN s (Ns-l); (5.37)
I,IES f./ES
cpe HHH cYMMapHaH CBH3b
1
F (8) = N aij; (5.38)
S .. . ES
1.,1
annpOKCHManHOHHaH xapaKTepHCTHKa
( afj)! ( aij- a iJ ) t
g (8) = liES + Il= 1 II E S . (5.39)
N s (N s -l) N2_N +Ns
nOKa3aTeJIb G (8, a) KOHKpeTH3HpyeT (4.20), (5.29), F (8)-
(4.29), (5.28), g (8) - (4.24) (5.30). CBH3H OO'beKTOB C ca-
MHMH COOOR He paCCMaTpHBaIOTCH. KaK H paHee, onTHMYM
(5.39) COOTBeTcTByeT MaKCHMYMY (5.37) npH nopore a, paBHOM
nOJIOBHHe cpe HeH CBH3H Me:>K Y OO'beKTaMH MHO:>KeCTBa S.
Bce YKa3aHHbIe KpHTepHH BhIpa:>KalOT TpeooBaHHe OO'be H-
HeHHH B MHO:>KeCTBe 8 HaHOOJIee CBH3aHHbIX OO'beKToB.
PaccMoTpHM qaCTHbIH CJIyqaH 3a aqH (5.37), Kor a a eCTb
(1, O)-MaTpHna KaKOrO-JIHOO rpacpa, a a = 1. B 3TOM CJIyqae
(5.37) npe CTaBJIHeT COOOM pa3HOCTb Me:>K,lJ.y peaJIbHhIM qHCJIOM
BHYTpeHHHx yr B 8 H MaKCHMaJIbHO B03MO)KHbIM Ns (Ns - 1).
51cHo, qTO 3Ta pa3HocTb BcerJ{a HenOJIO:>KHTeJIbHa H OCTHraeT
CBoero MaKCHMaJIbHOrO HYJIeBOrO 3HaqeHHH TorJ{a H TOJIbKO
Tor.na, KorJ{a. ail = 1 JIH JIIOObIX pa3J1HqHbIX i,j E S, T. e.
Kor,na S - nOJIHbIR nOJ{rpacp ,naHHoro rpacpa. 3TO 03HaqaeT, qTO
B aHHOM cJIyqae 3a aqa (5.37) 3KBHBaJIeHTHa 3a aqe 0 nOCTpoe-
HHH Bcex nOJIHbIX no rpacpoB aHHoro rpa<pa, KOTopaH, KaK
H3BeCTHO, HOCHT nepeoopHbIH xapaKTep [1451-
nOCKOJIbKY CBH3b Ka:>K OrO OO'beKTa C caMHM CODOH He pac-
cMaTpHBaeTcH, JIerKO YBH eTb, QTO 3aJ{aqa 0 nOCTpoeHHH no -
u
MHO:>KeCTBa nOJIHOCTblO COBna aeT C 3a aQeH 0 nOCTpoeHHH
pa30HeHHH C eJ{HHCTBeHHbIM HeTpHBHaJIbHblM KJlaCCOM (T. e.
pa30HeHHH, Bce KJIaCCbI KOToporo, KpOMe OJ{Horo, CO,ll.ep>KaT
pOBHO no O HOMY OO'beKTY). .l{eHcTBHTeJIbHO, ,lI.JIH TaKHX pa3-
OHeHHH Bce CJIaraeMbIe KpHTepHeB, KpOMe OTpa:meHHbIX B
(5.37)-(5.39) ,lI.JIH HeTpHBHaJIbHOrO KJIaCCa S, HYJIeBbIe.
CKa3aHHoe 03HaqaeT, qTO MHoroe H3 9 5.1 HMeeT OTHO-
weHHe K ,lI.aHHOM 3a aqe. B qaCTHOCTH, ,lI.JIH peweHHH 3a aq
(5.37)-(5.39) MorYT HCnOJIb30BaTbCH COOTBeTCTBYIO He MO,nH-
<pHKaUHH paCCMOTpeHHbIX TaM aJIrOpHTMOB KJIaCCH<pHKaUHH.
277
MbI OrpaHH1JHMCH 3,!{eCb TOJIbKO paCCMOTpeHHeM aJIrOpHTMa
nOCJIe,!{OBaTeJIbHOrO nepeMe eHH5I OT,!{eJIbHbIX OO'beKTOB (ne-
peHH.L{eKCaQHH), KOTOpbIM B ,!{aHHOM CJIyqae npeBpa aeTC5I
B aJIrOpHTM ,!{ooaBJIeHH5I HJIH y,!{aJIeHH5I OO'beKTOB H3 BbI,!{eJI5Ie-
Moro MHomeCTBa S m.
,.QJI5I Y.L{OOCTBa 0003HaqeHHH MHomeCTBO S oy,!{eM 3a,!{aBaTb
oYJIeBCKHM (1, -1)-BeKTopoM a = (at, ..., aN), i-51 KOMno-
HeHTa KOToporo xapaKTepH3yeT COCT05lHHe OO'beKTa i E m:
ai = 1 ,!{JI5I i E S H ai = -1 ,!{JI5I i $. S. Tor,!{a H3MeHeHHe
. u
3HaKa t-H KOMnOHeHT@ ai 03HaqaeT H3MeHeHHe COCT05lHHH
i- ro OO'beKTa: Y.L{aJIeHHe H3 S, eCJIH ai = 1, H ,!{ooaBJIeHHe B
S, eCJIH ai = -1, OnepaTop H3MeHeHH5I 3HaKa i-H KOMnOHeH-
TbI 0003HaqHM Ob TaK qTO Oi (a ) = (ai' a 2 , ..., ai-I' -ai'
ai + l' ..., aN).
BbIqHCJIHM npHpaIneHHe nOKa3aTeJI5I (5.37):
L\i = 0 (Oi S, a) -0 (8, a) = - 2a i (ail-a). (5.40)
iES
npH OnTHMaJIbHOM S ,!{JI5I Bcex i E m cnpaBe,!{JIHBO L\i 0,
qTO ,!{aeT CJIe,!{YID ee HeOOXO,!{HMoe YCJIOBHe OnTHMaJIbHOCTH
MHO)KeCTBa S no KpHTepHID (5.37).
ECJIH S OnTHMaJIbHO, TO ,!{JIH JIIDOOrO OO'beKTa i E m ero
cpe,!{H5I5I CB5I3b C OO'beKTaMH S He MeHbIlle a, eCJIH i ESt H He
OOJIbIlle a, eCJIH i $. S. 3TO YCJIOBHe ,!{OKa3bIBaeTC5I Ha OCHOBe
(5.40) TaK )Ke, KaK aHaJIOrHqHble YCJIOBHH B n. 5.1.5.
M3 Hero BbITeKaeT CJIe,!{YID HH npOCTOM aJlrOpHTM nOCTpoe-
HH5I S (B KJlaCCe BceB03MO)KHbIX nO,!{MHO)KeCTB). BbIqHCJIeHH5I
HaqHHaIDTC5I C a = (-1, ..., -1), T. e. S = 0.
Haxo,!{HM napy i, j, ,!{JI5I KOTOpbIX aij MaKCHMaJIbHO B a.,
ECJIH aij a, BKJIIDqaeM i H j B S. B npOTHBHOM CJIyqae npH
aij < a HeTpHBHaJIbHOrO S, onTHMH3HpYIDInerO (5.37), He CY-
ecTByeT.. ,[{aJIee ,!{ooaBJlSleM B S nOCJIe,!{OBaTeJlbHO OO'beKTbI,
HMeID He MaKCHMaJIbHYID cpe,!{HIDID CB5I3b C OO'beKTaMH, Y)Ke
BKJIIOqeHHbIMH B S, O Tex nop, nOKa 3Ta cpe HHH CBH3b He (
CTaHeT 1\1eHbIlle, qeM a. t
3TOT aJIrOpHTM COBna,!{aeT C aJIrOpHTMOM «CneKTp» t
(n. 5.2.5) B TOM MO,!{HcpHKanHH, Kor,!{a npHcoe,!{HHeHHe HOBbIX
OO'beKTOB ocy eCTBJI5IeTC5I no cpe,!{HeM CB5I3H C Y)Ke OToopaH-
HbIMH H B 3TOM CMbICJIe MO)KeT paCCMaTpHBaTbC5I KaK ero OOOCHO-
BaHHe.
H3JIO)KeHHbIH aJIrOpHTM MO)KeT npHMeH5ITbC5I II B TOM
CJIY1Jae, Kor,!{a, KaK 3TO qaCTO ObIBaeT Ha npaKTHKe, TpeoyeTCH.-
1JTOObI MHO:>KeCTBO S BKJIID1JaJIO HeKOTopbIe oa30BbIe OO'beKTbl.,
278
Boo6rne npH 3a):(aHHOM HalJaJIbHOM 8 MorYT HCnOJIb30BaTb a
C5I CPOPMYJIbI (5.40) ):(JI5I HaXO)K eHH5I Toro OO'beKTa i, COCT05lHHe
KOToporo CJIe,l{yeT H3l'vleHHTb. ECJIH k TOT OO'beKT, ):(JI5I KOToporo
k = max i > 0, TO CJIe):(yeT nepeHTH K MHO)KeCTBY OkS.
i
nponecc BbIlIHCJIeHHH MO)KeT ObITb opraHH30BaH Ha OCHOBe
MeTo):(a BTOpbIX pa3HocTeH [8] TaKHM oopa30M, tITOObI He ClIH-
TaTb Ka)K):(bIH pa3 i no HCXO):(HOH MaTpHne, a nepeClIHTbIBaTb
HX Ha OCHOBe 3HalleHHH i ):(JI5I npe):(bI):(yrnero MHO}KeCTBa S.
BOJIee Toro, 3HatIeHH5I i ,Z(JI5I nOJIYlleHHoro Ha ,l{aHHOM ware
MHO)KeCTBa OkS MorYT ObITb BhItIHCJIeHbI Ha OCHOBe 3HalleHHH
i llJI5I HCXO,l{HOro MHO)KeCTBa S CJIe,l{YlOrnHM oopa30M [9]:
t1 i (OkS) = i (8) + ik' (5.41)
r,l{e BTopbIe pa3HocTH l1 ik = i (OkS) - i (S) onpe,l{eJIe-
HbI <POPMYJIOH
A { 4a i ak (ai1 -a) (i =1= k);
L1tk =
-2 i (S) (i = k).
(5.42)
,[(eHCTBHTeJIbHO, no onpe,l{e.lIeHHIO OkS npH i =1= k
i (Ok S) = G (Oi Ok S, a) -0 (Ok S, a) =.: - 2rx i I, (ail-a),
i€Ok S
r,l{e nOCJIe,l{Hee paBeHCTBO BbITeKaeT H3 (5.40).
BbItIHTa51 OTCIO,l{a l1 i (S) (5.40), nOJIYtIHM
t1 ik = - 2a i [I, (aij - a) - I, (aij-a)]= 4a i ak (ai1 -a),
ie o AS ie S
liTO H TpeOOBa.lIOCb ):(OKa3aTb. B TO )Ke BpeM5I npH i = k
OiOkS = S, TaK liTO l1 i (OkS) = G (S, a) - G (OkS, a) =
= - kS.
3TO 38BepwaeT ,l{OKa3aTe.lIbCTBO (5.42).
I1cnOJIb30BaHHe cpOPMY.lI (5.41), (5.42) ,l{.lI5I nepeclIeTa 3Ha-
tIeHHH i npH H3MeHeHHH S cyrnecTBeHHO COKpaI..I.(aeT BbItIHC-
JIeHH5I, TaK KaK TpeoyeT .lIHIllb YlleTa e,l{HHCTBeHHoro ai 11, npH
Ka)K,l{OM i.
C.lIe):(yeT 3aMeTHTb, liTO npH H3MeHeHHH nopora a (HanpHMep,
B HTepaTHBHOM a.lIrOpHTMe annpOKCHManHH (c. 184» nepBbIe
pa3HocTH t1 i ,l{JI5I MHO)KeCTBa S MorYT ObITb nepeCtIHTaHbI no
C.lIe):(YlOrneif cpopMY.lIe:
t1 i (a') = i (a) + 2'(Xi N s (a-a'), (5.43)
279
t,l{e
N' = J N , i S ;
S '1\T s -I,iES.
<l>oPMYJIY (5.43) JIerKO BbIBeCTH H3 (5.40); ee HCnOJIb30Ba-
HHe n03BOJIHeT npH H3MeHeHHH nopora He Bhp.IHCJI5ITb 6. i 3aHO-
BO no HCXO,nHOH HHcpopManHH [9].
PaccMoTpHM Tenepb npHpa eHHe nOKa3aTeJIH (5.38):
Ft=F(OiS)-F(S)= (l,i ( F(S)-2 aij ) . (5.44)
I Oi S I · S
1£
Heo6xo,nHMbIe YCJIOBHSI OnTHMaJIbHOCTH (Pi 0) npH 3TOM
BbITJISI,nSlT CJIe,nYIOInHM 06pa30M.
CYMMapHaSl CBSl3b 06'beKTa i E m C 06'beKTaMH S npeBhl-
maeT cpe,nHIOIO CYMMapHYIO CBH3b B S, eCJIH i E S, H He npe.
BblIIlaeT ee, eCJIH i S.
9TH YCJIOBHH onpe,neJI5IIOTCSI B CHJIY (5.44) 3HaKOM pa3HocTH
F (8) - 2 aiJ; K03qxpHnHeHT 2 OTpa}KaeT r:rOT <paKT, liTO B
iES .
F (8) cpe,nHHH CBSl3b Ka}K,noro 06'beKTa C OCTaJIbHbIMH YIIHThl-
BaeTCH ,nBa1K,l{bI H3-3a CHMMeTpHqHOCTH a.
C<}JOPMYJIHpOBaHHbIe YCJIOBHH TaK}Ke npHBo,n5lT K aJIrOpHT-
MY nOCTpOeHH5I S THna «CneKTp», HO KpHTepHeM npHcoe,nHHe"
HHSI 06'beKTOB 3,neCb SlBJI5IeTC5I Y1Ke CYMMapHa51 CB5I3b HOBoro
06'beKTa C Y)Ke oTo6paHHbIMH. npHcoe,nHHeHHe 06'beKTOB npe-
KparnaeTCH, Kor,na Pi 0 ,nJI5I Bcex i S.
npHMeHeHHe onHcaHHbIX aJIrOpHTMoB K HaIIleMY npHMepy
weCTH 06'beKTOB, KaK HeTpy,nHo BH,neTb, ,naeT MHO)KeCTBO S =
= {5,6}. .
PaccMoTpHM Mo,nH<pHKanHIO 3a,naqH 0 Bw,neJIeHHH no,nMHO-
JKeCTBa Ha TOT CJIyqaH, Kor,na CTpOKH H CTOJI6UbI MaTpHnbI
J{aHHbIX MO}KHO BbI6HpaTb He3aBHCHMO [74, 10). npH 3TOM sa-
J{alla CBO,nHTCH K BbI60py no,nMHOLKeCTBa CTpOK H no,nMHOLKeCTBa
CTOJIOnOB, HaH60JIee COOTBeTCTBYIO HX ,nPyr .npyry. 3Ta 3a-
.na'-Ia paccMaTpHBaJIaCb TaK)Ke B [71] npHMeHHTeJIbHO K npOOJIe-
Me Bw,neJIeHH5I ,nBYX rpynn npH3HaKOB, o,nHa H3 KOTOpWX B
MaKCHMaJIbHOH CTeneHH onpe,neJI5IeT ,nPyrylO (CM. n. 4.2.1).
B 3TOM CJIyqae B KaqeCTBe HCXO,nHOH BwcTynaeT MaTpHna no-
u
Ka3aTeJIeH CB5I3H npH3HaKOB, a 3a,naqa CBO,nHTCSI K BbI,neJIeHHIO
u
np5lMoyrOJIbHOH no,nMaTpHnbI C MaKCHMaJIbHbIMH 3JIeMeHTaMH_
CcpOPMYJIHpyeM TOqHYIO OOInYIO nOCTaHOBKY 3a,naqH [71].
HMeeTcH npSlMoyrOJIbHa51 n X m MaTpHna BeJIHqHH wiJ(i =
= 1, ..., n; j == 1, ..., m). MHO)KeCTBO ee CTpOK 0003Ha-
'1HM I == {I, ..., n}, a MHO)KeCTBO CTOJIOnOB - J = {I, ...m}.
-
nYCTb (X, Y) - napa nO,l{MHOLKeCTB, TaKHX, liTO X /,
Y s;; J. 9Ty napy OY,l{eM xapaKTepH30BaTb npHMoyrOJIbHOH
(1, O)-MaTpHneH n X m r OTHOIlleHH5I X X Y:
r.. = { I, (i, j) E X X Y ;
Z} 0, (i, j) X X}7 .
OqeBH,l{HO, MaTpHua r nOJIHOCTblO xapaKTepH3yeT nO,Z(MHO-
}KeCTBa X s;; I H Y J.
nOKa3aTeJIb KalIeCTBa annpOKCHMaUHH MaTpHnbI W C
nOMO blO (a, )-MaTpHU C KapKaCOM r B COOTBeTCTBHH C Teo-
pHeH n. 4.1.2 HMeeT BH,l{:
[ *1 WI) J 2
(X Y) ) (w, r)2 + (w,l-r) _ _ j€Y +
g , = g (r =
(r,r) (t-r,t-r) IXIIYI
[ W iJ ] 2
+ (i. J}EXXY
mn-IXIIYI
9To nOKa3aTeJIb aHaJIOfHqeH (5.39). MO}KHO CCPOPMYJIHpO-
BaTb H HCCJIe,l{OBaTb aHaJIOfH KpHTepHeB (5.37), (5.38), HO MbI
STOfO ,Z(eJIaTb He OY,l{eM B Ha,l{e>K,l{e, lITO qHTaTeJIb CMOLKeT caM
npO,l{eJIaTb STY paOOTY no aHaJIOfHH C npe,l{bI,l{Y HM. 3,l{eCb
MbI XOTHM npO,l{eMoHCTpHpoBaTb B03MO>KHOCTb HenOCpe,l{CTBeH-
Horo HCnOJIb30BaHHH nOKa3aTeJI5I KaqeCTBa annpoKcHMaUHH
g, KOTOpOH B npe,l{bI,l{YInHX pa3,l{eJIaX He ObIJIO Y,l{eJIeHO ,l{OCTa-
TOqHOrO BHHMaHH5I.
PaCCMOTpHM aJIfOpHTM nOCJIe,l{OBaTeJIbHOfO nepeMe eHHH
001>eKTOB ,nJI5I MaKCHMH3anHM nOKa3aTeJIH (5.45).
Ha OOIneM lIIafe aJIfOpHTMa paCCMaTpHBalOTCH TeKYJnHe
MHo:meCTBa X I H Y s;; J, KOTopbIe HCnpaBJI5IIOTC5I C no-
u
MOIII.blO ,nBYX onepanHH: nepBa51 MeH5IeT COCTOHHHe TOfO SJIe-
MeHTa i E I, npH KOTOpOM npHpa eHHe t1g MaKCHMaJIbHO (H
nOJIO}KHTeJIbHO), BTOpa51 npH H3MeHeHHOM B pe3YJIbTaTe nepBoH
onepaUHH MHO>KeCTBe X MeH5IeT COCT05lHHe j E J, ,l{alO ero
MaKCHMaJIbHOe nOJIO>KHTeJIbHOe npHpa eHHe t1g no Y. nOJIY-
qeHHbIe TaKHM 06pa30M H3MeHeHHbIe X H Y 51BJI5IIOTC5I HCXO,l{-
HhIMH ,l{JI5I CJIe,Z(YIO efO lIIara. BbIqHCJIeHHH npeKpa aIOTCH,
eCJIH H3MeHeHH5I COCTOHHHH OT,l{eJIbHbIX i E I, j E J He npHBO-
Jt5lT K nOJIO>KHTeJIbHOMY npHpOCTY g, H paCCMaTpHBaeMbIe Ha
,lJ.aHHOM ware X H Y CIIHTalOTCH pe3YJIbTaTOM npHMeHeHHH
aJIfOpHTMa.
(5.45)
8
3,!{eCb, KaK H paHee B ,!{aHHOM nYHKTe, COCTOSlHHH 3JIeMeH-
TOB MHO)KeCTB I H J xapaKTepH3YIOTCSI 6YJIeBCKHMH (1, -1)-
BeKTopaMH a = (al, ..., an) H = ( 1' ..., m), ,!{JISI KOTOpbIX
_ { 1 iEX; A._ { 1, iEY;
ai - PJ -
--1 i X; -1,j Y.
H3MeHeHMIO COCT05lHHSI i-ro HJIH j-ro 3JIeMeHTa OTBeqaeT
H3MeHeHHe 3HaKa ai HJIH j Ha npoTHBOnOJIO>KHbIiL
,aJISI YCKopeHHSI BbIqHCJIeHHSI npHpOCTa g Ha Ka>K,!{OM mare
On5lTb BOCnOJIb3yeMcSI MeTO,!{OM BTOpbIX pa3HocTeH. npH 3TOM
6y,!{eM paCCMaTpHBaTb pa3HOCTH He caMoro nOKa3aTeJISI g, a
COCTaBJI5IIOIIIJiX ero 3JIeMeHTOB, TaK KaK Henocpe,!{CTBeHHoe
HCnOJIb30BaHHe g B CHJIY ero HeJIHHeHHOCTH He npHBO,!{HT K
u
ynpo eHHIO BbIqHCJIeHHH.
0603HaqMM qepe3 r i (I,O)-MaTpHny, nOJIyqeHHYIO M3 r
B pe3YJIbTaTe M3MeHeHM5I COCTOSlHH5I 3JIeMeHTa i E I. B COOT-
BeTCTBHH C (5.45) 3HaqeHHe g (r i ) onpe,!{eJI5IeTC5I cJIe,!{YIO MMH
COCTaBJI5IIOIIIJIMH ero BeJIHqHHaMH: (w, r i ), (r i , r i ), (w,
1 - r i ), (1 - r i , 1 - r i ), KOTopbIe HeTpY,!{Ho nOJIyqMTb Ha
OCHOBe COOTBeTCTBYID HX BeJIHqHH ,!{JI5I r. A HMeHHO cnpa-
Be,!{JIHBbI CJIe YIO He JIerKO npOBep5leMbIe COOTHomeHHSI:
(w, rl) = (w, r) -(Xi i ,
(5.46)
r e
i = aij,
j€Y
(5 . 47)
H
(r i , ri) =(r, r)-ai I y,. (5.48)
3HaSl (w, r), (r, r), I Y I, it no cpopMYJIaM (5.46), (5.48)
JIerKO nOJIyqMTb (w, ri) H (r 1 , r i ), nOCJIe qero onpe,!{eJISlIOT SI
OCTaJI bHbIe BeJI HqHHbI:
(w, 1- ri) == Wij -(w, r i ), (1 - r i , t - ri) == mn-(r i , ri).
i, j
(5.49)
B COOTBeTCTBHH C (5.46) i = - ai (w, ri) - (w, r»,
T. e. C TOqHOCTblO ,!{O 3HaKa i COBna,!{aeT C nepBOH pa3HOCTblO
BeJIHqHHbI (w, r).
TaKHM 06pa30M, ,!{JISI 6WCTporo BWnOJIHeHHSI nepBOH one-
panHH aJIrOpHTMa He06xo,!{HMO MCnOJIb30BaHMe BeJIMqHH (w, r),
(r, r), I Y I H i (i E I), KOTopble, TaKHM 06pa30M, ,!{OJI>KHbI
3anOMHHaTbC5I no pe3YJIbTaTaM BblnOJIHeHMSI npeJ{hI,!{y ero
282
......
rnara. TOfL{a no <popMYJIaM (5.46), (5.48), (5.49) onpe,n:eJI5IIOTC5I
BeJIHlIHHbI, COCTaBJI5IIO He g (r i ), H npOH3BO,l{HTC5I BbIlIHCJIe-
HHe g (r i ) no cpopMYJIe (5.45).
IlocJIe onpe,l{eJIeHH5I i E I, npH KOTOpOM g (r i ) MaKCHMaJIb-
HO, HeOOXO,l{HMO (npH g (r i ) > g (r)) H3MeHHTb COCT05lHHe i,
B pe3YJIbTaTe lIero H3MeHHTC5I HallaJIbHa51 HHcpopManH5I ,!{JI5I BTO-
pOM onepanHH. Oo03HallHM lIepe3 rj pe3YJIbTaT H3MeHeHH5I coc-
T05lHH5I j E J B MaTpHne r. AHaJIOrHlIHO npe,n:bI,n:y eMY Bbl-
lIHCJl5IIOTC5I 9JIeMeHTbI, COCTaBJlHIO He g (rj), no cpopMYJIaM:
(w, rJ) = (w, r)- j L\i, (rb rj) = (r, r)- j I X I; (5.50)
(w, t-rj)= Wij-(W, rj), (t-rjJ I-r}) =
i, i
= nm-(rb rj),
(5.51)
r,l{e
N= Wij.
i€X
(5.52)
3TO 03HallaeT, liTO ,!{J151 BbIlIHCJIeHH5I g (rj) (j E J) HCnOJIb-
3YIOTC5I (W, r), (r, r), I X I ' l1i (j E J), KOTOpbIe, BOOO e rOBO-
p5l, H3MeH5IIOTC5I nOCJIe BbInOJIHeHH5I nepBOH onepanHH (B CB5I-
3H C H3MeHeHHeM COCT05lHH5I i E I). A HMeHHO BMeCTO (w, r)
Ha,l{O HCnOJIb30BaTb (w, r i ), onpe,l{eJI5IeMOe CPOPMYJIOM (5.46),
BMeCTO (r, r) - (r i , r i ), onpe,l{eJI5IeMOe no cpopMYJIe (5.48),
MO HOCTb I X I 3aMeH5IeTC5I Ha I X I - ai, a l1i - Ha l1 -
- aiWij. 3,l{eCb BeJIHlIHHa - aiWij eCTb npHpa eHHe l1i npH
H3MeHeHHH COCT05lHH5I i E I, T. e. BTOpa51 pa3HocTb BeJIHqHHbI
(w, r).
TaKHM oopa30M, B pe3YJIbTaTe nepBOM onepaQHH nOJIYlIa-
IOTC5I H3MeHeHHbIM BeKTOp a (ai 3aMeHeHO Ha -ai) H HOBbIe
BeJIHlIHHbI (w, r), (r, r), I X I, l1i (j E J), COOTBeTCTBYIO He
3TOMY a. Tenepb no <popMYJIaM (5.50), (5.51) BhIlIHCJI5IIOTC5I
COCTaBJI5IIO He g (r j) ,l{J151 Bcex j E J, nOCJIe lIero MeHHeTC5I coc-
T05lHHe Toro j, ,!{JI5I KOToporo g (r j) MaKCHMa.HbHO (H npeBbIlilaeT
g(r i )). IlpH 3TOM MeH5IeTC5I BeKTOp ( j 3aMeH5IeTC5I Ha - j)
H B COOTBeTCTBHH C 9THM nepeClIHTbIBalOTCH (w, r), (r, r) no
<popMYJIe (5.50), I Y I 3aMeH5IeTC5I Ha J Y I - Pi, l1 i , (i E I) KOp-
peKTHpyeTC5I Ha BTOpylO pa3HocTb - jWij.
TaKHM oopa30M, OOlI(HH liar aJIrOpHTMa ,!{aeT B KalleCTBe
HCXO,!{HbIX ,l{JI5I CJIe,l{YIO ero Illara BeKTopbI a H , a TaK)Ke Be-
JIHlIHHbI (w, r), (r, r), lXI, I YI, l1 i (i E I), l1i (j E J). Ilo
HToraM BbInOJIHeHH5I nepBoM onepanMH MeH5IIOTC5I, KaK onHca-
283
HO, fX, (W, r); (r, t), I x I, AI (j E J), a no HToraM BTOpOM -
p, I Y I, i (i E J) H nOBTopHO - (w, r), (r, r).
nepBOHallaJIbHbIH BbIOOp MHO}KeCTB X H Y MO}KeT ObITb
c,neJIaH no-pa3HOMY. Y,l{OOHO B KalleCTBe HCXO,l{HbIX paCCMaT-
pHBaTb 0,l{H09JIeMeHTHbIe MHO)KeCTBa. llOCKOJIbKY, OlleBH,l{HO,
B KJIaCCe 0,l{H03JIeMeHTHhIX X H Y MaKCHMYM g (X, Y) COOTBeT..
cTByeT MaKCHMaJIbHOMY Wij, CJIe,l{yeT BblopaTb X = {i}, Y =
= {j}, r,l{e i H j -"- HH,l{eKCbI MaKCHMaJIbHOrO 9JIeMeHTa MaTpH-
l1.hI Wi}.
ECJIH W = (Wi}) - MaTpHna nOKa3aTeJIeH CB5I3H Me:m,l{Y
npH3HaKaMH MHO}KeCTB I H J (KoTopble MorYT COBna,l{aTb,
I = J), TO H3JIO}KeHHhIM no,l{Xo,l{ MOLKHO HCnOJIb30BaTb ,Z(JI5I
BbI,l{eJIeHH5I Tex cOBoKynHocTeR BXO,l{HbIX npH3HaKOB X H Bbl-
XO,l{HbIX npH3HaKOB Y, KOTopbIe npe,l{nOJIaraeTC5I ,l{aJIee aHa-
JIH3HpOBaTb B COOTBeTCTBHH C MeTO,l{aMH rJIaB 3, 4.
npH 3TOM MO:>KHO He orpaHHlIHBaTbC5I HaXO}K,l{eHHeM
nepBoro cpaKTopa (X, V), a paCCMOTpeTb MaTpHny OCTaTOqHblX
CB5I3eR W - Ar - J..t, r,l{e A H J..t onpe,l{eJI5IIDTC5I <popMYJIaltflI
(4.20) H (4.21), H npHMeHHTb K HeM nponecc BhI,l{eJIeHH5I napbl
MHo:meCTB nOBTopHO. TaKaH peaJIH3aUHH KalleCTBeHHoro aHa-
JIOr8 MeTO,l{a rJIaBHbIX KOMnOHeHT npH MHoroKpaTHoM H3BJIe-
qeHHH nap (X, Y) nOCTeneHHO HClIepnaeT BCID MaTpHUY w.
aTO CJIe,nyeT H3 Toro, liTO B KJIaCCe nap (X, Y) eCTb TaKHe,
KOTophIe BbI,l{eJIHIOT JIIDOOR 9JIeMeHT MaTpHUbI W, TaK tlTO ,l{JI5I
u
9Toro KJIaCCa OTHOIlleHHH npOXO,l{HT ,l{OKaSaTeJIbCTBO CXO,l{H-
MOCTH MeTO,l{a, ,l{aHHoe B n. 4.1.5.
5.2.7. BblR8Aenue CTPYKTYPbl OCH08HblX C8Rseu
PaccMoTpHM nOKa3aTeJIb (5.29) KalleCTBa pa3-
6HeHH5I R = {R 1 , ..., Rm} co CTPYKTYPOM x (npH <pHKCHpO-
BaHHOM a):
G (R, %) = (ai}-a). (5.53)
(s, t)€X IsR 8 16Rt
nYCTb R <pHKCHpOBaHO. nOCTaBHM 3a,l{aqy HaXO:>K,l{eHHH
TaKOR CTPYKTYPbI x (R), KOTOpa51 MaKcHMH3HpyeT (5.53) npH
Sa,l{aHHOM R no BceB03MO}KHbIM x. 9Ta 3a,l{atla HMeeT TPHBH-
8JIbHOe peIlleHHe.
KaK B 9 5.1, 0003HallHM lIepe3 A st = (aij- a) CYMMY
IERg IERt
CB5I3eH Me}K,l{Y Rs H Rt (3a BbllleTOM nopora). Tor,l{a (5.53) eCTb
G (R, ) = Ast, (5.54)
(5, t)€x
284
I
J
TaK liTO 3a,nalla CBd,llHtCSl K BbIOOPY TeX (S, t), KOTopble ,naloT
nOJIO)KHTeJIbHOe nplipa eHHe nOKa3aTeJIIO (5.54). HHbIMH CJIO-
BaMH, x (R) ,l{OJI)KHa COCT05lTb li3 Tex nap (s, t), ,nJI5I KOTOpbIX
Ast > O. TIpH Ast = 0 YlIeT CB5I3H (s, t) B X HHKaK He BJIHSleT
Ha BeJIHlIHHY G (R, x), TaK liTO TaKYIO napy MO:>KHO BKJIlOlIaTb,
a MO>KHO H He BKJIlOlIaTb B X.
CTPYKTYPY x (R) 6Y,l{eM Ha3bIBaTb cmPYKmypou, onpeae-
IlJleMOU pas6uenUeA! R.
OlleBH,nHO, liTO eCJIH (R*, x*) MaKCHMH3HpyeT G (R, a),
TO x* = x (R*). CJIe,l{OBaTeJIbHO, KpHTepHH (5.54) COCTOHT
B BbI60pe TaKoro R, npH KOTOpOM CYMMa nOJIO)KHTeJIbHbIX Be-
JIHlIHH Ast MaKCHMaJIhHa. Ho CYMMapHa51 BeJIHlIHHa Bcex Ast
KaK CYMMa BeJIUlIHH aij - a nOCTOSlHHa npH 3a,naHHOM a. Cne-
,nOBaTeJIbHO, npH MaKCHMH3aUHH G (R, x) ,nOCTHraeTC5I MaKCH-
l\tIYM a6COJIIOTHOH BeJIHlIHHbI OTpHuaTeJIbHbIX lIHCeJI Ast, paB-
HOM I Ast I, r,ne CYMMlipOBaHHe npOBo,nHTC5I no TeM (s, t),
,nJI5I KOTOpbIX Ast < O. TaKHM 06pa30M, o,nHoBpeMeHHo ,l{OCTH-
raeTC5I MaKCHMYM [531
m
0' (R) = I A 8tl.
s, t == 1
(5.55)
MbI ,l{OKa3aJIH, liTO eCJIH ,l{OnYCTHMbI JII06bIe CTPYKTYPbI x,
TO 3a,nalla MaKCHMH3aUHH (5.53) no (R, x) 3KBUBaJIeHTHa 3a-
,nalle MaKCHMH3aUHH CYMMbI (5.55) Mo,nYJIeH BeJIHlIHH no R;
npH 3TOM x* = x (R*). HHbIMH CJIOBaMH, npH onTHMH3aUHM
(5.53) OTbICKliBaeTC5I TaKoe pa36HeHHe R, npH KOTOpOM MaT-
pHua KJIeTOK a (s, t) = (aij - a) (i E Rs, j E Rt) 51BJl5IeTCH
HaH60JIee KOHTpacTHoii: nOJIO)KHTeJIhHbIe CB5I3H aij - a OT-
oHpalOTC5I B OCHOBHO 1 B O,nHH KJIeTKH, a OTpHuaTeJIbHbIe-
B ,l{Pyrlie. TaKHM o6pa30M, B O,l{HH KJIaCC nona,nalOT 06'beKTbI,
KOTopble 60JIee HJIH MeHee o,nHHaKOBO B3aHMO,l{eMcTBYIOT C ,l{PY-
fHMH 06'beKTaMH [53]. Ho npo O,l{HHaKOBOCTb B3aHMo,neMcTBHH
3,neCb MO:>KHO rOBopHTb JIHIllb B Cpe,l{HeM, B OTJIHlIHe OT 3a,l{a-
11M (5.27) C OnTHMaJIbHOM BeCOBOH CTPYKTypOti, r,l{e KJIeTKH
a (s, t) opraHH3YIOTC5I H3 Tex aij, KOTopbIe Bce 6JIH3KH K Ast
(H ,nPyr K ,l{pyry). ,[{eJIo B TOM, qTO nopor a B03HHKaeT B paMKax
3a,nallH annpOKCHMaUHH KJIeTOI{ O,nHOH H3 Bcero ,l{BYX BeJIHlIHH
ex (npH CHJIbHbIX CB5I3S1X) H P (npH CJIa6bIX CB5I3S1X) (CM.
n. 4.1.2, 4.1.3).
,[{o CHX nop MbI He <-pHKCHpOBaJIH KJIaCC ,nonYCTHMbIX
pa36HeHHH. AJIfOpHTM onTHMH3aUHH (5.55) 6Y,l{eT ,l{aH ,nJIH
MHO)KeCTBa Pm Bcex pa36HeHHH co CTPYKTypOti, pa36HBalO-
HX m He 60JIee lIeM Ha m KJIaCCOB. KaK YCTaHOBJIeHO B
n. 5.2.1, anpHopHoe orpaHHlIeHHe Ha lIHCJIO KJIaCCOB Heooxo-
285
.....
J\I1MO 113-3a TOrO, qTO HHalle OnTMMYM 0 (R, x) 0YJ\eT ,aOCTHr-
HYT Ha rpiHpe a-CXO CTBa.
PaCCMOTpHM arJIOMepaTHBHbIH aJIrOpMTM, Ha Kam,l{OM ma-
re KOToporo no .L{aHHOMY pa30HeHHIO R = {Rl' ..., R p } CTpO-
HTC5I pa30HeHHe Rst C nOMOInblO OO'be.L{HHeHHH KJIaCCOB R s H
R t (5, t == 1, ..., p; 5 =F t). 1JeMY paBHo npHpa eHHe nOKa-
3aTeJIH 0' (R) (5.55) npH OO'be,l{HHeHHH KJIaCCOB? OKa3bIBa-
eTCH [53 J,
0' (R) - 0' (Rst) = 2<D (s, t),
(5. 56)
r)1.e
1 p
CI> (s. t) ="2 U sign ASk -sign At1 I rnin (I ASk I. I Atk I) +
k=l
k=l=s, t
+ I sign Aks-sign A kt I min (I Aksl, I AktDJ +
+ min (A+, I A_ I), (5.57)
npHlIeM A+ (A_) - CYMMa Bcex nOJIO)KHTeJIbHbIX (oTpHna-
TeJIbHbIx) lIHCeJI H3 lIeTBepKH Ass, Ast, Ats, Attt sign x -
3HaK x, a I x I - aOCOJIIOTHaH BeJIHlIHHa x.
113 (5.57) CJIe,l{yeT, qTO MHHHMaJIbHOe YMeHbmeHHe 3Haqe-
HHH nOKa3aTeJI5I (5.55) npH OO'be,l{HHeHHH COOTBeTcTByeT MH-
HHMYMY <D (8, t) , TaK liTO OO'be,l{HHeHHe KJIaCCOB CJIe,l{yeT npo-
BO,l{HTb no MHHH!\1YMY <D (s, t).
,[{oKa3aTeJIbCTBO paBeHCTBa (5.56) OCHOBaHO Ha paBeHCTBe
J x + y I = I x I + J y I - 2 min (I x f , I y I), (5.58)
CnpaBe,l{JIHBOM ,l{JI5I JIIOObIX x H y, HMelO HX pa3HhIe 3HaKH.
,[{JIH .L{OKa3aTeJIbCTBa paBeHcTBa (5.56) oY,l{eM paccMaTpH-
BaTb MO,l{YJIH cYMMapHbIx CB5I3eH At; Me)K,l{Y KJIaCCaMH pa3-
OHeHH5I Rst, COCTaBJIHIO He 0' (Rst). ,[{JIH Y,l{OOCTBa KJIaCC
Rs LJ Rt pa30HeHH5I Rst 0003HaQHM Qepe3 Rsut. 3TO n03BO-
JI5IeT OCTaBHTb y Bcex OCTaJIbHbIX KJIaCCOB Rst Te HH,l{eKCbI l, q,
KOTopbIe OHH HMeJIH B pa30HeHHH R. PaccMoTpHM Bce B03MO)K-
HbIe CJIYQaH:
1) 1 =1= s U t, q =1= s U t;
2) 1 = s U t, q =1= s U t;
3) 1 =F s U t, q == 8 U t;
4) l=sUt,q=sut.
B cJIYQae 1 KJIaCCbI C HOMepaMH l, q B pa30HeHM5IX R H Rst
COBna,l{aIOT, TaK liTO Ai: paBHO Azq (cYMMapHaH CB5I3b l-ro
286
I'll
H q-ro KJIaCCOB B pa36HeHHH R) Ii TeM 60JIee I Ai I = I A/ q I.
CJIe.L{OBaTeJIbHO, 3TH BeJIHqHHbI B pa3HocTH (5.56) B3aHMHO
YHHQTO}KaIOTC5I H He <PHrYPHPYIOT B <D (s, t).
B cJIyqae 2 1 = s U t, TaK qTO A: = AS<7 + Atq. ECJIH
3HaKH Asq H Atq COBna.L{aIOT, TO I Ai J = I Asq I + I Atq I II,
CJIe.L{OBaTeJIbHO, BKJIa.L{ KJIeTKH (l, q) B JIeBYIO qaCTb (5.56) paBeH
HY JIIO. B <D (s, t) 3Ta KJIeTKa TaK:>Ke HHqerO He BHOCHT, TaK
KaK sign Asq - sign Atq = O. ECJIH :>Ke sign Asq =1= sign Atq,
TO B CHJIY (5.58)
, A I = I Asq 1+1 Atq 1-2 min (I Asq I, I Atq I),
TaK 1ITO I Asq I + 1 Atg 1- 1 A: I == 2 min (I Asq I, I Atq I),
T. e. (l, q)-KJIeTKa .L{aeT B 3TOM CJIyqae B pa3HocTH (5.56) BeJIH-
qHHY 2 min (I Asq I, I Atql ), BXO.L{51 YIO B 2<1> (s, t).
AHaJIOrHqHO paCCMaTpHBaeTCH CHMMeTpHqHa51 cHTya[(H5I
CJIyqaH 3.
B cJIyqae 4 HMeeM 1 == q = s U t, TaK 1ITO Ai = Ass+
+ Ast + Ats + Att. 0603HaqHM qepe3 A+ CYMMY nOJIO)KH-
TeJIbHbIX, a qepe3 A_ - OTpHnaTeJIbHbIX 1IHCeJI B qeTBepKe
Ass, ASb Ats, Att- OqeBH.L{Ho, I Assl + I Ast I + I Atsl +
+ , Att I = A+ + I A_I. C .L{pyroH CTOpOHhI, I A I =
= I A+ + A_ , = I A+ I + I A_ J + 2 min (A+, I A_ I ),
TaK 11 TO I Ass' + I Ast I + I Ats I + I Att I-I AI 1=
= 2 min (A+, I A_ I), 1ITO .L{aeT BKJIa.L{ KJIeTKH (l, q) B pa3-
HOCTb (5.56), TaK:>Ke yqTeHHbIH B <D (8, t).
3TO 3aBeprnaeT .L{OKa3aTeJIbCTBO paBeHCTBa (5.58).
npH npaKTHqeCKHX paCqeTaX XOp0I.I10 3apeKOMeH.L{OBaJIO.
ce6H KOM6HHHpoBaHHe arJIOMepaTHBHoro aJIrOpHTMa C aJIfO-
pHTMOM «nepeHH.L{eKcanH5I» (c. 256), npHMeHHeMbIM nOCJIe Ka)K-
.L{oro 06ne.L{HHeHHH KJIaCCOB. TaKoH KOM6HHHpOBaHH H aJIro-
pHTM «CTPYKTypa» ycneI.I1HO npHMeHHJICH .L{JI5I aHaJIH3a CTPYK-
TYPbI pH.L{a KOHKpeTHbIX 06neKTOB. 3.L{eCb He06xo.L{HMO npelK.L{e
Bcero COCJIaTbCH Ha pa60TW no aHaJIH3Y $YHKnHOHHpOBaHII5I
MOJIeKYJI5IpHO-reHeTHqeCKHX CHCTeM ynpaBJIeHHH, nO.L{bITOjl{eH-
HbIe B [70], a TaK>Ke Ha pa60TbI no aHaJIH3Y H cOBepIlleHcTBoBa-
HHIO opraHH3anHoHHbIX CTPYKTYP KpynHbIx npOMbIIllJIeHHbIX
npe.L{npHHTHH (CM., HanpHMep, [69]).
AJIrOpHTMbI BbI5IBJIeHHH CTPYKTYPbI MO)KHO HCnOJIh30BaTb
TaK:>Ke .L{JIH <t>aKTopHoro aHaJIH3a: CBH3H Me:>K.L{Y 06neKTaMH MO-
rYT 06n5lCHHTbCH He O.L{HHM, a HeCKOJIbKHMH <paKTopaMH, ne-
u u
penJIeTeHHe KOTOpbIX nopO:>K.L{aeT CJIO)KHbIH «HeperYJIHpHb1H»
BH.L{ CTPYKTYPbI. 3Ta B03MO)KHOCTb 6bIJIa BCKpbITa H ycnelIlHo
287
HCnOJIb30BaHa B. Jl. KynepIllToXOM H B. A. TpcxpHMOBbIM B
8H8JIH3e <paKTOpOB, onpe,l{eJIHIO HX npOCTpaHCTBeHHOe pacce-
JIeHHe )KHBOTHbIX (CM. [85]).
AJIrOpHTMbI, OCHOBaHHbIe 1Ia BbIHBJIeHHH pa30HeHHH (H
nOKpbITHH) C BeCOBOH CTPYKTYPOH, pa3paooTaHbI H ycneIllHo
npHMeHeHbI B aHaJIH3e pH.L{a MaTepHaJIOB H. D. MYlIHHKOM H
ero coaBTopaMH [8, 10, 74].
CTIe,l{yeT OTMeTHTb cYInecTBeHHoe OTJIHlIHe nOKa3aTeJIH (5.53),
npe,l{Ha3HalleHHoro .L{JI5I BbIHBJIeHH5I MaKpocTpYKTypbI, OT aHa-
JIOrHlIHOro nOKa3aTeJI5I (5.2) .L{JIH nocTpoeHHH pa30HeHH5I. 3a-
,naHHe 3HalleHHH nopora B (5.2) nOJIHOCTblO onpe.L{eJI5IeT perne-
HHe B KJIaCCe BceB03MO)KHbIX pa30HeHHH, oOJIa.L{alOInee p H.L{OM
,nOCTOHHCTB KOMnaKTHOCTH, OTMelleHHbIX B 5.1.1. B TO )Ke Bpe-
MH, 3a,l{aHHe nopor8 B (5.53) npHBO.L{HT K TpHBHaJIbHOMY pa3-
oHeHHIO co CTPYKTYPOH (rpacpy a-CXO):(CTBa) , TaK liTO nOJIY-
u
lIeHHe .L{eHCTBHTeJIbHO arperHpOBaHHoro npe.L{CTaBJIeHHH CBH-
3eM TpeoyeT anpHopHoro 3a,l{aHHH lIHCJIa KJIaCCOB In pa30He-
HHH. He cnacaeT nepexo,l{ K 3a.L{alle annpOKCHMaUHH (n.4 .1.2),
KOTOpbIH nOJIHOCTblO OCBOOO)K.L{aeT 3a.L{allY KJIaCcHcpHKaUHH
OT anpHopHoro 3a,l{aHHH napaMeTpoB: tIHCJIO KJIaCCOB onTH-
MaJIbHOrO pa30HeHHH co CTPYKTYPOH MO)KeT ObITb YMeHbilleHO
TOJlbKO C nOMO blO anpHopHbIX orpaHHlIeHHH (CM. 5.2.1).
B CBH3H C STHM B. A. TpOcpHMOBbIM H B. JI. KynepIllToxoM
OMJIa npe,l{npHHRTa nonMTK8 MO,l{H HKaUHH 3a,l{allH annpOK-
CHMaUHH B KJI8CCe pa30HeHHH co CTPYKTYPOH C nOMOInblO BBe-
.neHHH .L{onOJIHHTeJIbHOrO KpHTepHH, TpeoYIO ero, lITOOM CJIO>K-
HOCTb HCKOMOH CTPYKTypbI ObIJIa MHHHMaJIbHOH [99]. O,l{Ha-
KO npH STOM O.L{HH anpHopHbIH napaMeTp ObIJI 3aMeHeH ,l{PyrHM:
BMeCTO lIHCJIa KJIaCCOB npHIIlJIOCb B 51BHOH cpopMe 3a,l{aBaTb KO-
s<l$HUHeHT OTHOCHTeJIbHOH Ba)KHOCTH KpHTepH5I KalleCTBa
annpoKcHMaUHH H KpHTepH5I CJIO}KHOCTH CTPYKTypbI. MbI pac-
CMOTpHM 3,l{eCb .L{PyroH no.L{Xo,n K npOOJIeMe SJIHMHHHpoBaHHH
anpHopHhIX napaMeTpoB. nonpooyeM B 51BHOH <popMe OTpa-
3HTb B C8MOM annpOKCHMaUHOHHOM KpHTepHH TpeooBaHHe He-
TpHBH8JIbHOCTH nOJIYlIaeMoro pa30HeHH5I.
L(JIH SToro pa30HeHHe co CTPYKTYPOH (R, x) 6Y,l{eM xapaK-
TepH30BaTb He O.L{HHM, a ,l{BYM5I OTHOlIleHHSlMH, no O,l{HOMY Ha
Ka>K,nbIM qJIeH napbI (R, x): pa30HeHHe R = {Rl' ..., Rm}-
.. m
OTHOIlleHHeM 3KBHBaJIeHTHOCTH e = U Rs X Rs, CTPYKTYPY
5=1
X - OTHOlIleHHeM p = U Rs X Rt; COOTBeTCTBYIO He oYJIeB-
(s. t)Ex
CKHe (1, O)-MaTpHUbI ooo3HallHM e H r, TaK liTO eiJ = 1 +-+
+-+ (i, j) E 8, rij = 1 (i, j) E p. 3aAalla anIIpOKCHM8[(HH
.naHHOH MaTpHUbI a B HeKOTopOM KJlaCCe MaKpOCTpYKTYP Tor.n
6y eT HMeTb BHA 3a aqH MHHHMH3aUHH BeJlHqHHW
N
!!=la- A lr-A 2 e-J11 2 = (a,j- A lrij-A 2 e iJ-I1)2.
I, 1=1
(5.59)
<l>aKTHqeCKH MbI annpOKcHMHpyeM MaTpH Y a B KJIaCCe
MaTpHU c qeTbIpbMSI pa3JIHqHbIMH 3J1eMeHTaMH: J.t J1S1
(i, j) ($ pUs, + Al JISI (it j) E p - 8, J.t + A 2 JIR
(i, j) E 8 - p, J.t + Al + Az .nJIR (i, j) E 8 n p. npHpaBHHBaR
aA iJA iJA
HYJIIO qaCTHbIe npOH3BO HbIe a')..,l ' a ' a,.., , nOJIyqHM, qTO on-
THMaJIbHbIe 3HaqeHHR , A 2 , J.t npH 3a aHHOH nape (R, x) CBS!-
3aHb! CJIe YlOlliJIM oopa30M:
Al 'iJ + r iJ e,j == (ai j-IJ') riJ; (5.60)
I, / I. / I. J.
Al 'ijetJ + A2 eij = (aiJ-IJ')eiJ; . (5.61)
i, J Ii i I.
1-'= N(:-l) (atJ-A.lrlJ-A.2eij)' (5.62)
i, /
3,l{eCb, KaK H paHee, HarOHaJIbHbIe 3J1eMeHTbI MaTpHn CBS{-
3H He paCCMaTpHBaIOTCR, TaK qTO (5.62) npe CTaBJISleT!J. KaK
cpe Hee apH<pMeTHqeCKOe MaTpH bI OCTaTOqHbIX cBH3eH a-
- r - A 2 e. CJIe OBaTeJIbHO, cpe Hee 3HaqeHHe MaTpHUbI
a - r - A 2 e - f.t npM OnTHMaJIbHOM J.t paBHO HYJlIO. CMbICJI
Bcex BeJIHqHH, BXO,l{5I HX B (5.60)-(5.62), B TepMHHax pa3-
oHeHMR co CTPYKTYPOH (R, x) CJIe,l{YIOInHH. CYMMbI
rij = NsN t , eij= N: H rijeij= N:
I, J ( s, t)e x i, I s i, I se 1
(r e I - MHO:>KeCTBO HH eKCOB Tex KJIaCCOB Rs, JI5I KOTOpbIX
(s, s) E x) - xapaKTepHCTHKH onpe eJIeHHOCTH COOTBeTCTBYIO-
HX OTHOlIIeHHM. npaBaSl qaCTb (5.60) eCTb nOKa3aTeJIb
G (R, , 11) KaqeCTBa pa36HeHH5I co CTPYKTYPOH (npH nopore
!J.) (5.54), a npaBaH qaCTb (5.61) - nOKa3aTeJlb G (R, !J.) Ka-
qeCTBa pa30HeHH5I (5.2). .
10 SaKe 1743
289
jllil !l111I11I1 J! IJlIIL!l1l11111111111111111 III IIII IIIIIIIIIIIIIIIIIII' I, " III , I '" "" '"
113 (5.60) H (5.61) nOJIYlIaeM OnTHMaJIbHbIe 3HalleHHR A 1 H
"'2 (npH 3a,naHHOM !J.):
A1 =
m
N:G(R,)(,f.t)- N:G(R,J1)
s = 1 Sf:. 1
.
J
A2 =
m
N: N s Nt- ( NS ) 2
s = 1 (s. t)6" s61
NsNtG(R, J1)- N:G(R, X, 11)
(s. t) 6 K s:1
( N: ) NsNt- ( NS ) 2
s ( s. t) s '" ss 1
(5. 63)
.
npHMeM ,naJIee ,nJIH npOCTOTbI, liTO B (5.59) !J. = 0, T. e.
arinpoKCHMalIJISI a npOHCXO,l{HT B KJIaCCe MaTpHU Aif + A 2 e.
ECJIH 3a,l{aHHoe !J. =1= 0, TO npOCTO BMeCTO a CJIe,l{yeT paCCMaTpH-
BaTb a - fl.. Tor,l{a, paCKpbIBaH B (5.59) CKOOKH, JIerKO nOKa-
3aTb, liTO npH 3a,l{aHHbIX A1 H A2 MHHHMH3anH5I (5.59) 3KBHBa-
neHTHa MaKCHMH3anHH CJIe,l{YIO ero KpHTepHH:
HR, X)=A 1 G(R, x, ; )+ G(R, 2 )_
I.
- 1. 1 A 2 N: .
s€l
(5.64)
TaKHM oopa30M, pemeHHe 3a,l{allH CBO,l{HTCH (npH nOJlOLKH-
TeJIbHbIX Al II "'2) K COBMeCTHOMY npHMeHeHHIO KpHTepHeB Ka-
qeCTBa pa36HeHH5I H pa30HeHH5I CO CTPYKTYPOH(npH BeCOB X
K03<P<PHnHeHTax KpHTepHeB Al H A 2 ) H pa3JIHlIHbIX noporOBbIX
3HaqeHHH ( 1 H ; ). TIOMHMO 3TOro B (5.64) qmrypHpyeT qJIeH
N:, KOTOpbIH (c BeC01\1 A 1 A 2 ) TpeoyeT, lITOObl Te KJIaCCbl pa3-
I
OHeHH5I R, KOTopbIe HMelOT cy ecTBeHHbIe BHYTpeHHHe CBH311
(Ha ypoBHe 1 ), 6bIJiH nOMeHbllIe. TIpH oTpHn.aTeJIbHOM 3Haqe.
HHH KaKOH-JIHOO H3 BeJIHlIHH AI, A 2 COOTBeTCTBYIO HH KpHTe-
PHH HeOOXO,l{HMO MHHHMH3HpoBaTb. npH nOKa3aTeJIe (5.64)
OnTHMaJIbHOe pa30HeHHe (co CTPYKTYPOH), Booo e rOBopH, He
51BJI5IeTC5I TpHBHaJIbHbIM. .aJI5I ero nOCTpOeHH5I MO:>KHO HCnOJIb-
30BaTb KaKHe-JIHOO H3 onHcaHHbIX JIOKaJIbHO-OnTHMaJIbHbIX
aJIrOpHTMOB nOCTpoeHHH pa30HeHH5I (YlIHTbIBaH, liTO B G (R,
AI )
x, 2" CJIe,l{yeT opaTb )( = )( (R) H, CJIe,l{oBaTeJIbHO, 3Ha-
290
I
=+
t
qeHHe (5.64) onpe,l{eJl5leTC5I TOJlbKO pa30HeHHeM R). npH 9TOM
B CHJlY (5.64) npHparneHHe f J1erKO Bbl1JHCJl5IeTC5I KaK B3BerneH-
Ha51 cYMMa npHparneHHH nOKa3aTeJIeH G (R, x, a) H G (R, a),
KOTopble paccMoTpeHbl B COOTBeTCTBYlOrnHX pa3,l{eJIaX, nJllOc
npocTo BWqHCJl5IeMa51 ,l{OOaBKa, ,l{aBaeMa51 TpeTbHM ,l{HcnepCH-
OHHbIM qJleHOM N:. MbI He OY,l{eM onHCbIBaTb COOTBeTCTBYIO-
I
v
HX nOCTpoeHHH.
B CHJlY C,l{eJlaHHOrO paHee 3aMeqaHH5I Y1JeT 3a,l{aHHOrO !.t
CBO,l{HTC5I K aHaJIH3Y a - J..t, HJlH, qTO TO }Ke caMoe, K YBeJlH.
",.
qeHHIO 3HaqeHHH noporOB 2 B (5.64) Ha J..t. cl>OPMYJlbl (5.60)-
(5.62), (5.64) ,l{alOT OCHOBY HTepaTHBHoro aJIrOpHTMa BbIqHC-
J1eHH5I HCKOMoro pa30HeHH5I co CTPYKTYPOH no cxeMe: 3a,l{aH-
HbIe , "'2' J..t onpe,l{eJl5llOT J10KaJIbHbIH onTHMYM (5.64), Ha oc-
HOBe KOToporo BbIqHCJl5IIOTC5I HOBbIe 3HaqeHH5I AI' A2' J..t no
(5.60)-(5.62) .
cl>YHK[(HOHaJl (5.59) npHBO,l{HT TaKJKe K CJIe,l{YlOrneMY HTe-
paTHBHOMY aJlrOpHTMY nOCTpOeHH5I cyoonTHMaJlbHOrO pa3-
OHeHH5I co CTPYKTypOH, oCHoBaHHoMY Ha HCnOJlb30BaHHH MaT-
PH[( OCTaTOqHbIX CB5I3eH.
COrJlaCHO STOMY aJlrOpHTMY npH 3a,l{aHHOH MaTpH[(e Alf Bbl-
QHCJl5IeTC5I cyoonTHMaJlbHOe pa30HeHHe R, HaHJIyqIIIHM oopa-
30M annpOKCHMHpYlOrnee MaTpH[(Y OCTaTOqHbIX CBH3eH a-
- Alf, B CMbICJIe KpHTepH5I (4.19), TaK qTO A2' J..t onpe,l{eJl5l-
IOTC5I KaK cpe,l{H5I5I BHYTpeHH5I5I H BHeIIIH5I5I qaCTb CB5I3H ,l{J1H
R COOTBeTCTBeHHO. nocJle SToro ,l{J151 MaTpH[(bI a - A 2 e BbIQHC-
J151eTC5I CTpYKTypa pa30HeHH5I R KaK MaTpH[(a A l f + f.t (MaT-
v
pH[(a r xapaKTepH3yeT CTPYKTYPY CB5I3eH MeJK,l{Y KJlaCCaMH
R), OnTHMaJlbHO annpOKCHMHpYlOrna51 MaTpHQY a - "'2e TaK-
}Ke B CMbICJIe (4.19). JIerKo BH,l{eTb, qTO STa CTpYKTypa onpe-
,l{eJl5leTC5I CJle,l{YlOrnHM oopa30M. ,[{JI5I nOJIyqeHHOrO R =
= {Rl' ..., Rm} paCCMOTpHM m X m MaTpH[(Y CYMMapHbIX
CB5I3eH (no a - A 2 e) (Ast), r,l{e Ast = aij npH S =1= t H
ifRs f E Rt
Ass = (aii - "'2). 5Y,l{eM HCKaTb ,l{JI5I MaTpH[(bl (Ast) on-
l: fER 8
THMaJIbHYIO CTPYKTYPY (npH TpHBHaJIbHOM pa30HeHHH), qTO,
KaK Y}Ke rOBOpHJlOCb B n.5.2.2, CBOJ(HTC5I K npOOJleMe no,l{-
60pa TaKoro nopora a, KOTOpWH paBeH nOJIYCYMMe cpe,l{Hefi
BHYTpeHHeH H cpeJ(HeH BHernHeH CB5I3H B rpacpe a-cxoJ(cTBa
(no MaTpH[(e Ast) npeJ(CTaBJI5IlOrneM STY OnTHMaJlbHYIO CTPYK-
TYPY. nOJ(60p HCKOMoro a ocyrneCTBJI5IeTC5I HTepaTHBHbIM aJl-
ropHTMOM n. 4.1.2 (CM. TaKJKe 5.2.2) H rpa<t> a';CXOJ(CTBa Ha
(Ast) H eCTb HCKOMaH CTpYKTypa )( = {s, t} lAst> a} J(JI5I
10. 291
pa36HeHH5I R. Tenepb OCTaeTC5I TOJIhKO Onpe.L{eJIHTb Al KaK no-
u u u u
ypa3HocTb cpe HeH BHYTpeHHeH H cpe HeH BHeillHeH CB5I3H
aiJ - A 2 e iJ OTHOCHTeJIbHO (R, X), nOCJIe 1Jero HCXO Ha51 HH-
<popMaUH5I JIH CJIe,l{YIO ero Illara (AI H r) rOTOBa.
npouecc npO,l{OJILKaeTC5I ,l{0 Tex nop, nOKa 3HalleHHe KPH-
TepH5I (5.59) He nepeCTaHeT YMeHbIllaTbC5I.
COrJIaCHO H3JIOLKeHHOMY aJIrOpHTMY CHaqaJIa onpe,l{eJI5I-
eTC5I R, a Y)Ke nOTOM ero CTpYKTypa (no HeH3BeCTHOMY 3apa-
Hee nopory). OopaTHoe BJIHSlHHe CTPYKTYPbI Ha pa30HeHHe on-
pe,l{eJI5IeTC5I TeM, liTO R H eTC5I no MaTpHne CB5I3eH, H3 KOTO-
pbIX BbIlITeHO BJIH5IHHe CTPYKTypbI, T. e. BHYTpeHHHMH ,l{JISI R
OKa3bIBaIOTC5I KaK pa3 Te CBR3H, KOTopbIe !taK ObI He BXO.L{5IT
B CTPYKTYPY X.
J
3AKJlIOIJEHHE
PaCCMOTpeHHhIe B KHHre MO.L{eJIH H MeTO.L{bI 06p33YIOT KOC-
THK JIHHeHHOH TeOpHH aH3nH3a KaqeCTBeHH@X npH3HaKOB H
CTPYKTYp. HenHHeHHbIe, 60nee TOHKHe MeTO.L{bI ManO OTpa:>Ke-
HbI B pa60Te, HO CJIe.L{yeT HMeTb B BH.L{Y, qTO TOJIbKO Tenepb coG-
CTBeHHO H CT8nO nOH5ITHO, KaKHe MeTO.L{bI 51BJI5IIOTC5I JIHHeHHbI-
MH, 3 KaKHe - HeT. Pa3BHTHe HeJIHHeHHhIX MO,neJIeH KOHCT-
PYHpOBaHH5I H OnHCaHH5I npe,nCTaBn5leTC5I Ba:>KHbIM H nepCneK-
THBHbIM HanpaBneHHeM 3HanH3a ,naHHbIX. He MeHee Ba:>KHO Ha-
u
npaBneHHe YHHBepcanH38UHH MO.L{eneH C neJIbID O.L{HOBpeMeH-
HOH 06P360TKH npH3H3KOB, H3MepeHHbIX B IllK3JI3X pa3HbIx
THnOB, C coxpaHeHHeM HX cnenH<pHKH. I1MeIOTc51 H .npyrHe
Manopa3pa6oTaHH@e acneKTW aHaJIH3a .L{aHHUX, MHorHe H3
KOTOpbIX e e TOJIbKO npe.L{CTOHT qeTKO ccpopMynHpoBaTb.
Hap5l.L{y C TaKHMH, KOHCTPYKTHBHbIMH, 3aMeqaHH5IMH npH-
XO.L{HTC5I cnbIlIIaTb H npHHnHnH3JIbHhIe B03pa:>KeHH5I npOTHB
u
TeopHH aHanH3a .L{3HHbIX B TOM CMhICJle, KOTOpbIH npHH5IT B
.L{aHHOH pa60Te. C3MbIMH cepbe3HbIMH H3 HHX HaM npe,nCT8B-
JI5IIOTC5I cne.L{YIO He:
1. MO,lJ.eJIH aHaJIH3a ,naHHUX He CJIe.L{YIDT OObIQHOMY nYTH
MaTeMaTH3anHH HaYKH: CHaq3JI8 M3TeM3THqeCK351 Mo,nenb 51B-
JIeHH5I, a nOTOM Y:>Ke cnoc06bI o6pa60TKH .L{aHHbIX KaK MeTO,n@
H.L{eHTH<pHKanHH n3paMeTpOB MO.L{eJIH, BKJIIOQa51 npOBepKY
a.L{eKBaTHOCTH. KaK npaBHJIO, OHM He CBH3aHbI .L{a:>Ke C MaTeMa-
THKO-CTaTHCTHQeCKHMH MO.L{eJI5IMH nOpO:>K,neHH5I iI.aHHbIx. He
6Y.L{YQH OCHOBaHHWMH Ha MO.L{eJIH 51BJIeHH5I, MeTO.L{@ 3H8JIH38
.L{aHHbIX 06H33TeJIbHO HBJl5IIDTCH npOH3BOJIbHbIMH, 3BpHCTH-
QeCKHMH.
2. KOM6HH3TopHbIe 33.L{8QH KOHCTpYHpoBaHH5I K3qeCTBeH-
HbIX npH3HaKoB H CTPYKTYP MHor03KcTpeManbHbI, OHH BbIHY:>K-
,lJ.eHHO peIllalOTCH HeTOqHbIMH, JIOK8nbHbIMH MeTO.L{aMM, KOTO-
pbIe npM pa3JIHQHbIX YCJlOBHHX .L{31OT MHoro pa3nHQHbIX, HO
nOQTH paBHOneHHbIx no KpHTepHIO BapH3HToB, a 3TO npHBO-
u
.L{MT K npOH3BOJIbHOCTH npMHMMaeMbIX peIlleHHH.
3. I1cnOJIb30BaHMe MeTO.L{OB aH3JIH38 .L{aHHbIX npH BceH HX
CJlO:>KHOCTH H rpOM03.L{KOCTH npHBo,lJ.MT JIMllib K TpHBH3JIbHbIM
293
BbIBo,!{aM, KOTOpble HCCJIe,nOBaTeJIb HJIH PYKOBO,!{HTeJIb, eCJIH
OH XOp0I.I10 3HaeT 001>eKT, Mor ObI c,!{eJIaTb H TaK.
,[{JI5I pa300pa 3THX nOJIO:>KeHHH HeOOXO,!{HMO HanOMHHTb,
qTO MeTO,!{bI aHaJIH3a ,!{aHHbIX npHMeH5IIOTC5I Ha nepBbIX 3Tanax
n03HaHH5I, Kor,!{a TeopeTHlIeCKHe npe,!{CTaBJIeHH5I 0 CHCTeMe
BeCbMa paCnJIbIBlIaTbI H HeTOllHbI. Hx HCnOJIb30BaHHe no,!{pa3Y-
MeBaeT npHH5ITHe peI.I1eHHR (a cpOpMHpOBaHHe rHnOTe3bI-
TO:>Ke peI.I1eHHe) B CHCTeMe oopaTHOH CB5I3H «lIeJIOBeK-MaI.I1H-
Ha» Ha OCHOBe aHaJIH3a lIeJIOBeKOM BapHaHTOB, BbIpaoaTbIBae-
MblX MaI.I1HHOH no ero 3a,!{aHH5IM. MaI.I1HHa 3,!{eCb He 3aMeH5IeT
qeJIOBeKa H He ,!{aeT OKOHlIaTeJIbHbIX perneHHH, a JIHillb nOMO-
raeT npHH5ITb H OOOCHOBaTb perneHH5I, ynpo a51 H npenapHpY5I
,!{aHHbIe, oopa a51 BHHMaHHe H3 HaJIHlIHe Tex HJIH HHbIX CB5I3eiL
n03TOMY, npH3HaBa51 cnpaBe,!{JIHBOCTb nepBoro 3aMella-
HHSI, MbI oopa aeM BHHMaHHe Ha TO, qTO aH3JIH3 ,!{aHHbIX HaMe-
peHHO CTaBHT ceo51 B TaKoe nOJIO)KeHHe, npeTeH,nY5I Ha pOJIb
MeTO,!{OJIOrHH oopaOOTKH ,!{aHHWX B YCJIOBH5IX, Kor,!{a Mo,!{eJIb
paCCMaTpHBaeMoro 51BJIeHHH OTCYTCTByeT. Ho MO:>KHO JIH BOOO-
u
e rOBopHTb 0 HaYlIHOH TeopHH aHaJIH3a ,!{aHHWX B YCJIOBHSlX,
Kor,!{a HeT Mo,!{eJIH 51BJIeHHSI H, CJIe,!{OBaTeJIbHO, Bce ,!{03BOJIeHO?
MO:>KHO, eCJIH npaBHJIbHO nOCTaBHTb 3a,!{allY: TeOpH5I ,!{OJI:>KHa
3aHHMaTbCSI He OOOCHOBaHHeM npaBHJIbHOCTH Tex HJIH HHbIX
MeTO,!{OB oopaOOTKH, a BbI5IBJIeHHeM H OOOraIneHHeM CMbICJIa
3THX MeTO,!{OB H peaJIbHbIX ,!{OCTHraeMbIX C HX nOMO blO neJIeH.
B qaCTHOCTH, BbI5IBJIeHHe B3aHMOCB5I3eH Me:>K,!{Y pa3JIHQHbIMH
MeTo,!{aMH ooora aeT HX CO,l].ep:>KaHHe H, CJIe,l{OBaTeJIbHO, YJIYlI-
IIlaeT B03MO:>KHOCTH HHTepnpeTanHH, T. e. TeopeTHlIeCKoro
ocpopMJIeHHSI pe3YJIbTaTOB aHaJIH3a KOHKpeTHWX ,l{aHHWX.
CKa3aHHoe He 03HallaeT, liTO MeTO,l{bI aHaJIH3a ,l{aHHbIX
He MorYT paCCMaTpHBaTbC5I KaK MeTO,l{bI H,l{eHTHcpHKanHH MO-
,!{eJIeH. HanpOTHB, HCnOJIb30BaHHbIH MeTO,l{ HaHMeHbIIlHX KBa,l{-
paTOB MO:>KeT paCCMaTpHBaTbC5I H npH HaJIHlIHH MaTeMaTHKO-CTa-
THCTHtleCKOH MO,l{eJIH nOpO:>K,l{eHH5I ,l{aHHbIX, KOr,l{a HeOOXO,l{HMO
BbIHCH5ITb 3qxpeKTHBHOCTb H Ha,l{e:>KHOCTb nOJIYlIaeMbIX C ero
nOMO blO oueHOK, HO 3TO OTHOCHTCSI K ,l{PyrHM 3TanaM n03Ha-
HH5I 001>eKTa. ,[(aJIbHeHI.I1ee pa3BHTHe H3JIO)KeHHbIX nOCTpoe-
HHii, HeCOMHeHHO, npHBe,l{eT K pa3paOOTKe COOTBeTCTBYIO eH
u
MaTeMaTHKO-CTaTHCTHlIeCKOH TeopHH H, CJIe,l{OBaTeJIbHO, K JIYlI-
IIleMY nOHHMaHHIO CMbICJIa paCCMOTpeHHbIX MeTO,l{OB. Ho K Bon-
POcy ° B03MO:>KHOCTH HX npHMeHeHH5I Ha HaqaJIbHbIX 3Tanax
n03HaHH5I 3TO He HMeeT OTHOI.I1eHH5I.
C yqeTOM CKa3aHHoro CTaHOBHTC5I 51CHOR HenpaBOMepHOCTb
BToporo 3aMeqaHHH 0 npOH3BOJIbHOCTH nOJIyqaeMbIX pe3YJIb-
TaTOB. To, lITO OKOHqaTeJIbHble peweHHH <POpMHPYIOTC5I HCCJIe-
294
(
)
I
........
u
,!{OBaTeJIeM, a He MaIlIHHOH, n03BOJI5IeT eMY B COMHHTeJI bHhIX
CJIyqaHX CKOJIb yrO,!{HO MHoro pa3 nOBTOp5lTb paCqeTbI npH pa3-
JIHqHbIX YCJIOBH5IX. B KaqeCTBe BbIpaOOTaHHOM 3aKOHOMep-
HOCTH HCCJIe,!{OBaTeJIb <POPMYJIHpyeT TO oo ee, qTO co,!{ep-
LKHTC5I B pa3JIHqHbIX paBHOneHHbIX BapHaHTax. ECJIH }Ke Ba-
pHaHTbI He ,!{onYCKaIOT e,!{HHOH HHTepnpeTaUHH, TO MO}KHO YT-
BepLK,!{aTb, qTO paCCMOTpeHHbIe ,!{aHHble He ,!{alOT OCHOBaHMH
,!{JIH K8KOrO-JIHOO pelIIeHH5I. Booo e OKOHq8TeJIbHbIM KpHTe-
pHeM npHMeHHMOCTH Tex HJIH HHbIX MeTO,!{OB 3,!{eCb 51BJI5IeTC5I
B03MO}KHOCTb HCnOJIb30B8HH5I pe3YJIbT8TOB HX npHMeHeHH5I
u
,!{JI5I ,!{aJIbHeMlIlero pa3BHTH5I TeopHH H3yqaeMOrO 51BJIeHH5I.
TpeTbe 3aMeqaHHe aneJIJIHpyeT K cnOCOOHOCT5IM HCCJIe,!{o-
BaTeJIeH HHTepnpeTHpOBaTb nOJIyqaeMbIe pe3YJIbTaTbI. KaK
H3BeCTHO, cnOCOOHOCTH ,!{aJIeKO He O,!{HHaKOBbI. BbI,!{alO HeC5I
HCCJIe,!{OBaTeJIH cy ecTBeHHO npO,!{BHraJIH Bnepe,!{ TeopHIO H3Y-
u
qaeMbIX 51BJIeHHM npH HaJIHqHH caMbIX CKpOMHbIX TeXHHqeCKHX
cpe,!{CTB coopa H oopaOOTKH ,!{aHHbIX. n03TOMY 3,!{eCb eCTeCT-
BeHHO COCJIaTbC5I Ha onbIT paOOTbI, HaKOnJIeHHbIH B OOJIaCTH
HCnOJIb30BaHH5I MeTO,!{OB aHaJIH3a ,!{aHHbIX B HayqHbIX HJIH yn-
paBJIeHqeCKHX neJI5Ix. 9TOT onbIT nOKa3bIBaeT, qTO, KaK npa-
BHJIO, CnenHaJIHCTbI B TOM HJIH HHOH KOHKpeTHOH OOJIaCTH npH
H3yqeHHH CJIOLKHbIX CHCTeM HMelOT npe,!{cTaBJIeHHe JIHlIlb 00 HX
OT,!{eJIbHbIX qepTaX, KOTopbIe rHnOTeTHqeCKH MorYT ObITb CB5I-
3aHbI ,!{pyr C ,!{pyrOI\i CKOJlb yro,!{Ho pa3JIHqHbIMH cnocooaMH,
npHqeM KaLK,!{brH H3 TaKHX cnocoooB MOLKeT ObITb OoocHOBaH
TeMH HJIH HHbIMH cooopaLKeHHHMH. n03TOMY npHMeHeHHe Me-
TO,!{OB aHaJIH3a ,!{aHHbIx n03BOJI5IeT cy ecTBeHHo CY3HTb MHO-
}KeCTBO B03MO)KHbIX BapHaH'fOB, KOTopbIe npeBpa aIOTC5I Te-
nepb H3 npaB,!{onO,!{OOHbIX P8..3MbIIlIJIeHHM B paOOqHH HHCTPy-
MeHT ,!{aJIbHeHlllero H3yqeHIf51 CHCTeMbI. Bo MHorHX cJlyqa5lx
u
nOJIyqaeMbIe pe3Y JIbTaTbI nOMoralOT HCCJIe,!{oBaTeJlIO «BbIHTH»
Ha TaKHe KaTeropHH II CBH3H, KOTopbIe BOOOIne He paCCMaTpH-
BaJIHCb npH coope ,!{aHHbIX.
--
"qIlJII IIII I
lOLl
OpttA03IC8Hue
3JIEMEHTbi nHHERHOR AnrE5Pbl
OJ. BEKTOPHOE OPOCTPAHCTBO
nOJ( N-MepHblM se"mOpOM ()y,lteM nOHHMaTb nOCJIe,ltO-
B8TeJIhHOCTb HS N He O()SIsaTeJIbHO paSJIHqHhlX Be eCTBeHHhtX qHCeJI X=
c:::I (xt> == (Xt, ..;, XN). CYMMOa se"mop06 x CI::I (Xh .;;, XN) H Y"
c= (Yh ;;;, YN) HaShtBaeTCSI peSYJIbTaT nOpaSpSI,ltHorO CYMMHpOBaHHSI
% + Y c:::I (Xl + Y1, .;;, XN + YN); npOUS6eoeHue qHCJIa a Ha BeKTOp
% == (Xi, ;.;, x H ) onpe,lteJISIeTCSI nOKOMnOHeHTHbIM YMHO}l(eHHeM:
CZx:::o (CZXt, .;., aXN).
COBoKynHocTh BeKTopOB L, KOTopaSI BMeCTe co BCHKHMH X, Y E L
CO,ltepmHT H HX CYMMY x + y, a TaK}I(e npOHSBe,lteHHSI ax npH BeIneCT-
BeHHhtX a, HaSbIBaeTCSI /lUHetlHblM 6e"mopHblM npocmpancm60M.
BCSIKHA 3JIeMeHT x JIHHeAHOro npOCTpaHCTBa L MO}l(eT 6b1Th npe CTaB-
m
JIeH B BH,lte /luneilHoil "oM6uHaquu X c:I akxk HeKOTopbIX BbIAeJIeHHbIX
k=l
BeKTopOB Xl, .:;, Xm E L npH nOAXO,ltSI HX ah: COBoKynHocTb BeKTO-
pOB Xl, ..., Xm HaSbIBaeTCH /lUne(J,Ho-nesa6ucuMoil, eCJIH HH O,ltHH HS
HHX He SIBJISIeTCSI JIHHeAHOA KOM()HHaUHeA ApyrHX. ECJIH Bce BeKTopbI
HS L MorYT ObITb npe,ltCTaBJIeHbI B BH e JIHHeAHbIX KOM6HHanHA BeKTO-
pOB JIHHeAHO-HesaBHCHMoA cOBoKynHocTH Xl, ..., x m HS L, TO 9Ta COBO-
KynHocTb HaSbIBaeTCSI 6aaUCOM L, a qHCJIO m - pa3.MepnOCmblO npo-
cmpancm6a L. npH m = N npOCTpaUCTBO L COCTOHT HS BceBOSMomHbIX
N -MepHbIX BeKTopOB.
B OObItIHOM TpeXMepHOM npOCTpaHCTBe pOJIh 6a3Hca HrpalOT OCH JIIO-
60H CHCTeMbI Koop,ltHHaT. npOCTpaHCTBO HYJIeBOH pa3MepHocTH COCTOHT
H3 e,nHHcTBeHHoro 9JIeMeHTa - HYJISI 0 = (O, ..., 0). npOCTpaHCTBO
pa3MepHocTH 1 COCTOHT HS BeKTopoB BH,lta ax,. r,ne X - «l>HKcHpoBaH-
HbIHfrBeKTop, a a - JIlo()oe tIHCJIO, H HaSbIBaeTCSI npllMotl {ax}. npo-
cTpaHcTBo pa3MepHocTH 2 Ha3bIBaeTCSI n/lOC"OcmblO. B JII060M npo-
cTpaHcTBe nOJIOmHTeJIbHOH pasMepHocTH HMeeTCSI 6eCKOHetIHO MHoro
6asHcoB, HO Bce OHH COCTOHT HS O,ltHHaKOBoro KOJIHtIeCTBa BeKTopoB.
C"aAllpnOe npous6eoenue (x, y) ,ltBYX BeKTopoB x = (Xlt ..., x N )
H Y == (yt, .::, YN) - 9TO qHCJIO, onpe eJISIeMOe KaK CYMMa nOKOMno-
HeHTHbIX npOH3Be,lteHHA
N
X, y)== XfYt.
1- 1
(n. t)
296
-
CKaJUIpHblH KBa,ltpaT (x, x) c= !'xl BeKTopa x ()YAeM HaShlBaTb ero
I
OAunou H 0609HaqaTb d (X) == l X 1 2 . ,[lJIHHa BeKTopa x-y 6y,zr.eT Ha3b1BaTb-
CSI paccmOJUl,ueM MeQY X H y:
N
d (x, y) ==1 "%-g j1c:l (x,- Yd l .
1- 1
(n .2)
9Ta TepMHHOJIOrHSI HeCKOJlbKO paCXO,ltHTCSI C OObIqHOH, Kor a eB-
KJlHAOBCKOH HOp OH x HaSbIBaeTCSI KopeHb KBa,ltpaTHbIH U x II c:::s Y (x, x),
8 eB KJlH,ltOBC KHM paCCTOHHHeM Me}l(,zr.y X H Y - HopMa II X - Y 11 ==
= Y d (X, y). TaKHM oopaSOM, B aHHOH KHHre no,n paCCTOHHHeM no-
HHMaerCH KBa paT eBKJIH,nOBCKoro paCCTOHHHSI, KOTOpbIH He HBJIHeTCH
MeTpHKOH B OObIqHOM CMbICJIe, T. e. He Y OBJIeTBOpSIeT CTaHAapTHblM aK-
CHOMaM MeTpHKH [87]. McnoJIb30BaHHe HMeHHO TaKoro paccToHHHSI BeCb-
Ma y,nOOHO npH CPOPMYJIHpOBKe H peweHHH sa,ltaq annpoKcHManHH O HHX
BeKTopoB PyrHMH, TaK KaK KaqeCTBO annpOKCHMa1I.HH BCIO Y B aHHoii
KHHre Ol(eHHBaeTCH cYMMaMH KBa,ltpaToB pasHocTeH KOMnOHeHT, T. e.
KaK pas KBa,npaTaMH eBKJIH,l\OBCKHX paccToSIHHH. B tIaCTHOCTH, Teo-
peMa nHcparopa B ,ltaHHbIX TepMHHax MO}l(eT 6bITb CcpOPMYJIHpOBaHa
CJIe,ltYIOIn"HM o6pasoM. llJIH JI106h1x opTOrOHaJlbHblX x H y CnpaBe,ltJIHBO
paBeHcTBo
I x-y 1 2 =1 x 12+1 y 1 2
(n.3)
r,lte I x - y r , I x 1 2 , I y II CYTb CKBa,zr.paT runOTeHYSbI» H CKBa.ApaThI
KaTeTOB» COOTBeTCTBeHH .
BeKTopbI x H g HaShlB810TCSI opmo2onaAbnblMU, eCJIH (x, g) = O.
,l:{JIH BCSIKoro JIHHeHHOro npOCTpaHCTBa L paSMepHOCTH m MHO}l(eCTBO
Bcex opTOrOHaJlbHbIX eMY BeKTopoB, T. e. TaKHX X, qTO (x, y) = 0 AJISI
JII060ro y E L, oopasyeT JlHHeHHOe npocTpaHcTBo paSMepHocTH
N - m (CtIHTaeTCSI, tITO BeKTopbl x E L N -MepHbIe).
PaCKpbIBaSI CKO()KH B (n.2), npe,l\CTaBHM paCCTOSIHHS qepes CKa-
JlSIp Hble npOH3Be,zr.eHHS:
d(x, y)=lxl l +ly\I_2(x, y).
en .4)
CKaJIHpHOe npOH3Be eHHe BeKTopoB X, y xapaKTepH3yeT eBKJIH OB-
CKYIO ,ltJIHHY npoeKUHH O,ltHOrO BeKTopa Ha OCb Pyroro. BeKTop eAH-
HHqHOH JlHHbI Ha3bIBaeTCSI HOpMup06aHnblM. CKaJISIpHOe npOHSBe,l\eHHe
HOpMHpoBaHHbIx BeKTopoB (x, y) paBHO KOCHHYCY yrJIa Me}l( Y npSIMbIMH
{ax} H {ag}, npOXO SI HMH tIepes X H y.
_ 1 N
0603HaQHM tIepe3 X == N 1: xl cpeAHlO1O apH<pMeTHqecKYIO KOMno-
leal
1 - -
HeHT BeKTopa x. BeJIHqHHa N (x-x, g- g) HaShlBaeTCH l ICost/Jt/Juquen-
1 BbIqHTaHHe qHCJla X H3 BeKTopa X nOHHMaeTCSI KaK BbltIHTaHHe
HS Bcex KOMDOHeHT x.
297
..nOM KO pUal{UU BeKTopOB x H y, a KOCHHYC yrJla Me)K.ny {a (x-X)} H
{a (y-y)} -K03if1if1Ul{UeHmOM KOppe/lfll{UU 'C (x t y) Me)K Y x H y:
N
(Xi-X) (Yi-Y)
I == 1
't(x,y)= I
N N
i 1 (Xi-X)'I 1 1 (Yi- Y )
(x-x, y - y)
V(x-x, x-x) (y-y, y -y ) ·
(n.5)
CJIe,nyeT HMeTb B BH.ny, tJTO B CTaH apTHbIX MaTeMaTHKO-CTaTHCTH-
qeCKHX paCCMOTpeHHHX N - STO 06'heM BblOOpKH, a COOTBeTCTBYIOInHe
KO.9<p<pHl(HeHTbI HaSblBaIOTCSI BblOOpOtJHblMH [46].
n.2. MATPHU.bI
nYCTb Xl, ..., x n - HeKOTopaH cOBoKynHocTb N-Mep-
HbIX BeKTopOB xk = <x:). Oopa3yeM .Mampul{Y X = (x ) paSMep-
HOCTH NX n, BSSIB B KatJeCTBe ee CTOJIOU.OB 9TH BeKTopbI, TaK tJTO xk -
3TO k-H cToJI6el MaTpHl(bI X. OT,neJIbHble N -MepHbIe BeKTopbl paCCMaT-
pHBalOTCH KaK MaTpHUbI N XI, T. e. CTOJIOl(bI.
nYCTb a = (ak) - npOH3BOJIbHbIH n-MepHblH BeKTOp. no npOH3-
Be eHHeM X a nOHHMaeTCSI JIHHeHHaH KOMOHHal(HH CTOJIOl(OB xk, r e
n .
pOJIb KO.9<p<PHUHeHTOB HrpalOT KOMnOHeHTbI BeKTopa a: X a = akxk.
k=l
HeTPY HO BH eTb, tJTO KOMnOHeHTaMH BeKTopa Xa SlBJIHIOTCH CKaJIHp-
Hble npOH3Be eHHH COOTBeTCTBYIOInHX CTpOK MaTpHl(bl X B BeKTopa a.
MaTpHl(Y X MO)KHO paCCMaTpHBaTb KaK onepamop, npeoopa3YIO-
InHH n-MepHble BeKTopbl a B N -MepHble BeKTopbl X a. MHO}l(eCTBO Bcex
3HatJeHHH .9TOrO onepaTopa L (X) = {x I x = Xa, a - JII060e}, otJe-
BH HO, eCTb JlHHeHHOe npocTpaHcTBo BceB03MO)KHblX JIHHeHHblx KOMOH-
HaUHH BeKTopOB Xl, ..., x n . Pa3MepHOCTb npOCTpaHCTBa L (X) HaSbI-
BaeTCH paHZOM MaTpHl(bI X. PaHr COBna aeT C MaKCHMaJIbHblM KOJIHtJe-
CTBOM JlHHeHHO-He3aBHCHMbIX CTOJIOUOB, a TaK}Ke C MaKCHMaJIbHblM KO-
JIHtJeCTBOM JIHHeHHO-He3aBHCHMblX CTpOK MaTpHUbI X.
nYCTb A = (aki) H B = (b is ) - Be MaTpHl(bl pasMepHocTH
m X n H n X 1 COOTBeTCTBeHHO. 0603HatJHM n-MepHble CTOJI6ubI, 00-
pa3YIO He MaTpHl(Y B, tJepes b S (5::::1 1, ..., l). flpou38eiJenueM AD
Ha3b1BaeTCH MaTpHua m X 1 co cToJI6uaMH Ab s . MHaqe rOBopSl,
(k, S)-H .9JIeMeHT MaTpHl(bI AB paBeH CKaJIHpHoMY npoHsBe eHHIO
n
k-CTpOKH MaTpHUbI A H s-ro CTOJIOua MaTpHl(bI B: ahibis. MaTpHua AD
1=1
OCYIneCTBJIHeT npe06pa30BaHHe I-MepHblx BeKTopOB B m-MepHble C no-
MO blO nOCJIe OBaTeJIbHOrO OCYIneCTBJIeHHH . npeoopasoBaHHH A H B.
Oo03HatJHM tJepe3 AT MaTpHUY nxm, nOJlYtJeHHYIO mpancnonupo8a-
nueM A, TaK tJTO CTpOKaMH MaTpHI1b1 AT HBJIHIOTCH CTOJIOl(bI MaTpHUbI A.
Onepal1HSI TpaHCnOHl1pCBaHRH MO}KeT npHMeHSlTbcH H K pe3YJlbTaTY npo-
HsrseJ,(eHHSl MaTpUl(, npHtJeM CnpaBe,AJIHBO paBeHcTBo (AD)T = DTAT
298
-
CKaJIHpHOe npOH3Be,neHHe (x, Y) BeKTopOB x, Y MomeT paCCMaTpHB8Tb-
CH KaK npOH3Be,neHHe xT Y MaTpHil,bI x T pasMepHocTH 1 X N H MaT.
PHUbI y pa3MepHOCTH NX 1.
MaTpHl1a Ha3bIBaeTCH 1\,8aiJpamnotl, eCJIH tIHCJIO ee CTpOK COBna,lta-
eT C tIHCJIOM CTOJI0I10B. 06pamflafl Atampu'1a A-I ,nJlH KBa,npaTHoH MaT-
PHUbI A onpe eJIHeTCH paBeHCTBOM AA-I = A-IA = E, r,lte E =
= (eij) - TaK Ha3blBaeMaH eiJunUtlnafl .Mampu'1a; ee HarOHaJlbHbIe
3JIeMeHTbI paBHbI e,nHHHl(e, a OCTaJlbHbIe - HYJlIO, T. e.
{ I,
eij = 0,
i=j;
i =f=j ·
OqeBH HO, AE = EA = A JlH JII060H A. 06paTHaH ,nJIH A MaT-
pHua CYll.\eCTByeT Tor a H TOJIbKO Tor,na, Kor,na ee paHr paBeH n, T. e.
Bee CTo.,,6UbI H CTpOKH JIHHeHHO He3aBHCHMbI.
feoMeTpHtIecKH A-I peaJIH3yeT oopaTHoe npeoopa30BaHHe JlH one-
paTopa A, TaK tITO cyInecTBoBaHHe A -1 paBHOCHJlbHO B3aHMHOH O,ltHO-
3HatIHOCTH onepaTopa A. BblqHCJleHHe A-I CBO,nHTCH, no CYTH eJla, K
pelIIeHHIO CHCTeMbl n X n ypaBHeHHH A Y = E, HMelOll.\eH n 2 HeH3BeCT-
HbIX 3JIeMeHTOB MaTp Hl(bI Y.
0.3. C06CTBEHHbIE 1JHCJlA H BEKTOPbI
,
9JIeMeHTaMH MaTpHl(bl S = X TX HBJIHIOTCSI CKaJlSIp-
Hble npoH3Be eHHH (X S , xt) CTOJl()nOB MaTpHUbl X, a 9J1eMeHTaMH MaTpH-
D.bI SX = XX T - CKaJlSlpHbIe npOH3Be,neHHH CTpOK MaTpHl(bI X,
npHtIeM ST = S H SX T = SX, T. e. 3TH MaTpHl(bl CHMMeTpHtIHbI. B TOM
CJlYtIae, Kor a BMeCTO X S paCCMaTpHBalOTCH HX l1eHTpHpOBaHHble H HOp-
xs-xs
MHpOBaHHble BapH8HTbI ZS= ..r _ _ ' 0()pa3YIOIl.\HeMaTp H I1Y Z,
r (XS-X S , XS-X S )
CTOJIOl1bl KOTOpOH HMelOT e HHHtIHYIO ,nJlHHY H HYJleBbIe cpe,nHHe, npOH3-
Be eHHe ZTZ npe CTaBJlHeT COOOH He qTO HHoe, KaK MaTpHIJ.Y K03«l>CPH-
l(HeHTOB KOppeJIHl(HH.
MaTpHl1bI CKaJIHpHbIX npOH3Be,neHHH Ha3bIBalOTCH MampU'1aMU
FpaMa H OOJla aIOT MHorHMH nOJle3HbIMH CBOHCTBaMH, KOTopble y 06HO
CPOPMYJlHpOBaTb B TepMHHax COOCTBeHHblX tIHCeJl H BeKTopOB.
nYCTb A - KBa,npaTHaH MaTpHIJ.a pa3MepHocTH n X n. HeHYJIe-
BOH BeKTOp x = (Xl' ..., x n ) Ha3bIBaeTCSI C06Cm8eftftblM 6e1\,mOpOM MaT-
pHl1bl A, cooTBeTcTBYlOrnHM co6cm8eflnO.MY tlUCIlY A, eCJIH cnpaBe,nJlHBO
paBeHCTBO Ax = AX. TaK KaK BMeCTe c x 3TOMY paBeHcTBY Y OBJIeTBO-
pHIOT Bce BeKTopbl ax, TO npHMeHeHHe onepaTopa A OCTaBJIHeT npHMYIO
{ax} HeH3MeHHoH, JlHWb YMHO}KaH ee «TOtIKH» Ha A.
Kam aH MaTpHIJ.a n X n HMeeT pOBHO n c06cTBeHHblx tIHCeJl, cpe-
H KOTOpbIX, Booorne rOBopH, MorYT ObITb KOMnJIeKCHbIe, a TaK}Ke O,nH-
HaKOBbIe BeJIHtIHHbI. LJHCJlO HeHYJIeBblX co()cTBeHHblx tIHCeJl paBHO paH-
ry MaTpHl1bI. COOCTBeHHbIe tIHCJIa MaTpHl( rpaMa HeOTpHl(aTeJlbHbI H Be-
rneCTBeHHbl. 0003HatIHM c06CTBeHHble tIHCJla MaTpHl(bl fpaMa A tIepes
A1 A 2 >-- ... >-- Am > 0, m n (ocTaJlbHble cooCTBeHHbIe tIHCJla-
HYJleBble), a COOTBeTCTBYlOrnHe HM coocTBeHHble BeKTopbI-C 1 , c 2 , ..., cm.
3TH BeKTopbl MorYT ObITb B3HTbI JlHHeHHO-He3aBHCHMbIMH, npHtIeM JlH
As =1= At coocTBeHHbIe BeKTopbI C S H c t 00H3aTeJlbHO opTOrOHaJlbHbl:
(C S , c t ) = o.
299
lllllllllllllq1l11111111111111111111111111111111111111111111111111111111111111111111111111111111
------
,L{JUI npOCTOThI ,ltaJIee OrpaHHtIHMCSI CJIYtIaeM, Kor a m == n, npH-
qeM cpe,ltH COOCTBeHHblX tIHCeJI At. > A 2 > ... > An HeT o,nHHaKOBbIX.
HopMHpyeM CO()CTBeHHbIe BeKTopbl oS TaK, tITO()bI OHH HMeJIH e H-
HHtIHYIO JIHHY. Tor a (cS, d) paBHO HYJIIO npH 5 =1= t H 1 npH 5 = t.
3TO OSHaqaeT, tITO ,ltJIH MaTpHUbI C, CTOJIOnaMH KOTOpOH CJIY}KaT BeKTopbl
, cnpaBe JlHBO paBeHCTBO
c T C= CC T z=E,
(n .6)
KOTopoe 03HaQaeT, tITO c T SIBJISIeTCSI 06paTHoii ,ltJISI C. feoMeTpHtIeCKHH
CMblCJI: MaTpHl(a C ocy eCTBJlSIeT npe06pa30BaHHe nOBopOTa. 0603Ha-
qHM qepes
A = Diag (AI, ..., An)
(n.7)
MaTpHu.y n X n, HarOHaJIbHble 9JIeMeHThI KOTOpOH paBHbI As (5:::;:' 1,
..., n), a OCTaJIbHbIe - HYJIIO. Tor a no onpe eJIeHHIO co()cTBeHHbIx
BeKTopoB
AC = CA.
(n .8)
YMHO}l(aSI 9TO paBeHcTBo cnpaBa Ha C , nOJIYtIHM
n
A=CAC T = AsC s csT,
s== 1
(n.9)
r,ne c'c'T nOHHMaeTCH KaK npOH3Be,neHHe MaTpHl(bI N X 1 (cTOJIOu.a) Ha
MaTpHl(Y 1 X N (CTPOKY), pe3YJIbTaTOM tIero SIBJlHeTCSl MaTpHu.a c
(i, }.)-M 9JIeMeHTOM c1 c j I
0.4. AHAJlH3 fJlABHbIX KOMOOHEHT
0603HatIHM tIepe3 aI, ..., an c06CTBeHHbIe BeKTopbI
MaTpHl(bI S = XTX, COOTBeTCTBYIOll\He c06CTBeHHbIM tIHCJIaM Al >
> A 2 > ... > An- YMHO}l(aH paBeHCTBa XTXa s = Asa! CJleBa Ha X
y()e}K aeMCH, tITO BCKTOpbI Xa S HBJlHIOTCH COOCTBeHHbIMH BeKTopa-
MH MaTpHl(bI XX T , COOTBeTCTBYIOll\HMH TeM JKe COOCTBeHHblM tIHCJlaM
As (5 = 1, ..., n). npH 9TOM eCJIH JlHHa as paBHa a sT as = 1, TO ,nJIH-
Ha Xa S paBHa As. TaK tITO HOpMHpOBaHHbIH c06CTBeHHbIH BeKTOp ZS
MaTpHl(bI XX T, COOTBeTCTBYIOll\HH c06CTBeHHOMY tIHCJlY As, MO}KeT 6blTb
npe CTaBJleH COOTHOlIleHHeM
"lj- S s
J' As Z = Xa ·
(n .10)
0003HatIaH qepe3 Z H A MaTpHl(hI C HOpMHpOBaHHbIMH CTOJIOuaMH
ZS Has. nOJIYtIHM
1
ZA2 =XA.
(n. 11)
t
. r,ne A 2 = Diag (V , ..., ).
300
-
YMH01K8H (n.ll) cnpaBa Ha AT, HMeeM npe,ncTaBJIeHHe X tIepe3
co()cTBeHHbIe BeKTopbl MaTpHU XTX H XX T :
I
X=ZA2 AT = V As zSa sT .
5
(n .12)
YMHO}l(aSl (n.ll) CJIeBa Ha ZT, nOJlYtIHM npe,ncTarsJIeHHe co()cTBeK-
HbIX QHCeJI:
I
2 T
A =Z XA.
-
(n.13)
BeKTopbI ZS Ha3blBalOTCSl Z/laBnblMU 1\,OMnOnenmaMU MaTpHl(h1 X-.
MaTpHua rJIaBHbIX KOMnOHeHT Z nOJlYtIaeTCH B pe3YJIbTaTe nOBopoTa
XA MaTpHUbI X C nOCJle,nYIOInHM H3MeHeHHeM MaclIITa6a:
s 1 s
z = ,j- Xa ·
y As
(n . 14
,ltJIH OTbICKaHHH rJIaBHbIX KOMnOHeHT MO}l(HO BOCnOJIb30BaTbCSI
cneAYIOIneH HTepaTHBHoH npol(e YPoH.
Ha nepBoM lIIare OTbICKHBaeTCH c06cTBeHHaH TpoHKa (At, Z1, a l )
C MaKCHMaJIbHbIM c06cTBeHHbIM tIHCJIOM Ha OCHOBe MHoroKpaTlforo
npHMeHeHHH npeoopa30BaHHH, aBaeMoro JII060H H3 MaTpHIJ. S = XTX,
SX == XXT, X. ,l:{JIH onpe,neJIeHHOCTH paCCMOTpHM MaTpHUY S = XTX,
KOTopaSI O()bItIHO HMeeT HaHMeHblIlYIO pa3MepHocTb cpe H nepetIHC-
JIeHHbIX.
npOl(eCC HatIHHaeTCH C npOHSBOJIbHOrO n-MepHoro BeKTopa a (0).
Ha t-M lIIare no 3a aHHOMY a (t - 1) BbltIHCJISIIOTCSI a (t) H A (t):
A (t) == ya T (t-l) SSa (t-I), a (t) =Sa (t-I)/A (t). (n.15)
- M01KHO nOKa3aTb, tITO A (t) At H a (t) a l npH t '00, eCJIH
Al > A 2 --
n03TOMY nOCJIe OCTH}KeHHH Heooxo,nHMOH TOtIHOCTH B BbltIHCJIeHHH
a l , QTO O()bltIHO onpe,neJlHeTCH no COBna,neHHIO a (t) H a (t + I) B npc....
,l{eJIax saAaHHoH norpelliHOCTH, M01KHO BbItIHCJIHTb r JlaBHYIO KOMnOHeH-
TY ZI no cpopMYJIe (n.14): Zl = Xal/ .
Tenepb AI' Zl, a l onpe eJIeHbI H npoH3Bo HTCH nepexoA K MaTpHl(e
OCTaTOtIHblX cBH3eH
Sl=S-Al a 1 alT.
npHMeHHH K KOTOpoii npOl1eCC (n.15) HatIHHaH C npOHSBOJIbHOrO a (0)"
JIHHeHHO-He3aBHCHMoro OT ai, nOJIYtIHM cJle,nYIOll\YIO TPOHKY: A2' Z2, a 2 .
npH HaH,neHHbIX P TpoHKax (As, as. ZS) (s == 1, ..., p) npoue c
(n.15) npHMeHHeTcH K MaTpHl(e
"
S"=SP-I-ApaPaPT =S- Asasa sT ,
s == 1
HatIHH3H C npOH3BOJIbHOrO a (0), JIHHeHHO-HesaBHCHMoro OT ai, ..., aPe
JrOT npouecc npHBe,neT K Ap+l' a P + 1 , ZP+l, eCJIH Ap+ > Ap+2.
30r
(n . 16)
-
n.5. CKAJISIPHOE npOH3BEAEHHE MATPHil
H CJI EA
MaTpHnbI n X m MO}KHO paCCMaTpHBaTb KaK nm-
MepHbIe BeKTopbI, tITO aeT B03MO}KHOCTb npHMeHHTb K HHM BeKTopHbIe
onepaUHH. ECJIH X H Y - Be MaTpHnbI o,nHHaKOBOH pa3MepHOCTH,
TO HX CKaJIHpHOe npOH3Be eHHe (X, Y) = :E XisYis = Tr (X Ty),
I,s
r e Tr (A) 0003HatIaeT C/leo MampUl1bl A, CYMMY ee ,nHarOHaJlbHbIX .9JIe-
MeHTOB Tr (A) = ai i.
i
M3BecTHo, tITO CJIe,l{ MaTpHII.bl paBeH CYMMe ee COOCTBeHHbIX tIHCeJI:
Tr (A) == A5'. n03TOMY JlHHa MaTpHII.bI X KaK BeKTopa
I X f2=(X, X)=Tr (X T X) (n.17)
paBHa CYMMe COOCTBeHHbIX 3HatIeHHH MaTpHII.bI XTX.
,l{JISI CJIe a MaTpHU cnpaBe JlHBO CJIe YIO ee COOTHOlIIeHHe:
Tr (AD) = Tr (BA). a.(n .18)
0.6. AH«I>«I>EPEHI\HPOBAHHE
npaBHJIa
LLJIH cHMMeTpHlIHbIX MaTpHII. HMelOT MeCTO CJIe YlOinHe
HcpcpepeHUHpoBaHHH:
o (zT A) 0 (zT Az)
=A, =2Az,
oz OZ
iJ (zT z) =2z. (n .19)
oz
n.7. npOEKI\USI
nYCTb z - npOH3BOJIbHbIft N-MepHbIH BeKTOp, a X-
npOH3BOJlbHaH MaTpHII.a N X n. flpOeKI1Ufl Z Ha npOCTpaHCTBO L (X)
onpe,neJlHeTCH KaK TaKOH BeKTOp cp (z), ,nJlH KOToporo pa3HocTb Z -
- cp (z) opTOrOHaJlbHa BceM x E L (X). 9TO onpe,neJIeHHe Q()ooina-
eT cTaH,napTHoe npe,ncTaBJleHHe 00 opTOrOHaJIbHOH npoeKnHH B OObItIHOM
reOMeTpHtIeCKOM npOCTpaHCTBe: OTpe30K Z - cp (z), Jle)Ka HH Me*,ny
Z H ee npOeKII.HeH cp (z), ,nOJl}KeH ObITb nepneH,nHKYJIHpeH (opTorOHaJIeH)
o()JIaCTH, Ha KOTOpylO ocy eCTB.,1IHeTCH npoeKTHpoBaHHe.
OKa3bIBaeTcH, onepaTop npoeKTHpoBaHHH (npoeKmop) MO)KeT
()bITb npe CTaBJleH MaTpHneH
PX=X(XTX)-lX T , (n.20)
T8K tITO cp (z) = Pxz. npH 3TOM MaTpHna npoeKTopa Px Ha L = L (X)
onpe,neJlHeTCH TOJlbKO npOCTpaHCTBOM L H He 3aBHCHT OT BH a MaTpH-
UbI X, nOpO)K,nalO eH 3TO npOCTpaHCTBO.
npOH3BOJlbHaH MaTpHil.a P 3a,naeT npoeKTOp Tor,na H TOJlbKO Tor,na,
Kor,na OHa CHMMeTpHtIHa P = pT, H uoeMnOmeftmna, T. e. PP = P.
,[lJI'H npoeKTopa P MaTpHil. a E - P TaK:>Ke sa,naeT npoeKTOp, HO Ha op-
TOrOHaJIbHOe K L npOCTpaHCTBO.
MO)KHO nOKa3aTb, tITO Pxz - caMbIH 6J1H3KHH K Z BeKTOp npOCT-
paHCTBa L (X), T. e. PxZ HBJIHeTCH pelIIeHHeM 3a atIH annpoKcHMaUHH
302
-
'.-
BeKTopa z C nOMO blO JIHHeHHbIX KOMOHHaIJ.HH CTOJlOl(OB MaTpH-
UbI X:
\z-Pxzl<Iz-yl
JIH Bcex OTJIHtIHbIX OT Z BeKTopOB y E L (X).
npoeKUHH Pxz npHHa,nJle}KHT L (X) H, CJIe OBaTeJIbHO, MO}l(eT
6bITb nOJIYtIeHa YMHO}KeHHeM X Ha nO XO Sl HH BeKTOp a. B CHJlY
(0.20)
z=P x z=Xa,
r e a=(XTX)-lXTz.
(n.21)
n.8. AnnpOKCHMAI\HSI ,nAHHbIX H fJlABHblE
KOMnOHEHTbI
PaCCMOTpHM sa atIY 06 OTbICKaHHH TaKoro npocTpaH-
CTBa L = L (Z) sa aHHoH pasMepHocTH p, r e Z - MaTpHl(a
N X p, CTOJlOl(bI KOTOpOH 00pa3YlOT oasHc L, tIT06bI npoeKl(HSl PzX
MaTpHl(bI X Ha L 6bIJIa KaK MO}KHO OJIH}Ke K HCXO,nHOH MaTpHIJ.e X.
3 ecb no npOeKl(HeH PzX nOHHMaeTCH MaTpHl(a, CTOJIOnbI KOTOpOH SIB-
JISIIOTCH npoeKl(HHMH COOTBeTCTBYIO HX CTOJlOl(OB MaTpHl(bI X.
,ltJISl aHaJlHTHtIeCKOrO HCCJIe OBaHHH, nOCKOJIbKY Pz onpe eJlHeT-
CH TOJIbKO npOCTpaHCTBOM L, BbIoepeM Z TaK, qTOObI ee CTOJl6l(bI 6bI-
JIH opTOrOHaJlbHbI H HOpMHpOBaHHbI (opmonopMup06annbl). Tor a
nOCTaBJIeHHaH 8a atIa annpOKCHMal(HH MO}KeT 6bITb YTOtIHeHa KaK sa a-
tIa OTblCKaHHH MaTpHl(bI Z pa3MepHOCTH N X p, KOTopaH MHHHMHSH-
pyeT CYMMapHYIO norpeIIlHOCTb
I X-Pz X (2 =Tr [(X-Pz X)T (X-Pz X)] (n,22)
npH YCJlOBHH
ZT Z=E. (n.23)
PaCKpblBaH B (0.22) CKOOKH, nOJlYtIHM, tITO MHHHMHSal(HB (0.22)
npH YCJlOBHH (n.23) 3KBHBaJIeHTHa MaKCHMHSal(HH
Tr (ZT XX T Z)
(0.24)
npH TOM }Ke YCJlOBHH.
PemeHHe CCPOPMYJlHpOBaHHOH sa,naQH aeT MaTpHl(a Zp nepBbIX
p rJlaBHbIX KOMnOHeHT MaTpHl(bI X, npHQeM onTHMa..'IbHOe SHatIeHHe
p
(0.24) paBRo CYMMe nepBbIX p c06CTBeHHbIX tIHCeJI As, a OnTHMaJIbHOe
s=1
SHatIeHHe norpemHocTH (0.22) - cYMMe nOCJIe HHX c06CTBeHHbIX ilK-
n
ceJI :E As (p < n).
s=p+l
npH p = 1 3a,natIa CBO,nHTCH K OTblCKaHHIO TaKoH npHMoH {az}, lITO
CYMMapHoe OTKJlOHeHHe CTOJlOl(OB X OT HX npOeKl(HH H8 3TY npSIMYIO
I
6h1JIO MHHHMaJlbHO. npH 9TOM Pz = zzT, TaK KaK (zTZ)-l= Izli =1
B CHJIY (n.23), H nOKa3aTeJlH (n.22) H (n.24) HMeIOT CJIe YIO HH BHJt:
(X:- Zi (z, X S »2, (z, xsr . (0.25)
i,s S
303
nepBbIH H3 HHX ,nonYCl<aeT npOCTYIO reOMeTpHtIeCKYIO HHTepnpe-
Tal(HIO B TepMHHax 06'beKTOB - CTpOK MaTpHl(bl X: OH paBeH CYMMe
paCCTOHHHH (T. e. KBa,npaTOB eBKJJH,ltOBCKHX paCCTOHHHH) OT 06'heKTOB
KaK n-MepHUX BeKTopOB O npHMOH {az}.
BTOpOH nOKa3UBaeT, qTO rJIaBHaH KOMnOHeHTa ZI = Xa 1 /-VX;:- eCTb
TaKaH JIHHeHHaH KOM6HHal(HH CTOJl()UOB MaTpHl(bI X, KOTopaSI MaKCR-
14H3upyeT CYMMapHYIO ,nJlHHY HX npOeKI.{HH Ha 9TY KOM6HHal(HIO. ECJlH
CTOJlGU.U x 8 l(eHTpHpOBaHHbl H HOpMHpOB8HHbI, TO BeJlHtIHHbI (z, X S ) HMe-
lOT CMbICJI K03cp<pHUHeHTOB KOppeJIHl(HH npH3HaKOB H annpOKCHMHPYIO"
meH HX JIHHeHHOH KOMoHHal(HH.
. 'l(am,naH cJIe YlOmaH rJIaBHaH KOMnOHeHTa zP npe CTaBJIHeT CO..
ooA TaKYIO JIHHeHHYIO KOMoHHaUHIO CTOJIOl(OB MaTpHl(bI X, KOTopaSI on..
THMH3HpyeT TOT }Ke nOKa3aTeJIb (n.25) npH ,ltOnOJIHHTeJIbHOM YCJIOBHH,
qTO Z BbIOHpaeTCH opTOrOH8J1bHO npe,nbI ymHM rJIaBHbIM KOMnOHeHTaM.
AHaJIorHqHbIM 00pa30M MaTpHl(a Ap, cToJI6l(aMH KOTOpOH CJIY}K8T
P nepBUX COOCTBeHHUX BeKTopOB al, ..., a P MaTpHltU S= XTx, HBJlH-
"eTCH pelIIeHHeM 3a atIH O() OTbICKaHHH TaKOH MaTpHl(bI A pa3MepHOCTH
n X. P C OpTOHOpMHpOBaHHUMH CTOJl()l(aMH, KOTopaSI MHHHMH3HpyeT
1
J
d (X T , P A XT) = I X T -P A XT 1 2 ,
(n .26)
HJIH, tITO TO )Ke caMoe, MaKCHMH3HpyeT
Tr (AT XT XA),
(n.27)
.
npHtIeM OnTHMaJIbHUe 3HaqeHHH 9THX nOKa3aTeJIeH paBHu TeM )Ke CYM"
.MaM co()cTBeHHbIx tIHCeJI&
. MaTpHl(bI Ap H Zp pemalOT 3a,ltatIY MaKCHMH3altHH
Tr (ZT XA)
(0.28)
Ha MHO}l(eCTBe MaTpHl( A pa3MepHOCTH n X p H Z pasMepHocTH NXp
C JlHHeHHO-He3aBHcHMbIMH HOpMHpOBaHHUMH CTOJIOl(aMH, npHtIeM
T P-
Tr (Zp XA p ) = YAs.
s=1
(n .29)
B tIaCTHOCTH, nepBaSI c06cTBeHHaH TpoHKa (AI' aI, ZI) HBJIHeTCH pe-
weHHeM 3a atIH MaKCHMH3al(HH CKaJIHpHOrO npoHsBe eHHH
(z, Xa)=zT Xa
(n.30)
no' BceB03MomHUM HOpMHpOBaHHUM Z H a, npHqeM -V At. eCTb MaKCHMaJIb-
HQe ,3HatIeHHe nOKaSaTeJIH (n.30).
TaKHM 0()pa30M, rJIaBHbIe KOMnOHeHTU alOT pelIIeHHe sa atI an-
npOKCHMal(HH 3a,ltaHHOH MaTpH1I.bI X B npOCTpaHCTBax MeHblIIeH paa-
!depHOCTH, He3aBHCHMO OT Toro, KaKHM HS nOKaSaTeJIeii (n.22), (n.24),
(n.26), (0.27), (n.28) Ol(eHHBaeTCSl KatIeCTBO annpOKCHMal(HH.
. , PasMepHocTb p annpoKcHMHpYlOmero npocTpaHcTB8 BU()HpaeTCSI
. p n
no BeJlHtIHHe E As/ As. CMUCJI KOTOpOH pacKpuBaeTcSI nOKaSaTeJIBMH
. ., 8==1 8=1
(n:22) H (n.24). tIeM ()JIH}I(e ORa K e,ltHHHne, reM TOtIHee annpOKCX-
au.H I. n09TOMY O}KHO sapaHee 3a aTbCH HeKOTopOH BeJlHqHHOH
< 8 < 1 H C nOMomblO 8JIrQPHTM8, HSJIO}KeHHOrO B n.4, nOCJIe,ltOBa-
80'1
--
.
TeJIbHO HaXO HTb rJIaBHbIe KOMnOHeHTbI zs. To 3HatIeHHe p, npH KOTOpOM
p P n
OTHoweHHe As/Tr (X TX) = :z 'As/ :z As BnepBbIe npeBSOH eT 1-8,
8=1 5=1 8==1
rapaHTHpyeT AOCTH)KeHHe He06xo HMOH TOtIHOCTH annpOKCHManHH.
n.9. KAHOHH1JECKHH AHAJlH3
PaCCMOTpHM CJIe,nYIOInYIO 0600 eHHYIO 3a atIY annpo-
KCHMal(HH. nYCTb X H Y - MaTpHl(bI paSMepHOCTH N X ml H NX m2
COOTBeTCTBeHHO. nonbITaeMCH annpOKCHMHpOBaTb BeKTopbI X = Xa
npOCTpaHCTBa L (X) BeKTopaMH Y = Yb npOCTpaHCTBa L (Y). HHbI-
MH CJlOBaMH, 6y,neM HCKaTb TaKHe X E L (X) H Y E L (Y), !{otopbIe
MHHlfMH3HPYlOT paCCTOHHHe d (X, y) = I X - Y 1 2 . npOCTpaHCTBa L (X)
H L (Y) 3aBe,nOMO CO ep}KaT 0, KOTOpblH HBJIHeTCH HX OomHM 3JIeMeH-
TOM. q TO()bI HCKJIlOtIHTb 3TO TpHBHaJIbHOe pemeHHe HCXO,nH H3 paBHO-
npaBHOCTH BeKTopOB X H y, nOTpe()yeM, tIT06bI OHH HMeJIH O HHaKOBYIO
e HHHqHYIO ,D;JIHHY.
TaKHM o()paSOM, 3a atIa COCTOHT B TOM, tITO()bI HaHTH TaKHe a H b,
YAOBJIeTBOpHlOmHe YCJlOBHHM
aTXTXa=l, b T yTYb=I, (n.31)
KOTopbIe MHHHMH3HPYIOT
d (Xa, Vb) =1 Xa- Yb J2=a T X T Xa-2a T X T Yb+b T yT Vb, (n .32)
HJ1H, tITO TO }l(e caMoe, MaKCHMH3HPYI01 KOCHHYC yr JIa Mem,uy X = Xa
H y = Vb:
(X, y)=x T y=aTXTYb.
PeweHHe 3TOH 3a atIH onpe,u.eJlHeTCSI YCJIOBHHMH
PxY=AX; PyX=AY,
(n .33)
(n .34)
r,lte A - HCKOMoe MaKCHMaJIbHOe SHatIeHHe (n.33).
Y CJlOBHSI (n.34) HMelOT HarJIH HhlH reOMeTpHlIeCKHH CMblCJI. Pac-
CMOTpHM nJlOCKOCTb, nopO}K aeMYIO BeKTopaMH X H y. DHa nepeceKaeT
npOCTpaHCTBa L (X) H L (Y) no npSlMbIM, nopO}K aeMblM 3THMH BeKTopaMH,
npHtIeM npoeKTopbI He BbIBO SIT H3 3TOH nJIOCKOCTH, TaK tITO OHa OpTO-
rOHaJIbHa H L (X), H L (Y), a 3HatIHT, H HX nepeCetIeHHIO L (X) nL (Y).
aTO COOTBeTcTByeT 06b1QHOH TpeXMepHOH KapTHHe: HaHMeHblIIHH yrOJI
Me}l(,ltY BeKTopaMH BYX nJlOCKOCTeH, aHaJIOrOB L (X) H L (Y), OCTH"
raeTCSI Ha BeKTopax, nepneH,ltHKYJlSlpHblX JIHHHH nepeCeqeHHSl .9THX
nJIOCKOCTeH.
· nO CTaBJIHSI COOTHoweHHH (n.34) O HO B ,ltPyroe, nOJlYQHM
Px Py x='A 2 X; Py Px y='A 2 y, (n.35)
TaK tITO HCKOMbIe X = Xa H y = Yb HBJIHIOTCH CO()CTBeHHbIMH BeKTO-
"p aMH MaTpHl( P Xy == PXPy H Pyx;:::::l PyPX' COOTBeTCTBYlOmHMH HX
MaKCHMaJIbHOMY COOCTBeHHOMY tIHCJIY 'A".
'I C06CTBeHHbIe BeKTopbI MaTpHl( PXPy H PyPX' COOTBeTCTBYlOmHe
O,ltHOMY H TOMY }Ke c06CTBeHHOMY tIHCJlY, Ha3bIBalOTCSI /CanOnU'I,eCKUMU
ltJUl X H Y, a HX CKaJIHpHbIe npOH3Be4eHHSl (KOCHHYCbI ,)'rJIOB) - /Ca-
nOfl,U'I,eC/cUMU ICOppe/lfl/{UflMU.
305 .
COOTBeTCTBYIOInHe BeKTopbI a H b HBJISIIOTCH COOCTBeHHbIMH BeK-
TOp8MH KBa,npaTHbIX MaTpHU
pXY =(XTX)-lXTy(yTY)-lyTX; pYX=
== (yTy)-lyT X(XTX)-lXTy, (n.36)
COOTBeTCTBYIOInHMH TO My )Ke caMOMY co6CTBeHHOMY IJHCJlY.
0603HatIHM A = (XTX)-lXTy, B = (yTY)-lyTX; Tor a, OtIe-
BH,nHO, pXY AD, pYX = BA. H3 ABa = ")'a CJle yeT (BA) (Ba) :::I
= A 2 (Ba), TaK tITO Ba HBJIHeTCH co6CTBeHHbIM BeKTopOM BA H, CJIe-
,nOBaTeJlbHO, OTJIHtIaeTCH OT b JIHWb HOPMHPYIOInHM MHO)KHTeJleM:
b = aDa, r e a BbloHpaeTcH TaK, tITOObI BblnOJlHHJIOCb YCJlOBHe
bTyTYb = 1. AHaJIorHqHo C TOIJHOCTblO ,no HOPMHpYlOmero MHO}KHTeJIH
a = Ab. 3TO MO}l(HO HCnOJIb30BaTb ,nJIH Toro, tITOObI npH pelIIeHHH saAa-
tIH KaHOHHtIeCKOrO aHaJlH3a HCKaTb coocTBeHHbIe BeKTopbI TOJlbKO O HOH
H3 MaTpHl( PXPy, PyPX' AD, BA, BbIoHpaH TY, KOTopaH HMeeT HaHMeHb-
lIIYIO p a3Mep HOCTb.
I(OJIHtIeCTBO HeHYJIeBbIX co6cTBeHHbIx QHCeJI MaTpHU PXPy, AD
O HO H TO }l(e: HX pOBHO m = min (ml' m 2 ), npHtIeM Bce OHH nOJIO}l(H-
TeJIbHbI H He npeBbIllialOT e HHHl(bI. nepeHYMepyeM HX B nopH Ke y6bI-
BaHHH 1 A. Af ... A, _1 H 0003HatIHM tIepe3 X S H yS KaHO-
HHtIeCKHe BeKTopbI, cooTBeTcTBYlOmHe c06cTBeHHoMY tIHCJIY A:. npH
As =1= At COOTBeTCTBYlOmHe KaHOHHtIeCKHe BeKTopbI opTOrOHaJIbHbI Pyr
p yry:
(x S , yt) = o.
0603HatIHM tIepe3 X p H Y p MaTpHl(bI N X p, cToJl6l(aMH KOTOpbIX
CJIY}KaT nepBbIe p KaHOHHtIeCKHX BeKTopOB X S H yS COOTBeTCTBeHHO. 3TH
MaTpHltbI AalOT pelIIeHHe CJle,nYlOmeH 3a atIH: HaHTH B npOCTpaHCTBax
L (X) H L (Y) TaKHe CHCTeMbI JIHHeHHO-He3aBHcHMbIX HOpMHpOB8H-
--' --'
HbIX BeKTopOB, oopa3YIOInHe MaTpHl(bI X H Y pa3MepHocTH N X p,
-., --' --'
)tJISI KOTOpbIX MHHHMaJIbHO pacCTOHHHe d (X, Y) = I X - Y 1 2 .
TaKHM 06pa30M, KaHOHHtIeCKHe BeKTopbI X S H yS 06pa3YIOT OpTO-
HOpMHpoBaHHbIe 6a3HCbI l npocTpaHcTB L (X) H L (Y), MaKCHMaJIbHO
OPHeHTHpOBaHHble APyr OTHOCHTeJIbHO Pyra: KOppeJIHl(HH (xS, yS)
npHHHMaIOT MaKCHMaJlbHO B03MO}l(HbIe 3HatIeHHH As (5 = 0, 1, ..., nt-I).
fI09TOMY BenHtIHHbI As MorYT 6bITb HCnOJlb30BaHbI JISI Ol(eHKH CBSISH
Me>KAY 9THMH npocTpaHcTBaMH HJIH nOpO}K aIOInHMH HX MaTpHuaMH
X H Y.
CJIe yeT OTMeTHTb, tITO eCJIH L (X) n L (Y) He CBO,nHTCSI K 0,
TO JII060H BeKTOp z E L (X) n L (Y) ,naeT KaHOHHtIeCKYIO napy z, z,
COOTBeTCTBYIOInYIO MaKCHMaJIbHOMY co6CTBeHHOMY tIHCJIY 1. ,[{eHcTBH-
TeJlbHO, Pxz = z, PyZ = Z AJIH Z E L (X) n L (Y), TaK tITO
Px Py z=z H Py Px z=z
TpHBHaJIbHbIe peweHHSI z E L (X) n L (Y) He HeCYT AOnOJIHU-
TeJIbHOH HH«l>opMal(HH 0 CBHSH Me}l( Y X H Y H AOJI}KHbI HCKJIlOtIaTbCSI
HS aHaJIH3a BMeCTe C COOTBeTCTBYIOmHM COOCTBeHHbIM tIH JIOM 1. ECJ1H
I
j
\
1 TOtIHee, )tJISI npocTpaHcTBa MeHblIIeH paSMepHocTH peIJb HAeT
o nOA6aSHce.
306
......-1
.::Lo\..
,
paSMepHOCTb L (X) n L (Y) paBHa p -< m, TO Ao = Ai a; c::: A p -l c::I
E::: 1, H HeTpHBHaJIbHblH KaHOHHtJeCKHH aHaJlH3 HaqHHaeTCH JIHlIIb C
Ap, x P , yp.
,[{JIH OTblCKaHHH KaHOHHtJeCKOH KOppeJIHI1HH Ap H COOTBeTCTBYIOInHX
KaHOHHtJeCKHX BeKTopOB x P , yP (p = 1, 2, ;.., m - 1) CJIe yeT HCKaTb
MaKCHMaJlbHbIe COOCTBeHHble tJHCJla H COOTBeTCTBYIOInHe co6CTBeHHbIe
BeKTopbI JIIOOOH H3 OCTaTOQHbIX MaTp HI.(:
p-l
P p P ,,2 S sT.
XY= XY- I'vsX Y ,
s=O
p-l
P XY p _ p XY ,,2 s sT.
- - I'vsa a ,
s= 0
(n .37)
,
\
p-l
pYX p = pYX _ A b S b sT .
s=O
n.1o. 060611tEHHA 06PATHASI MATPHU.A
B cJIYtJae, Kor a MaTpHua A pa3MepHocTH n X n HMe-
eT paHr m < n, oopaTHoH MaTpHl(bI A-I He CYIneCTByeT, TaK qTO ypaB-
HeHHe Ax = b pa3pellIHMO He ,nJIH BCHKoro b, a TOJlbKO JlH bEL (A).
A
Tor a MO}l(HO onpe eJlHTb TaKYIO MaTpHUY A-, tJTO BeKTOp x = A- b
HBJIHeTCH pelIIeHHeM ypaBHeHHSI Ax = b npH bEL (A), a npH
b fE L (A) MHHHMHsHpyeT OTKJIOHeHHe I Ax - b pa BeKTopa Ax OT b.
3Ta o606U1/!nHo o6pamnall Mampu'1a A- onpe eJIHeTCH CJIeAYIOInHM
o()pa30M.
BbI()epeM B A m JIHHeHHO-He3aBHCHMbIX CTOJIOI.(OB, KOTopbIe o()-
pasYIOT MaTpHI1Y B pa3MepHocTH n X m. Bce CTOJlOl.(bl as MaTpHUbI A
HBJIHIOTCH JIHHeHHblMH KOMoHHal.(HHMH B/s CTOJIOI.(OB MaTpHl.(bI 0, r,ne
IS - m-MepHbIe BeKTopbI, onpe,neJlHeMbIe H3 pelIIeHHH ypaBHeHHH as =
= Rls. TaKHM oopa30M, A = RP, r.n;e P - MaTpHl.(a co CTOJI()l.(aMH
IS (5 = 1, ..., n), npHtJeM paHrH MaTpHU B, F paBHbI m. Tor a o6o()-
IneHHO oopaTHaH MaTpHl.(a A- onpe.n;eJlHeTCH paBeHcTBoM
A - = pT (ppT)-l (B T 8)-1 B T . (n .38)
npH 9TOM AA- = B (BTB)-B T eCTb npoeKTop Ha L (A), a
A-A = pT (ppT)-lP - npoeKTop Ha L (AT), TaK tJTO AA- Be,neT
ceog KaK e }lHHtJHaH MaTpHl(a JlH BeKTopOB x E L (A): Jlg HHX
AA-x = x, a A-A - KaK e,nHHHtJHaH JlH L (AT).
,[lJIH MaTpHI.( fpaMa HMeeT MeCTO npe,ncTaBJleHHe (n.9) A =
n
:=: AsCsC sT . ECJlH npH 9TOM MaTpHl.(a A HMeeT m HeHYJIeBblX CO()CT.
s=1
BeHHbIX qHCeJl, TO
m 1
A-= -c s csT.
s = 1 As
(n .39)
JI HTEP A TYP A
1. A r a Hoe r SI H A. f. COUHaJlbHbIe HCCJIe,nOBaHHH H KOJlHtJeCTBeH-
Hble MeTO bI.- B KH.: I(OJIHlIeCTBeHHbIe MeTo,nbl B COl{HOJlOrHH. M.,
HaYKa, 1966, c. 5--=33.
2. A ii B a 3 SI H C. A., B e }I( a e B a 3. H., C Tap 0 B e p 0 B O. B.
KJIaCCHcpHKal{HH MHorOMepHbIX Ha()JIIO.n;eHHH. M., CTaTHcTHKa,
1974. 240 c.
3. A H S e p M a H . M. A., B P a B e p M a H 3. M., P 0 SOH o-
S P 1I. H. MeTO,n nOTeHUHaJlbHbIX CPYHKl{HH B TeopHH ooYtJeHHSI
MamHH. M., HaYKa, 1970. 384 c.
4. A n pee $I H 10. ,[{. AJiropHTM nOCTpoeHHH KJIaCCOB no MaTpHue
paccToHHHit- MalIIHHHbIH nepeBO H npHKJla HaH JIHHrBHCTHKa,
1966, BbIn. 9, c. 72-79.
5. A p K a A b e B A. f., B P a B e p M a H 3. M. OoYtJeHHe MalIIHHhI
KJIaCcHcpHKanHH o()'beKToB. M., HaYKa, 1971, 192 c.
6. B e }K 1(. TeopHH rpacpoB H ee npHMeHeHHH. M., MlI, 1962.
319 c.
7. B 0 H rap ,It M. M. npO()JIeMa Y3HaBaHHH M., HaYKa, 1967, 192 c.
8. Bop 0 A K H H A. M., M Y tJ H H K M. B. npHOJIH>KeHHblH aJiro..
pHTM annpoKcHMaUHH rpacpa, oCHoaaHHbIH Ha MeTo,ne BTOpblX paa-.
HocTeit- ABToMaTHKa H TeJIeMeXaHHKa, 1977, N2 4, c. 114-120.
9. Bop 0 A K H H A. M. H ,np. BbI,neJIeHHe KOMnJIeKCOB npoeKTHbIx
paoOT KaK OO'beKTOB aBTOMaTH3al{HH npol{ecca npoeKTHpoBaHHSI
npOMbIlIIJIeHH blX npoHsBoACTB. - ABToMaTHKa H TeJIeMeXaHHKa,
1978, N2 5, c. 97-105.
10. Bop 0 A K H H 1I. M. AJlropHTMbl nocTpoeHHSI arperHpoBaHHOH
CTPYKTYPbl pa30HeHHH JlH B3BelIIeHHbIX rpa<poB.- B KH.: Mo,ne-
JIH arperHpoBaHHH COUHaJlbHo..3KoHoMHtJecKoM HH<popMaUHH. Ho-
BOCHOHpCK, H3,n. M3HOnn CO AH CCCP, 1978, c. 71-81.
11. Bop 0 ,n K H H <1>. M., M H P K H H B. r. 3MnHpHqeCKOe onHcaHHe
B COl{HOJlOrHH.- B KH.: MaTeMaTHKa H COUHOJlOrHH. HOBocHoHpcK,
H3 . H3HOnn CO AH CCCP, 1972, c. 3-41.
12. B p. a B e p M a H 3. M. MeTo bI 9KCTpeMaJlbHOH rpynnHpoBKH na-
paMeTpoB H 3a,naQa BbI,neJIeHHH cyrnecTBeHHbIx <l>aKTopoB.- ABTo-
MaTHKa H TeJIeMeXaHHKa, 1970, N2 1, c. 123-132.
13. B P a B e p M a H 3. M. H ,np. ,[{HarOHaJlH3aUHH MaTpHUbI CBH3H H
BbISlBJIeHHe CKpblTbIX <paKTopOB.- B KH.: npOOJleMbI pacllIHpe-
HHH B03MO>KHOCTeH aBTOMaTOB. M.. H3A. Mny AH CCCP, 1971,
- BbIn. 1, -c. 42-79.
14. B P a B e p M a H 3. M. HAP. lIHHrBHcTHtJecKHH no,nxo,n K 3a,naQe
oopa60TKH ()OJlbllIHX MaCCHBOB HH<popMaUHH.- A"BTOMaTHKa H Te-
JIeMeXaHHKa, 1974, N2 11, c. 73-88.
15. B Y p K 0 B B. H., f p 0 n n e H B. O. Pa3pe3bI B CHJIbHO CBH-
saHHbIX rpa<pax H nOTeHUHaJlbI nepecTaHoBoK. - ABToMaTHKa H
TeJIeMeXaaHKa. 1972, N2 6, c. 111-119.
308
-
a
,......\
16. Ban H H K B. H.t LJ e p B 0 H e H K H C A. 51. TeopHSI p8cnosHa-
BaHHH oopa30B. M.t HaYKa, 1974. 354 c.
17. Bop 0 H H H 10. A. BBe,neHHe Mep Cxo,nCTBa H CBH3H ,ltJlH peWeHHJI
reoJIoro-re0<pH3HtJecKHx 3a,natJ. ,[lAH CCCP, 1971, T. 199, N2 5,
c.l011-1014.
18. B bI X 8 H ,n Y JI. 1(. HeKOTopbIe np06J1eMbI TeopHH aHaJIH3a ,naH-
HbIX. - Tpy,nbI TaJlJlHHCKOro nOJIHTeXHHqeCKOrO HH-Ta, N2 366,
1974, c. 3-15.
19. r a B p H JI e 11 10. H. COl(H8J1hHO-9KOHOMHQeCKOe nJIaHHpOBaHHe.
CHCTeMl?! H Mo,neJlH. M., 3KoHoMHKa, 1974,175 c.
20. r 0 JI hAM a H P. C. HeKOTopbIe BonpOCbl nOCTpoeHHSI JIOrHtJeCKHX
Mo,neJIeH. - B KH.: YnpaBJleHHe H HH<popMaUHH. J10rHtJeCKHe Me-
TO,nbI HarH038. BJla,nHBOCTOK, H3 . MAny ,[lBH AH CCCP,
1974, BbIn. 14, c. 103-111.
21. r Y C e H HOB r. A. MccJle,ltoBaHHe K8tJeCTBa KJIaCCHcpHKal(HH
B CJIYtJae HeKOJlHtJeCTBeHHbIX npH3HaKOB.- B KH.: npHKJIa,ltHOH
MHorOMepHblH CTaTHCTHtJeCKHH aHaJIH3. M., HaYKa, 1978, c.61-68.
22. r Y T 11 H H 11. B., M Y tJ H H K H. B. CTaTHCTHlIeCKHH aHaJlH3
KaqeCTBeHHblX ,ltaHHbIX.- B KH.: OnbIT HCnOJIb30BaHHSI 8BM B
HCCJIe,nOBaHHHX KYJIbTypbI. M., H3,n. MocKoBcKoro rocy apcTBeH-
Horo HHCTHTYTa KYJIbTypbI, 1976, c. 261-287.
23. .n}K H H H K. J10rHKa B CTaTHCTHKe. M., CT8THcTHKa, 1973. 128 c.
24. ;a M H T P H e B A. H., )I( y p a B JI e B 10. H., K p e H-
,It e JI e B <t>. n. 0 MaTeMaTHtJeCKHX npHHI1Hnax KJIaCCH HKanHH
npe,nMeTOB H HBJleHHH. - ,[lHcKpeTHbIH aHaJIH3. HOBOCHOHpCK,
H3 . HM CO AH CCCP, 1966, BbIn. 7, c. 3-12.
25. .n 0 p 0 <p e 10 K A. A. AJlrOpHTMbI ooYtJeHHH MallIHH pacn03HaBaHHIO
oopa30B oe3 yqHTeJlSl, OCHOBaHHbIe Ha MeTo,ne nOTeHI1HaJlbHblX «l>YHK-
UHH.- ABToMaTHKa H TeJleMeXaHHKa, 1966, N2 10, c. 78-87.
26. ,l:{ 0 p 0 cp e 10 K A. A. AJirOpHTMbI aBTOMaTHtJeCKOH KJIaCcHcpHKanHH
(0030p JIHTepaTypbI).- ABToMaTHKa H TeJIeMeXaHHKa, 1971,
N2 12, c. 61-85.
27. .n p 0 6 blIII e B 10. n., n y x 0 B B. B. AnnpoKcHManHSI HetJeT-
KHX oTHoIIIeHHH.- B KH.: 8MnHpHtJecKoe npe cKa3aHHe H pac-
n03HaBaHHe oopa30B (BbIqHCJlHTeJlbHble CFCTeMbI, BbIn. 76). Ho-
BOCHOHpCK, H3,n. 11M CO AH CCCP, 1978, c. 75-82.
28. ,l:{ 10 P a H H., O,n e JI JI n. I(JlaCTepHbIH aHaJIH3. M., CTaTHcTH-
Ka, 1977. 128 c.
29. E JI H C e e B a H. H., P Y K a B H III H H K 0 B B. O. fpynnHpoB-
Ka, KOppeJlHUHH, pacno3HaBaHHe oopa30B. M., CTaTHcTHKa, 1977.
, 144 c.
30. )I( Y K 0 B C K a SI B. M., M y tJ H H K 11. B. <PaKTopHblH aHa-
JlH3 B COUHaJIbHO-3KOHOMHqeCKHX HCCJle,nOBaHHHX. M., CTaTHcTHKa,
1976. 151 c.
31. )K Y P a B JI e B 10. H., I< a M H JI 0 B M. M., T y JI H r a-
HOB Ill. E. AJlropHTMbl BbIqHCJIeHHH oueHOK H HX npHMeHeHHe.
TaIIIKeHT, <t>aH, 1974.
32. 3 a r 0 p Y if K 0 H. r. MeTo bI pacno3HaBaHHH H HX npHMeHeHHe.
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\ 33. 3 a r 0 p y if K 0 H. r. 3MnHpHqeCKOe npe,ncKa3aHHe. HOBocH-
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34. 3 a r 0 p y H K 0 H. r. , 3 a C JI a B C K a H T. H. (peA.) n pH Me-
HeHHe MeTO OB pacn03HaBaHHH 00pa30B B COI1HaJlbHbIX HCCJIe,nOBa-
HHHX. HOBOCH()HpCK, HaYKa, 1968. 195 c.
309
35. 3 a .n e JI. nOHHTHe JIHHrBHCTHtJeCKOH nepeMeHHOH H ee npHMeHe-
HHe K npHHHTHIO npHOJlH:>KeHHbIX pellIeHHiL M., MHp, 1976, 165 c.
36. 3 a C JI a B C K a SIT. 11. H ,np. Pa3BHTHe CeJlbCKHX nOCeJleHHH.
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37. 3 bI K 0 B A. A. TeopHH KOHeqHbIX rpacpoB. HOBOCHOHPCK, HaYKa t
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38. I< a M e H C K H H B. C. MeTo,nbI H Mo,neJlH MHoroMepHoro IIIKaJlHpO-
BaHHH (0630p).- ABToMaTHKa H TeJleMeXaHHKa, 1977, N2 8, c.
118-156.
39. I< a p 3 a HOB A. B. 8KOHOMHbIH aJlrOpHTM HaXO}K,neHHH 6HKOM-
nOHeHT rpacpa. - Tpy,nbI TpeTbeH 3HMHeH llIKOJlbI no MaTeMaTHtJec-
KOMY nporpaMMHpOBaHHIO H CMe:>KHbIM BonpocaM, M., H3,n. 3MH
AH CCCP, 1970, BbIn. 2, c. 70-74.
40. I( a p H a n P. <I>HJlOcocpcKHe OCHOBaHHH cpH3HKH. M., nporpecc,
1971. 380 c.
..i1. I< e M e H H ,l{:>K. f., C H e JI JI ,l{:>K. JI. KHoepHeTHtJecKoe MO,ne-
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42. K e H ,n a JI JI M. ,[l>K., C T h 10 apT A. CTaTHCTHtJeCKHe BbIBO bI H
CBH3H M., HaYKa, 1973. 900 c.
43. K e H a JI JI M. ,l:{:>K., C T b 10 apT A. MHorOMepHbIH CTa-
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44. R e H A 9 JI M. ,[{:>K. PaHrOBbIe KOppeJlHl(HH. M., CTaTHcTHKa,
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45. K 0 B a JI e p tJ Y K B. 51. 0 UHKJIHtJeCKHX lIIKaJIax.- BbltJHCJIH-
TeJlbHbIe CHCTeMbI. HOBOCHOHpCK H3,n. MM CO AH CCCP, 1975,
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46. K paM e p f. MaTeMaTHqeCKHe MeTO bI CTaTHCTHKH. M., MHp,
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47. K Y JI boa K C. TeopHH HHcpopMaUHH H CTaTHCTHKa. M., HaYKa,
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48. I( Y n e p III T 0 x B. JI. 0 pOJlH nopora cymecTBeHHOCTH HH,nH-
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49. K Y n e p III T 0 x B. JI., M H P K H H B. f. Ynop otJeHHe B3aH-
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50. K Y n e p III T 0 x B. JI., M H P K H H B. f., T P 0 ct> HMO B B. A.
CYMMa BHYTpeHHHX CBH3eH KaK nOKa3aTeJlb KaqeCTBa KJlaCCHcpHKa.
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MeTO,n HaHMeHbIIIHX KBa,npaTOB B aHaJlH3e KaqeCTBeHHbIX npH3Ha-
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52. I( Y n e p III T 0 x B. JI. , n 0 JI H m Y K JI. H. , T P 0 ct> H-
MOB B. A. I(03<P<PHl(HeHTbI KOppeJl l(HH CBH3eH Me:>K,ny OO'beK-
TaMH. - B KH.: ,Mo,neJlH arperHpOBaHHH COl(HaJlbHO-3KOHOMHtJe-
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53. K y n e p III T 0 x B. JI., T P 0 cp HMO B B. A., AJlrOpHTM aHa-
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54. JI a H K aCT e p n. TeopHH MaTpHU. M., HaYKa, 1978, c.
280.
310
--.1
r
55. JI 6 0 B f. C., ROT 10 K 0 B B. H., M a H 0 X H H A. H.
00 O,nHOM anropHTMe pacn03HaBaHHH B npOCTpaHCTBe pa3HOTHn-
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H3,n. 11M CO AH CCCP, 1973, BbIn. 55, c. 108-110.
56. JI 6 0 B r. C. , K 0 T 10 K 0 B B. 11., M a III a p 0 B 10. n. MeTO,lt
o()HapY}I(eHHH JIOrHqeCKHX saKoHoMepHocTeH Ha 9MnHpHtJecKHx
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11M CO AH CCCP, c. 29-41.
57. JI () 0 B f. C. JIorHqeCKHe CPYHKnHH B 3a atJax 9MnHpHtJecKoro npo-
rHo3HpoBaHHH. - B KH.: aMnl"pHqeCKOe npe,lJ.CKa3aHHe H pacno3-
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58. JI H T B a K B. f. 00 ynopH OqeHHH OO'beKTOB no npeAnOqTeHH-
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59. JI blOC P., f a JI a H T e p E. nCHXo<pH3HQeCKHe llIKaJIbI. - B KH.:
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60. M a a M H r H A. B. npOBepKa 3HaqHMOCTH cTaTHCTHqeCKOH rH-
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KynHocTH pas()HeHHH Ha KJlaCCbI.- B KH.: CTaTHcTHtJecKHe MeTOAbI
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62. M a JI e H BOa. CTaTHCTHqeCKHe MeTo,nbl 3KOHOMeTpHH. M., CTa-
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64. M e JI JI e p <1>., K a n e K K H B. POJlb 9HTponHH B HOMHHaJIhHOH
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65. M H P K H H B. f., M y tJ H H K M. B. feoMeTpHtJecKaSI HHTepnpe-
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66. M H P K H H B. f. AHaJIH3 KatJeCTBeHHblX npH3HaKoB. M., CTa-
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67. M H P K H H B. f. Mo,neJlHpOBaHHe MHoroMepHoH COIJ.HaJlbHO-
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68. M H P K H H B. r. H p. IlIKaJlbl ynopH OQeHHH.- B KH.:
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69. M H P K H.H B. f., I( Y n e p III T 0 x B. JI., T P 0 cp HMO B B. A.
o MO eJlHX arperHpOB HHH ,nHCKpeTHOH HHcpopMaUHH.- B KH.:
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70. M H P K H H B. f., P 0 H H C. H. fpacpbI H reHbI. M., HaYKa,
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71. M H P K H H B. f., P 0 C T 0 B U e B n. C. MeTo,n BbI,lteJIeHHH
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HaHOnn CO AH CCCP, 1978, c. 107-112.
311
72. M H P K H H B. f., q e p H bI A 1I. B. 06 HSMepeHHH t>.JIHSOCTH
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74. M Y q H H K H. B. AHaJIH3 SKcnepHMeHTaJIhHbIX rpacpoB.- ABTO-
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75. 0 P JI 0 B A. M. npHKJIa,ltHaSl TeopHSI HSMepeHHH.- B KH.:
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76. 0 C H n 0 B r. B., A HAP e e B 3. n., MeToAbI HSMepeHHSI B COnHO-
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77. n a T p Y r H H 10. A. O() H3MepeHHH KatIeCTBeHHbIX npH3HaKoB.-
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78. n JI 0 T K H H A. A. YCTOHQHBOCTb pa30HeHHSf KaK KpHTepHH on-
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79. non H In Y K 1I. H. BbI()OP0tIHbIH aHaJlHS cBH3eH Me)KAY o()'heK-
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B KH.: Mo,neJIH arperHpoBaHHH COI1HaJIbHO-.9KOHOMHQeCKOH HHcpOp-
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80. n 0 JI H Y K JI. H. nOCJIe,noBaTeJIbHbIH aJirOpHTM CTPYKTYPHOH
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81. n 0 JI T e p 0 B H q B. M. OllTHMaJIhHble pa30HeHHH npoH3Bo,nCT-
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82. n 0 JI Sf K JI. X. HeKoTopbIe COOTHOlIleHHSI Me1K,ny MeTOAOM pac-
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84. n H p H a K. 06 OnTHMaJIhHOM lIIKaJIHpOBaHHH HOMHHaJlbHbIX
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85. P a B K H H 10. C., nTHUbI JIeCHOH SOHbI npHo6bH. HOBOCH()HpCK,
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86. P a H C K a SI H.H., T e p e x H H A. T., <t> P e H K e JI h A. A.
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87. P 8 0 C. P. JIHHeHHbIe CTaTHCTHlleCKHe MeToAbI H HX npHMeHeHHe.
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88. P e H x a p T P., III M e H K a JI B. HeKoTopbIe TeOpeTHqeCKHe
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90. P 0 C T 0 B U e B n. C. K aHaJIHSY CXO CTBa Mep CBSISH Me1K Y
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91. P 0 C T 0 B U e B n. C. AJiropHTM nocTpoeHHSI THnOJIOrHH ,ltJISI
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B KH.: MO,lteJlH arperHpoBaHHSI COUHaJIbHO-9KoHoMHtJecKoH HH-
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98. T P 0 cp HMO B B. A.t MO eJIH KaqeCTBeHHoro cpaKTopHoro aHa-
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-
...-
- -
CONTENTS
Preface ..
Introduction
. . . . . . . . . . . . . .
.... ..... ....
5
8
Chapter 1.
MODELS for REPRESENTATION of QUALI-
TATIVE VARIABLES
1.1. Variables as mappings. . . . . . . . . . . . . . . . 16
1.2. Terms of relations and matrices .. 28
1.3. Classifications and distributions. . . . . . 39
. . . .
Chapter 2.
ANALYSIS of DISCRETE DISTRIBUTIONS
2. I. Lo g Ii n ea r mod e Is. . . . . . . . . . . . . . . . . . . 57
2.2. Association coefficients for cross-classifications. . 89
Chapter 3.
ANALYSIS of OBJECT-ATTRIBUTE TAB-
LES
3.1. Linear approximation of boolean tables. . . . " . . . . 112
3.2. Regression by nominal variables. . . . .. . . . . 131
Chapter 4. r
TERMS of INTE RRELATION MATRICES in
DATA ANALYSIS
4.1. Factor construction problems ............ 172
4.2. Problems of description and evaluation of association
among qualitative variables ............... 205
Chapter 5.
ANALYSIS of ASSOCIATIONS STRUCTURE
AMONG OBJECTS
5.1. Classification of a set of interrelated objects. . . . . . 230
5.2. Problems of structure revealing .. . 259
Conclusion. .. ............. 293
Supplement.
ELEMENTS of LINEAR ALGEBRA.
References . .
... .........
. 296
. . 308
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