ON THE SIGN CHANGES IN THE CENTROID METHOD

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TRU Mathematics 12（1976〕

ON THE S I GN CHANGES IN THE CENTROID METHOD

Yosiro皿B舩A and Kazuo FUKUTOMI （Received April 23， 1976〕ノ 1． Introduction． The factor extraction is the fundamental procedure in factor analysis， and㎜y methods have been proposed． Among th㎝the centroid method was developed at the early stage of the study of factor analysis． Neverthless， few atte皿pts have been made to exanline its adequacy． In the centroid method， the sign dhanges have to be performed at each stage of factor extractions， and the rule for choice of them seems to be to a certain extent ar｝）itraxγ． Several rules for dhoice of sign（；hanges have been proposed （Thurstone ［4］，P． 151 ），but it is diffi印1t to understand their substantial meanings． The aim of this paper is to make the character of sign change clear and to give the more adequate rule for the dhoice of them・ We consider the following factor analysis mode1， x＝Lf≠θ．もロ where x andθare rand㎝vectors of p c（皿ponents， fエs a rεmd㎝vector of k components， L is aρ＞k matrix〔P＞k）・We assume that E［f］＝e， E［e］＝ 0， E［fe，］＝ 0， E［ff，］＝工 and E［eeっ is diagona1． Let O ＝ E［竺’］ and V＝E［ee「］， then we have c ＝ LL，「← v ． ¶enote that L is the colmnon factor皿atrix and the above decompositi㎝ of．’C is unique（see Tumura G FUkutorni［5］）．The ith diagonal e1㎝ent of L・…the c・一・・i・y・f・・、，・㎝・t・d by珍 The factor extractlon， the est皿atlon of the factor space，1s based on the salT4）1e covariance matrix A． First， the estimate m of the dimension of the factor space is deteエmined by a certain procedure， and then m factors are extracted fTom A． Next， the initial apProx迦tions of テ_{communalities are guessed and they are adjusted by the iteratlon process．} Here we also used the iterative procedure血the c㎝troid methos． On the k 61

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62．

_{it・rati㎝PT㏄ess，・㎝・v・1・・5．・f e・tim・ted・騨1itie・．・砿坤es bec㎝・} to be invariant〔see R」ktitomi［2］）． This prdb1㎝1shall be dealed with in 』the next section and a theor㎝shall be presented．． Secti°n 3 are d叩t己t・・giv・．th・．qUpirica1 result・by artifigial experlments on p（rpulatlo 1 covariance matrices， and tke prdblem㎝the choice of sign changes is discussed． It has been reported that， in the case k：＝1，Thurstone’s cc肋Plete method could obtain the・true solution，工ndependently of the initial apprσx迦ations［2］． In this paper， we shall deal with the case必＝2．「 2・The invariance of estimates on iteration process’of the centroid method． Now we sha11 give the iteratiye fqnmala ’of the．c飽troid hethod． Let

l蕊㌶「蕊゜㌶1三：1：。：瓢蒜、鵠留．bl認1遮1，

・ωb・t一・・iX， deri・・d f・㎝鋤・sub・ti加・血・弓伍ラ血・・i・・d・・gO皿・ cells． ” The iterative fomiula’of the centroid method can be written ’「

lllre ，、、， a，。rd2「露ζ二1ノ蕊1き：ll蕊ll；霊一．

i・th・ρ・。・mat・返砲・se抽・・1・m i・av・ct・T rep・6・enting th・・ig・・Ch・ng・ for extraction of theゴth factor at the nth stage of iteration proce§s （s6e Lawley 【3］）．．．．』 After・a・number・f iterative p・・ced碇es》々ωmay b。 invari。nt， i．。．

・ω一・佃刀一…・』・…t・・b・， Q， Now we・h…9・・r・』中．・㎝・ernin・

the invariance on the iteration process．肥O肥M．S・PP・呂θ』伽け雇sゼ9η・乃卿・mカr4四r西1．ノi・fix・d after カ72θ 」VカバZ 乞力e㌘ation．（i）ヱア．there exist合 α栩order veeto？C that Qe＝θ．． then E… ．・．・．・．：・「．一・〒多．．．・弓（冗＋η＝乃…ω rn≧〃）」⑳nt ・．・〔ii）ガ輪’θθ鋤・・rp．・・d・・ V・・妨・e伽蝿一㌢蝿ゴ「・、㎝4β ave・θonstαηtsμηθg祐αZ カ0 0 九 then ，

・2弓⑭一β2㌍η一・2・…ω一β2弓ω ・・≧・，一

』・ピ・．・ρ・漉垣厄力吻・r◆ 1

1’ROOF…SUb・tituting the c・nditi・n．（i〕．・f Th・・r㎝垣t・①，w・ hqv・＿ n

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、 ON［［HE SIGN（：HANGES IN TMil．（a㎜OID ME［［H｛）D 63 2rn＋ヱノ h i ・ Using the conditiOn hξ「n＋v

。軸「％佛ωQ戸Q’Aω。．。

一一z ．。’θ’Aω。．．一一阜（i）once more， we have

＝醐ω。．ヨz（n）．

一砂一z t 1n the case of （ii〕，we have 。2h8（n≠2ノ．β2hそ「・子1／ z ∂ −r㏄．。β。．ノ’Aωg触「n）Q戸Q’Aω〔。。．．β。、．）．

−t−∂ ．づ“∂

By the salne way aS the case （羊〕，w『have 。2ヵ4「n＋o一β2頚η≠カー。蜘ωr泌，一．β。、，） t 9 − −z 二9 」。2。！Aω。．．β2ぬ「nノ。． −t −t −∂ −J 1 ＝α2hぞrnノ．＿ β3hそ白n）．・ z ． ∂ ・一、 We remark that the conditions of Theor㎝． depend on only the 「・・tru。t。。e。fQ， b。t。。t。。Aイ”）．・t i，．ca11。d th。伽α鋤力。α。。． In the iteration method， the assuranc百of mique convergence is essentia1， but if the invariant case occurs， we could not expect to obtain the mique solution． For exaiXple， consider the case that p＝5 andη2＝2． The e matrix always satisfies one of． the conditions of Theorem， and then the unique solution can not be obtained， unless Q keeps unfixed to the last． 3．Empirical results of the artificial experi皿ents on population COVarlapce matr1CeS． In this section， three sets「of results of the artificial experiments are presented． We rpade the covarianCe matrices O fr㎝the population ． factor matrices Z｝s given in Table l such that ． 0＝LL，チV。 where V＝J− diag立辺，．

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64 ．Y．㎜㎜K・㎜

The initial apProximations used for the experiments were the largest value in eaCh colum of offdiag C， unities， or somet㎞es a玲itraワ典es 91ven・ We used the co呵）1ete methOd， or its modification （described later） for the choice of sign changes and according to the purpose of experiments， the fixed Q matrices were given in advance． The iteration process were continued mtil the differences in all the estimated co］nmumalites between 加・・uccessi・・iterati…r㎝・i・・ess th。n・3・10−6 i皿。。9。i加d。．［he computations were done on either IBM 370 0r FACOM 230−28． A．Experiment 1． First， the experiments were carried out on the covariance matri）（derived fr㎝L， in oTder to ex輌ne whether we could obtain the true solution independently of initial apProximations by the coJnplete method． In these experiments， if we extracted the true number of factors， i．e． m＝2， the true solutions were always obtained and if we extracted the excess factors， say栩＝3二’the true solutions were also obtained， accompanying one specific factor． The results in the latter case are shown in Table 2． Two different initial approximations − unity and the largest value of offdiagonals −were used． In one case， the specific factor was added to Variable 6 and in the other case， to Variable 4． ’ihrough this experi皿ent， no invaria皿t case occurred． B． Experiment 2． Next， a series of eXperi皿ellts were carried out on the salne covariance皿atrix by usi㎎the sign change matrix G’一｛−1−ll｝｛｛1｛｝・ This Q satisfies the condition 〔ii） of Theorem， i．e．1et c，＝r1／2デ1／2ノ， then鍵三2チZ2・ By the use of the above Q， the difference between the estimated com㎜ality of Variable l and that of Variable 2 should be invariant on the iteration process．・・t・、2−・iω一弓「°」・S・vera・麺・・i・・ap…x迦・ti・n・…h・h・d・fferen・ v・1u・・f dヱ2 Were used・Th・re・ult・血Tab1・3・h・W th・t・11 the c・nverg・d communalities but ones mder co賦rai皿t seem to be close to the true values． Consider a measure of difference betUeen the true cmmalities and the converged ones such that

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ON THE SIGN〔HANGES IN THE 〔llrm｛〕ID MET［HOD 65 TABI正i l Popuユation Factor Matrices

variable

1

L

2 3 五 4 五 5 五

12345678

．8 ．7 ．6 ．5 ．3 ．3 ．4 −．2

03556564

● ● ● ■ ● ● ● ．．6 ．0 ．4−．2 ．3 ．3 ．2−．4 ．1−．5 ．5 ．7 ．3−．6 ．4 ．5 ．8 ．7 ．6 −．4 ．2 −．1 ．7 ．2 一．1 ．2 ．3 ．7 ．6 ．5 −．3 ．8

∨5425643

． ● ● ・ ■ ・ ● ・ _●

75443564

■ ● ● ● ● ■ ● 一．5

64﹁一ム﹁7ダ7’6

■ ● ● ● ● ● ・ TABLB 2 Results of函）eri皿ent l

Converged〔㎞㎜alities Using Different Initia1 Approximations

、 Three Factors Extracted， Involving One Specific Factor

variable

True

Co㎜．

Initial

Approx．

Converged

Co㎜．

Initial

Approx．

Converged

Co㎜．

12345678

．64 ．58 ．61 ．50 ．45 ．34 ．52 ．20 ロコ −占¶←−占−﹁⊥1占−¶⊥ _．6400 ．5800 ．6100 ．5000 ．4500 ．4487＊．5200 ．2000 ．56 ．57 ．57 ．55 ．48 ．43 ．54 ．18 ．6400 ．5800 ．6100 ．5108★ ．4500 ．3400 ．5200 ．2000 ＊ This value involves the variance based on the specific factor．

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66 _{Y．TUMJRA AND K． FUKU［1］（測旺} ． d『一Σr弓一考元2。．i≧3

曲ere￡…ar・the・・nv・rg・d・・皿㎜・ities・・． ’

、，t。1

_{ﾇ。蒜e。1；llrl、ご：還芸、鑑票1、三隠。罐麗゜ns}

ご蒜。leごh㌫翌霊二「．1？、6；1−°’5 d m fact’

Suppose thatρ is fixed in （1〕， then the right side of （1） can be w。itten in the functi。n。f h4（n）。 t

『考「刷一魂ω・hl「n）・…・・；ωか

。。一蒜lie「潔彦｝1㍗’°n°f△2轡》・t二2vllue°f

ρ △ん…佃ヱノーΣ ゴL−1 （llfliz）・・iω・…h2「n）1・ 9

1：：：鐘灘翻i も霊㌧

lA， j・1 i・nece・sary・f°r c・nverg・nce・t・th・t・u…1uti…nd if X、 i・ close−to Unity， the．convergence may be quite slow． R・mark r lf㌧＝1・th・皿i・lu…1uti㎝…1d・・t b・・カt・in・d・lt can be proved that the condition of［lheorem is sufficient for that the B matrix has the eigenvalues equal to one， but not necessary − for instance， in the case that p＝8 andη2＝4， the B matrix corresponding to the following Qmatrix has eigenvalues eqUal to one，、ρ’＝

1111

一 ⇔

1111

↑ ↑

1111

︹一

1111

一

1111

↓

1111

F

1111⊥

一 C．EコCI）eriment 3． In order to examine the relation between the magnitude ・f㌧and th・n血b・r・f iterati・n・ t・c・nv・・g・nce・we caπi・d・ut th・ experiments on the covariance matricesl derived’fr㎝five factor matrices in Table 1， using various Q matrices． The results are shown in Table 4， illustrating clear relati㎝ship．［［1ie correlation coefficient between the

n皿ber・f iterati・n・and th・v・1…f1”1−Uf・r 48 P・i・・in Tab1・4is

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釘

．s合白§詔冨石句お。8肩ぢ宕ωΦ貞

ON THE SI《刑《］H低NGES IN THE（正蜘01D METHOD ト・。も．昌芯．ひNも． NH⑰O． ●Hも． 9・O． H∩・O．口も． nHひ一．㊨ON吟．寸O蕊． 8ぼ．ザ @ P ︵ooO寸 ●●O●． Hoo∩ひ． HO∩∩．ひ・oひH．岩借． HO曽柏． HO頃㊨． oo_ﾐひ㊨． ●ひO●． ●φ●oo．吟∩●∩．ひ曽ON．ま叶町トか∩∩． Nひ享〆トOO吟． OHH●． N崎ひト． HめR． OO．畏．㌫訳．吟Oひひ．㊨含㊨．吟パO吟．ひ∩H●．トOoo●．ト8寸．トのON．なH吟． oo垂ﾐ． ooﾐ寸㊨． “OO吟．㊨パH●． oo 盾盾ﾐ吟． oo 盾盾_ﾐ吟． NトひH．

89．

dO㊨φ． 9吟㊨． ●ひひ㊨．台O●． oo 盾潤_“． oo 潤怐． ogH．

．

マ∩． ●．含． O●． N崎． Noo． o吟H．吟．㊨両．崎㊨．含． “． oo㊨． ◎ooO．目

t吟㊨吟●トoo

‘

8

ー N．1

§︼6﹄婁

W椙口■儒パ ℃ 8一婦﹀

鷺君曇

口§︼6包身

冨

?` ρ§冶苔目’§°§君§同

二§Φ桐苔嘗88柏担昌ー8芭和§8

N甘Φ肩輪8封桐o日ヨ㊤躍め田曽

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68 _{Y．㎜RAAND K．㎜}

TABI湿4 Results of IIXperiment 3

Nロ血）er of Iterations USi㎎Vaエious Sign Cha㎎e M包trices

…1脇豊ct°「

Sign Change

Matrix

Max． Eigenvalue Value of Number ofof B MatriX lQ t1孟 ’Q l Iterations

1 Ll

2 3 4 5 6 7 8 9 10

11

一十十十十十十一十十十十一一一一

十十十十十十十十

MC4

_−●■一

_¥十十十

十十十十．十十十一一一一

¥十十十十

十十十十十一十一一一

¥十十十十十

一十十十十十十一十十十十一一十一十十十十十十十一一一

¥十十十十十

十十十十十十十十

一一一

¥十一十十

十十十十十十十十

〇

_{一一一十十十十十}

十十十十十十十十

一一

¥一十十十十

十十十十十一十一

一十十十十十十十

十十十十十十十十

一一十十十十一十

．6448 ．6830 ．6901 ．7379 ．7606 ．7787 ．7874 ．8144 ．8675 ．8824 ．9766 41．47 78．15 79．57 69．22 34．57 92．16 55．95 78．15 66．59 64．00 46．24

17

24 24 30 32 34 37 40 56

61

249

12 L2

13

14

15 十十十十十十十十

〇

一十一十十一十一

十十十一一十一十

十十十十十一十十

十十十十一十一十

十十一十十一十十

十十十十一十一十

十十十十十一十一

．6229 ．6276 ．6352 ．6849 83．17 57．76 46．24 51．84 23 23 25 26

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ON THE SIGN（HANGES IN THE（㎜〔）ID MEIHOD 69 T《BI正 4 （ Contillued ）

・b・鷲i。豊ct°「5認。㍗ge

Max． Eigenvaユue Value of ’N輌r of ofβMatrix lQ，1工’Ql Itdrations

16，五2

17 18 19 20 21 22 23

十十十十一十十十

十十一十十一十十

十十十十一十一十

十十十÷十一十十

十十十一一十一十

十十十十十十十十

十十十十一十一十

十十一十十一十一十’

¥十十十十十十

十十一十十一十一

十十十十一十十十

十十十十十一十一

十十一十十一十十

十十十十一十十十

十十十十十十十十

一一一

¥十一十十

．6856 ．．

V151

．7512 ．7806 〔7909 ．7998 ．8227 ．9072 74．56 46．24 66．59 51．84 78．85 57．15 44．62 27．04 28 27 34 36 43 45 50 86 24

_@L3

25 26 ・27 28 29 30

十十十●十一十十

一十十十十十●十

十十十十十十十．十

c

十十十一一一十一

十十十一十■十一

十十十十十十一十

十十十一十十十十

一一

¥十十十一十

十十十一十十十十

一十十十十十一十

十十十十十十十十

十十十一十一十一

十十十一十一十十

十十十十十十一十

．5838 ．5840 ．5955 ．6325 ．6748 ．7478 ．7659 170．04 212．58 158．26 170．56 159．77 154．26 139．71 19 18 19 23 25 26 36

31 L4

32

一十十十十十十十

十一一一十十十十

、，‘．1

十十十十十十十十

r＋＋＋＋一一一一．8290 ．8813 58．98 81．00 55 69 ，

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冊

Y．㎜AAND K． FUKUTua

TABLE 4 （C（mtinued）、

恥鷲畿ご瓢㍗e．竪㌶謬ue悟舗

㎞ber of

It6fations

33 L

4

鮪 35 36 37 38 39 40 ．” “＋＋’{‘i’＋：，・ナ＋“㍗’ 一．十十十一十一一

一十十十十十一十

十、十十一十十1十十

十十十十十十十十

一．一十十十十●一

一十十十十十十十

：＋1＋一一十十十、十

十十十十十十十十

c

．一斗＋＋＋十一：一

十十十十十十十十

一十一十十十’一一

十十十十十十十十

や十十十十一一十

十十十十十十十十

一’＋＋＋＋rr＋一．8974 ．9026 ．9282 ．9299 ．9419

．9M4

．9565 ・82．81 ． 104．04 ．69．89 ’・V6．04 ．、P06・92 71．．57 く 73．27 ●9891 ’ 51．55 79

93

109

126

129 156 167

610

，

41 P5

42

43

44

45

46 47 ’48 49 ，‘ L 』一十十十十十十十、・ ¥’十十十一一十一十十十十十十十十

MC4

t’一一＋＋＋一、＋

一十十十十十十十

・十・十十一十一，十一

十十十十十十十十

一十一一十十一十

十十十十十十十十

r−一一十十十一．十

十十十十十十十十

丁＋一＋■一’＋．一＋

十十十十十十十十

MC3

ド’・’ C十一十十十・一十．十十十十十十十十一●一

¥十十十十

十十十十十十十十

c

−十十十十十十十

r9648 ．9658 ．9710’ ．9769 ．9855 ．9865 ．9873 ．9923 1．0000 も 41．99 41．99 36．72 42．77 ．43．56 ・43．56 57．15 58．06 79．92 225 236 Z7S’ 337

490

528

532

804

’sLC

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ON THE SIGN（HANGES IN THE（醐〔）ID M匡冊α〕 0．9964（No．49羽as the invariant case and㎝itted）・ 1￡t the nuロiber of pairs with diffbrent sign eadh other of eight rσws in Q be denoted bγv● It seems that ill tlle case v＝4 （「p／2）， the magnitude ・f㌔i・怜th・・甑11』・e case・・1・r v・2 i・th・㎞・ri・nt case・In廿亘s case， by adj usting the sign Clunges so as to increase v up to 4， we could not only avoid the卵ariant case， but make convergenCe speedy． In Table 4，』σmeans theρdeteエrmined by the c㎝troid method and MC30r MC4 means that t｝鴇ρ dete曲ed by the complete皿ethod was modified so that v＝3 0r v＝4 respectively． 4．Discussi㎝． rh加ura 3θカαZ．16］ have given theα呵pirical Tesults of experiments on the POPulation covariance matrices by using the maxi皿ロ皿 1il［el血ood method， suggesting tllat if the true㎜由eτof factors is㎞㎜，ぴet随solution could be obtained ind⑤end㎝tly of initlal approxlmati（ms． But this can not be proved strictly． In this stu己y， we hほd the si皿ilar 臼㎎〕irica1 τesults』by using the c㎝troid method， if the invariant case can be avoided， though it can be proved excqpt the case k：＝1・ Anderson 8 RUbin［1】stated曲t the c㎝troid method am）rox迦ate the P・in・ip・1・㎝P。・㎝t鵬麺・㎞1・t th・ゴth・・1um。f e b・・㎞・te鋤9ゴ and let the residual matrix after extracti㎝theゴー2th factor be denoted

by％・佃奄・・㎝・肋血Ψest頃・h・血。i㏄。f・ign change f°τ曲5血

血・t・rs・ロ・噸蜘南お・・㎎・a・・P。・sib1㌔皿・P・・cedn「e卿m典te

盤芸三，°；ぱe＝11㌶1’4°：誌竃蕊蕊t〒、・u ’、I

correSP（mding to theρby the co姐Plete met婦are ratheτ large・ The results血Table 4 shcms that the criterion of the c（mplete皿ethod is not best and it needs the above modificati（m freqUently． In the case k≧3， the procedure for mOdification is very c（卿1icate and． its effect is still questionable．．［1］工2］【3］［4］［5］工6】㎜〔氾S Anderson，T．W．6R山bin，H．：Statistical inference in factor analysis・ Proceedi㎎s of 3rd Berkeley Sy皿甲osium， V，（1956），111−150．，TRU Math．9，_{FUkut㎝i，K．：On the adeqμacy of factor extractions} （1973〕， 119−136． Lawley，D．N．：Astatistical exanlination of the c㎝troid method， Proc．Roy．Soc．1］din． A64，（1955）， 175−189．． Thurstone，L．L．：M㎞1tiple factor analysis， Univ・of Chicago PTess，（1947）・． Tumura，Y．ξFU㎞tc皿i，K．：On the identificati㎝in factoT analysls， Rep．Stat．App1・Res．，JUSE， 15，（1968）， 6−11． ’i、mura，Y．， Fukutomi，1（． G Asoo，Y．：On the uぼ亘que convergence of iterative procedures in factor anlysis， TRU Math・4，（1969），52−59・ 71

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72

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ON THE SIGN CHANGES IN THE CENTROID METHOD