Title Multivariate Statistical Analysis of Speech Sounds(Dissertation_全文 )

Author(s) Tabata, Koh-ichi

Citation Kyoto University (京都大学)

Issue Date 1973-11-24

URL https://doi.org/10.14989/doctor.r2408

Right

Type Thesis or Dissertation

Textversion author

Kyoto University

」χ４ｆ

MULTIVARIATE　STATISTICAL　ANALYSIS

　　　　　　　　　　　　　　OF

　　　　　　　　　SPEECH　SOUNDS

　　KOH-ICHI　TABATA

Department　of　Information Science

　　　　　　Kyoto　University

　　　　　　　March,1973

-･

'1゛"

ヒ1‐乙

MULTIVARIATESTATISTICALANALYSIS

OF

SPEECHSOUNDS

DOC

1973

10

電気系

KOH-ICHITABATA

｀DepartmentofInformationScience

KyotoUniversity

March,1973

PREFACE

VJhenweexplainphenomenainthenatureand,furtﾆher,tryto

establishlawswhichgovernthem,itisnecessaryasoneofthe

guidingprinciplestoworkaccordingtoapattern

1-

observation,

qualitativeexplanation,modelestablishmentandthenquantitative

explanation.However,itisinsomeaspectsdoubtﾆfulwhetherwe

canbringlighttocharacteristicsofspeechsoundonlybythat

patﾆtﾆern.Ideallywemayhavetowaituntil"Advancedinformation

science"becomesdeveloped.Underpresentconditionsitseemsthatﾆ

wehavenochoicebuttoproceedwitﾆhthestudyofspeechsounds

accordingtotheguidingprinciplementionedabove｡

Now,thephysicalcharacteristicsofvoiceofeachindividual

aredifferentfromthatofeveryother,andwecannotexpectthat

thephysicalaspectoftwoutterancesofthesamewordbyonespeaker

isidenticalevenunderthesamecircumstances｡

Therefore,ifwearegoingtofindacertaincharacteristic,

wehavetoconfirmitqualitative:lyorquantﾆitativelyonthebasis

ofsufficientdatainordertoassertthatthediscovery1Snever

accidental.

Thisresearchwasperformedfromsuchastandpoint

lappliedmultivariatestatisticalanalysistospeechspectraobtain-

edbyphysicalobservation,andthenclarifyvariouscharacteristics

ofspeechsoundonthestatisticalfoundation｡

Tobebrief,themethodofmultivariatestatisticalanalysis

isconcernedwithmulti-dimensionalvectorwhosecomponentsarenotﾆ

independentofeachother.Ifweassumethatcomponentsofspeech

spectrumarethoseofmulti-dimensionalvector,itﾆbecomesauseful

meansforspeechanalysis.Althoughthevalueofmultivariate

analysisofvarianceaboutpracticalapplicationshasbeenneglected

despit二ethemathematicalresults.ofitstheory,Itwasveryuseful

inthisstudv.Thisfactrevealsthatthepracticalvaluewasfor

thefirsttimecertifiedhere.Further,thisstudycontributesto

thetheoryofmultivariatestatisticalanalysisaswellasits

practica:Lusage｡

Characteristicsofspeechsoundsclearedherearemainlyconcern-

ingco-articulation,individualdifferencesofspeakersandcharactﾆer-

isticsofconsonants,eachofwhichhasbeenunknownuptonowand

isinterestingtospeechscience｡

Formantfrequencyhastraditionallybeenusedinthiskindof

research.Thisthesisgivesusacluewhichwillreleaseusfrom

thetraditionalresearchmethod,whichhasgovernedwitﾆhaniron

handandconfineduswithinsomelimits｡

Atﾆtheendofpreface,wewillinsist,"Thisstudyhasbeen

developedasaconsequenceofseriousref:Lectiononthetraditional

wayofresearchinwhichresearcherswereinahurrytotryspeech

recognitionwithouteχaminingsufficientlybasicacoustic-character-

isticssuchasco-articulation,individualdifferencesofspeakers,

etc..

Weusedacomputertﾆhroughouttheprocess fromthe

collectionofdataofvoicetotheinvestigationofanalvzedresults.

Ifwedidnothaveacomputer,thisstudycouldnotbeaccom-

plished."

-2-

Koh-ichiTabata

March,1973

PREFACE

GLOSSARYOF

CHAPTER:L

2.3

CHAPTER

2.6

3

3.1

3.2

3

3

3

3

4

4

3

4

5

6

1

2

CONTENTS

1<£>

288

111

20

20

20

21

22

25

27

27

29

33

34

37

37

3

4

4

4

8

0

4

5

46

50

50

工nt

2.2

2.2.1

2.2.2

2.2.3

2.2.4

2.4.1

2.4.2

CHAPTER2MultivariateStatisticalAnaly

2.1Introduct

ofVari

MultivariateDistribut

Introduct

NormalMultivariateDistribution･････････････丿･･･････

ProbabilityEllipsoid‥‥‥‥

ConcentrationEllipso

SamplesfromtheMultinormalPopulation‥‥‥‥‥

MultivariateLinearRegression‥‥‥ ● ● ● ● ● ●

2｡4MultivariateAnalysisofVariance‥･‥‥‥･‥‥･‥‥‥‥･‥･

TheLikelihoodRatio

2.4.3MethodofMultivariateAnalysisofVariance…30

2.5TwoProposalsforMultivariateStatisticalAnalysis

Principal-ComponentAnaly

Facilit:iesforSpeechAnalysisandSpeechSpectrum

NonreverberantRoom,Microphone,Taperecorderand

Pre-emphasis

Observat

1/4-OctaveFilterAnalyze

SpeechSpectra…‥

On-line,RealTimeDataInputSystﾆem

SegmentationofConnectedSpeechbyVisual

CHAPTER4AnalysisofJapanesevcvUtterancesofFiveMale

工ntroduct

EstablishmentofExperimentalObjectsandFactors51

-3-

4

4

3

4

4.5

4

4

4

4

CHAPTER5

CHAPTER

6

7

8

9

lCN

inin

5.3

5.4

5.5

5.6

5.7

6

6.1

6.2

6

6

3

4

6.6

700

66

LinearModelandMultivariateAnalysisofVariance53

RelationbetweenNormalizedCriterionり'and

DiscriminationScore …………………………56

GeometﾆricRepresentationofMulti-factor

Distribut

]:ntroduct

Conc:Lus

Introd

5

6

6

6

9

3

6

8

70

72

74

75

n4″り

詣88

89

93

95

96

102

104

105

108

111

ComparisonWitﾆhPrincipal-ComponentAnalysis･･････

Discussionaboutﾆ

AnalysisofVar

4.10Conclusion

AnalysisbyFormantFrequency

RegressionEstimatebyUsingFinalVowelsand

AnalysisofJapanesecvcvUtterances‥‥‥‥‥‥‥,‥‥‥‥‥74

MultivariatﾆeAnalysisofVarianceforFour-Factor

DesignwitﾆhRepeatedMeasurements

EstablishmentofExperimentalObjectsandFactors82

ResultofAnalysisofVariancefor･Four-FactorDes

Examinationof

EvaluationofInteract

AnalysisofJapanesecvandvvUtterances,･･････‥‥‥‥95

MultivariateAnalysisofVariancesforDivided-Type

Two-FactorDesign‥‥‥‥

Two-FactorDes

Divided-Type

Discuss

Conclus

ResultofAnalysisofVarianceforDivided-Type

6.5RelationbetweenNormalizedCriterionり'and

DiscriminationScore

GeometricRepresentationofAnalysisofVariancefor

-4-

CHAPTER7

7.1

7.2

7.3

7.4

CHAPTER8

ACKNOWLEDG

REF

Three-DimensionalRepresentationofJapanesePhonemes113

工nt113

114

116

118

122

128

131

135

137

138

139

146

147

156

165

cvcvUtterancesandtheirSpe

ConsonantSpe

ﾌ.6Conclus

Conclusi

ClassificationfromtﾆheViewpointﾆofMannerand

PlaceofArticulat

ﾌ.5ExpressioninThree-DimensionalSpace

OTHERREFERENCES

THEAUTHOR'SPAPERSRELATEDTOTH工STHES工S

APPENDIXATheAsymptoticDistributionoftheLikelihoodRatio

TestﾆCriterioninMultivariateAnalisisofVariance

forFour-FactorDesignwithSingleObServation.‥‥‥141

APPEND工XBTheIntersectionProducedbytheEllipsoidandthe

APPEND工XC

APPEND]:XD

APPEND]:XE

APPEND工XF

StraightLineorthe

VarianceofFactor

Measur

TheVectorSpaceNormalizedbytheResidualR.‥‥‥149

TheKolmogorov-SmirnovOne-SampleTest.･･････････････････150

TheAsymptoticDistributionoftheLikelihoodRatio

TestCriterioninMultivariateAnalysisofVariance

forFour-FactorDesignwithRepeatedMeasurementsin

whichA11InteractﾆionsAreDiSregarded.‥‥‥‥‥‥‥‥‥152

APPENDIXGTheAsymptoticDistributionoftheLikelihoodRatﾆioTest

CriterioninMultivariateAnalysisofVariancefor

Divided-typeTwo-FactorDesignwithRepeated

APPENDIXHMatrixofSumsofSquaresandCrossProducts

inSubs

-5-

/1

/rL

ＣV

VV

CV

vcv

cvcv

＼ (t)

Speechsound

･●

●

●

●

●

●●･一＠一

･一

＠一

Matrix

A(pxq)

A(pxq+1)

A゛

A‾1

国

tr(A)

rank(A)

lp

j'

･す

一1

･･

●

●

●

●

●一●一

●●

･

●

GLOSSARYOFSYIIBOLS

Phonemicsymbol.

Phoneticsymbol.

Consonant.

Vowel.

Phonemesequence

Phonemesequence

Phonemesequence

一･

一一＠一

Phonemesequence:

Consonant-Vowel.

Vowel-Vowel.

Vowel-Consonant-Vowel.

ConsonantﾆｰVowe:L-Consonant

-Vowel.

Thei-thoutputof20-channelfilteranalyzer.

pxqmatrixA.

px(q+l)matrixA.

ThetransposeofmatrixA.

TheinverseofmatrixA.'‘

ThedeterminantofmatrixA.

ThetraceofmatrixA.

TherankofmatrixA.

p-dimensionalunitmatrix.

LetR(pxp)beasjnnmetricpositivedefinitematrix,

thentheorthogona:LmatrixTeχistssuchthat

λ0

F~LL

T=R

-6-

T

j

p

oλ

f±

0

.X°

Z

へy

ぷ

一Se･

｀゛here入1>･･･,λ (>O)aretheeigenvaluesofR-･･〃

Define

汗

as

p

follows:

jo呵

瓜o

lけ

まiSalsoasymmetricpositivedefinitematrix,and

lによ.診.

R‾亀(Ry)‾1

Zeromatrix

Randomvector,probabilityanddistribution

ｅ一

Ｓ●

ｅ●

Pr{E}

pHeIh}

e(x)

μA

N(m,�)

N(μ,Λ)

0

X2

･S･Ｓ

X(IXp)=(xi,X2,x;:p-dimensionalrandomvector

,･(pxl)りFI

謡光二竃二謡二三鶯三二ﾆ

●Ｓ

一ｅ

･一

ExpectedvalueofZ

Expectedvalueofぷ

･●一

Covariancematrixof､z:

μ=e(x)

Λ=e(x-μ)'(r-μ)

Theunivariatenormaldistributionwithexpected

valuemandvariancea^.

Thep-dimensionalnormaldistributionwith

expectedvalueμandcovariancematriχΛ.

Zerovector｡

Chi-squaredistribution｡

-7-

Ap

W(Λ,P,r)

W

X

e(X)

χ.

Zi●

χ..

Q

一一

Ｓ･

･Ｓ

Chi-squaredistribut:ionwithp?egreesoffreedom.

Wishartdistribution.TheSumiEllji゛Uいwhere

p-dimensionalvectﾆorui'sareindependently

distributedaccordingtoN(0,Λ),hasW(A,p,r)

distribution. riscalledthedegreesoffreedom.

Theleft-handsidevariateisdistributedaccording

totheright-handsidedistribution.

Forexample, z~N(0,Λ)

Samplesfromthepopulation

ｅ一

一一

･･

一一

Ｓ一

●･

]』

一一

Iχ･~一χ ●

一幽

χ一＠･

●

ip

X np

j

●

● Thedatamatrix.

wherexlj°゛'゛xnaretherealizatﾆionsofp-dimensional

randomvector.

Expectedvalueofx一･ e(x)=

』

ThemeanvectorofXi(i=l,･‥,n)

Themeanvectorofぶり(i=l,--゜゛a･≪＼

e(X,)

･I･

eぐ4)

一･ Z=

j

―一n

j°:L,‥

1

correspondingtothesuffixj:Xj.= -b

ぶ

,b)

bΣ一一一

j

Themeanvectorof･≪^ij(i°19¨゜9a;j°1゛'゜゜゛

1abz..=冨ふぶり

りj

SS

jb

Totalvariance.(Matrixofsumsofsquaresand

crossproducts.)Forexample,

ab

1=1j

EI(゛ij

゛‥)(゛ij-゛･.)

-8-

●

S

〈μ〈Λ〈B

《〈B

〈〈A

〈〈μ

A

AB

A.

Q

Oj

"Qi

R

~R

･9

wherezij(i°111°･,a;卜1,･･･,b)arethesamplevectors.

Thesamplecovariancematrix.Forexample,

S°

lab

゛..)'(x,,-゛¨)'

wherezij(i°1゛'゜゜9a;j°19･･,b)arethesamplevector.

ThemaximumlikelihoodeStﾆimates

el

Ｓ●●Ｓle

Themaximumlikelihoodestimateofexpectedvalueμ.

ThemaximumlikelihoodestimateofcovariancematrixΛ.

ThemaximumlikelihoodestimateofparametermatrixB.

Themaximumlikelihoodestimateofμ,A,Bunderthe

nullhypothesisH.

Analysisofvariance

Se

S･S9･･

●一

一･

一一

･Ｓ

FactorA.

工nteractionfactorbetweenfactorAandfactorB.

Thei-thtreatment(level)offactorA.

Totalvariance.(Matrixofsumsofsquaresand

crossproducts.)

Thei-thvarianceobtainedbythebreakdownofQ.

Thei-thvariancenormalizedbytheresidualR.

-Q-°R焉iR斗9wherenisthesamplesize.

Residualvariance.(Matrixofsumsofsquaresand

crossproducts.)

n

士RR‾1°I,wherenisthesamplesize.

-9-

μ

αi

'"ij

"ijk

BＺ

Ha

L(∂)

(Mω

LL

入

l£J

り

りi

β

j

･一

＠一

一･

ゆ･

Ｓ●

一●

Thegenerallevelvector;

ThemaineffectvectorscorrespondingtoAi,Bj

treatments(levels),respectively;

Theinteractionvectorcorrespondi叱to(Ai･Bj)

treatment,and;

Theresidualvector･;

inthefo:Llowinglinearmodel:

*ijk°μ十αi十βj十rk十りjk

Parametermatrix(unknown).

Designmatrixofexperiment,orMatrixofknown

vectorszainregressiontheory･

Testofhypothesis

一＠

●●

一一

一●

ｅ･

II

●一

･一

AnullhypothesisforfactorA.

Alikelihoodfunctionwithm-dimensiona1

parameterd.

Ln=mL(0

Lω゜;;2:L(∂),whereωiSthesubspaceofΩ

入=-μ:LikelihoodratiocritﾆerionfortheLΩ

hypothesisHO:∂.丘ωagainstthealternativeH:∂e,Ω-ω

ZZJ･こ 鴎帚:Likelihoodratiocriterionwhichisa

monotonicfunctﾆionofλ,wherenisthesamplesize･

LikelihoodratiocriterionbyBox,whichisdistributed

asaChi-squaredvariateasthesamplesizetend

toinfinity･

NormalizedcriterionofV:

V'=

. ¶

(Valueof1Ssignificantlevel0tIχ I､testcorre-

spondingtothedegreesoffreedomofiり)

-10-

兌1

聡十聡

そ

{刈 H}

･e一

Others

･･一･

聡=rank(Qi)

刻十ii2=n-rank(R),wherenisthesamplesize.

isamemberof-

Asetofz'swhichsatﾆisfytheconditionH

-11

CHAPTER1

]:ntroduction

Thespeechsoundisoneoftherepresentativemeansofinfor-

mationcommunication.Amongthevariouskindsofinformationconvey-

edbyit,thereareindividualityandemotionalexpression,andso

on.Themostnotableofallisphonemicinformation.Itseemsquite

asimp:Lephenomenonthatwecancommunicatevocallywitheachother

becausewehavebeenaccustomedtotalkingand:Listeningsincechild-

hood.Physically,however,itisamiracle.Thestﾆructureofthe

vocalorgansofeachspeakerisdistinctﾆfromthatofothers,and

nobodycanuttertheverysamevoiceintﾆhephysicalsense.Further-

more,thecharacteristicsofacousticsystemorelectro-acoustic

systemwhichinterlinksthespeakersandtheauditorynervesof

listnersareremarkablydifferentaccordingtotheirenvironment･

Consideringthatthevoicesofvariouspersonsareidentifiedexactly

thesamephonemedespitetheirbeingphysica:Llymu:Ltlfarious,(that

is,theyarealldifferentphysically),itSe゛゛lsimpossibleto

explainvoiceinformationinaphysicalfield.Thismayleadusto

believethatitisgovernedbyanabsolutelydifferentfield.If

weassumethisdifferentfieldtobethe"Informationscientific

field,"wemayhavetowaituntil"Advancedinformationscience"

becomesdevelopedinordertofullycomprehendit｡

Inthepresentsituation,thereisnootheraltﾆernativebutthe

physicalobservationofphenomenainordertoexplainandtoestablish

thelawswhichgovernthem.Itisorthodox,butwouldbehopeful

totrytoapproachtheessenceoftheproblembyrepeatingthe

cycle-observation,qua:Litativeexplanation.modelestablishment

andthenquantitativeexplanation｡

Atfirst.letamulti-dimensionalvectorcorrespondtospectrum,

-12-

atdesiredtimepoint,obtainedfromspeechsignalsthroughphysical

observation.工notherwords,frequencycomponentsofspectrumare

consideredascomponentsofmulti-dimensionalvector.Postulating

varioustypesofmodelsbasedonthespectrumvectorsthusobtained.

wecantrytoexplainthecharacteristicsofvoicequalitativelyand

quantitatively｡

Thecharacteristicsofsoundhaveessentiallyanaspectrelated

toprobabilitysinceavoiceappearsinmultifariouswaysgoverned

bychanceasdescribedabove.Therefore,tﾆhemodelisessentially

aprobabilityone｡

Deviationsbetweentheactualdataandthispresumedmodelare

duetoprobabilitychancesaswellasimperfectionsinthemodel｡

I'Thatismostﾆimportanthereisthattointroducethemodel

doesn'tmeanImmediatelytoconfirmthecharacteristicswhichweare

attemptingtoexplainwithit.Toconfirmthem,athoroughexamina-

tﾆionisneeded.Unlessthequantity,whichrepresentsacertﾆain

characteristicexplainedbytﾆhemodel,issufficientlylargerthan

theresidualerror,wecannotprovetheexistﾆenceofthecharacter-

istic.Besides,wecannotﾆassertthevalidityofthepresumed

modelitself.Toguarantﾆeeit,theintroductionofstatistical

procedurestestofsignificancybecomesapressingproblem｡

Mu:Ltivariatestatistﾆicalanalysiswasadoptedforthisstudyas

ausefullanalyticalmethodwhichsatisfiesthisrequirement.

Briefly,multivariatestatisticalanalysis1Sastatisticalmethod

dealingwithmulti-dimensionalvectorwhosecomponentﾆSarenot

independentofeachother.Wecanschemeconsiderablyfreelythe

formationsofmodelsbasedonthevector,andgivetﾆhemmathematical

basesnecessaryforthetestofsignificancy｡

Thespeechspectrumisoneofimportantmeansforexpressing

speechsignals.Almostphysicalinformationofspeechsignals1S

reflectedinit.Ascharacteristicsofthefilteranalyzer(whichare

representedby"Windowfunction"1naviewpointof"Short-time

Fourieranalysis")bearontheobtainedspectrumcomponents,it1Sa

-13-

problemwhatcharacteristicsshouldbeadopted.Nevertheless,

spectrapreservemuchmorecontentsofprimarysignalsthanformant

frequenciesd0.Wecananalyze,infact,onlyvowelsbyusingformant

frequencies,whilebyusingspectrawecananalyzebothvowelsand

consonants｡

Evidently,therespectivefrequencycomponentsofthespeech

spectrumarenotindependentofeachother.Therefore,thecomponents

ofthemulti-dimensionalvector,definedbytheabovecorrespondence,

arealsonotindependent.Todealwithaprobabilityeventwhich

hasthissortofvectorasarealization,multivariatestatistical

analysisisasuitablemethod｡

Thoughmultivariateanalysisofvarianceappliedtﾆothisstudy

hasbeendevelopedinafieldofmultivariatestatisticalanalysis,

ithasbeenapttobeundervaluedaboutpracticalapplicationsin

spiteofthemathematicalresultsofitstheory｡

However,thismethodwasrecentlyfoundtobeextremelyusefu:L,

sincewecouldclarifyconsiderably,bythismethod,patternsofCO-

articulationandofdifferencesbetweenindividua:Lspeakerswhich

werenotunderstoodquantitativelysofar.Thismeansthatthe

practicalworthofthemethodofmultivariateanalysisofvariance

was,forthefirsttime,confirmedhere｡

Wedesignedvariousmodelsandexperimentstoclarifyproperties

ofspeechsounds.Tointroduceandproveexpressionsofthemodels

ofmultivariateanalysisofvariance,nolesslaborwasspent｡

Meanwhile,twomatters,intheoryaswellas'inapplication,

werefoundtocontributetomultivariateanalysisofvariance:

(1)Notioaofdirection,besidesnotionofamountindicatedgenerally

bytheanalysisofvariance,wasnecessarytounderstandtheeffects

oftherespectivefactorsofanalysisofvariance,andoneofits

techniquesofexpressionwasdescribed;

(2)amodel0ftheanalysisofvariancefor"Divided-Type"modelwas

proposed.Thisisanup-to-datemodelinwhichwedevelopananalysis

ofvariancefortwo-factordesignbydividingoneoftwofactorsinto

14-

morethantwoportions.

Thismodelwasintroducedtodoresearchonpatternsofsome

tﾆypesofco-articulations,andthenformulated.

Weshouldnowexplainthestandpointofthisstudyfroma

viewpointofspeechscience.

Amongthestatisticalcharacteristicscomparativelywellstudied

uptothistime,therearedistributionofspeechamplitudelevel,

distributionoflong-timespectrum,on-offrateofspeech,and

fundamentﾆalfrequency.Butmostofthesecharacteristicsareconcern-

edwitﾆhwhatiscalled"acousticcharacteristics."Therearesome

statisticaldataonthelinguisticcontentsofspeechconcerning

vowels.butfewconcernedwithconsonants.Especially,quantitative

explanationsonco-articulatﾆionarequitefew.Althoughalmosta11

attemptstoexplainitareconductedbyformantfrequencies,those

whichhavesufficientstatistﾆicalfoundationsarefew.

ThereportofBroadandFertig(,1)StateSthatthepatternsof

influencesofvariouskindsofinitial0rfinalconsonantsonthe

vowelregionofcvcsyllables(sequencesofconsonant-vowel-consonant:

Onlythevowe:L/I/wasused)arequantitativelydisplayedbyananaly-

sisofvariance(univariate).Ithasastatisticalground,sothat

itﾆisquiteinteresting.

Thefirst,secondandthirdformantsarestillseparatelydealt

withasunivariate,respectively,intheirreport.However,since

thereisnoguaranteethattheseformantsareIndependentofeach

other,andsinceitisnotclearhowmanypartsofsoundinformation

theformantsshare,theirreportisnotsufficienttoexplainCO-

articu:Lation.

Wecarefullystudiedtheprocessoftheirresearch,andthen

plannedfurtherresearchwhichmustgobeyondit.

Wedirectlywrestlewiththeproblemofco-articulationInclud-

ingconsonantsaswe1:Lasvowels,usingspeechspectrumwhichisa

simpleexpressionofphysicalspeechinformationasmentionedabove.

-15-

And,wealsotriedtoevaluatetﾆheindividualdifferencescausedby

differentﾆspeakers･

Wegainedtheotherresultsconcerningtherelativespatial

distributionofconsonants

whichhasfourfactors

andfina:Lvowel

three-dimensionalrepresentationof

-16-

consonants.Eventhoughspatialrepresentationbasedonactual

speechsoundhadearlybeentriedonvowels,thespatialrepresent-

ationofconsonantshadbeencomposedbyphonetician'sinsightso

far.Physicalandanalyticexpressionsbasedonactﾆualspeech

soundare,forthefirsttime,triedhere.

Wewillstatetheframeworkofthisthesisattheendofthis

chapter｡

工nChapter2,wegiveasummaryofmultivariatestatistical

analysisusedinthisthesis,andanexplanationofthemultivariate

analysisofvariance.Wealsotouchalittleontwotheoretical

proposalsforthemultivariateanalysisofvariance.

Acomputerwasusedinthisstudythroughouttheprocess

fromtheCo:Llectionofdataofspeechsoundtotheinvestigationof

theanalyzedresu:Lts.Thisstudycouldnotbeaccomplishedwithout

thecomputer.InChapter3,wetouchonthemeansofspeechana:lysis

includingthiscomputersystem,onspeechspectraandonthedefinition

ofvectorscorrespondingtospectra.Alsowedescribethemethod

ofsegmentationadoptedhere｡

InChapter4,weperformmultivariatestatisticalanalysis,

speaker,initia:Lvowel,consonant(nasa:L)

onavcvword(vowel-consonant-vowel)spokenby

fiveadultmales,andmeasuretheamountandthedirectﾆionofthe

co-articulationsandthespeaker-effect.Bymultivariateregression

theoryweinvestigatethecorrelationbetweenfina:Lvowelandeach

spectrumsectionofword.Principal-componentanalysiswasalso

performedandcomparedwithmu:Ltivariateanalysisofvariance･

Thedifferencebetweenspectralinformationandformantinformation

isa:Lsoconsidered..･.･･..･

Chapter5dealswithanalysisofvarianceforfour-factordesign

withrepeatedmeasurementsonwords―C V,C2V2‘―spokenbyanadult

male.AssigningfourfactorstoCI゛V1゛C2andV2irespectively,we

observeconsiderablyindetﾆailtheirinteractionsaswellastheir

maineffects.Particularly,wemeasurethedepthofco-articulations

howfartheinf:Luenceofonephonemeextendstotheothersin

phonemicsequence｡

Chapter6zTocomparepatternsofco-artIculationswitheach

otherintwotypesofphonemicsequences-C-V2(consonant-vowel)

andv1-V2(vowel-vowel);wescrutinizethemagnitudeofthe

influenceoftheprecedingconsonantCandvowelV10nthefollowing

vowelV2.Forthatpurpose,weextendthemodel0fusualanalysis

ofvariancefortwo-factordesign｡

Chapter7:Weanalyzenineconsonants,andobserverelatﾆions

oftheirrelativedistﾆributioninatﾆhreedimensionalspace.Asa

result,thephonemesarenearlyrepresentedwithatriangularprism.

whichisstatistica:Llyguaranteedfromvariousaspects.Theexperi-

mentofthischapterwasperformedtogetherwiththeexperimental

schemeofChapter5｡

SofarastheAppendicesareconcerned,wedescribemathematical

deductionsandproofswhichwillberedundantinthetext.

-17-

CHAPTER2

MultivariateStatistica:LAnalysis

2.1工ntroduction

Multivariatestatisticalanalysisdealswithatheoryof

statisticsinwhich;

(:L)Whenweconsiderpcharacteristics(p>l)concernedwithan

indiv工duala,theirrespectivemeasuredvaluesareexpressedbythe

componentsofvector

z(z(1xp)゜(xαls'¨9xαp) (2.1)

whichisarealizationofap-dimensionalrandomvariable.

(2)Theobservedvaluesonnindividuals{cc=l.･･･,n)aresummarized

inthedatamatrix

Xlp

^np

｢--'―一一

(2.2)

whoserespectiverowsareindependentlydistﾆributedaccordingtoa

p-dimensionalprobabilitydistribution.

Notethat゛CZandxg(a.6°19"･,n,a≠B)ai°eIndependentofeach

other.whilethecomponentsof･>^ax(xiandx(xj(恥j°19°･･.p.i≠j)

arenotindependentofeachother.ThevariatesXrtl,･･･9x(zpare

dependentamongthemselvessothatwecannotsplitoffoneormore

fromtheothersandconsideritbyitself.Thevariatesmustbe

consideredtogether.:

-18-

Anotﾆherimportantcharacteristicofmultivariateanalysis

arisesfromadivergenceofinterestﾆbetweenmathematicianand

statistician.Thenaturalinclinationofthemathematicianis

towardsgeneralization.Givehimaresultforonevariateandhe

inquiresaftﾆertheresultfortwo;givehimthatandheinquires

aftertheresultforp.Thestatistician,ontheotherhand,is

continuallystrugglingtoreducethedimensionsofhisproblem.In

multivariateanalysisheusuallyhasembarrassingprofusionsof

variatesandhisobjectistomakepassmallashecan.(Hestill

prefersnaslargeaspossible.)Discriminantanalysis,forexample.

triestoreducetheproblemofdistinguishingbetweenmultivariate

populationstothescaleofasinglevariate.(2)

Now,thissortofdifferenceofstandpointisobligedtocause

adifferenceofevaluationofcertaintheories,themostintensively

influencedoneofwhichwasthemultivariateanalysisofvariance｡

Thetheorywasestablishedbyextendingthestatisticalconcept

andmethodoftheunivariatecasetothemultivariatecasedirectly･

Unfortunately,remarkableapplicationofthemu:Ltivariateanalysis

ofvariancetopracticalcasehasnotbeenfounddespitethefact

thatitstheorywasoneofmagnificentresultsinmathematicS53)

TherewereevensomeauthoritieswhodenieditspracticaluSagj4)On

thecontﾆrary,inthisstudy,thismethodwasveryusefulfordiscover-

ingseveralnewfactsonspeechsound.Thisrevealsthattheprac-

ticalvalueofmultlvariateanalysisofvariancewasconfirmedsub-

stantlallyforthefirsttimehere.Asimilaritymaybepointedout

abouttheregressiontheoryinthecasewheredependentvariatesare

multivariates｡

0ntheotherhand,multivariateanalysishasaweakpoint.

Namely,itisdifficulttoproceedwithsystematicanalysisexceptin

thecasewherewecanassumeanormaldistributiontobethefunda-

mentaldistribution.But,weareveryfortunateinthisstudybecause

1tispossibletoassumethenormaldistribution,aswecanunderstand

itinthesectiondescribinganalysisofresidual.Respective

-19-

componentsofprobabilityvectorinthisstudydonotrepresent

numerizedvaluesofvarioustypesofconceptsasinpsycho:Logy,but

physicalvalueswhichhavethesamedimension,thatis,thefrequency

componentsofspectrumdistribution｡

Wewillgiveanoutlineofmultivariatedistributionfirst,in

thischapter,followedbytheregressiontheory,analysisofvariance.

andprincipal-componentanalysis.Wewilltouchalittleontwo

proposalsformultivarlateanalysisofvarianceinatheoretical

aspect.

2｡2MultivariateDistribution(5)(6)

2.2.1NormalMultivariateDistribution

Ifthedensityfunctionofap-dimensionalrandomvector.x;(1xp)

is

(27r")"^IAけ 収-μ)が(x-U)'} (2.4)

whereμisagivenp-dimensionalvector,andAisagivenpxp

symmetricpositivedefinitematrix,xissaidtobedistributed

normally,ordistributedaccordingtothep-dimensionalnormal

distributionN(μ,Λ).Then,μiscalledtheexpectedvalueofthe

vectorX,andΛiscalledthecovariancematriχ.

μ=6(.χ;)(2.5)

A=e(x-μ)'(x-μ).(2.6)

2.2.2ProbabilityEllipsoid

lf硝九分)isdistributedaccordingtoN(μ!A),(‘l;-μ.)rl(゛‾μ)

20-

hasaχ^-distribution(Chi-squareddistribution)withpdegreesof

freedom.

Letχ;((x)bethenumbersuchthat

Pr(χ;≧χ7)(o,)}=(19

whereχ;hasaχ^-distributionwithpdegreesoffreedom.

Thus,

Pr{(jc-μ)Λ‾1(.x;-μ)'jχ^(a)}=1-a.

Inthep-dlmenslonalspaceofZ,

(z-μ)Λ‾1(z-μ)'£Xp(o･)

(2.7)

(2.8)

(2.9)

representsthesurfaceandtheinteriorofae1:Lipsoid■whosecenter

isμ.TheshapeoftheellipsoiddependsonΛ‾1,andthesizeon

χp(a)forgivenΛ-1Theinterioroftheellipsoid(2.9)is

consideredtocontain(:L-a)×100%ofthepopulationandisca:Lied

the"Probabilltﾆye:L:Lipsoid."

lfΛ‾1°Ip,forexample,(2.8)saysthattheprobabilityIs

1べxthatthedistancebetweenZandμislessthanぶ<?,(a),whereIp

isthep-dimenslonalunitmatrix.

2.2.3ConcentrationEllipsoid

LetμandΛ=e(Z-μ)゛(Z-μ)betheexpectedvalueofthep-

dimensionalprobabilityvectorj:(1Xp)andthecovariancematrix.

respectively･

Then,

(x-μ)Λ‾1(z-μ)'=P+2 (2.10)

1scalledtheconcentrationellipsoidcorrespondingtoX,whereA1S

-21-

thepXpsymmetricpositiv!E!definitematrix.Thevolumeofthe

concentrationellipsoidisequa:Lto

pp

(p+2)T伊

Γ(号+1)

(2.11)

Nowweconsidernewp-dimensionalprobabilityvectory,whosedensity

is

g(μ)=

r(-|-p+i)

固{‘(r+2)号訪,

0,

(μ-μ)Λ‾1(μ-μ)゛≦p十2,

otherwise.(2.12)

Then,itcanbeprovedthate(y)=μ,e(μ-μ戸(ZI-μ)=A.

Therandomvectoryisuniformlydistributedintheinterior(andon

thesurface)oftheconcentrationellipsoid,whilethedensityofμI

isOattheoutsideoftheellipsoid｡

Theconcentratﾆione1:Lipsoidisconsideredtobethegeometric

expressionofdistributionofzwithexpectedvectorμandcovariance

matrixA.

2.2.4SamplesfromtheMultinomialPopulation

Let^1.‥･,Xjjberealizationsofp-dimenslonalrandomvariable

distributedaccordingtothemultinormaldistributionN(μ,Λ).

Thesesamplevaluescanbesummarizedinthedatamatrix

ゴ~

｢万

X11°゜゜Xlp

●●●●●●●

xnl°●●

xnp

]

(2.上3)

ThedensityfunctionofthejointdistributionofX,゜¨゛znis

22-

f(ZI･¨゜･zn:μ･/1)゛

一

一

1

(2π)}叩|/1け1 呻緋j

l(゛i‾″)j‾1(゛i‾″)勺

(2.14)

whereM'(pxn)゜(μ≒¨',μ･)andtrindicatesthetraceofmatrix,.

becauseXj,･･･ﾀxnaremutuallyindependent･

Thefunction(2.14)isalsoconsideredtobethelikelihoodof

observatﾆion(2.13),fromwhichwecanconcludethatthemaximum

likelihoodestimateofthemeanvectorμisμ=Z..

Wherex.゜圭£Ziisthesamplemeanvector,andthatthemaximijinni=IA1

likelihoodestimateofthecovariancematrixAisΛ=-Qwheren

n

q=Σ

1こ1

(zi-z.)゛(zi-z.)

isthematrixofthesumsofsquaresandcrossproducts

ThematrixQcanbewritteninthefollowingform;

where

Then,

Q=Y'Y

Y= [ﾚﾝ]

rank(Q)=rank(Y'Y)=rank(Y)=inin(n-l,p),

because

n

ふ(Xj-X゜)≒R‘｀;i‾゜;'゜0°

Distributionofthesamplemeanvector

(2.15)

(2.16)

(2.17)

LetZ.=否iﾐyl°xibethesamplemeanvectorofnindependent:

observatﾇionsZifromthemultinormalpopulationN(μ,Λ).

-23-

(2.20)Q:(pXp)=X'AjX

Thenx.isalsoamultinormalrandomvariablewithparametersμ

and(l/n)A.

e shartdistribution

ThematrixQdefinedby(2.15)canbewrittenasthesumsofthe

productsofn-1independentp-dimensionalrandomvectorswiththe

conmondistributionN(0,Λ).Ingenera]L,anysymnietﾆricpositive

definitematrixQofquadraticandbilinearforms,whichcanbe

transformedtothesum

且‘'?“i (2.18)

whosep-dimensinalvectors"iareindependentlydistributedaccording

tothedistribut:ionN(0,Λ),issaidtohavetheWishartdistribution.

ThedensityfunctionofQis‥

W(Λ,P,r)=

2‾

1

‘

圈1(r-p-1)

{‾iﾄA‘IQ}･(Tり),(2'1?)

｢nr[-|-(r+i-i)]

exp{緋tΓΛ

ifQisapositivedefinitematrix.

皿j‾p(p‾1)IAI

Cochran'sTheorem

Forthesamplematrix(2.15),r=n-1.Weshallrefertoras

thedegrees-of-freedomparameteroftheWishartdistribution;together

withAitspecifiestheformofthedensityfunction｡

Ifp°1andΛ=1,theWishartdensitybecomesthatoftheChi-

squareddistributionwithrdegreesoffreedom.Wishart-dlstributed

matricespossessmanyofthepropertiesofChi-squaredvarlates.

SupposezαisdistributedaccordingtoN(O,A)independentlyof

゛6((゛邦)゛SupposethematrixAj(nxn)usedintheforming

-24-

(i°1,‥･,m)

isofrank■■i.andsuppose

x゛x゛Q1十゜¨゛十%i (2.21)

wherex゛(pxn)゜(ぷll゛‘",x').(Note:工n°AI十'゜'十Am')

ThenanecessaryandsufficientconditiontﾆhatQi(i=l,･･･,in)is

distﾆributedas

n

ﾑ "a '"a' (2.22)

whereuばisdistributedaccordingtoN(0,Λ)independentlyofaβ

((がB),andQiisdistributedindepe゛dentlyofQ-j(i≠j)isthat

Corollary

工fr

ri+r2+'‘゜+rm°n°(2.23)

ilp,thenQiisdistributedaccordingtotheWishartdistributionW(A,p,ri)

2｡3llultivariateLinearRegression.(5)

Inunivariableleastsquaresweconsiderscalardependent

variablesxh¨'9xndrawnfrompopulationswithexpectedvaluesare

zlβ,*･･,znβ,｢espectively,whereβisacolumnvectoroftheI

componentsandeachoftheZぼ1Sarowvectorofiknowncomponents.

Undertheassumptionthatthevarianceineachpopulationisthe

same.theleastsquaresestimatesofthecomponentsofβare

&(か(1)゛(£j(゛'≒)‾1(j

l句≒')゜

-25-

Ifthepopulationsarenormal,thevectoristhemaximumlikeli-

hoodestimateofβ(β=6).Undertheassumptionofnormality,the

maximumlike:lihoodestimateof(y2is

a2゛｀lぷ1(%‾゛(j)2'

Thealgebraofthemultivariatecaseisessentiallythesameas

thatoftheunivariatecase.

InthemultivariatecaseXn,1Savector.βisreplacedbya

matrixBand(J2replacedbyacovariancematrixΛ.

SupposeXl゛･‥ﾀ*nareasetofnobservations,°x;Cχ(a°1ﾀﾞ≒n)

beingdrawnfromN(z(zB゛Λ)゜Ordinar:i｀:lythevectorszoc(with兌

components)areknownvectors,andthePXpmatrixΛlandthe

兌xpmatrixBareunknown.If

X'(pXn)゜(･ぢ,¨',ぷぶ),Z゛(兌xn)゜(ZI≒¨'9Zぶ)｡

(2.24)

thenwecansummarizedtheaboverelationinthefollowingequation:

e(X)=ZB (2.25)

Assumingn2p十又｡andtherankofZtobe£,themaximumlikelihood

estimateofBis

{B =(ZIZ)‾1

andthatofnAis

(Z'X)

nA=I(X-ZB)1(X-ZB).

(2.26)

(2.27)

IftherankofZislessthana,theinversematrixofz'zdoes

notexist.工nsuchacase,(2.26)isnotcorrect,butthemaximum

likelihoodestimateBsatisfies

-26-

Z'ZB=z'χ (2.28)

and(2.27)holdsstilltrue･

Themultivariateanalysisofvariancestatedinthefollowing

sectioncanbeobtﾆainedfromthemodel0fthefixedcoefficientsofthe

multﾆivariatelinearregressiontheory.

2.4llultivariateAnalysisofVariance

2.4.1Introduction

Themethodofmultivariateanalysisofvariancewasobtainedby

generalizingthemethodofunivariateanalysisofvariancetothe

caseofvectorvariates.Therefore,asthebasicconceptisthe

sameasthecaseofunivariate,wearegoingtoexplaintheconcept

briefly.｀

Factor residualandmodel

Analysisofvarianceisoneofthemeansfortheanalysisused

inthedesignofexperiments.Thegoal0fthedesignofexperiment

istoelucidatetheactionsofasetofvariouscauseswhichinfluence

thecharacteristicvaluesshowingtheresultsofexperiment.

(1)工ntheexperimentweartificiallysetuptheseveralconditions

oftreatmentsforeachcause-whichiscalledfactor-and

comparetheeffectsofthefactorsonthecharacteristicvalueswith

eachother.

(2)Thedataofthecharacteristicvaluesshowingtheresultofthe

experimentarenotconstanteveninanidenticalsituation,butare

accompaniedwithsomeaccidentalerrors.

(3)Toexplainthephenomenaoftheresultsofexperiment,amodel

1Spreparedaccordingtothefactorssetupat(1)andtheerrors

indicatedat(2).

-27-

!:neveryexperiment,whethertheexperimentissuccessfulor

not1Sdeterminedbywhethertheconditionsoftreatmentspicked

upforcomparisonareadequate.Thatiswhy(1)isimportant.

Then,itiS十aprobl万゛万i71万howtochoosefactorsandlevels.

Factors:Thecauseswhicharepickedupandcomparedamonga

setofvariouscauseswhichareconsideredtoinfluence

thecharacteristicva:Luesoftheresults.

Levels:Theconditionswhicheachfactﾆortakes.(Theyarealso

calledcategories.)

Accordingtothenumberoffactﾆorspickedupinanexperiment.

theexperimentiscalled"one-factorexperiment","two-factorexperi-

ment","three-factorexperiment"andsoon.Taking"k-factorexperi-

ment"asanexample.ineachcaseofallthecombinationsofkfactors

weexperimentrtimes(r≧1)undera:LItheconditionsbeingkept

constant.Particularly,whenr>l,theexperimentiscalled"k-factﾆor

experimentwithrepeatedmeasurements"inwhichwecanobtaininfor-

mationontheinteractionaswellasonthemaineffect(described

later).Thetypeofthemode:ﾚof(3)isdifferentaccordingtothe

numberoffactorsandaccordingastherepetitionexistsornot.

Anexperimentcontainsgenerallysomeunmanageableerrors,as

insistedat(2),evenwhenitisperformedinconsiderablyaccurate

conditﾆions.Unmanageableerrorsshouldbeessential:3-ydistinguished

frommistakeandfailure.Thecauseoftherandomerrorsisgeneral¬

1ycalled"Error-factor"thataccompanythedatahoweversuccessfully

anexperimentisaccomplishedwithoutanymistakeandfailure.The

error-factorisasetofinfiniteminutecauseswhichcouldnotbe

managedintheexperiment.

Besideserrors,thereisanotionof"Residual"mainlycaused

fromthewayofobservinganexperiment.Thedifferencebetween

theactualvalueofthephenomenonandthevaluewhichcanbeexplain-

edbythemodel(thatis,theresidualpartwhichwecannotexplain

-28-

bythemodel)iscalled"Residual"whenweexperimentwiththefactors

chosenat(1)andthemodelsintroducedat(3)inordertoobservea

certainphenomenon.Aserrorsandresidua:LsappeartogetherWitﾆhin

themodelofcertaintype,botharegenerallycalledresidual.

StatisticalDecision

Nowletusconsidertodecidestatisticallywhethertheeffect

(influence)ofcertainfactorappearsintheresultsofexperiment

ornotﾆ.Then,wemustneveradmittheeffectonlybythefactthat

thedifferencebetweenthemeanvaluesofthedatawhichbelongto

therespectivelevels(forexample,AiandA2)ofthefactorseems

apparentlylarge,becauseexperimentsareinevitablyaccompaniedwithresidual.工ftheresiduaﾕislargerthanthedifferencebetweenmeanvaﾕues,it"becomesdifficulttodetectthedifferencesincetheyare

lostﾆinnoise,thatis,residual.1nthatcase,itisinsignificant

t:oconsiderthedifferenceofmeanvaluesitself.Wemustconclude

thatsuchfactorhasnoeffect.Meanv

valuesareconsiderablylargerthantheresidua:L,wecanconfirm

sufficientlythedifferenceofthemeanvalues,theeffectofthe

factor,andthedifferencebetweenAiandA2isstatisticallysignlfi-

cant｡

Thus,statisticaldecision(so-namedstatisticaltest)is

characterizedbythewayofthatwearesuretﾆOcomparecertain

characterswiththelargenessofresidualwhenwecomfirmstatist!-

callythecharacters.Thisisaveryimportantpoint.Furthermore,

instatisticaltest,weprovidequantitatﾆivelyadegreeofcertainty

forjudgingthesignificancy｡

ThesignifIcancyofeachfactorandthevalidityofmodelare

notrecognizeduntiltheseprocessesarecompleted.

2｡4.2TheLikelihoodRatioTest(7)

WeWi1:Ldescribeaboutﾆthelikelihoodratiotest,whichwas

usedinthisstudy,amongvariousmethodsofstatisticaltest.

-29-

LetΩbeam-dimensionalparameterspace,andωbethepartial

regionin･theΩ.Likelihoodratiocriterionfor･thetestofthe

statisticalhypothesisHo:∂ﾐωagainstthealternativehypothesis

H:∂eΩ-ω,isasfollows:△ll

λ=

max

∂､ぞω

-

max

∂4Ω

L

L

(∂)

(∂)

-

一

一

Lω

-LΩ

(2.29)

whereL(∂)=!f(X:∂)iSthelikelihoodfunction(Xisthedatamatrix

asinEq.(2.13),∂:m-dimensionalparameter).Inthelikelihoodratio

tﾆestgweobtainsamplingdistributionofλunderthehypothesisHO･

ThecriticalvalueλaischosensothatPr{入く入{ZIHO}゜a,whereais

thepreviouslygivensignificancelevel°{入:Oく入<λCz}iScalledthe

critica:Lregion.IfO(λ(λ(χ91:ejectthehypothesis｡

工nactualtestweoftenperformatestequivalenttothelike-

llhoodratiotestbyuti:Lizingrathersimplestatistics,whichhas

amonotonicrelationtoλ,forthetestcriterionthanλitse:Lf.

2.4.3MethodofMultlvariateAnalysisofVariance(8)

Wewilltaketheanalysisofvariancefortwo-factordesignwith

repeatedmeasurementsinordertoshowtheprocedureofanalysisof

variancesimply.Considertwofactors

provided,

-30-

AandB,andassumeA1~

(2.30)

AaandBi'｀'Bbtobetheirlevelsorcategories.(SeeTable2.1)

Whenlet*ijkbethek-thobservationunderthei-thtreatment(level)

offactorAandthej-thtreatmentoffactorB,wepresumethe

linearmodel

゛ijk(1)(p)゜μ十αi十ﾉ?j十‘rij十≪ijk･

(i°1,゜゜゜,a,j°1,゜゛゜,b,k°:L,"゜,r)

and

abぶ

1“i乱気βj

bΣF=り

aΣだ

= '■ij一

一 0,

^ijk-NCO･A).

Table2.1Multivariateanalysisofvariancefor

two-factordesignwithrepeated

measurements.

FactorB

B,Bj Bb

≪

』

A.

χUI

Z11｢

●

●

●

*Ibl

Xibr

A,

●

●

●

りi'

りj｀

Zり｢

●

●

●

Aa

Jfau

.χ;alr

●

●

●

Xabi

.χ;abr

(2.31)

(2.32)

Where,d^represents"Maineffect"ofthetreatmentkj_.βjalso

represents"Maineffect"ofBj,and^ijrepresentstheeffectspecific

tothecombinationoflevelAiandlevelBjﾀthatisthe"Interaction"

betweenfactorAandfactorB.Wedetermineμtobeindependently

distributedaccordingtothep-dimensionalnormaldistributionN(O,A).

Thetotalvariance,namely,thematrixofsumsofsquaresand

abrcrossproductswiththesamplemeanZ¨゜゜轟石呉ぶぶり.maybe

-31-

representedas

abrQ(pXp)゜ぶ石ぶ(゛ijk゛…)'(゛ijk゛…)

where'denotesthetransposedmatriχ.

ThebreakdownofthetotalvarianceQbecomesas(2.34).

Where,

(2.33)

ab

Q°昌(゛i¨‾゛…y(゛i¨‾‾゛…)十jUI(゛゜j゛‾゛…)な゜j'‾孔¨)

ab

十且jFI(‘゛ij'‾゛i¨‾゛゛j゛十゛…y(゛ij'‾゛i･.‾゜゛;･j°十゛…)

abr

十iEjylぷ(゜｀;ijk‾゛ij゛)でijk‾゛ij.)-Qi十Q2十Q3十R'(2‘34)

1br1arlr

゛i¨゜言石ぶ1゛ijk'゛'j'ﾆ盲且ぶ゛ijk'゛ij.゜7ぶ゛ijk-

Now,letusconsiderthelikelihoodratiotestofthehypothesis

thatHo:tfl=･･゜゜(Za°O'

ThemaximumofthelikelihoodfunctionLinSection2.4.2is

‰゜(211jl)11>n|頑今eぶ(“1pn)

(n=a･b･r).WhereAisthemaximumlikelihoodestimateofA:

andnA==R.

(2.35)

Also,themaximumofthelikelihoodfunctionLunderthehypo-

thesisthatHo:(Z1°‘゛゛゜゜゜(Za゛Oistrue,is

‰゜(2好ipnjl門nA]^･xp(弓pn) (2.36)

《WhereAisthemaximumlikelihoodestimateofΛunderthehypothesis

H,andnA°Qi十｀R'∧

Weutilizethestatistics,whichisamonotonlcfunctionofthelike-

lihoodratiocriterionλ,

z･=(Lω

-LΩ)

11

一一

辿ﾘ

-

-

IrI

|Qi+叶

-32-

(2.37)

forthetestcriterion.Itisgenerallydifficulttoobtainthe

exactdistributionof7｡.However,sinceQi~で耳“i'≪i(≪i~N(0,Λ))

holdsifthehypothesisistrue,andR~W(Λ,p,n-ab)holdsalways,

themomentof･lumaybeobtainedexactlysothatwecanlearnits

asymptotﾆicdistributionbyBox.(9)

N3°elyy

=-{n-jと2-(p十聡+1)/2}1og

J

(2.38)

ﾄﾉﾔ勺-(pｻけ1)/ｻ1゜(巴斗斗｀･

isdistributedasaChi-squaredvariateWitﾆhp兌1degreesoffreedom

3二二で二ぶ二こ二二ご乱

｡,｡(,｡-｡｡｡)

Table2.2Multivariateanalysisofvariancefor

two-factordesignwithrepeated

measurements.anddegreesoffree･

domofQi.Degre･offreedomof

Risn-ab(n=abr),瓦十島=ab.

Factor Effectvector Qijl:訟m欧fl゛e‘

A αiQi a-1

B βj Q2 b-1

AB Ti] Q3 ab-a-b+1

(Thedetailedprocedureofthedemonstration1Sunderstandableinthe

explanationofsimilarmodel0fAppendixAorG.)

2.5TwoproposalsforMultivarlateStatistical

AnalysisofVariance

Inthisreport,twomatteres,intheoryaswellasinapplica-

t:ion,werefoundtocontributetomultivarlateanalysisofvariance.

-33-

Theseare:

(1)Notionofdirection,besidesnotionofamountindicatedgeneral-

1ybytheanalysisofvariance,wasnecessarytounderstandthe

effectsofrespectivefactorsofanalysisofvariance,andoneofits

expressingtechniqueswasdescribed.

(2)Amodel0f"Divided-type"analysisofvariancewasproposed.

Thisisanup-to-datemodelinwhichwedevelopananalysisof

variancefortwo-factordesignbydividingoneoftwofactorsintomore

thantwoportions｡

ThelatterwasIntroducedtodoresearchonpatternsofsome

typesoftheco-articulationsandthenformulated｡

Thedetaileddescriptionof(1)isinSection4.5,andthat

of(2)isinSection6.2.

2.6Principal-ComponentAnalysis(2)(10)

SupposewehavepvariablesXl,････*･jXp,eachofwhichis

observedonnindividuals.工fwewritexαiforthea-thobservation

onthei-thvarlate,thea-thobservatﾆionvector.x;ぼisshownas

fo1:Lows;

゜X;α゜(xα111°゜IX(Zp)(2,40)

Theobjectofcomponentanalysisistoeconomizethenumberof

variates..Todothis,weshallseekforlineartransformationsof

type∧--づで.I｀'●･･

p

Yai°jE“ijzαj i=l,2,...,p. (2.41)

]

theyα19¨゜≫yofp゛Wehavetheneffectedagenuinereductioninthe

-34-

dimensionsoftheproblem,thewholecomplexofvariationsbeing

expressibleinmくpvariates.Butthisisexceptional.Whereitis'

notpossibleweshalltrytﾆocarryoutanapproximatereductionin

thissense:weshallchoosethecoefficients“ijsothatthefirst

ofournewvariatesyαlhasaslargeavarianceaspossible;weshall

tﾆhenchoosethesecondyα2soastobeuncorrelatedwiththefirst

andtohaveaslargeavarianceaspossible;andsoon.工nthisway

wetransformtﾆonewuncorrelatedvariateswhichaccountforasmuch

ofthevariationaspossibleindescendingorder.Itmaybefound

t:hatthefirsttwoorthreeofthesevarlatesaccountfor"nearly"

thewholeofthevariation,say850r90%,andthecontributionof

theotherp-2orp-3issmall.Wecanthensaythatthevariation

isrepresentedapproximatelybythefirsttwoorthreevariatesand

infavourablecircumstancesmaybeabletoneglecttheremainder.

Now,whenwepresumeZ(Z((Z゛19¨゜,n)tobeobservatﾆ10nvector

●asmentionedabovetoexplainmoreindetail,wedefinethatsample

meanvectorZ･,andsamplecovariancematrixSare,respectively,

z｡(lxp)=

S(pxp)=

ごぷ

｀

とj

l(゛i‾゛゜)'(゛i‾゛')

(2.42)

(2.43)

Here,letXi>λ2>'¨>入p>ObethepeigenvaluesofSande19e29‥･9

ep･ﾝ.‘bethecorrespondingeigenvectors,wheree,e,'°1seje:'°O(

i≠j).Thei-thcomponentyαi°(Xfjrx.)ej'of

H(i=(yぼ19¨゜9y^p)=(･%-ぷ.)(ei',"≪,ep')

iscalledthe"i-thprincipal-component"ofthesamplevector･χα,the

varianceofwhichisequalto入i°(ycciandy(Zjareuncorrelatedwith

eachother,(i≠j))

Then,wecanmeasureby入i/t17(S)ho｀ヽ7muchthei-thprincipal-

componentisimportantsince入1十゜¨+λp°tﾆr(S)‘Ifweassumethat

-35-

itispossibletoneglectthe(m+1)-th,･･,thep-tﾆhprincipal-

components,

m且

ぐ λi/tr(S))x100% (2.A4)

ofthetotalvariationofZα((Z°!,･･･,n)maybeexplainedbythe

1-st,･･･,m-thprincipal-components｡

Therefore,thecomponents(thenumberofwhichisp)ofthe

orgina:LsamplevectorzcZcanberepresentedwithaSma:Llernumber

ofprincipal-components(thenumberofwhichism)thanp,thatis,

thecomponents(thenumberofwhichism)ofthenewsamplevector

Va.･

-36-

4

CHAPTER3

FacilitiesforSpeechAnalysisand

SpeechSpectrum

3.1Introduction

Ourideasontheequipmentforspeechanalysiswillbediscussed

intﾆhischapter.Thespeechsoundsutteredinasimplifiednon-

reverberantroomareana:Lyzedthroughthe20-channel:L/4-octave

filter-bank,andtheoutputsarerectifiedandthensmoothed.The

smoothed20-channelsignalsaresimultaneous:1-ysampledatintervals

of10msec,arrangedontﾆhelineofonechannelbythemultiplexer,

andguidedintotheAD-converter.ThedigitalphonemedataAD‘convert-

edarecompiledonmagnetictapesthroughthecomputer.(SeeFig.3.1.)

A1:Lthecontrolsmentionedaboveareconductedbyon-lineand

inrealtime.Itis,ofcourse,alsopossibletocompilethesignals

recordedbyt!letaperecorder.Thesoundsthusobtainedareatfirst

embossedontheline-printerinordertosegmentthestationaryor

theboundarypartsoftﾆhesyllables.Bycontrollingtheapparent

shadesofeachpointbychangingthelettersonthepaperforthe

line-printer,thespectraofutterancesarepresentedonitﾆ(the

expressioni&similartotheSonagram)｡

Theresultsofthesegmentﾆationthroughthevisualobservation

areputintothecomputerbythetypewriter,andmemorizedwiththe

speechsoundspectra｡

AnalysisofvarianceisperformedinFORTRANlanguage.Asa

considerablylargeamountofdataisneededforoneexperiment,the

diskpackfilesystemisusedonthecalculation.

-37-

MIC:Microphone

TRこTaperecorder

P-E:Pre-emphasis

F-A:Filter-Analy多er

MPX:Multiplexer

AD:AD-converter

LP:Line-printer

MTごMagnetictape

DISK:Diskpack

20-channels

Computer

NEAC2200/200

ヶ∠

○○ DISK

Fjg･;3.1On-line,realtimedatainputsystem.

Typewriter

3.2NonreverberantRoom,Microphone,Taperecorderand

Pre-emphasisCircuit

Thenonreverberantroomofourlaboratorywillbedescribed.

0ntheinnersurfacesoftheauditory-examlnation-room(modelAT-5E),

whichism万j1万debyRionCo.Ltd.(2meterscube),wehavestickedpieces

ofglass-woolthethicknessofwhichis20cm.Itis,namely,a.

simplifiednonreverberantroom.Asaresultofroughmeasurement,

wefoundthesound-insulation-effectwasmorethan30dBabovelOOHz.

38-

Λlthoughwedidn'tmeasuretheechoes,wecouldscarcelyhearthem.

ThemicrophoneusedhereismadebySonyC0.Ltd.(mode1ECM-52)

工tisthe"electret"condensermicrophone,thatisasimplified

condensermicrophone･

ThetaperecorderusedhereismadebyTEACC0.Ltﾆd.(ModelR-

341C),andusuallyusedatthebroadcastﾆingstation.工tscharacter-

isticsagreewithBroadcastingTechnicalStandardinJapan(BTS

5313).

Apre-emphasiscircuitwasincludedinordert0improveto

signal-tﾆ0-noiseratioathighfrequencies.Thefrequencycharacter-

isticofthepre-emphasiscircuitisshowninFig.3.2,andwemade

thecircuitasFig.3.3byusingtheoperationalamplifier(analog

integratedcircuit).

dB

0

-10

uicn

-20

二.ﾆｰp

J〆y

////

〆///

//

//

-

/100 200 500 1K 2K 5K

Fig.3.2Frequencycharacteristicofthepre･emphasiscircuit

(breakpointfrequency=1.6kHz).

-39-

10K

Hz

IN

A:TA7502M(Toshiba)C,=500PF

R,=200Kぶ?

50K300K

Fig.3.3Pre-emphasiscircuit

3.31/4-OctaveFilterAnalyzer

OUT

十

-

The1/4-octavefilteranalyzeriscomposedofthreeparts

frequenciesofthefiltersincrease:･byafactor

-40-

/linorder.

(1)1/4-octavebandpass-filter-bank(MadebyRionCo.Ltd.,Model

SA-11)

(2)rectifiers(homemade),and(3)smoothingcircuits(homemade).

Wewill,now,explainabouttheseparts.

(1):L/A-OctaveBandpass-Filter-Bank

1/4-octavefilter-bankisthefilter-bankin‘whichthecenter

Thefrequenciesusedinthisresearchare210,250,297,354,420,500,

595.707,841.1000,1190.1410.1680.2000.2380.2830.3360.4000.4760and

5660HZ,andthenumberofthefiltersis20.

Theattenuationcharacteristicsofthefi:Ltersareasfollows:

(1)Theattenuationvalueatthefrequencywhichis1/8-octavefar

fromthecenterfrequencyis3±ldB.

(2)Theattenuationvalueatthefrequency,1/4-octavefar,ismore

than12dB.

UTU3

(3)Theattenuationvalueatthefrequency,one-octavefar,ismore

tﾆhan35dB｡

Thefrequencycharacteristicsofthefiltersaredescribedin

Fig.3.4.ThecircuitcompositionisasFig.3.5.Thelinearityof

theinput-outputoftﾆhefilterismorethan40dB.

dB

0

-10

20

30

40

j万ﾘ j 万 A 7 X

⑥仙j 坏 1

ﾉ/

ﾉ

ﾘ ｽ y ﾘ 1

V

ﾂﾞﾉ

坦ぶﾉﾘj A ＼

K

A//

ﾗ X ぐり∧

y

ぐぐ

言冷万万 y ＼

K

＼

X

＼

ﾉ⑤

万万ｸ y泣

ﾐ 万

言回1 ぐ ｽ く ＼

＼

X

＼

で

＼

＼

＼

＼ﾚ1002005001K2K5K

Fig.3｡4Frequencycharacteristicsofthe1/4-octavebandpassfilter-bank.

-41-

10KHz

INCi

Li

Fi&3.5Filtercircuit

OUT

(2)Rectifiers

Theoutputwavesofeachfilterarefu:Ll-wave-rectified.To

guaranteethelinearityoftheinput-outputsufficiently(namely,in

ordertoobtaintheoutput-valuesinproportiontotheinput-amplitudes

evenwhentﾆheamplitudesbecomesmall),wemadethecircuitasFig･

3.6byusingtheoperationalamplifier(analogintegratedcircuit)･

Asaresult,thelinearityof40dBwasguaranteed｡

,A:TA75Q2M(Toshiba)

20KeD:1S306(NEC)

Fig.3.6Fullwaverectifier.

OUT

(3)SmoothingCircuit

Asmoothingcircuitisnecessarytosmootheachfull-wave-

rectifiedsignal.Foreachchannel,weprovidedthesecond-order

-42-

lowpass-filterwhosecut-offfrequencyis40Hz,asthesmoothing

circuit.Uemadetheactive-filterasFig.3.7usingtheoperational

amplifier(analogintegratedcircuit).

Thefrequencycharacteristicoftﾆhelowpass-filterisasin

Fig.3.8.

E,

Ri=11.5Ki?

R3=7Kj?

R4=17.2Kぶ?

C2=0.8μF

C5=0.16μF

Fig.3.7Smoothingcircuit.

Transferfunction;

E2-Hωo2-=一一-一一一一E,s2+ぼωos+ωo2

一

一

A:TA7502M(Toshiba)

| H=1.5

α=√2

fo=言=40Hz

OUT

+

-

-l/RiRa

s2C2C5十sC5(1/RI+1/R3+1/R4)+1/R3R4

-43-

ureo

dB

0

-10

20

30

40

‾‾'べ

へへ犬

Xゝゝ

XXN

ぺ

｀戈

＼

､

＼曳

＼＼

＼＼

X

10 40 100 400

Fig.3.8 Frequencycharacteristicofthelowpassfilter

(smoothingcircuit).

3.4On-line,RealTﾆimeData工nputSystem

Hz

Wehaveanon-linerealtimedatainputsystemwiththecomputer

NEAC2200/200showninFig.3.1.Thesignals(20channels)

Speechspectra

-44-

whichpassedthroughthe1/4-octavefilteranalyz-

eraresampledatIntervalsof10ms,andthenAD-converted,and

arrangedonthelineofonechannelbythemultiplexer..Theaccuracy

oftheconversionis10bits(andsignbit).Timerequiredfor

AD-conversionis50ys.

TheprocessoffilingtheAD-convertedspeechdataonthe

magnetictapesofNEAC2200/200isperformedbyon-lineandin

realtime.

3.5SpeechSpectra

Iftheoutputsofa1/4-octavefilteranalyzerwith20channels

(theoutputsof(3)inSection3.3)thisisnamed"Theamp:Litude

outputﾆsofthe20-channel1/4-octﾆavefilters"

≪(t)゜b21(t)+b22(t)十゜¨十b2p(t)

-45-

arerepresentedby

(3.1)

b,(t)･b2(t)･‥゜･bp(t)(p°20)inorderofincreasingfreque゛cy,they

denotespeechspectrumattimet(bi(t)≧O).And･thele昭th‾

ofthevectorof(bi(t),b2(t),≪≪゜,bp(t))isconsideredtobe

theinstantaneousamplitudeofspeechsoundattimet.Schematic

representationofaVCV(vowel-consonant-vowel)syllablesequence,by

usinginstantaneousamp:Lltudea(t),isseeninFig.3.9.Asseenin

tﾆhefigure,theinstantaneousamplitudeofspeechsoundcorresponds

tﾆotheintensityofspeechsoundateverymoment,varyingapproximate-

lyover60dBrangeeveninausualconversation,anditissurelyone

oftheimportantpieceofinformationonspeechsound.Inthisstudy,

however,wearetogetridoftheinformationoftheinstantaneous

amplitudeinsteadofpayingattentionparticularlytotherelative

intﾆensitybetweenfrequencycomponentsofspeechsoundspectﾆrum.In

otﾆherwords,normalizethesquaresumofspectrumcomponentsat1｡

Meanwhile,theauditorysensefortheIntensityofthespeech

soundissaidtobeproportionaltothelogarithmofphysical

intensityofsound(Weber-Fechners'Law)｡

Takingaccountofthesetwopoints,inthispaperspectrumxCt)

attimetisdefinedasfollows:

where,

dB

0

α(t)

-30

60

゛(t)゜(Xi(t),X2(t),.."Xp(t)),

xi(t)=1og｣ﾁﾞ{ま=1ogﾌﾌｳtｱﾂﾞｻｹT

Vowel

0 50 100 150 200ms

Fig.3.9Schematicrepresentationofavcv(vowel･consonant-

vowe!)wordbyusinginstantaneousamplitude研い.

3.6SegmentationofConnectedSpeech.

byVisualObservation

(3.2)

250

t

Tosegmentaconnectedspeechutterenceintodiscreteparts

suchasphonemes-isknownastheproblemofsegmentationof

connectedspeech.Automaticsegmentationofspeechsoundisoneof

theresearchsubjectsoninformationprocessingforhumanspeech,

andIsalsoadifficultproblemasyetiinsolved.Sincehumanabilitﾆy

-46-

isconsiderablyfavorabletotreatinformationastothissortof

problem,wearetosegmentvisually.Todothatﾆ,thespeechsounds,

recorded･onthemagnetictapesoflthecomputer,areoncetypedouton

theline-printer.Typeoutthreekindsofinformations,whichare

presentedbelow,inordertoacquireaccurateresults.Thevalue.

introducedfromtheoutputsofthefilteranalyzer

butnotx(t)ofEq.(3.2)

b(t)=

b1(t)゛゛¨bp(t)゛

(3.3)

(3.4)

,isusedhere.(SeeSection3.5.)

十‥･十(bp(t2)-b

(1)Theinstantaneousamplitudeofspeechsound:

α(t)=/bl(t)十‥.十bj(t)

(2)Themagnitudeoftimedifferentiationofspectrumvector

(b,(t:)･゛゜'･bp(t)):

(b1(t2)-bl(t

t2-tl

(3)Shaderepresentationofspectrum(bi(t),･･゜゛bp(t))゛

Although(1)or(2)hasbeenadoptedfortrial0fautomatic

segmentationofthespeechsoundbyutilizingthevariationofits

value,itisactuallyimpossibletoperformexactlythesegmentation

onlyonit.However,ontheoccasionofsegmenationbyvisualobser-

vation,theseareconsiderablyreferentﾆial.

(3)meanstoobservedirectlytheintensityofspectrum

distﾆrlbution.工tissimilartotheSonagram,andveryfavorablefor

analysis.Controltheapparentshadeofeachpointbychanging

lettersontheline-printer.Adoptapieceofapparentlydarktype

forthepointﾆcorrespondingtointensivespectrumcomponent,whilea

pieceofthintypeforweakcomponent.

Usingthesethreesortsofinformationdetﾆerminevisuallythe

stationaryortheboundarypartsofthephonemes;andputthetimet゛S,

correspondingtothem,intothecomputertythetypewriter.Theseare

¥●●●j●●filedwiththeoriginalspeechsounddata.

-47-

X100

b(t

300

*＼

*＼

yt‥

＼,

●●●●

/゛

●●●●●●♪｡｡

(ljb

●●●●●

*'*-*'*ぺ本｀*j

●●●●●●●●●●●/

●●●●●●!*

/

｡

y

｡･｡

＼

●

300nns本一本-*一水´

*

d

1000

a(t)

≫―'

/゛

●●●●●●●●●500

150ms/

寒-χ一¥:-*-*一本一本

AA/AA//////｡

/'////A/A*3A)f()xAA//A/

//l＼^/^^^^｡7)^^f…^^l＼/a/

?'.AAAAAAr^ax≫r'K*A//A/

AAAAAAAhttp://www.fAA/A/

AAAAAA>f*'5f?xift")fifAAAA/

AAA*FAA!f*-f'>f!iX)ft5x≫AAAAA

ﾝﾝﾝ‘'XXご'心心X"I"‘'ﾝﾝﾝ＼"

AAAi(xXH#rKJX≫cl)(A≫AAA*

y≫x*xx≪)f｡i);jK>ff)n)A≫AAA*

AA≫*)*j<≫>

ヽ＼AA≫≫≪>f≫f)f?i<Ax*≫*AAAx-

/AAA/A///AA////////ヽ＼

////AA

///////AA

A//////AA

A/////AAA

/A//AAA*AA

////AifKAA/AAxX＼)(･≫/

AAifAAAxtJe'ifAA≫xill;xxX/

#*■x■AAltf)S)(■AAA*x■･^^;^(｡^(^(■/

水***^*Oi)*/＼Af＼*x--ni***/

xx)nf≪^tt>*f＼f＼'＼^^9.fi'**/

水*****c｡is≫≫AAAi<.x.r*t.v≫≫)f/

)(*HK)(X(^;e"<AAAi(K^f^y^f^(x/

x≫xxxtffcjJxAAAifxXi'^^Ak/

A*)(≪)(xc;yxAAAAAxx≫A)fA

A*＊*^(*ui^≫AAAAA)f**A≫A

ﾝﾝ*ﾝﾝ**をﾝﾝ＼へ＼｀､'本水｀'､｡､'

ヽ＼AA/A/AAA//////////ヽ'

＼

へ

0

4KHz

2K

ﾍﾍ

1K

500＼

＼＼ﾍﾍ

ﾍ＼>>>>>

＼へ

＼

＼

250

t350300ms

00

1t-

1

'I

ｔ.

09

10･*-'

CM200150

t7t5

100

1

t3

50

Fjg｡3｡10Shaderepresentationofthespeechspectra.a(t),and6(t)bytheline-printer.

andthes^mentationbyvisualobservatioaThewordis/deba/thatweactually

providedforanalysisofChapter5｡

-48-

Fig.3.10showsanexampleofhowtotypeoutontheline―printer

andthesegmentationbyvisualdecision

1≦･bi(t)

1≦bj(t)<10

10

10

くb

争

<bi(t)

102くb･(t)≦10‾251

10‾2‾くbi(t)≦103

≦1000,

t:hewordis/deba/that

/

V

*<SJ

(3.5)

(3.6)

-49-

weactuallyprovidedforanalysisofChapter5.

AdjusttheintﾆensitﾆyofspeechsignalswhenthesignalsareAD-

convertedsoastomakethemaximumvalueofspectrumcomponentbi(t)

below1000.1fthevalueobtainedbyAD-conversionislessthan1

b:(tﾆ)く19itisalteredthroughacomputﾆertobi(tﾆ)゜1.Hence,

andthedynamicrangeofAD-conversionbecomes60dB.

Theassignmentoftypeforshadeexpressionofspectrain

Fig.3.10isasfollows.

-----…･-------･----･--･--…･―blank

■At)<10ﾍ.--..…...…….-........_...

- - I

CHAPTER4

AnalysisofJapanesevcvUtterancesofFive

ﾆHaleSpeakers

4.1Introduction

Thereasonthatwemakesyllablesequences

vowel(CV),vowel-consonant-vowe:L(VCV),etc.

consonantandvowel

articu:Lation

-50-

consonant-

anobjectofthe

basicanalysisoftﾆhespeechsoundsisthatthesephonemes

arenotutteredindependently.Itis,of

course,basicallynecessarytoinvestigatethecharacteristicsof

thephonemesutteredindividually.However,itismoreactualto

investigatethecharacteristicsofeachphonemeinthesyllable

sequencesdescribedabove,since,particularlyinJapanese,consonants

areseldomutteredIndividuallyﾀbututteredrespectivelyinthe

formofthesyllablewhichaccompaniesavowel｡

lieinvestigatedthecorrelationsofthesephonemesco-

andtheindividua:Ldifferencesbetweenvarious

speakers,makingthesyllablesequencesofvcvtypeanobjectof

ana:Lysis｡

Althoughformantfrequencieshavebeenutﾆi:Lizedtraditionally

forthiskindofresearch,wehavebeenkeenlyfeelingrestrictions

onitsana:Lysis,asmentionedinChapter1,andinsufficientpoints

initswayofexplainingtheco-articulation.工nthisstudy.

however,ithasbecomepossibletodealdirectlywiththespectraof

consonantsaswellasthoseofvowelsbyintroducingamethodof

multivariateanalysissothatwecouldbereleasedfromtherestric-

tions.Furthermore,weinvestigated,thistﾆime,notonlytheCO-

articulationsbutalsotheInfluencesofspeakers.Thisfactmay

notbeseeninanyotherresearch.

Now,inthischapter,v^econsiderdthecomponentsofspectrum

distributionsofavcvwordatvarioustimepointﾆsasthecomponents

ofmu:Lti-dimensionalvectorsatfirstﾆ;performedmultivariateanalysis

ofvariancewithfourfactors speaker,initialvowel,consonantﾆ

andfinalvowel;analyzedthechracteristicsofco-articulationsand

individualitﾆiesofspeakers;andtreatedthemquantitatively.'

Asaresu:Ltofcomparingthevaluesofthefactor-effects

(obtainedfromtheresultsofthisanalysisofvariance)withthe

discriminationscoresofeachfactor,notionof"Direction"aswell

as"Amount"wasfoundtobenecessaryforexplainingthesefactor-

effectsforthefirsttime.Wealsoclarifiedthedifferencebetween

tﾆhisanalysisandtheprincipal-componentanalysis｡

Others,wemaderesearchesfor,arethedifferencebetween

informationincludedinspectraandthatincludedinformants,and

thecorrelationbetweeneachsectionofvcvwordandfinalvowelby

t:hemethodofthemultipleregressiontheory.Wechosenasalsounds

forC,inthischapter,inordertomaketheanalyseseasy.Thisis

becausenasalsoundpossessesthebeststationarinessamongconsonants.

4.2EstablishmentyofExperimentalObjectsandFactors

SupposeaViCV2(vowel-consonant-vowel)wordlike/ame/.

Choosingoneof/a,i,u,e,o/fortheinitialvowelViandthefinal

vowelV2,respectively,andoneof/in,n,o/fortheconsonantC,we

makeallthecombinationswiththem.Then,75kindsofwordsare

equipped.Wemade,further,375wordsasmatﾆerialsfortheanalysis

byasking5adultmalestﾆouttertﾆhese75kindsofwordsinthesimpli-

fiednonreverberantroom.Schematicrepresentatﾆionofv1CV2word

byamplitudeisseeninFig.4.1..9pointsofstationaryortransition

partﾆsofeverywordarechosenanddenotedt=tl,･･･,t9,inturn,upon

visualobservation.Presumethespectrumcomponentsateachtimeto

-51-

bevectorcomponents,andvectorscorrespondingtothetimettobe

x(t).

apmiTduiY

-一単一二

t. t2t3t4tst6t7t8t9

Fig.4.1Schematicrepresentationofv,cv.wordand

definitionofti.

t=tl:stationarypartofv1,

t=t3:boundaryofV,-C,

t=t5:stationarvDartofC,

t=t7:boundaryofC-V,.

t=t9:stationarynartofv2.

Ifiiseven,ti=(ti+i十ti-i)/2.

t

Weperformedmultﾆivariateanalysisofvarianceforfour-factor(

speaker,Vi,CandV2)designvithsingleobservationasTable4.1upon

assortingthevectorswhichcorrespondtothesametimetfroma11

materials.

Table4.1Multivariate｡alysisofvarianceforfour-factordesignwithsingle

･observatioa

Factor Level Maineffect No.oflevels

A:speaker Ai αi i=1~a(a=5)

B:V, Bj βj j=1~b(b=5)

C:C Ck rv k=1~c(c=3)

D:V2 Di ∂l I=1~d(d=5)

-52-

ΛsdescribedinSection3.5ofChapter3,iftheoutputsofthe20-

channel1/4-octavefilter-bank(20filterswhosecenterfrequencies

cover210upt0｡5660Hz)areassumedtobeb.(t)･b2(t)･‥･･bp(t)

(p°20),inorder,b.(t),･..,bp(tﾆ)representﾆthephonemespectﾆraat

timet.Aftﾆernormalizingthesquaresumofthesecomponentﾆsat1,

weestablishedp-dimensiona:LvectorZ(t)bytakingt:he:Logarithmof

itscomponents.Namely,wedefinedp-dimensionalvector

゛(t)゜(Xj(t),.."゛p(t))゛ith

Xj(t)=log戸=

よ

=L=

S

=

E2

=一一√匹(4.1)

whichwouldbeusedfortheanalysis.Theamplitudeoutputsofthe

filteranalyzerwereAD-convertedatevery10ms,thenputintothe

computerbyon-lineinrealtime.Thespeechspectﾆrumpatternsshaded

bychanging:Letterswereplottedontheline一printer,andthenmarked

eitherboundarypointsorstationarypartsuponvisualobservation,

asinSection3.6.

4.3LinearModelandMultivariateAnalysisofVariance

Supposealinearmodel,whichhasthefourfactorsmentionedin

Sectﾆion4.2,ateveryt(゜tﾆ19°･･jtg)forvectorsx(t)゜(xi(t),..゜9

Xp(t))thatrepresentthespectra(T゛able4.1)

Xijkl(t)゜μ(1)十a-(t)十βj(t)+rk(t)十∂I(1)+11jkl(1)

(1≦i≦a,1≦j≦b,1≦k≦c,1≦l≦d)一-

abc?げ

1βjﾏﾔ

'ijkl‾N(o･A)9

-53-

dぶ δ1°0,

Λ=Λ(t).

(4.2)

(4.3)

(4.4)

αi(t),βj(t),r^Ct)anddi(t)representthei-thmaineffect

offactorA,j-thoffactorB,k-thoffactorCand兌-thoffactorD,

respectively.Wedetermineμ(t)inordertosatﾆisfythecondition

givenbyEq‘(4°3),andassume゛ijkl(t)tobeindependentlydistributed

accordingtothep-dimensionalnorma:Ldistribution.

Letting"1"representtransposedmatrix,thebreakdownof

thetotalvariance0(pxp)(matrixofsumsofsquaresandcross

products)becomesasinEq.(4.5).

(Wedisregardtﾆaslongasthereseemstobenomisunderstanding.)

abcd

Q゛ぶ亙ぶ1F1(゛゛;ijkl‾゛‥‥)゛(゛ijkl"゛‥‥)

゜QI十Q2十Q3十(れ十R,

where

QI=

and

====

23J-

QQQR

aΣ貨乞公がΣ心丑一

･1･Jkl｡1

X....こ

Z･j‥゜

(4.5)

(゛i‥.-z‥‥)1(zi･‥-z‥‥)

(゛･j･.‾゛‥‥)゛(゛･j･.-゛‥‥)

(z.･k.-z‥‥)゛(z‥k.-z‥‥)

(X...I-z‥‥)i(z‥･I-X....)

Cd

1=1k=iぷ(゛iikl‾゛J...゛.j..゛‥k.‾゛...1+3゛‥‥)゛

゛(゛iikl‾゛i‥.-゛･j‥-゛‥k.-X..･1+3x....),

膳縁･聶

俗士

■*^ijkl'zi'¨゜占

j

t

l

J

I

A

･^ijkl'

dlabdlabcぶ

1゛ijkl'゛¨k°゛涵Jぶ1ぶIF1゛ijkl'゛･･･'abcふぶkl?ijkl゛

PLQajQajQi*.representthevariances(Matricesofsumsof

squaresandcrossproducts)correspondingtofactorsA,B,CandD,

respectively,andR(Matrixofsumsofsquaresandcross

products)doestheresidual｡

Now,wesha:LIdevelopatestofthehypotﾆhesisforfactorAat

timet(゜th･...tg)thata:LItheeffectsofAiareequal(thereisno

effectofA);

HA(t):α1(t)゛‥‥.゜ぼa(t)=0.

-54-

(4.6)

V7ecantestthehypothesissinceitispossibletoprovethatthe

likelihoodratiocriterion

ﾘ゜{n-£2-y(p+£^+1)}log!≒も旦L (4.7)

isdistributedasymptoticallyaccordingtoχ^-distributﾆionwithpjと1

degreesoffreedomundertheconditionn=a･b･c･d,又,1=a-1,

聡十几2°8十b十c十d-3,n-a-b-c-d+3≧p-｀゛hennissufficiently

large.(SeeAppendixA.)WeobtainedtestﾆcriterionVforeach

factoratt=tl,...,tgbasedonthematerialsmentionedabove(Fig.

ﾑ.1).Asthedegreesoffreedomofthefactorsaredifferentfrom

eachother(SeeTab:Le4.2)

Table4.2D^reesoffreedomofQi;

detrreesoffreedomofRisn-a一b-c-d+3.(n=abcd)

Factor A B C D

Qi Qi Q2 Q3 Q4

瓦 a-1 b-1 C-1 d-1

p瓦 P(a-l)lp(b-1) p(C-1) p(d-1)

瓦十瓦 a+b十c+d-3 a十b十c十d-3 a+b+c十d-3 a+b+c十d-3

andsoaretﾆhevalueofViosignificantlevelofx2testdifferentfromeach

other,wesignifiedthenormalizedcritﾆerion

v' -

Valueofり

significantlevelofχ2testcorre-;

spondingtothedegreesoffreedomof1ﾘ)(4.8)

inFig.4.2.FromFig.4.2,itcanbesaidthat

(1)Theeffect(maineffect)ofthespeaker-factoristheぐLargest

amongfourfactorsatthestationarypartofnasalconsonant:

(2)Theeffectofthevowel-factoristhelargestatthestationary

partofvowelamongfom:ヽ'factors.

(3)Althougharesultdoesnotinsistthataneffectofv2(VI)

-55-

is･observedatthestationarypartofVi(V2)(り'<!),itmaybe

necessarytoschemeamoreaccurateexperimentinordertoassert

it.

(4)AtthestationarypartofVi(V2),theeffectoffactorCis

observedbutitissmallerthanthatofspeaker-factor･

(5)TheeffectoffactorCismaximumatthepartofCamonga11

sections.TheeffectofCatV:,islargerthanthatofCatV;.

｡nU0U3JU3pOZl[UUUO|^

30

20

10

1

tl t2

Fig.4.2

t3t4t5t6t7,‥t!L･t9

Multivariateanalysisofvariancefor

four-factordesignwithsingleobservation.

4.4Relationbetweennormalizedcriterionxﾇ'anc!

DiscriminationScore

｀ThelargerthenormaﾕizedcriterionV',forfactorAisthan:L,

themore∧(ziり.(i°1~a)differgreatlyfromeachother･-

Hence,whenacertain‘X(oneofzijkl(t))isgiven.itmaybeeasier

-56-

todiscriminateinwhichcategory(tﾆhatcorrespondst0levelAiinthe

caseoffactorA)xbe:Longs.Wetﾆrieddiscriminationineachcaseof

allfactorsattﾆ(=tﾆ1,...,t9)byusingthedistancewhichis"basedon

quadraticforminordertoinvestﾆigatetherelatﾆionbetﾆweenV'and

discrimination.Forexample,wecalculatedthemeanvectorATi...(t)

andtﾆhesamplecovariancematrixS;(t)forthecategorywhichcorres-

pondstoeachlevelA:(i°1p･･,a)offactorAattimet,usingx's

whichbelonginthatcategory･

Xj...(t)=

Si(t)=1

-bed

1bcdbedぶぶEXijkiCt) (4.9)

bcd石ぶ

111F

1(‘゛ijkl(t)-Xi...(t))'(゛ijkl(t)-゛i‘..(t)).

(4●10)

Assume,now,thatwediscriminatethatgiven゛(t)(oneof°゛ijkl

belongsinthecategoryAiwhere"i"makes

(t))

(X(t)-Zi‥.(t))Sﾏ(t)(Z(t)-Zi‥.(t))1+10glSi(t)|

(4●11)

minimumamongi=l,°･･ﾀa.

Ifeverycategoryhasthenormaldistributionandfrequenciesof

occurrenceofeachcategoryofAI~Aaarethesame,thenotionof

discriminationofEq.(4.11)correspondsthat"Thecategory,inwhich

theprobabilityoftheoccurrenceofgivenj;(t)isthelargest,is

Ai."(Bayes'theorem.)Thediscriminationscoresforeachtimeand

eachfactorobtainedbythisjudgementarepresentedinTable4.3.

Thediscriminationscoresofthefactors,whoseeffectswerethe

largestinanalysisofvarianceateachtime,arenearly100%,and

thatofthefactors,whoseeffectsarethesecondlargest,arenotﾆ

sounexpected,too.AlthoughC-effectsatthestationarypartof

(七゛ty)｀ct-t^yふぐヽnasa:L^andatthebot!ndarybetweennasalandvowelareconsiderably

!"･rl'j八smallcomparingwiththelargesteffect,weshouldnoticethatthe

-57-

0.5 301●2510

Normalizedcriterionμ/

Fig.4.3Discriminationscorpsvs.normalizedcriterionμ/

-58-

discriminationscoresofthenasalsare95.5%and89.1%,respectively

Table4.3Disariminationscoresineachfactor(%).

FactorNo.of

categoriestlt2

1

t3

1

t4 t5 t6 t7 t8 t9

A:speaker 5 100'99.8i99.811001100 99.899.8 99.5 98.7

B:Vi 5 10010096.5179.5'69.9 64.5 55.7l

52.853.1

C:C 3 69.6ﾌ3.1j85.1 95.2 95.5 95.5 89.1180.377.91

D:v. 5 54.7 53.3 58.1 64.0 72.0 82.7 96.011001

100

Wevjillspeakfurtherofthefactthatthereistherelation

describedinFig.4.3betweenthediscriminationscoresobtainedhereand

~

thenormalizedcriterionり≒Thus,itisfoundthatv'hasclose

relationswiththediscriminationscorethoughV'meansoriginally

thestatisticsfortestingthehypothesis.

100

90

8070

(心)S8SUOIJBUimUOSIQ

60

50

4.5GeometﾆricRepresentationof:iulti-FactorDistributﾆions

AssignifiedinSection4.4,itissupposedthatthedirections

ofthedistributﾆionsofvariancesofeachfactormaybedifferentﾆ

fromeachotherbythereasonthatthediscriminatﾆionscoresofthe

secondandthethirdlargestﾆfact･ors(whichpossessconsiderablysmall

valueofeffectascomparedwiththelargesteffect)donotbecome

worse.AsitdealswithratiolQi十判/|r|,thatis,theratioof

thevarianceofeachfactortotheresidualvariance,inanalysisof

varianceasdescribedbyEq.(A.7),wecanobserveonly,sotospeak,

therelativelargenessofthedistributionsofeachfactor｡

Therefore,weschemegeometricinterpretatﾆionofthedistribution

asfollowsinordertoclarifytherelationbetweentﾆhedirectionsof

distributionsofeachfactor｡

Atfirst,ifwegenerally:Letμ,Λbetheexpectedvectorand

tﾆhecovariancematriχoftheprobabilityvectﾆorofZ(1xp),respec-

tively,wemaythinkthat

(z-μ)Λ‾1(.x;-μ)゛=p+2 (4.12)

expressesgeometricallythepatternofthevarianceofX,whereEq

(4.12)representsaconcentrationellipsoidforZasmentionedin

section2.2.3.

Inthischapter,wearegoingtosignifythevarianceofzby

utilizingthefollowingellipsoid(4.13)whichissimilartothe

aboveellipsoid(4.12)(similarityratio1/√iこ<-2).

(z- いか1(z- μ)'=1. (4.13)

WhereぶandAarethemaximumlikelihoodestimatesofμandA,

respectively.(LetZI･'‥9Xnbesamplevectors.then

μ=X.=nRli

―一n ATj,andxぺyい乱か.゛i‾゛.)'(゛i-゛.).)

-59-

Thereasonisthat(4.13)iseasytounderstandnumericallybecause

thedistancebetweencenterμandthepointofintersection(produced

bytheellipsoidrepresentedbyEquation(4.13)andtheprincipal

axis

八

eigenvectorcorrespondingtoeigenvalueofΛ)isjust

ヽ一

べifoneofeigenvaluesofxisa;(i=l~p).(SeeAppendixB).

Next,asshowninAppendixC,Qi十Rcanbethoughttoexpress

thevarianceoffactorA,andi-(Qi十R)canbealsothoughttoexpress

thecovariancematrixoffactorA.

V'e,further,projectthevarianceQI十Rontothenewvectorspace

obtainedbynormalizingtheoriginalvectorspacebytﾆheresidualR.

Supposethenonsingularlineartransformation

X→

_主

i=z(R/n)2, (4.14)

whereRistheresidualvariance,andn(=abcd)issamplesize･

(Zandiarevectorsintheoriginalspaceandthenewspace,

respectively.)

Then,thecovariancematrix士(QI十R)istransformedto

i-(Ql+R)=r"を(QI+n)f‾1

(SeeAppendixD.)

SothatRitselfbecomes

尨

n゜R‾

1RR‾

士

゛Ip

(4.15)

(4.16)

(Notethat･(QI十R)R‾lisalsoconsideredtobeanothernormalization,

butitisnotalwayssymmetricmatrix,sothatitisunsuitablefor

geometricexpression.)

WewillcontinuetodiscussEq.(4.15).AsresidualmatrixR

issymmetricandpositﾆivedifinite(provided,n-a-b-c-d+3≧p.See

60-

ﾀ .

1-l-･-1･rq.(A.7)ofΛppendi芦Λ),P.2'exists.WhereR2R2°RandRistﾆhe

inversematrixofrび.(SeeGLOSSARYOFSYMBOLSastothedefinition1､

ofR^).^'atrixQiissynunetricanditsrankisa-1(provideda-1≦p.See

Section2°2-4),andR2isalsosymmetricandrealnonsingular...1_.L

Accordinglyn^Oi?.2becomessymmetﾆricanditsrankisa-1.Hence,

.1tfollowsthattheeigenvaluesof

-1-_-Lz゛

areOi>O2>゜゛゜>(:ya-1>O(withprobability1)and(ya°‥.゜op°0

Thentheeigenvaluesof

ぎ1'(QI十R)R‾1al=λa' (4.17)

become入1>λ2>･‥>λa-l>l,入a°...゜入p=:L.Because,入=0+1isshown

fromthefact

Also,

λa'=[R

斗特りけ

p]α゛゜R

､_1

((h十R)R‾7

‾礼IR‾

y

α゛十lpα1°oa'十a'=(a+l)a'

=λ1･入2･･●●入a-1(4.18)

Whenjlisnamedthemaximumeigenvalue,andtﾆheeigenvectorOi

correspondingtoitisprovisionallynamedthefirstprincipalaxis

oftheellipsoidwhichisexpressedby

心i (61ﾅli)]-lil=i[r"2"(Qi+R)R2fl?=1 (4.19)

andwhichrepresentsthevarianceoffactorA,y/χ‘iandOiare"consider-

edtobetheamountandthedirectionofthesubstantialproportion

ofthevarianceoffactorA,respectﾆively･

-61-

Attq:ﾌ(attheC-V2boundary),themaχimumeigenvaluesandthe

firstprincipalaxesofspeaker-factor(Qi十R),C-factor(Q3+R)

~~andv2-factor(C八十R)･werecomputed,thentheresultsbecame(2.42,

a1)ﾀ(:L.429Cj)and(2.5^,di),respective:1-y.Theseareillustratedin

Fig.4.4｡

a1_

C1

di

Vz-factc〕r

Fig.4.4Geometricrepresentationofmulti-factor

distributions(f=t7).

TheIntersections,madebythreeellipsoids(thatarerepresentedby

~~i[言(Qi十R)]‾lX'=1(i=l,3,4))andthreeplanestﾆhataredeter-

minedbya,&Ci,Ci&dianddj&Ojarealsoillustratedinthefigure.

(HowtoobtaintheintersectionsisinAppendixB.)

TheangleebetweenvectorsXandUisdefinedby6ふcos'x‘が･

Theanglebetveen,｢Olandc.is93,andsoon.Besides,

匹に

甲

i(-4?R)‾1i'゜ilj'4F°11

-62-

fortheresidualH.Fromthisdescriptﾆion,itcanberealizedthatthe

principalaxesoft二hevariancesofthesethreefactorsmeetnearly

atrightangleswitheachotﾆher:Similarly,thedistributﾆIonalrela-

tionsbetweenthespeaker-factor(Q*,十R)andC-factor(Q,十R)at

t°ts(thatisthestationarypointofC),andthatbetweenspeaker-

factor(Qi十R)andVz-factor(Q'4十R)att=t9(thatisthestationary

pointofv2)areasFig.4.5.Fromthisfigure,wecanunderstandthat

thediscriminationscoredoesnotdecreasesincethedirectionsof

thevariancesaredifferentfromeachothereven･iftheamountof

varianceisslight｡

Fig.4.5Geometricrepresentationofmulti-factor

distributions(t=t5,t=t9).

4.6ComparisonWitﾆhPrincipal-ComponentAnalysis

Klein,PlompandPolsexpressvowelsbyusingthefirstfour

principalcomponentsofaprincipal-componentanalysisandarguethe

distributionofvowelsandspeaker'sindividualities,regardingthe

-63-

amplitudeoutputsof18-channel1/3-octavefiltersasthecomponents

of18-dimensionalvector.(Theydealtwith600utterances

-2十Q

1'aｺ匹Fふ

日U=

J=lk=II=1

1abcd

abcd,?i^ふふ(゛ijkl"‘X..")゜0

-64-

12

kindsofvowelsbypronouncedby50malespeakers,)However,their

explanationsarenotdirectbecausetheyconsiderprojeCtﾆionof

vowelsontheplanedeterminedbytheprincipalaxesofaprincipal-

componentanalysiswhichmakeneitherthevowel-factornorthe

speaker-factormaximum.

Weobservedthefollowinginordertoclarifylhoweach

factorisexpressedbyaprincipal-componentana:Lysis.

TotalvarianceQofEq.(4.5)isexactlythesameasthesamp:Le

covariancematrixSwhichisusedinaprincipal-componentanalysis.

Namely,Q°nS,wherenisthenumberofx's.Astheeigenvectorsof

Sarearrangedinorderoflargenessoftheeigenvaluescorresponding

tﾆothem,andarenamede.,e2,‥･9respectively,theinnerproductby

i(゜゛ijk1(t)-゛･...(t))andCmmakesthem-thprincipalcompone吐of

X(eiei'=1,eiej･=0.i≠j).Ifthe2-dimensionalvectormadeby

thefirstandthesecondprincipalcomponentsisrepresentedbyAT,X

denotestheorthogonalprojectionofionei-egplane.(Thevariance

explainedbythefirstandsecondprincipa:Lcompenentinthiscaseis

83Zofthetotalvariance.)ThebreakdownofthetotalvarianceQby

usingisimilarlytoEq.(4.5)isas

(4.20)

whereQj,Rhavethesreletn:nFasQiandRinEq.(4.5)do,andp=2

工nthiscase,x....=-瓦石Jぶljl1J1占iijkl゛0'because

-Similartotheprecedingsection,Rrepresentstheresidualvariance･

--VarianceoffactorA,forexample,canberegardedasQi十R.And

letiぞlil=1,i【ｻ?;-(可1十瓦)]‾lil=1betheellipsoidswhichdenote

.~な･

thesevariances,respectively･

TheillustrationsoftheellipsoidsaredrawninFip｡4.6.

eiSpeaker

-factor

Fig.4.6Principal-componentanalysis

(t=t5,t=t,).

e2

Oneofthatdenotesspeaker-factor(り'1十R),factorC(Q3十R),and

residualRattﾆ=t5,andtheotherdenotesspeaker-factor(可3十i)｡

一--factorv2(Q4十R)andresidualRatt=t9.ComparingFig.4.6with

Fig.4.5,itisunderstoodthatthefactorswhichhavethesecondand

lesslargestﾆeffectsarelostinthevarianceofthefactorwhich

hastﾆhelargesteffect,andthattheratiosofthesevariancesto

theresidualvariancearenotsofavorable｡

Tobebriefastocharacteristicsoftheprincipal-component

analysisinthisstudy,whatﾆismainlyexp:Lainedisaboutthesuperior

factor,butitisdifficultfortheinferiorfactorstobeexplained

sufficiently.Itﾆisjustlike,"Theweakbecomethevictimofthe

strong.μ

-65-

-･･-

4.7DiscussionaboutResidual

Bythemodel0fEq.(4.2),itispresumedthatEq.(4.4)

Eijkl°zijkl-μ-(zi-βi-7k‾δl(4.21)

hasthep-dlmensionalnormaldistributionN(0･A).lfμ･町･戸j･rk･

∂larereplacedbymaximumlikelihoodestimatesj:････9Zi･･･Xt....･･･>

and･゛...{-x(Eq.(A.2)ofAppendixA),then^iiklreplacing^ijkl

WJ-.●●7-一匹.I●/'‘゛″‾'‾:'I‘"~-----･.-し●l………77'｀‾･･･‥

ｿﾞﾌJ……"'‘してこいj………､し……….･1-

■'ijkl°'系jkl‾゛i…‾゛･j‥‾゛‥k.‾゛…I+3乱…ﾄ(4.22)

isdesiredtobedistributedaccordingtoN(0,Λ).Now,wearegoing

toinvestigatethedistributionalpatternofeachcoordinatecomponent

upontransformingcoordinateinorderthatΛ=(1/n)Rbecomesa

diagona:Lmatrix.WeutilizedA,maximum:likelihoodestimateofΛ,

insteadofΛwhichisunknown.

Arrangetheeigenvalues'ofΛinorderof:Largeness:denotethem

byai^,...,a^p;and:Letrl""゛'■prepresenttheeigenvectorswhich

correspondtotﾆhem.

Ifthem-thcomponentof

恥jkl°(JZijkl:'･l'"'゛･'^ijklV)

issupposedtobe^m,iikl■Thesecomponentsareuncorrelatedwith

Λeachothe:°.If^ijkl‾N(o･Λ)･

Ym,ijkl/(Jm~N(0,:L):Singledimensional

standardnormaldistribution.

isobtained.since

八

,

'

ym.ijkl-'ijkl^m'~N(0,‰Λlml)=N(0,(y2m).

Theobservedcumulativefrequencydisti°ibutionSN(X)ofym,ijkl./(Jni･

plottedOlbnormalprobabilitycoordinates,becomes,forexamp:Le,as

Fig.4.7,whent°t7andm°1.ThedistributionshowninFig.4.7is

-66-

匈

iswell-approxir.atedbytﾆhenormaldistribution.

99

9 0

7 0

5 0

3 0

1 0

{%}uoijnqujsipXouanbaaiaAHBinmnopoAjasqQ

1

● ●

/

//

ｿ

ﾀﾞ

KS-test

.-'

5亙

g

-

1

/

,////ﾀﾞ

//●/ ●

ﾂﾞ//

J//

ﾂ/

/

/X´

/

うぐ

ﾉｼ/

/

~//

/

/

∧-3 2

Fig.4.7

-1012

ylか1

Observedcumulativefrequencydistribution

ofthefirstcomponentoftheresidual.(t=t7)

3

Toexaminewhethertheseobservedcimiulativefrequencywereobtainedfrom

thepopiilationhavingthenormaldistributionornot.wewillutilized

KS-test(Kolmogorov-Smirnovone-sampletest).(SeeAppendixE.)

LetFo(X)denotethecumulativefrequencydistributionfunction

ofthestandardnormaldistributionN(O,1),thatis,thestraight

lineofFig.4.7,andthebandofFo(X)±D(z(fromthemathematical

tableofKS-test,thecriticalvalueDOC°7ZwhenN°375,level0f

-67-

significanceCX°5%)isbetweentwodottedlinesinthesamefigure.

Thefactthat‘the:LinedenotingSN(X)doeSnotcrossbeyondthe

dottedlinesrevealsthat^m,ijk1/Cmhasthestandardnomaldistri-

butionN(O,1).Fig.4.8showstheresultofobtainingDforeachymﾀ

ijkl/びm(゛゜1~p)att°t7andofcomparingthiswithsignificant-

levelvalueDa'EachvalueofDissmallerthanDsothatallof

theseseemstohaveunivariatenormaldistributions.Although

^ijklisnotalwaysdistributedaccordingtop-dimensionalnormal

distﾆributionevenifeverycomponentﾆof痢jklhastheunivaritenormal

distribution,theassumptiononresidualZijklmaybealmostﾆappro-

priate,consideringfromtheseresults.

Q

%

10

U0HBIA9P

mnuiixem

CriticalvalueofKS-test

1 5

Da=7%

10

Them-thcomponent

(α=0.05)

15

一 一

20

Fig.4｡8MaximumdeviationDofeachcomponentandthecriticalvalue

ofKS-test

4｡8AnalysisbyFormaりtFIでequency

m

Ueanalyzedthesamespeechmaterialsinawaysimilarto

Section‘A.3,considering‘thefirst,thesecondandthethirdformant

frequencies(represented･by‘･Fi,F2andF3,respectively)asthree-

dimensionalvectors(P=3).''FormantFrequencyExtractionbyainverse

FilterandMomentCalcu:Lation';(12)reportedbyNakatsuiandSuzuki

wasusedforextractingtheformantsfromthespectra,obtained

throughthe1/4-octavefi:Ltersmentionedbefore.Todescribethe

accuracyofthismeasurement,theerrorofeχtractingthefrequencies

f -

-68-

inthecaseofthesyntheticspeechsoundislessthan3.6Z.(13)

However,wesometimesreliedonvisualinspectationsinceitisnot

alwayseasytﾆoextractintﾆheactualcasewhichcontﾆainsnasaﾕizedvowels｡

Formantfrequenciesobtﾆainedintﾆhesamedatamentionedaboveat

t(=ti,t3,t7,t9),andwecarriedoutﾆmultivariateanalysisatthese

tines.(Only/a,u,o/areusedasobjectsforv2inthecaseof

formant.V1&Carethesameasbefore.)Theillustrationofthe

ellipsoid,tﾆhatﾆrepresentsvarianceoffactﾆorinthemeaningof

Sectﾆion4.5,isasFig.4.9whent°t7,t9･

V2-£ictor

1aA｡‐

~~Q4+R

t-=:t7

Fig.4.9

Speaker

-factor

還

-69-

di

Speaker-factor

Geometricrepresentationofmulti-factordistributionsin

thecaseoftheformantfrequencies(t=t7,t=t9).

Keanwhi:Le,Table4.4givesthediscriminationscoresobtainedby

formantinthemeaningofSection4.4.BycomparingFig.4.9with

Fig.4.4andFig.4.5,itﾆispossibletoexplainthereasonthatthe

discriminationscoreforeachfactorInthecaseofusingformants

becomesworsethaninthecaseofusingspectra｡

Generally,itcanbesaidthatformantﾆisrathera"vocalic

factor.'^Theinformatﾆionsonanyfactorotﾆherthanvowel-factor

arebeingdecreasedascomparedwiththecaseofspectrumdistribu-

tion.

Teble4.4Discriminationscoresineachfactor(%):

byformantfrequencies.

FactorNo.of

categoriestl t3

1

t7 t9

A:speaker 5 43.1 45.3 56.0 46.2

B:Vx 5 96.0 84.0 30.7 24.0

C:C 3 37.3 49.3 64.4 49.3･

D:V2 3 37.3 39.6 77.3 99.6

4.9RegressionEstimatebyUsingFinalVowelsand

AnalysisofVariance

Intheanalysisuptothesectionabove,thespectrumsections

atdifferenttimepoints(t=ti~19)havebeentreatedasindependent

sectionsofeachother.Here,oneaspectﾆofthecorrelationbetween

thespectrumsectionsatvarionstimepointsinvcvwordsisexamined｡

Supposetheequationsofmultivariatelinearregressionateach

timetinordertoanalyzesimilarlytoSection4.3aftere:Liminating

theinfluenceoffinalvowelfromeachspectrumsection｡

LetZ(X(t9)(astationarypartoffina:Lvowel)representaknown

vector,andXn(t)observationvector;

presume

'X°a (t)(1xp)=17(t)+･を(t9)･B(t)十e^(t)(t=ti,...,t8)

ε

(X (t)~N(O,A),B(t):p゛pmatrix;Λ゜Λ(t);

-70-

(4.23)

andestimateV(t)andB(t),where°raisoneofぷiikl^anda°1゛"9n゛

Onlv/a,u,o/areusedasV2inthissection.(Vi,Carethesameas

inSection4.3.).Pence,n=a･b･c･d=5×5×3×3=225.Supposingx゛(pxn)

゜(xUt),...,xn'it)),Z'(p+lxn)=(zi',22≒‥.,ぢ),andZ¢(1×p+l)=(:L,*a

(t9)),vdt)andB(t),maximumlikelihoodestimatesofD(t)andB(t),

respective:Ly,areobtainedfromEq.(U.23).

t)

t)

=(Z゛Z)‾1(Z゛X) (4.24)

(SeeEq.(2.26)ofSection2.3.)

Byusingthese令(t)andB(t)･esti°ate゛(t)from･゛゜(X(t9)illeach゛)゛d･

oneofthesameVCVwordsusedinEq.(U.23),andlet£(t:)bethe

estimateof厦t).

Theresultsofthemultivariateanalysisofvarianceforthediffer-

&

enceZ

(X(t);

lo,(l)yx'ojt二) 礼収)゜゛Cz(tﾆ)一令(1ﾆ)-゛a(tﾆ9)冷(t:)

(similarlvtoSection4.3)isdrawnasFig.4.10.(Dottedlinesof

ﾐthesamefigureindicatetheanalysisfor･｀゜(t)itself.)

Asaresult,V2andthespeaker-effectdecreased

Therfore,

●＠

4(1)theeffectofCbecamethelargestﾆintheregionbetweenCand

V2amongalleffects:

(2)tﾆherelativevalueofeffectVitoothereffectsincreasedat

partVi.

Thisrevealsthatintensivecorrelationisseenbetweenv2and

speaker-effect.

(Thecorrelation[canonica]Lcorrelation)betweenvectorsZand

yhascloserelationswiththeregressiontheorybetweenぷandy｡)

-71-

(14)

15

1 05

｡nU0U911J0pazitEuuoM

1

Vi---

V一十-C→V

゛≒

へ｀ヽ

､

-〃〃--

仁ど

"""

1

1

/

/

11

1111

/

/

1

/

1

"

/

-･

/

/

/

/

/

/

/

- 一 一

/

C四--

i-

/

/

｢≒

『゙｀

I

//

｀χχ

'

ドﾄ

ｰｰ

ﾚド

'

/-―'

χ1？

^I"*"/

4//

ゝ

ゝχ

kゝ---

･ 四

一一- 一 一 一 -

t

tl t2 t3t4t5t6t7 t8t9

Fis4.10Comparisonbetweentheanalysisof

varianceofXaandthatofXa｡‘

ldisthedifferencebetweenZα

anditsregressionestimateby｡χ:a(t9)

4.10Conclusion

Consideringthespectrumcomponentsasthecomponentﾆsofmulti-

dimensionalvector,weperformedmultivariateanalysisofvariance

insectionsatvarioustimepointoftheViCV,utterancewithfour

factorsVi,C,V2andspeaker;andcomparedtheamountsofthe

effectsofeachfactorwithinthemselves.Thenweinspectedthe

relationofthevarianceellipsoidsofeachfactoralongtheir

principalaxes;signifiedthatnotionofdirectionaswe:LIasamount

isnecessaryforexplaningtheeffectsofeachfactor;andcompared

theseanalyseswiththeprincipal-componentanalysis.Anotherthing

weinvestigatedbythemethodofregressionestimatewastherelation

72-

betweenfinalvovrelsandeachsectionofwords.Furthermore,weper-

formedsimilaranalysisonthebasisofthree-dimensionalvectors

whichconsistoftheformantfrequenciesextractedfromthesame

materialsasabove,andcomparedtﾆhesewiththecaseofspectra.

Theresultsconcerningspeechsoundsareasfollows:

(1)Speaker-effectﾆisconsiderab:Lylarge,whileconsonant-effectis

notsolarge.However,thedirectﾆionsofthreedistributionsof

tﾆhesetwoeffectsandvowel-effectmeetatnearlyrightangleswith

eachother:

(2)Intensivecorrelationisseenbetweenvowelandspeaker-factor:

(3)Inthecaseofformantfrequency,theinformationsonanyfactor

otﾆhertﾆhanvowel-factﾆorarebeingdecreasedascomparedwiththe

caseofspectrumdistribution.

-73-

CFAPTER5

AnalysisofJapanesecvcvUtterances

5.1工ntroduction

Oneofthesubjectsofthisthesisistoconsiderquantitatively

theco-articulationsofutterance.工ntheprecedingchapter,v/e

experimentedalonpthesubjectabovefromvariouspointsofhypothesis

basedonvcvsyllables.'｡

Howevertheconsiderationintheprecedingsectionwasinsome

aspectsinsufficientbecause:

(1)ItwasdifficulttoinvestigatethecharacteristicsofCO-

articulationitselfbecausetheinfluenceofspeakerwasnixedas

weinvestigatedsimultaneouslytﾆhedifferencecausedbychanging;

speakers:

(2)Becauseweadoptedthemodelwithoutrepetitionsfortheanalysis.

wecouldnotclarifytheinteractionbetweenprecedingandfollowing

phonemes:

(3)Thenumberofthephonemesequencewastoosmall.'Andthekind

ofconsonantsweretoolim:ited.

:Therefore,weplannedanexperimentparticularlyemphasizing

muchimportancetoconsiderationofco-articulations:

(1)7echpseonlyonespeaker.Thereasonwasthatifwetreated

simultaneouslyvoicesofvariousspeakersinordertoinvestigate

tﾆhedifferencebetweenspeakersalltogether,itbecametroublesome

tofindoutthecharacteristicsthatwerepurelydependentﾆonc0.-

articulationsincethecharacteristicswerelostinthefluctuation

causedb'ythealterationofspeaker,thatwas,noise.Instead,as

a!consequenceoftheforegoing,theresultofthatanalysisinclined

-74-

tocontaincertaincharacteristicsdependentonthearticulational

habitﾆSofthesoeaker:

(2)･Inorder･toclarifytheinteractionsbetweenprecedingand

followingphonemes,weappliedamodel0fanalysisofvariancewith

repeatedmeasurements.りtlaineffect"indicatesthedegreeofthe

effectofphonemeitselfwhichisindependentofcontext,and

'‘Interaction"doeshowmuchtheeffectisdependentontheconteχt.

Interaction:Effectspecifictothecombinationsoftwo(ormore

thanthat)ofprecedingandfollowingphonemes･

Precedingandfollowingphoneme:Notonlyonesclosetoeach

otﾆherbutalsoseveraloftﾆhoseskippingoverafewphonemes.

(3)1･7emadeuseofwords0fCVCV-sequence.Thereasonisthatthe

scaleoftheexperimentistoolargeifweusemorethanfivephonemes

inconsiderationofinteraction,andthatthenumberoftwo-syllable

wordisfrequentinJapanese.(Cvrepresentsonesyllable.)Nine

consonantswereprovidedfor.C.I

Inthebeginningofthischapter,wedescribedthemultivariate

analysisofvarianceforfour-factordesignwithrepeatedmeasurements.

Althoughitﾆwasconsiderablyhardtoproveraathematicallvthemodel,

Mr.Kubo,astudentofKyotoUniversity,co-operatedinit.

Fromnowon,wewi].1describetheexperimentindetail.Consider-

ationoftheresultsoftheexperimentincludesexaminationof

residualsdistributionandevaluationofinteractions.

Ast二heexperimentofthischapterwasprojectedwiththatof

Chapter7,suchresultsofSection7.4shouldbeshownhere.

5.2llultivariateAnalysisofVarianceforFour-Factor

DesignwithRepeatedMeasurements

-75-

Themodel･oftheexDerimentaldesipnofthischapteris':Multi-.

variateanalysisofvarianceforfour-factordesignwithrepeatedme-

asurements."]:tisimpossibletoinvestigatetﾆheintﾆeractionsbetween

eachfactorbymultivariateanalysisofvariancewitﾆhsingleobser-

vation,whileitﾆbecomespossibletodoifweusetheanalysiswith

repeatedmeasurements.However,thelaborforprovingmatﾆhematically

themodelincreasesconsiderab:lyascomparedwiththecaseofthe

previouschapter.工nthemodeloftheprecedingchapter,thelongest

expressionconsistsof16terms:whi:Leinthischapter,thelongest

consistsof256terms｡

Mr.MasatoshiKubo,astudentofKyotoUniversityco-operatedin

accomplishmentoftﾆhemathematica:Lproofofthemodelofthischapter.

Sothatonlytheresultofitisdescribedhere,referingtohis

graduationthesisfort1!ebachelardegreeofKyotoUniversitysoas

torealizetheproofyL5)

designwithrepeatﾆedmeasurements.

Themodelisdefinedasfollows(tisomitted):

゛ijklm(1)(p)゜g

十(zi十βj+rk十∂1

generallevel

maineffect

+Sij+ζik十ηil+∂jk十λjl'1μkltwo-factorinteraction

十μijk十pijl十"ikl十^jklthree-factorinteraction

+91jklfour-factorinteraction

十■^ijklmresidual

(5.1)

where･1≦i≦a,1≦j≦b,1≦k≦c,1≦l≦d3゛d1≦!11≦e.(SeeTable5.1.)

Theconstantsa,b,canddarethenumbersoflevelsofthefactor

A,B,CandD,respectively.Theconstanteisthenumberof

-76-

repetﾆitions.

Table5.1Mu】tivarjateanalysisofvariancefor

four-factordesignwithrepeated

measurements.

Factor Effectvector Qi

I

J

A (Zi Q.

B βj Q2

C rk Q3

D δ1 Q4

U

AB りj Q5

AC ぐik Q6

AD ﾌ7il Q7

BC ∂jk Q8

BD λjl Q9

CD μkl Qio

oh

ABC りjk Q11

ABD ρ甲 Q12

ACD OiU Q13

BCD りd Q14

|| ABCD Vijkl Q15

Thegenerallevelvectoric(lxp)isdeterminedinorderthat

effectvectorsoC;(lxp)-‘?ijkl･(lxp)satisfythefollow!昭conditions:

a

Σ

1二1

a

Σいb胃

a-=.

^ij-

∂jk°

b

Cβ戸O'ぶ

1rk°O'

l)ajWjiro'ぶ

1ぐik

Cb_CDJ

I

∂jk゛o'胃1

λ

｡Σ㎞

jl°

d且 *i=o

aΣ出0｡

一に

ら '?il°0

Cd

I゛O'ぶ1μkl°IRμkl゛0

-77-

ab

iEμijk°z;

aC

ぶ(7iklﾆkﾐFI

μijkcab

語りjk°o'yijlﾌﾞPu＼=

d

"ikrぶjy

d

Σ

1=1

^ijl=o

bcd

ikl-o'jlJjjkFぷ1'J'^'=l?.'■jkl』0

abcdぶ

191jkにjlJlfijkげぷlfijkげぶ1?ijkにO‘

Besides,assumethat

Zijklm‾N(o･A)･

(5.2)

(5.3)

thatiSﾀ^ijklmisassumedtobeindependentlydistributedaccording

tothep-dimensionalnormaldistributionN(0,Λ).

Thebreakdownof･totalvariance.

Let"1''denotethetransposedmatrix,thenthebreakdownof

totalvarianceQ(pxp)(matrix6fsumsofsquaresandcross

products)isasfollows:(SeeTable5.1.)

.Q=Qi十Q2十Q3十Q4十Q5十Q6十Q7十Q8十Q9十(卜o

十Qll十Q12十Ql3十Q14十Q15十R,

where上

abcde｡,.

Q°ぷ

1

石ぶ

I

F

lr

E

I

(゛ijkhll‾y7¨‾j.)'(゛7ijkllllう゛¨゜゜')ﾘ

蜀y㎜-'-か'--a●･-aW¶a

Q1°bcde

i=i(゛i…'‾゛…‥y(゛i…'‾゛¨…)'

bQ2=acde2

JJ

Q3=abdeΣ

Q4°abcedΣμa

1(゛'j…‾゛…‥)べ゛'j…‾゛…‥)

1

(Z‥k‥-Z…..)'(x..k..-J:…‥)

1

1(゛¨小‾゛…‥y(゜゛;…1°∇‘゛…‥)'

b

Q5°cde且仏(Xij…‾゛i….‾゛･j…十゛…‥)"

(゛ij…‾゛i….‾X.j‥.十゛…‥)･

aCQ6°bdei=ik=i゛i'k¨‾zi‥‥‾ぷ‥k‥十z…‥y

'(zi.k‥一zi….-z‥k‥十z…‥),

ad

Q7°bce且ぶ1(゛i‥I°‾zi･.･‥‾z･‥･I.十X)'

-78-

(5.4)

Here,

Q8=ade

Q9°ace

●(

bcΣΣ(

j二ik=i

●(

ぐdふ

b石

Xi.･l.‾･Xj―X･‥l.十ぷ).

゛･jk‥‾゛･j…‾゛･･k‥十x…‥)'

゛･jk.--゛･j…‾゛‥k‥十z…‥),

゛･j小‾゛･j…-J:...1.十z…‥)'

(゛･j･1･‾゛･j…-゛…1,十z…..).

CdQio=8beJ

I

ぶ

1(ぷ‥kl‘‾z‥k‥‾ぷ‥･I.十z…‥)

'(χ‥kl.-z..k‥-z‥.I.十a･…..)

abc

Qii=de

Q12°ce

i

y

l石ぶ

1(゛ijk"゛ij…‾゛i-k-.~゛'jk･.十゛i….十゛･j･‥十゛‥k‥‾゛…‥y

'(゛ijk･.‾゛り…―X:･k･.‾゛･jk･.十゛i….十゛･j…十゛･･]r..-X…･.)

abdぶふF

I(耳j'1°‾゛ij･..~X:.,‘‾゛'j小十゛i….十゛'j…十゛…1'‾゛…･.)'

･(゛ij･l.‾゛ij…‾゛i･･1.‾゛･j小十゛i….十ぶ･j…十゛…I.‾゛‥…)･

acd

Q13°beぶ1ぶ1ぶ1(゛i゛kl°‾゛i-k"゛i‥I°‾゛¨kl°十゛i…'十゛¨k¨十゛‥小‾ぷ¨…)'

Qi5=e

'(zi

十ぷ･jk‥十ぷ･j･l

X=

z…..y

bcd

Q14°8e胃ぶ1ぶ1(゛゛゛'jkl.‾ぷ･jk‥-ぷ

i･･I.‾゛･･kl.十゛i….十゛･･k･.十z…l.-x…..).

･j小‾゛･･kl-十゛･j…十゛‥k‥十゛…I.-z…‥)'

'(゛･jkl･‾゛･jk･.‾゛･j小‾゛‥kl丿゛･j…十゛‥k‥十゛…1.-*･･…)

abcdふ

j=ik=i

ぶ

1(゛ijkl'‾゛ijk‥‾りj小‾゛i-kl°‾孔jkl°十゛ij…‾ﾄr;.],..十゜r;...

'(゛ijk!･‾゛iik""^ii･I.‾゛i･kl.‾゛･jkl.十゛ij…十゛i･k･.十゛i･･I.十゛･jk･･

十゛･j小十■r..ki.-゛i….‾゛･j…‾゛‥k‥‾゛…I.十゛……)'

abcdeR≒万万;

1

石ぶ

I

E

I

ぶ

1(゛ijkh‾゛ijkl°y(゛ijklm゛ijkl‘)'

1bcde.labce

゛i°‘¨゜i召7ふぶEIぶ1゛ijklm*゜゜゜'゛"小abcei=iJUIぶ1｢ぶ1゛りkh'

-79-

1cde12ce

町…cdek=iぶ1Jjijkh'゜゛゜'゛゜j゛に゜77ぶぶJ1゛ijklm'゜"'

゛ijk‥゜士

jよ

゛ijklm'T'

lbeyi･kl-゜百胃Jごijklm'゛"

^ijkl-゜‾ｼjl･'^ijklm°

Thenaximumlikelihoodestimatesofmaineffectsandinterac

心4;゜X,

(Zi°Xi....―X°･‥‥

八βj°ぷ･＼...-x°…‥'

八rk゛Z‥k‥‾Z‥‥.･

I

〈δ

八*ij

八ぐik

八ηil

〈θ

一

一

こ

一

一

X...I.-X..一一●1

'x'ij…‾.x;i….‾z･j…十ぷ…‥'

zi･k‥‾zi‥●●‾z‥k‥+ぷ>

Xi･.l.-Xi･‥.‾Z‥●I.+.X7‥･‥9

jk゛X､jk･.‾ぷ･j･･.‾z･･k･.十z…･.･

2j

八

I゛z･j小‾z･j…‾z‥小十ぶ…･.'

μkl一

一

=k‥~

〈μ

z‥kl.‾z‥k‥‾z…1.+x…‥･

゜x;ijk‥‾zij…‾zi･k‥‾ぷ･jk‥十zi…汁ぷ･j…十ぷ･･k‥‾ぷ…･.･

^ijl°xij°1.‾zij…‾xi･小‾‘x;･j･1.十zi….十z･j‥.十ぶ…I.‾‘x;……･

"ikl°‘x;i･kl.‾'x;i･k･.‾zi‥I.‾z‥kl.+zi･‥.+ぷ‥k･.+x..･i.-x..･‥ﾀ

rjkl°'r･jkl.‾χ･jk･.‾z･j小‾z･･kl.十ぷ･j…十ぶ‥k･.十ぷ‥･1.‾ぷ……･

Pijkrzijkl.‾.x;ijk･.‾zij･l.-Xi･kl.‾x･jkl･

十･^ij..ヽ十Jri･k‥ﾅzi･･1丿ぷ･jk･.十ぷ･j･!.十.x;･･kl･

‾zi….‾z･j…‾z･･k･.~x..･I.十ぶ･●…

-80-

(5.5)

Thelikelihoodratiotestfornull otﾆhesis.

Letusconsideratestofthehypotﾆhesisforthemaineffectof

factorAthatalltheeffectsofAiareequal(thereisnoeffectof

A):

HA(t):α1°゜"゜αa･･f0, (5.6)

1･7ecantestthehypothesissinceitispossibletoprovethat

thelikelihoodratiocriterion

ﾘ={n一又,2一一y(p十几1十:1)}log聯亙(5.7)

isdistributedasymptoticallyaccordingtoX^-distributlonwithpjと1

degreesoffreedomundertheconditions-n=a･b･C･d･e,

聡゜a-1,又･1十又･2°a°b゛c°d,andn-abCd≧p-whennissufficiently

]Large.ThedegreesoffreedomsoftherespectiveQiareshownIn

Table5.2.

Table5.2DegreesoffreedomofQi;乙十島=abed,and

degreesoffreedomofRisn-abed.(n=abcde)

Factor Qi jl

A Qi a-1

B Q2 b-1

C Q3 C-1

D Q4 d-1

AB Q5 ab-a-b+1

AC Q6 ac-a-c+1

AD Q7 ad-a-d十1

BC Q8 bc-b-c十1

BD Q9 bd-b-d+1

CD Qio cd-c-d+1

ABC Q11 abc-ab-ac-bc+a十b十c-1

ABD Q12 abd-ab-ad-bd+a+b十d-1

ACD Q13 acd-ac-ad-cd十a十c十d-1

BCD Q14 bcd-bc-bd-cd十b+c+d-1

ABCD Q15 abed-abc-abd-acd-bcd+ab+ac十ad+bc十bd十cd-a-b-c-d十1

-81-

5.3Establishmentof･ExperimentalObjectsandFactors

Schemedthefollowingexperimentalproiectonthematﾆerialof

wordspronouncedintheformoftwo-syllablewords-CiViCoV,

(consonant-vowel-consonant-vowe:L) byamaleadu:Lt.

Namely,assumingfourphonemes-Ci,Vi,Co,V2tobe

factors,letthemcorrespondtofourfactors-A,B,C,D-for

analysisofvariancementionedinSection5.2.Nineconsonants/

p,t,k,b,d,g,m,n,o/wereusedforCiandCo,andthreevowels/a,e,o/

forViandv2.ThesenineconsonantsareclassifiedasTable7.10f

Chapter7consideringfromtheviewpointofmannerandplaceof

articulation.The･‘reasonforchoosing:/a,e,o/forvowelswasthat

wecanhaveobservationinallposslb:Lecombinationsoflevelsfrom

eachfactorbyeliminating/i,u/since/tl.tu.di.du/donotoccurin

modernJapanese.゛

Amaleadultpronounced,threetimes,eachwordofthetotal

combinations-729kinds-ofCi,C2=/p,t,k,b,d,g,m,n,o/and

Vi,Vz=/a,e,o/atrandom.Thosewereutteredinthesimp:Lifiednon-

reverberantroomwithinoneday,andthetotalnumberwas2187.(

a°c°9,b=d=3.e=3)

WeaccentedalltheCiVimoradespitethefactthattherearea

10tofmeaninglesswordsInthecombinationsabove.

Furthermore,thesespeechdataarethesameasthoseinChapter

7.

X･7ewillgiveadefinitionofspeechspectraandprocedureof

segmentationbe:low(detaileddescriptionisinChapter3.)

VThenletbi(t)'゛‥‥'bp(t)(p°20)represent,inorder,a『』1it゛de

outputsof20-channelK-octavefilters(bankof20filterswhose

centerfrequenciescover210upt05660Hz)attimet,theyare

consideredtorepresentspeechspectraatthatlt:ime.Afternormali-

zingthesquaresumofthesecomponentsat1,weestablishedp-

dimensionalvectorx(t)bytakinglogarithm･ofitscomponents..･･.j.･●

Thatis.一一‥

-82-

Xi(t)=log{bi(t)ｿ

ｽﾞ‾Uyで‾こ

} (5.8)

Wedefinedp-dimensio°1vectorx(t)゜(x,(t)･‥･･り(t))withEq.(5.8)

whichwouldbeusedfortheanalysis(p°20).Amplitudeoutputsof

thefilteranalyzerareA-Dconvertﾆedatintervalsof10ms,then

putintothecomputerinrealtime.0nthepaperoftheline-printer

werepresentﾆthespeechspectrumpatternsoftheinputwithwhichwe

observedtodeterminethestationarypartsortransitionones.

Thisstudydefinestimestl.t2,,tl3asFig.5.1,correspond-

ingtothestationaryortransitionpartsineachword.

apnjiidiuv

tl

CI

ﾙ

V,

ﾙ

C2

→

V2

t2 t3t4t5t6t7t8t9tiotilt12t13

Fig.5.1c,v,cNiwordanddefinitionofti.

t=tl:stationarypartofc,,

t=t3:boundaryofCi-V..

t=t5:stationarypartofV,,

t=tT:boundaryofM-C3,

t=t9:stationarypartofC2,

t=tn:boundaryofCj-Va.

t=tl3:stationarypartofv2.

Ifiiseven.ti=(ti+1十ti-i)/2.

-83-

t

5.4Resu:LtofAnalysisofVarianceforFour-FactorDesign

Veperformedanalysisofvarianceateacht(=t3,tit,t5,t6,t7,t9.

t11,ti2,ti3)accordingtotheprocedureexplainedinSection5.2.

(t=t,isomittedTaecausethestationarypartofc,ishandto"bedefined‘)臨φ√ぶぷlomputedrespecti々elythetestcriterionfoy?he?死竺芒'sisthattherearenoeffectofeachfactorornointeraction‘between

eachfactor.Wenormalizedりbythevalueofsignificant:Levelas

thefollowingequation,becausethedegreesoffreedomcorresponding

tomaineffectsandInteractionsaredifferentfromeachotherand

soarethevaluesof＼1osignificantlevel0fχ^testdifferentfrom･each

other.(SeeTable5.2)

X)1=

(vaﾕueof＼%significaよitlevelofχ testcorre-

spondingtothedegreesoffreedomoiヽり)(5.9)

Thei1:Lustrationofvaluefortestcriterionりlnormalizedaboveis

asFig.5.2.工nthefigure,Ci,Vi,C2,V2representmaineffects,and

VlC2,C2V2thetwo-factorinteractions,andV1C2V2thethree-factor

interactions,andsoon.

Thefollowingisrevealedfromthisfigure:しI'

(1)工nthestationarypartsofeachphoneme,themaineffectofthe

phonemeismaximum:

(2)Theinteractionbetweentwocontiguousphonemesis,ofcourse,

smallerthantheeachmaineffect,butlargerthanthemaineffect

ofanyphonemeotherthanthetwocontiguousphonemes:

(3)Thetwo-factorinteractionbetweenthetwophonemes,thatare

notjustcontiguoustoeachother｡isconsiderablysmallerthanthe

interactionoftwocontiguousphonemes.Therefore,theinfluence

(onco-articulation)specifictothecombinationsofthephonemes

whichputafewotherphonemesbetweenthemisSma]｡1:

(4)Thereishardlyanyinteractioninthecaseofmorethanthree

-84-

こゝ

200

100

50

2 0

1 0

UOUOJUJ(X)ZI[tUUi()Ｚ

5

2

t3t5t7tot11

Fig.5.2Multivariateanalysisofvarianceforfour-factordesign

withrepeatedmeasurements.

-85-

t13

t

factors.i･7edidnotillustrateanythree-factorinteractionwhose

り゛iSlessthan1,butonlytheeffectsofthree-factorinteractions

ofthree―phonemeV1C2V2andCiViC2.(whicharecontiguoustoeach

other)aresignificantﾆ:

(5)(1)and(2)aboveshowthesubstantialproportionofthetota:L

variance.Accordingly,inthecaseofadoptinganroughmodelupon

omittingconsideratﾆionofinteraction,wehaveonlytotakeaccount

oftheeffectsofthelustprecedingandfollowingphonemes.V'emay

notthinkof'theeffectofthephonemesapartfartherthanthem.

(6)Influencesoftheprecedingandfollowingvov;elsofViandv2

onC･2arealmostthesame.(SimilarinFig.4.2and7.4.)Thisdoes

notcorrespondwiththeconclusionofareportobtﾆainedbyanexDeri-

mentofauditoryperceptiononthematerialofsyntheticspeech

SOUndS.(16)Thereportinsistsontheimportanceofinfluenceofthe

followingvowelV2upontheC2｡

Theremaybeadifferencebetweenphysicalandauditorycharac-

teristicsofco-articulation.

5.5ExaminationofResidual

Inthemodel0fthefour-factorexDerimentofthischapter,

■*ijklm~‘N(O･Λ) (5.10)

isassumedfortheresidualtermX..asinEc.(5.3).Now,ifwe-1!klm'

makeuseofthemaximumlikelihoodestimateofeffect-vector

ofEq..(5.5)astheeffect-vectorofEq.(5.l),theactualresidualis

ぷijklm･^ijkl(5.11)

whichisdesiredtobedistributedaccordingtoN(0,Λ).Eyexactly

-86-

(j)iioiincjujsinXouonbojjOAnuiniun｡7)己｡j()

99.9

99

mo

99

70

50

OOOin

3C^J―I

1

0.1

ﾉ

1

/

j

ﾉ

言/

z/KS' est

5§

ｽﾞ/

/ｹ

が差ﾀﾞ

//

ﾌﾞ/

ｿﾞ汐

.ｹ´〃〃//

/

ﾉ //

/ﾌﾞ1/

/

1

1･

ﾉ|

3 -2-1-012

y./o1

Fig5.3observedcumulativef｡quencydistribution

ofthefirstcomponentoftheresidual(t=t,,)

-87-

3

thesameprocedureasinSection4.7､weillustratethepatterns

ofdistributionandKS-Testoftﾆheresidualz‥-ぷ‥as1､1]klmijkl.

Fig.5.3andFig.5.4.FiR.5.3denotesthedistributionofthefirst

principal-componentoftheresidualatt°13､andFig,5.4doesthe

maximumdeviationofthefirstuptothe20thprincipal-componentﾆat

t°13.Judgingfromtheseresults､theassumptﾆionoftheresidual

Zijklmisnearlyproper.

%

321

nuoquiAap

UiniUIXBTAJ

CriticalvalueofKS-testn-=2.9%foc=5%)

15101520

Them-thcomponent

Fig.5.4Maχimum.deviationDofeachcomponentandcriticalvalueofKS-test.

-88･-

4

●

5.6Evaluatﾆionof!nteraction

AsitwasascertainedbytheanalysisinSection5.4thatthe

amountofthetwo-factorinteractiondoesnotﾆeχceedthatofthe

mainfactor,V7eare,now,goingtoconsiderhowtheresultofthe

analysisiswhenwedisregardtheinteraction.Usingtﾆhesamedata

asusedinSection5.4,provideamodelforanalysisofvariancefor

four-factordesignwitﾆhrepeatﾆedmeasurementsinwhichalltheinter-

actionisdisregarded.(SeeEa.(5.1)andTable5.3.)

Table5.3Multivariateanalysisofvarianceforfour-factordesignwithrepeatedmeasurements

inwhichaUinteractionsa･disregarded,anddegreesoffreedomofQi･

DegreesoffreedomofRisn-a-b-c-d+3(n=abcde),and

乙十島=a+b十c十d-3.

Factor A B C D

Maineffectvector 哨 βj ric δl

Qi Q. Qj Q3 Q4

瓦:訟ご認(,, a-1 b-1 C-1 d-1

where,

゛ijklm゛μ十(Zi十βj十rk十∂I十^Uklm･

ab

Σ(Zi°IΣβj゛i二1j=1

cd

°0,

k=ll=1

(5.12)

(5.:L3)

:p-dimensionalnormal

!ijklm‾N(O･Λ)distribution.(5.14)

NowwehaveabreakdownofthetotalvarianceQforthismodel.

abcdeq°ぷ

1且ぶ

1ぶlnΞ

1(゛iiklm゛…¨)'(゛ijklm゛…¨)

=Qi+Q2十Q3十Q4十旦,

QI°bcdeぷ1(゛i…゛‾゛….J'(Xi…゜‾゛…･.)

b'

Q2°acdejE(゛゜jよ‾゛゜゜¨゜)'(゛.j...゛).

Q3=abde JI(゛..]c..-X…¨)I(゛¨k‥‾゛¨…)'

-89-

(5.15)

where

Q4°abce

aΣ『

R｢

dX(x...,‘‾‘x゛…..)'(x..゜1‘‾z¨…)'

bcdeふ

k

Rぶ

l

j

l

(Zりklm""^l･･"‾り'‥‾‘X;¨k‥-Z･･･..+:iぷ･….)

゜(ぶりklm‾Zi….-X.j7‾Z･･k･.‾X･･･1f3Z…･･)

X=

1-abc

abc

砿jlΣ

Q1,‥.･,Q4areexactlythesameasthoseofEq.(5.1↓),andRisexpres-

sedas

R=Qs 十Q6十‥.十Q15十R (5.16)

byQ5,Q6,‥･,Ql5andRofEq.(5.1↓).･Namely,makinguseofnewresidual

Rwhichisthesumofalltheintﾆeractionsandtheoriginalresidua:LR,

analyzethemaineffectsofQ1~ら.Forexample,thetestof

hypothesis‾HA:α1°‥゜゜(な゜0canbeconductedbyusingthe

factthat

り=ln一見2一手(p十見1+1)}1og拍およjkL (5.17)

isdistributedasymptoticallyaccordingtoχ^-distribution,｀゛hose

degreesoffreedomisp見lundertheconditions一見i=a-l,見1十几2=

a十b十c十d-3,n-a-b-c-d+3≧p,n=abcde.(SeeAppendixF｡andTable5.3)

Theresultofcalculatingforeachfactorof'C1,Vi,C2,V2ateach

timetisasthedashedcurvesinFig.5.5.Herein,

り↑゜(5.18)り

(Valueof＼%significantlevelofxtestcorre-

spondingtothedegreesoffreedomofV)

Besides,themaineffectsofCi,Vi,C2,V2inthecaseof･considering

interactionsaredrawnforcomparison.(Solidcurves.)

Thisrevealsthatthemaineffectsdonotvaryoutstandinglyeven

wheninteractioniS･supposednottotakeplace.AsseeninFig.5.2｡

despitethefactthattheinteractionswerenotsosmallascompared

-90-

200

100

50

2 0

1 0

｡nUOUOJUOIJOZIIUUIJONJ

5

2

1

..､し､.｣｡|し1''‾ ｀｀ゝI' -ヽ,,二ヽ4

/~∽~-ぺ

/ﾉ´V,ゝゝ

/ゝ

/ゝχ

χ

L. /I

･ へ゛゛/.″

,1｀＼..゛｀ヽ､/ぺ｀､χ/'｀V,″

｀､｀y7χ

k

ｽﾞﾉX///tゝX

＼■/'W7fいぺ

＼c/ 'X＼//ヽ､へ

＼/でｼ/″iNへ､Vゝ

ゝ

ゝゝ

ｿ//

ﾉﾚIN＼ /.'｀､ -

/,｀＼ /,ヽ へ

/!ヽX //χ へ

/･｀ヽ＼ //｀ヽ ､へ/1゛｀ゝうぐ/// ----

W

nj Xゝ

ゝゝ

ﾉ

1

7ﾉ

/べ＼

u//■･■■ぐ〉心

○ﾉゝべ

ゝ

ゝゝゝ

ゝ

W -

J

〆ﾉ

●-Sが-〆〆〆

f/

/

/●,●

/

t3 t5

1

t7 t9 t11

Fig.5.5Multivariateanalysisofvarianceforfour-factordesign

withrepeatedmeasurementsinwhichallinteractions

aredisregarded(dashedcurves一一一一).Solid

curves()indicatethemaineffectsofFig.5｡2･

-91

t13

withthemaineffects,thesemaineffectsdidnotchanpeso

greatlyevenwhenalltheinteractionswerepooledtotheresidual.

Accordingly,itispredictedthatthedirectionsofspatialdistri-

butionofthemaineffectsaredifferentfromthatoftheintﾆeractions.

Toclarifytherelationbetveenthedirectionsofdistﾆributionsofthe

maineffectsandinteractionsbythesameprocedureasinSectionl

4.5,wetrytoexplainthedistributionsgeometrically･

Q3+R:VariancebvthemaineffectoffactorC2.

(!latrixofsumsof･squaresandcrossproducts)

Q4十R:Variancebythemaineffectoffactorv2･

(MatrixofSt!msofsquaresandcrossproducts)

Q10十R:VariancebyC2V2-interaction.

(llatrixofsumsofsquaresandcrossproducts)

R:ResidualVariance.

(Ilatrixofsumsofsquaresandcrossproducts)

Threee1:Lipsoidsfori(=3,4,10)expressedbyanequation

i[÷(Qi+R)]‾lil=i(ft(Qi+11)11‾もX=1

andtheresidualellipsoidby釧士R']~'x'=xx'=l

(5.19)

areillustratedinFig.5.6.Inthefigure,thecrosssections

producedbythethreeplanes,whichareproducedbythefirstmain

axesofthethreeellipsoidsexpressedby

i[まｰ(6i+11)]‾lil°1(i°3,4,10),-

aredrawn,too.

AccordingtoFig,5.6,thedirectionofthemainpartofdistri-

butionofC2V2-interactioni13nearlyatrightangleswiththatofC2-

main-effeetﾆandV2-inain-effect.犬

Performanceofanalysisofvarianceuponthenewresidualto

-92-

whichtheinteractionarepooled(asconductedinthissection)cor-

respondstospatialshrinkoftotalvarianceQinthedirectionof

distributionoftheinteraction,buttheoperation-shrinking

doesnotinfluencetotallymuchondistﾆributionofthemaineffect

sincethedirectionofdistributﾆionoftﾆheinteractionisnearlyat

rightangleswiththatofthemaineffects.Hereby,agreatchange

maynotbeseeninﾘ゛ofFig.5.5｡

c,v.

.VZ＼

Fig.5.6

c.

Geometricrepresentationofdistributionsofv2-effect,

C2-effectandtheirinteractionCaV^(t=tu).

5.7Conclusion

V'echoseonlyonespeakerbecauseitwastroublesometofind

outthecharacteristicswhichwerepurelydependentonco-articula-

tionこandwhichwerelostintheincreasingfluctuation-noise-

causedfromchangingspeakerifwetreatedvoicesofvariousspeakers

atonce｡

Performedmu:Ltﾆivariateanalysisofvariancewithrepeated

measurementsonfourfactorsCi,Vi,C2,V2,rakingtﾆwo-syllable

words/CiViC2V2/anobjectofanalysis.!‘7einvestigatedtheinterac-

tionvjhichindicatestheextentoftheeffectspecifictothecombi-

nationsofmorethantwoofC1,Vi,C2,V2,besidesthemaineffect

-93-

whichindicatestheextentoftheeffectofCl,Vl,C2,V2themselves｡

Theresultofanalysis:

(1)Themaineffectofthephonemewhichweremarkedwas

thelargestamongalleffectsatitsstationarypart:'･

(2)Theinteractionbetweentherema万rkedphonemeandthephoneme

whichwasjustcontiguoustoitwaslargerthanthemaineffectof゛

othernon-contiguousphonemesthoughitwas,ofcourse,smallerthan

maineffectsoftwocontiguousphonemesthemselves.

(3)These(1)and(2)signifythemainpartsofthetotalvariance･

Therefore,whenwetakeacertainphoneme,wemayhavetotake

intﾆoaccountoftheinfluences―maineffects―justprecedingand

followingphonemes,butdonothavetoconsidermaineffectsof

phonemesfartherthanthem.'Ifwe:Intendtoconsiderthelattereffects,

wehavetoconsidertheinfluencetwo-factorinteraction一一specificto

thecombinationoftheremarkedphonemeandjustprecedingone(O｢

゜followingone｣priortoconsideringthemaineffectsoffarther

phonemes｡

Thetwo-factorinteractionisnotsosmal:L,buttoneg:Lect

itinanalysisofco-articulation.doesnotyieldasignificant

differenceintheresultsofthemaineffect,becausethedirection

ofdistributionofthetwo-facterinteractionisdifferentfrom

thatofthemaineffects.(ThisfactStﾆrengthenstheanalysis

model0fChapter4.)

-94-

CHAPTER6

AnalysisofJapanesecvandvvuttﾆerances

6.1工ntroduction

Nopaperseemstoverifythehypothesisthattheinfluenceof

consonantonvowelinc-vseauenceislargebuttheinfluenceof

precedingvowelonfollowingvowelinv-vsequenceisfurtﾆherlarger｡

Theeχperimentofthischapterwasschemedtoobtainoneanswer

tothisproblem.Inthescheme,weprovidedsyllab:Le-sequencesof

t:ypeCV2(consonant-vowel)andV1V2(vowel-vowel),investigatedthe

influenceofConv2andthatofVionV2,andcomparedonewiththe

other.Thespeakerislimitedtoonemaleadultﾆ｡

Nowwecannotuti:Lizetheusualmodelofanalysisofvariance

inthiscase.工ntheusualmodel,forexample,ink-factormodel.

t:hetreatmentswhichinhereineachofallkfactorsaresimulta-

neouslycarriedoutsurelyineachexperiment.工nthischapter,C,

Vl,V2standforthefactors.However,evenifweschemeanalysisof

varianceforthree-factordesignontheassiffliptionthatthesetﾆhree

arethefactors,theydonotactﾆalltopethertowardspeechspectraZ･

Itisimpossibletoschemeanalysisofvarianceforthree-factor

design,because/cv./and/ViV,/areuttﾆeredasdifferentwords｡

Meanwhile,itispossibletoschemeatwo-factorexperimentfor

CandV2,andanothertwo-factorexperimentforViandv2.Asthere

is,however,noconmunityintheresidualsofthecoupleofexperi-

mentsintﾆhiscase,anditisimpossibletocomparethevaluesobtain-

edbythecoupleofanalysisofvariance,wecannotcomparetheamount

oftﾆheinfluenceofConv2andthatofv10nv2｡

y･'etriedtoeχtentthemodel0fanalysisofvariancefortwo-

factordesigntooneofthemethodswhichmustbeusefultoresolve

-95-

theproblemofthischapter,standingonthismatter.Wewilltempo-

ra:Llynametﾆhemodelsuggestedhereanalysisofvariancefor"Divided-

Type"two-factordesign.

6.2 MultivariateAnalysisofVarianceforDivided-TypeTwo-FactorDesign

AsmentionedinSeとtion2.4.3,theusualanalysisofvariancefor

two-factordesignwithrepeatedmeasurementsisasfollows(Table6.1).

Considertwofac‘torsA,EandpresumetheirlevelstobeA1‾"a>and

Bi゛Bi^,respectively°VJhenwelet^iji゛‥"゜x;ijr

observations(thenumberofwhich･isr)obtainedfrom

combinationsof(Ai,Ej),thelinearmodelisassumed

thecondition

“゛1≦i≦a,

1≦j≦b,

や■■■

1≦k<r.ヶ

where,

arjjkClxp)=μ十鴫十βj゛''ij+゛ijk

aΣ［

(Zi=0 ら｡Σ戸

^jk

a

ﾃﾞ0'.Σnj=

1=1

bΣ釦

rij=09

~.N(0,A):p-dlmensiona:Lnormal

distﾆribution.

bethe

the

tobeunder

(6｡1)

(6.2)

(6.3)

Dividethefactc?rA(whose1叩91sareAI~Aa)illtoAandj2)如

thismodel(Table6.2).Namely,assumeやheleve1Sofjl)tobe

-96-

Table6.1Multivariateanalysisofvariance

fortwo-factordesignwith

repftatedmeasurements.

Table6.2Multivariateanalysisofvariance

fordivided-typetwo-factordesign

withrepeatedmeasurements.

-97-

(6.9)

AI‾"ai9andthatofAtobeAal.1‾Aa(1≦ai<a).

Accordingtothisdivision,weintroduceanewmodelhereinafter.

*iik=μ十μ(t)十αi(tﾘβj十..(t)十γり(t)+61jk

o巾:1ｹ≒ﾚ

Undertheconditiona2°a‾a19(t°1,2)

atμ(1)+32μ(2)こo'

C3Σ一｢

印y"

aΣ心

bΣり

｣一

~j

6Σ坤

aΣこ

i=ai+i脂

=0

bΣ

Jﾆ

･1べ

1

βに0'

)十a2r(F=0

ｼﾄか卜o･

りk ~N(o,A):p-dimensionalnorma:Ldistri‘bution｡

(6.4)

(6.6)

(6.7)

TherelationonparametersbetweenEq.(6.1)and･Eq.(6.4)isas

(1し1″‾元tl

即ΣＥ

川1町二石

引Σ図

μ(2)_1a{z?}=αi‾μ(1)(1≦i≦31)'(z12)ﾆαi‾μ(2)(1り‾1≦i≦3)'

a2iﾆal+1'

r･.謬し上£r

IJ.Ja2i=al+1

ra)=γij‾｢(2J

IJ

ij.rWﾆ'uJ

(al+1≦i≦a)

(1≦i≦aべ

(6.8)

and8stoμ'βjor゛ijk'itisthesame8SEq'(6.1).

HereinﾀthemeaningoftheparametﾆersofEq.(､6.4)isasfollows:

(1)μ(1) ee

(2)α(1)i:

(3)(Z(2)i:

(4)β丿

(5)ざ1)js

(6)r(1)ij

(7)r(2)ij

11aineffectcorrespondingtodi゛xsionAorfactorA

llaineffectofleve:LAiwhichbelongstojl).

MaineffectoflevelAiwhichbelongstoぷ).

MaineffectofBj.l

工nteractionofぶ)andBj･

Interaction‘of･B･･and:LeveﾋLAiwhichbelongsto即).

InteractionofB

l

andlevelAiwhichbelongstoA.

-98-

1)

.TJgwill,then,trytobreakdownthetot:alvariance.Inthe

modelofEq.(6.1),thetotﾆalvarianceisbrokendowntﾆotﾆhethree

metricesofsumsofsquaresandcrossproductsformaineffectsαi,βj

andinteractionT'iandoneresidualmatﾆrixasindicatedinEq.(2.34),

whileinthenodelofEq.(6.4),thebreakdownbecomes

abr

Q(p勺)へEI

ふ

k

E

I(゛りk‾゛√゜゜)'(゛りk‾゛“‘)

=Q1十Q2(1)十Q2(2)十Q3十〇4十Q5(1)十Q5(7仙R.

両㈲

1223J-

0c､QO､Q

=br

rrrr

bba

====

Q5(1)=｢

Q5(2)゜r

i

R

where.

2

Σり引ΣＪaΣ・JbΣＪbΣＪ

3t(x?!.-X...)へ(X!t!.-X...)

(Xi..-ぷ宍.丿(x-^..-X...),

1゛1

(X･j.‾ぷ･¨丿(X°卜-X...)

1

27?一

tl

･j

≪"ΣTi

ac-c

゜ﾔ1

It(X＼‘;.-z｀ご‥-x･j'十ぷ‥'y

･(刈.∠χ(jt.ら;.j.←ﾄぷ‥.)

ぷ

Z

いJfjj.X＼･.一鶏.+x呪.)'

･(xり.-X;･.一刈.+jt･)

j

ｼ゛ij°‾゛i‥‾鶏

ﾆx?..y

rΣ淵

bΣJ

･j

aΣゞ=

:L

--゜゜‘abr

(1)

●●●

x. ●●

岬

りj

一

一

一一

一一

1

aΣが

'II

ai

Σ

l

b7ｔ一b[-J

j=1

l

rﾏﾑ一一r？乙

bΣｺりΣ

りjk

1

rΣ一一

k

rΣ一一

k

^ijk"

りjk･

耳jk

Xi jk

(?》‥=12‘ぷ.･･a2bΓμal+

Z･j

(6.10)

bC

1j=1k=1りk

a｢

=

ﾕΣΣzりk･

ari二1k=1

fjpH -

a2｢

a｢

i=ai+1k

E

I゛X7りk

-99-

aibriΞ1

1

ごｰﾆbrに

L

1

rΣ一一

k

aμ

-

Thesematricesofsumsofsauaresandcrossproducts(representedby

Qjcorrespondtothep.eaninps(1)~(7)ofeacheffectofF(!.(6.9)

asfolloV7S:

Qi●

●

●一

切％

●一

j0

9｀

0.一

″

5-ぐ

(7)

(2)･sQ2(か:(3)･Q3:(4)･q4:(5)'Q5(1):(6)

(6.11)

V7hereRrepresentstheresidualmatrix.

Assuming,forinstance,thatH:μ(1)゜μ(2)'゜0(DivisionofA(1)

andA(2)iSnotSignifiCant),WeCantestRsinceitispossibleto

provethat

V={n一見2--}(p十見l+1)}1ogJjhに.^J

(6.:L2)

isasymptoticallydistributedaccordingtox2-distributionwhose

degreesoffreedomisp£i,where£1°1,ll十兌2°ab,n-ab≧p,n°abc.

VJecanalsotesthypothesesconcerningothereffectsinthe

sameformofthecriterionasEq.(6.12)byusingQ:andtheresi

R.(SeeTable6.3,andAppendixG.)

Table6.3Multivariateanalysisofvariancefordivided-typetwo-factordesignwithrepeated

measurements,anddegreesoffreedomofQy).DegrleesoffreedomofRis

n-ab(n=abr),andぷ1十ぷ2=ab.a1十a2=a.

(i)ofEq.(6.9) Factor Effectvector 卵) 瓦=訟つ(も片≒

(1)A(1)-λ2) μ(l) Q. 1

(2)I A(1) (Zi(1) Q?) ai-1

(3) A(2ﾄﾞ (Zi(2) 好 a2-1

(4) B βj Q3 b-1

(5) (A(l)-j2))B rj(｀) Q4 b-1

(6) A(1)B 7'ij(1) 必) aib-ai-b十1

(7) A(2)B r..(2) /破) a2b-a2-b+1

-100-

6.3Establishi"entofObjectsandFactors

forExDerinent

17edrewmaterialsfromthewordsofCV?(consonant-vowel)and

VlV2(vowel-vowel)syllabieutﾆteredbyamaleadult.

Aswechosearbitraryoneof13kindsofconsonants/p,t,k,s,h,

b,d,g,m,n,g,r,z/,andarbitraryoneof5kindsofvowels/a,i,u,e,o/

witﾆhwhichweprovidedasmanycombinatﾆionsaspossible,therewere

90kindsofwords(90=13×5+5×5).Thetotalnumberofmaterials,

providedbyutterinseachwordfivetimes,was450.

Although/ti,tu,di,du,zi,si/amongCV2-combinationsarepro-

nouncedbyphoneticsymbol[t/i,tsu,3i,zu,3i,/i]essentiallyin

Japanese,respectively,theywerepronouncedaccordingtothephonet-

icsymbols[ti,tu,di,du,zi,si]inorderthatwecanhaveobservations

inallpossiblecombinationsof:Levelsfromeachfactor.

Vedeterminedthreepointscorrespondingtoboundary,transition

partandstationarypartﾆofeachwordbyvisualobservation,andnamed

themt=ti,t2,t3(SeeFip.6.1.).Analysisofvariancefordivided-type

twofactordesign,mentionedinSection6.2,wasperformedforeacht,

consideringtheSDectrumdistributionsoftﾆ1>t2,t3asvectors.The

factorswereassignedasA(1)゛VI,A(2)゜C,B=V2.Consequently,ai°5,

a2°13,b=5,c=5.Define

x.(t)=log(

bjCt)-

沁aΣ四

ｿ

), (6.13)

^^;hichisthei-tﾆhcomponentofp-dimensionalvectorX(t)=(XI(t).

,x(t))attimetﾆonthebasisoftheamDlitudeoutputs

bi(t)･‥･･b*.(t)of20-channel1/4-octavefiltersatthattime

(p=20).

-101-

が

apiijiidLuv

aprnndxuv

V,

tl t2 t3

V2

tlt2t3

Fia6.1cv.andv,v.words｡ddefinitionofti.

t=t,:boundaryofC-V2orVi-V2,

t=t3:stationaypartofV2･

t2=(t3十tl)/2.

6.4ResultofAnalysisofVarianceforDivided-TypeTwo-FactorDesign

Analysisofvariancewasperformedateacht(=ti,t2,t3)toward

thefactorassignmentdescribedinthesectionaboveaccordingto

procedureeχplainedinSection6.2.T･7ecomputedthecriterionﾘ

forthenullhypothesesconcernedwithseveneffectssignifiedinEI｡

(6.9),respectively.Asthedegreesoffreedomsforeacheffectare

notthesame(seeTable6.3),thecriterionVwasnormalizedbythe

-102-

correspondingvalueof1SsignificantlevelofX^

equatﾆion.

りl=り

test,asthefollowing

test

(6.14)

correspondingtothedegreesoffreedomv)

Thenormalizedcriterionﾘ゛iSillustratedinFig.6.2.Symbols,in

thefigure,correspondtoeacheffectofEq.(6.9)asfollows.

(C-V1):(1),Vi:(2),C:(3),V2:(4),(C-VI)V2:(5)

VlV2:(6),CV2:(7).

50

000

4CO(N

ヽnUOUDJUDpOZIIUlUJOM

10

jC;>

ぐV2

V,

ﾐ;

(C-V0V2･

"〆〆〆

(6.15)

〆

tlt2t3

Fig.6.2Multivariateanalysisofvariance

forthedivided-typemodeL

V,=/a･i･u,e,o/l

C=/p,t,k,s,h,b,d,g,m,n,o,

r.z/.

V2=/a,i,u,e,o/.

-103-

t

Theillustrationreveals:'

(1)Thewayofinfluenceoftheprecedingphonemeisconsiderably

and｡significantlydifferentduetowhethertheprecedingphonemeis

avowel0ra･consonant｡

(2)Theeffectofprecedingvowelintheregionoffollowingvowel

islargerthanthatofprecedingconsonant,butmakesnogreat

difference.1.｡.Iﾄ

6.5Re:lationbetweenNormalizedCriterionｿﾞand

DiscriminationScore

Accordingtotheresult(1)oftheSection6.4,thereisa

possibilitytodiscriminatewhethertheprecedingphonemeisvowelor

consonantonlybythespectruminformationofthestationarypartof

followingvowelsincetheinfluenceoftheprecedingphonemeon

followingvowelbecomesconsiderablydifferentbetweenthecases

onecase;precedingphonemeisvowel,andtheother;consonant.

-104-

Therefore,wetrieddiscriminationadoptingtﾆhedistancebasedon

quadraticform.(SeeSection4.4forthemeansofthediscrimination

whichisexactlythesameasthemeanshere.)

Thediscriminationstriedareasto:

(1)Vrhichistheprecedingphoneme,consonantorvowel?,

(Numberofcategorles=2)丿.｀

(2)T'Jhichisthefollowingvowel,/a‘/,･/i/,/u/,/e/or/o/?,

(Numberofcategories=5)ｲ･

using'onlyspectruminformationxCtj)atthetimet°ti･

These(1)and(2)arecorrespondingto(1)and(4)ofEq.(6.9),

respectively･

Abovewasconductedrespectivelyateacht(゜tl,t2,t3).t°tl

indicatestheboundarybetweenprecedingandfollowingphonemes,and

t°tろthestationarypartoffollowingvowel.Fromtheresu:Ltofthe

discriminationpresentedinTable6.4,thefollowingweresignified:

(1)Thetendencyonthediscriminationscoreissimilartothatof

V゛ofFig.6.2:

(2)1｡7ecoulddiscriminatﾆewith99.8%correctnesswhetherthepreceding

phonemeisvowel0rconsonantonlybythespectruminformation

inthestationatypartoffo:Llowingvowel.

Table6.4Discriminationscores(%)｡

FactorNo.of

c°!tegonesh t2 t3

V,―C 2 99.1196.9

|

99.8

V2 5 96.2 99.8 100

6.6GeometricRepresentationofAnalysisofVariance

forDivided-TypeModel

＼'etriedaspatialexpressionofvarianceellipsoidindivided-

typeanalysisofvariance.Thisdiffersalittlefromthespatial

expressionoftheusualanalysisofvarianceIntroducedinSection

4･5｡

ThevarianceoffactorAf1)isregardedasQ2(1)十R,andthatofjj

as02(2)十R,buttheseareconsideredinthespacenormalizedby

residualR.FromthediscussionofAppendixD,thevarianceofA

andA(2)isasfollows(n=abr):

1

-n

･1

-n

2

(Cぐ

(O'

ぐ

(1)十ii)=ぐ士(Q2(1)十R)R‘士,

2(2) +R)=r"士(Q2(2)-ﾄR)R°士

Sincethecentersofthevarianceellipsoids,which

-105-

(6.16)

(6.17)

and士(Q24)十R)signify,areontheendsofthevectorsofmaximum

like:Lihoodestimates

;;(1)=x(1!‥-z...3

;誤)=z(2!‥-z….

(6.18)

(6.19)

obtainedfromvectorsμ(1)μ(2)ofEq.(6.A),respectively(SeeAppendix

1ぶ(l),nG),thecentersofthevarianceellipsoidssignifiedby一手(Qj

and÷(02R十貫)arerespectivelyontheendsof

j(1)=(z(IL.-z‥.)(R/n)‘士,

ぴ)=(x(21‥-z‥.)(R/n)'士.

(6.20)

(6.21)

FromEquations(6.20)and(6.21),theconcentrationellipsoids

offactorA(1)andA(2)are,reSpecrively,

(i-i(1))[十(a2(1)+ii)]-1(i-i(1))･=p+2

(i一夕)[十(62°)+ii)]-1(

wherep=20.(SeeSection2.2.3.)

i-;y))l=p+2

(6

(6.

22

23

Assume!:heeigenvectorscorrespondingtothemaximumeigenvalue

ofthematrices･ofEq.(6.16)andEq.(6.17)tobe61,62>respectively･

Also,as

~(1)+aj(り=o (6.24)

;;(1)and;i(2)areononestraightline.Presumethatthevectorwhich

representsthislineisa｡

ﾚTheillustrationoftheintersectionsproducedbythetwoplanes

-106-

(determinedbya,ろ1andbva,62)andtheconcentrationellipsoids

ofEq.(6.22)andEa.(6.23)(inthissection,A(1)=VI,A(2)=C,B=V2)iS

asI^ip.6.3(onlyt=t3).

c:上(司2)

:十(郁1)

4) ﾉ恥

Fig.6.3Geometricrepresentationof

distributionsforthedivided

-typemodel(t=t3).

V,=/a,i,u,e,o/,

C=/p,t,k,s,h,b,d,g,m.

n,i),r,z/.

α

4)

･S.

ItisrecognizedthattheinfluencesofprecedingconsonantC

andprecedingvowelViwereseparatedatthestationarypartof

followingvowe:LV2.

-107-

6.7Discussion

Toinspecttheresultsoftheanalysisuptothesectionabove,

wechangedthedivisionoffactorA,andcompareditwiththedivision

inthesectionabove.

LettingPrepresentﾆthegroupofvoicelessconsonantﾆS/p,t,k,s,h/

(31°5),andTvoicedsounds/b.d.g.m.n.o.r.z.a.i.u.e.o/(a2=13),

theanalysiswasperformedundertheconditﾆionj1)=P,か)=T,B=V2,

providedthatAI)and瓦2)arediviSionSoffactorA.Theresultsof｡9

performingsimi:LaranalysisonSection6,4,6.5and6.6areasFig･

6.4,Table6.5andFig.6.5,respectively･

Itisunderstoodbyconparingtﾆheseresultswitheachother

thatthedivisionofC-Viissignificant.

Table6.5Discriminationscores(%).

FactorNo.ofcat^ones

tl t2 t3

P-T 2 97.1 87.1 79.6

V2 5 96.2 99.8 100

-108-

ゝ

こゝ

UOUDJUODDZITUUIJOI^

50

40

30

20

10

PTぐ

-

2

ヽ､､_ど認否1.

TV2PV,

tlt2t3

Fig.6.4Multivariateanalysisofvariance

forthedivided･typemodel.

P=/p,t.k.s,h/.

T=/b,d,g,m,n,n,r,z,a,i,u,

e,o/,

Vj=/a.i,u,e,o/･

-109-

T:古(辺2)fi)

ﾉ

如し

a

Fig.6.5Geometricrepresentationofdistributionsforthe

divided･typemodel(t=t3)･

P=/p,t,k,s.h/,

T=/b,d,g,m,n,g,r,z,a,i,u,e,o/.

-110-

&2

6.8Conclusion

Theresultsofanalyzing450words(whosetypesarecvandvv)

uttﾆeredbyamaleadultareasfo:Llows:

(1)Thewayofinfluenceoftheprecedingphoner!eonthefollow-

inpvowelisconsiderablyandsignificantlydifferentduetowhether

theprecedingDhonemeisvowel0rconsonant.

(2)Themaineffectofprecedingvowelintheregionoffollowing

vowel‘islargerthanthatofprecedingconsonant,butmakesnogreat

difference.

(3)Vecoulddiscriminatewith99.8%correctnesswhetherthepreceding

phonemeisvowel0rconsonantonlybyspectruminformationinthe

stationarypartoffollowingvowel｡

Theusualmodelforanalysisofvariancecouldnotbe

utilizedintheana:Lysisofthischapter.Instead,weintroduced

newlya'Divided-type':modelforanalysisofvariancefortwo-factor

design.＼-!etﾆriedtoexpressspatiallytheresultofanalysisof

thismodelinaparallelwithit,andconfirmedtheresultabove.

-Ill-

り2項欠

CFAPTER7

Three-DimensionalRepresentationofJapanesePhonemes

7.1工ntroduction

!･7ehavesofartriedtoexpressspeechsoundonacoordinate

planeorinthree-dimensionalspaceontﾆhebasisofparametersobtain-

edbyanalyzingspeechsound,andtoobservetﾆheirmutualrelations.

F1-F2formantplane,inwhichthefirstandthesecondformant

frequenciesareregardedasparameters,isamostpopularexampleof

them.Klein,PlompandP01S(:L1)havetriedtoplot12kindsofvowels

of50speakersinspacebyspectralinformation.Consideringspectﾆrum

representedbytheamDlitudeoutputsof18-channel:L/3-octavefilters

as18-diinensionalvector,tﾆheyperformedtheprincipal-component

analysis.They'theninsistﾆedthatifpresentvowelsontheplane

determinedbythefirstﾆtﾆwoprincipalcomponents,itsconfiguration

issimilartotheconfigurationinF1-F2forinantplane｡

Thusbyrepresentingphonemesoundbyrelationofspatialdis-

positﾆion,itispossibletogivesupport,socalledinphonetics.

toac:Lassifiedtableofphonemesandtoacardinalvowelfigurefrom

aphysicalpointofviewiftheresultscoincide.Moreover,there

wouldbemuchmoreprofitableaspectsincetﾆhemeasureofdistance

wasimported｡

I^enitcomestoconsonants,however,expressingexamplesin

thiswaywouldberare.Themainreasonistﾆhatitisnoteasierto

analyzeconsonantsthanvowels,fortherearesomedifficultieson

catﾆchingspectraofconsonantsattheirinstantofexplosioninsuch

caseasplosiveconsonants｡

Thisstudytookninephonemes/p,t,k,b,d,g,m,n,andD/asa

preliminarytothistrial0nspatialexpressionofphonemesincluding

-113-

consonants.Veexpressedthesephonemesinthethree-dimensional

spaceafterdefiningtheirspectra.Itisspontaneouslyindicated

thatthisexpressionhasaprofoundstatisticalmeaning｡

Furthermore,thischapterusedthesamedataasChapter5and

partiallytheresultsofChapter5becausebothexperimentswere

designedatthesametime.

7｡2cvcvUtterancesandtheirSpectra.

Itisactuallyrarethateachphonemeisindividuallyuttered.

Par!二icularlyinJapanese,itisconnnonforconsonantstobeuttered

withvowelslike/ka/.liepayattention,here,toC2oftwo-syllable

CiViC2V2(consonant-vowel-consonant-vowel)inordertoconsider

consonantswhichareplacedinmuchmorecomplicatedcircumstﾆances

thaninamonosyllableCV｡

Using/p,t,k,b,d,g,m,n,0/forCi,C2and/a.e.o/forVl,V2,

anadultmanspokeeachwordofallthecombinations―729kinds

threetimesatrandom.HeaccentedalltheCiVi-moradispitﾆe

-114-

thefactthattherearea10tofmeaninglesswordsinthecombinations

above.Thesoundswereutteredinthesimlifiednonreverberantroomp

withinonedayandthetotalnumberofwordswas2187.Thereason

forchoosing/a.e.o/asvowelswasthatwecanhaveobservationsin

a1:Lpossiblecombinationsoflevelsfromeachfactorbyeliminating

/iandu/since/tl,tu,di,anddu/donotoccurinmodernJapanese･

Furthermore,thesespeechdataarethesameasthoseinChapter5.

Wewillgiveadefinitionofspeechspectraandprocedureof

segmentation,below(detaileddescriptionisinChapter3).

Whenletbi(t),.....,b(t)(p°20)represent.inordeらamplitude

outputsof20-channel1/4-octavefi:Lters(bankof20filterswhose

centerfrequenciescover210upt05660Ez)attimet,theyare

consideredtorepresentspeechspectraatthattime.Afternormal-

izingtﾆhesquaresumofthesecomponentsat1,weestﾆablishedp-

dimensionalvectorx(t)bytaking:Log;arithir.ofitscomponents･

Thatis,

Xi(t)゜:Log{bi(t)//1♂(t)}J=lj

(7.1)

!'7edefinedp-dir゛ensio°1゛ectoi°x(t)゜(xi(t)･‥･･Xp(t))゛ithEq･

(7.1)whichwouldbeusedfortheanalysis(p=20).Amplitudeoutputs

ofthefilteranalyzerareA-Dconvertedatintervalsof10ms,tﾆhen

putintothecomputerinrealtime.0nthepaperfortheline-printer

werepresenttﾆhespeechspectrumpatternsoftheinputwithwhichwe

observedtodeterminethestationarypartsortransitionones.

ThisStﾆudydefinestimest1,ぢ2≫･･･≫tl3asFig.ﾌ.1,correspond-

inptothestationaryortransitionpartsineachword.

oprmidiuY

tl t2 t3t4 t5 t6t7t8t9trotilt12t13

Fig.7.1c,v,c,v.wordanddefinitionofti-

t=t,:stationarypartofc..

t=t3:boundaryofc.-V.,

t=t5:stationarypartofV.,

t=t7:boundaryofV,-C2,

t=t9:.stationarypartofC2,

t=tn:boundaryofC2-V2,

t=t13:stationarypartofv2･

Ifiiseven.ti=(ti+l十ti-i)/2.

-115-

t

7.3ConsonantSpectﾆra

Itisaimportantproblemwhetherconsonantswithindifferent

wordsshowcharacteristicsdeterminedindependentlyoftheirphonemic

environmentornot.工fthecharacteristicsofconsonantsdifferin

accordancewiththeircircumstances,wecannotphysicallymeasure

thembutonlybyauditorypsychologicalmeans.Analvsisofvariance

･●|●●ilforfour-factﾆordesignwithrepeatedmeasurementsonCiViC2V2Word,

inChapter5,answersthisquestion.LookingatFig.5.2inSection

5.4,itispossibletopersuadeourse:LvesthattheeffectofC2is

sufficientlysignificantascomparedwiththeresidualatt=to,t,,,

etC･,andthatthereexistdefinitephysicalcharactﾆeristicswhich

areindependentofthespeechconditions.

Todealwiththeargumentabove,wewillpresentagaintﾆhe

expressionofthelinearmodelEq.(5.1)usedinChapter5.namely,

wepresumealinearmodelrepresentedbythefollowingexpression

forspectrumvectorattimet=ti.

U

j(=1~b)thelevel0fthesecondfactorVi,k('=:L~c)theleve:Lof

thethirdfactorC2,1(=1~d)thelevel0fthefourthfactorv2,

respectively,andm(゜1~e)representstherepetition(a=d=9,b=c=e

=3).Omittingt,

xijklm(1xp)゜a;

4‘“けβけ7i,゛∂l

generallevel

maineffect

ｷsij+ぐik+ηil+∂jk+λj1ヰμ■kltwo-factorinteraction

+ｼﾞijk+pijl+･ikl+'■jkl

+91jk1

+xijklm

-116-

three-factorinteraction

four-factorinteraction

residual

(7.2)

Determine

Σ

i二1

Σ

i=1

(Zi

^ij

--

a

Σ

J二

b

Σ

j=1

一一

J

a

Σ一一I

I

b

==Σ

j二l

b

£

β

j

=Σε

i=1

so astosatisfytheconditionofthefollowingequations.

dl;.=Σ･J.=0.

.a

り゜ぶ1ぐilf―2ぐikづﾐｮ1ηa=l1'i>

λj11が1λj1ﾍｷﾞﾗA'd

‾ぷ/ki‾0

ｿijk°

b

2/jki

Pijkl

bC゛ぶjk°ぶ

1

∂jk

abda

μijk°ぶ/ijl°ぷ/ii.1ﾆFIpり1二Σﾐjikl

Cd°Jjjk11?F7jkl゛0

bC(1

°j弓ざりkにぶ?りklﾆぶ/ijkl°0

dΣ

'^ikl

(7.3)

Zりklm~N(0,A):p-dimensionalnormaldistﾆribution.(7.4)

AsweseeinFig.5.20ftheanalyzedresu:LtinSection5.4,

aspectrumofanysectioninautterancedoesnotrepresentone

phonemeonly.Influenceofotherarticulationally-comblnedphonemes

alwaysintervenes.Therefore,it1Sinsufficienttoconcludethat

spectraぶりklm(t)measuredwithintheregionsofconsonantsrepresent

spectraofconsonants.

Att°tilwhenspectraofconsonants/P,tﾆ,k,b,d,g,m,n,o/are

definedasmentionedlater,tﾆheeffectofthevowelV2issoover-

poweringthattﾆhecharacteristicsofconsonantmaybelostinthe

vowelifxりklmitselfisusedasspectrumofconsonant.

Wewilladoptmaineffectvectﾆorr.(k=i~c)ofC2inthelinear

model0fEa.(ﾌ.2)asspectrumofconsonant.γkisavectorwhich1S

determinedonlybythek-thconsonantofC2,and1Savalueindepend-

entofitsphonemiccircumstances.工notherwords,7krepresentsa

spectﾆrumwhichisregardedasapurecomponentofconsonantC2after

removinganycomponentconcernedwithotherphonemesexceptC2from

Zりkim゛Althoughtheproblemiswhetherobtainedrl｀4mustbe

sufficientlyandsignificantlydifferentfromeachotheraswellas

tﾆheyareseeminglydifferent,Chapter5clarifiesthispoint.

Wewillnowconsideratwhattimepointtwehadbetteradopt

へ(t)aSaconsonantvector.Fig.5.2saysthattheeffectofC2

-117-

ismaximumatﾆ∧themiddleoftheprecedingvowelandthefollowingone,

thatist°t9.Itseemsreallymaximumonthewholeofthenine

consonants,butwhatwecanonlyclassifyintﾆhistimepointare

mannerofarticulation(!･!1,112,M3ofTable7.1)andnasalsounds(

/m,n,o/):wecanseldomdovoicelessstops(/p,t,k/),(It‘willbe

clarifiedinthefollowingsection.)

Accordinglywewanttoselecta.timepointatwhichthesenine

phonemeshaveequallybalanceddistributionevenifthemaineffect

becomessmall.WepresumedmaineffectvectorsofC2-factoratthe

boundaryofC2andv2(t=t11)tobespectraoftherespectiveconso-

nants.

Actually,maximumlikelihoodestimateof7k

7k°｀･k●.-j;‥‥

isused,(SeeEq.(5.5))

(7.5)

7.4ClassificationfromtheViewpointofllannerand

PlaceofArticulation

Consonants/p,t,k,b,d,g,m,n,D/areclassifiedfromtheviewpoint

ofmannerofarticu:Lationorplaceofarticu:Lationasdescribedin

Table7.1.(17)Weeχaminedwhetherthiskindofclassificationwas

alsoref:Lectedinthespectrumdistribution.C2isinterpretedin

twowaysmannerofarticulation(M-factor)andplaceofarticu-

lation(Pこfactor) asTable7.1.

-118-

AssumeMI~M3levelsforll-factorandPI~P3levelsforP-

factor,respectively.TheseM-factorandP-factorareassignedto

thefirstfactorandthesecondone,respectively,oftheanalysisof

varianceoffour-factordesignofEq.(7.2)orEq.(5.1).AndViandv2are

assignedtothethirdfactorandthefouth,respective:Ly,asbefore.

Table7.1Classificationfromtheviewpoint

ofmannerandplaceofarticulation.

p.Labial

P2 Alveolar

P3 Palatal/Velar

Ml Voicelessstop

M2 Voicedstop

M3 Nasal

liedidananalysisofvarianceontrio-sequence~V1C2V2(disregarding

CifromC1V1C2V2sequence)whichhasrepetitionscorrespondingtothe

numberofconsonantsCi(nineinthistime),regardingM,P,Vi,V2as

fourfactors｡

yeconsider,foreχample,utterancesof/pame,tame,kame,bame.

dame,game,mame,name,Oame/tobeutterances/ame/repeatednine

times.工nthiscaseweusedthefirstutteranceoutoftﾆhree-repeated

utterancesofeachcombinationinCiViCaV,words(2187/3=729data)｡

Normalizedcriterionﾘ'.inthesamewayasinSection5.4,is

showninFig.7.2.TheeffectﾆSofmannerandplaceofarticulation

aresufficient:1-yseparated.EspeciallyamountﾆSofthebotheffects

atpartv2arealmostthesame,andinteractionsbetweenbothare

smal:Latthatpart(lioreindetail,interactionsatv2arelargerin

placeofarticu:lationthanmannerofit)｡

ToverifytheadequatenessoftheclassificationinTable7.1,

performanalysisofvarianceforfour-factordesignofχ,Y,Vi,V2based

uponclassificationmixingelementsofmannerandplaceofarticulation

intentionally.TheresultsareshowninFig.7.3｡

Thus,wecanseethattheclassificationaccordingtoTable7.2

1Snotgood,becauseinteractionofx,Yislargefarbeyondthemain

effectsofthem.

-119-

為

70

50

3 0

uouajuDpgzijBUUoj^

10

t5t7t9til

Fig.7.2Multivariateanalysisofvarianceforfour-

fectordesignwithrepeatEdmea万s万urements.

VI,V2=/a,e,0/,

M=(MI,M2,M3):manner.

'-P=(PI,P2,P3):place.

-120-

t13

t

ゝ

以

70

50

3 0

U0U3JUDpOZIJBUUOJsJ

10

t5t7t9tilt13

Fi&7.3Multivariateanalysisofvarianceforfour-

factordesignwithrepeatedmeasurements.

VI.V2=/a,e,o/,

χ･=(χhχ2,χ3),

Y=(Y,.Y,,Y3).

-121-

t

Furthermore,whythemain

Table7.2Somesortofclassification.effectofViisSma:Llerthanv2is

thatweassumedC1V1C2V2wordtobe

-ViCoV,.工ndespiteofthefact

thattheinfluenceofC1=/p,t,k,b.

d,g,in,n,0/onViisconsiderably

large,wedisregardedit.This

increasednoiseatpartVi,which

accordinglymadetheratiotothe

termofresidualworse｡

Itwouldberelativelyeasyto

classifyinmannerofarticu:Lation

sincetheeffectofitwasenoughlargeattimetg>as"clearinFig･

7.2.So,weexaminedtheeffectsofconsonantﾆSbe:Longingtoeach

mannerofarticulation(Mi,M2,M3),respectively.Schemeanalysisof

varianceforthree-factordesignwithrepetitioncorrespondingtothe

numberofCi(nine),assigningthetﾆhreefactorstoVi,C2,V2intrio-

sequence~V1C2V2disregardingCifromC1V1C2V2.Modelsforthree-fac-

tordesigncanbedefinedinthesamemeansasEq.(5.1)~Eq.(5.3)｡

Theresultsofperforminganalysisofvarianceforthree-factﾆor

designrespectivelyonthreecasesofC2-C2°/p,tﾆ,k/,C2°/b.d,g/.

andC2=/m,n,0/'-arepresentedinFig｡7｡4｡

ItisunderstandablefromFig.7.4thatwecandistinguishnasals

/m.n.O/att=t9,butabsolutelynotvoicelessstops.Meanwhile,all

thethreecasesarealmostequallydistinguishableatt=tll｡

Aboveisexplainedwhyweadoptedγkatt=tllasconsonant

spectra.

7.5ExpressioninThree-DimensionalSpace

Accordingtotheprecedingsectﾆion,wewillexpressthenine

-122

｡nU0IJ31U3|)OZ!|BIIUO;＼I

ヽ4U()U.>}U3pOZiri-'tUJOM

｡/＼UOUOimpOZIIBlUJOM

ts

(1)C2

t5

(2)

t5

=/

t7

t9 tu t13

t

p.t.k/;Voicelessstop.

t

9t11 t13

t

C2=/b,d,g/;Voicedstop.

t7t9 t11 t13

t

(3)Cz=/m,n,o/;NasaL

Fig.7.4Multivariateanalysisofvarianceforthree-factordesignwithrepeated

measurements.VI,V2=/a.e,o/.

-123-

consonants/p,t,k,b,d,g,m,n,o/withvectors

AA∧A∧AAA∧A∧7'1'｀'り=rp゛rt゛rk'rb゛rdJg'rmJ7n･h

(maximumlikelihoodestimatesby2189wordsatt=tll).

Nowwewouldliketoprovidealessthanthree-dimensionalspace

inwhichwecanwellunderstandthebehaviorofmannerandp:Laceof

articu:Lation.工fwecanhavethespacelikethis,wehaveonlyto

projectorthogonal:1-yrespectivevectorsinthe20-dimensionalspace

onitssubspacewhosedimensionislessthanthree.

Let≪ini≪m2representeigenvectors(puttingtﾆheminorderof

largenessofthecorrespondingeigenvalues)ofcovariancematrix:

Sm゛

十('■ptk-'■ptk十''bdg'''bdg十rl°9゛rIIII11)

whichconsistsofthreevectorsof(7.7),thatis.

'■ptk°zp十r§十rk

ﾀ

''bdg,゜jjｾﾞ:‘?Pj!≒'^mnn°jLΞﾋﾞjrﾆjL!･

(Note:''ptk十ﾘbdg十mno°O)

(7.6)

(7.7)

Theextentofseparationconcernedwithmannerofarticulationmust

berelativelyfavorableinthesedirections.

Likewise,supposethat°P1,°P2areeigenvectorsofcovariance

matrixSpwhichconsistsofthreevectorsof･

^pbm°り)十rM十rm

9''tdn°3十7'nﾀ''kgO°

rk十

3/

(Note:りbln十''tdn十''kgo°0), (7.8)

thentheextentofseparationconcernedwithplaceofarticulation

mustbealsofavorable.

Projectorthogonal:Lythe20-dimensionalvectors''p~'"oon

itsthree-dimensinalsubspacewhichisgeneratedbythreevectors

≪mi,≪piand･≪P2.Anarbitraryvector(throughtheorigin)that

-124-

belongstotﾆhesubspaceWIヽgeneratedby{amb°P1.≪p,}.is

representedas

xlaml十x2apl+x3ap2=iA(7.9)

provided,A'(px3)=(Om,≒apl≒ap2゛),X=(xi.χ2,X3).

LetF,)AtobetheorthogonalprojectionofrponW1.Sincerp'-ぢAisat

rightanglestoaarbitraryvectorjcAbelongingtoU1,

iis

r

p

O°(rp‾ ﾃﾞPA･iA)゜(7pA'-rpAA'･i)

arbitrary:then

A'°rp AA',thatis.〉r～

r

p

A'(AA゛j'1

Table7.3r=(7'1,rj.T3)which

istheorthogonalprojection

ofr(1xp)onWi.(p=20)

V｢ γ1 r2 r3

p 4.53 -4.20 -3.2?

t 483 3.48 -0.85

k 3.70 -2.75 1.07

b 1.16 -1.11ﾚ1.92

d 1.00 6.75 1.68

g 0.78 -0.83 4.48

m -5.69 -1.76 -1.55

n -5.83 3.95 -0.81

D -4.48 -3.53 1.17

(7.10)

(7.11)

WemadeTable7.3byobtain-

ing'■p-'"o

-125-

orthogonal

projectionsofrp゛roonWi

throughaboveprocess.The

resultofexpressing'■p-'-o

intﾆhree-dimensionalspace,whose

coordinateaxesare"mi.≪P1

and°P2becomesasFig.7.5.

A11thevectorsinFig.7.5are

movedparallelalongtheaxis

°mi,inordertomakeiteasier

tounderstand.Sothatactual

originisrepresentedbyG.

Thisrevealsthatﾆnine

phonemescomposenearlyatriangularprism,clarifyingrelation

betweenmannerandplaceofartieulation.

/k/

/g/

llI

―･―

/p/

＼

＼

/D/

am,

OP2

3.65

11-

＼

/1

/

/

1

卵

3｡85

/

/

/

/

--

--

75°

-皿-〃"

ぺ

(y

/""

へ

S

90°

//

＼

＼

＼

Residual

/n‘｀

ゝゝゝ

ゝχ＼

/

/

/

X

ゝ

ゝ

/

/

//

/

/

/

Fig.7.5Three-dimensionalrepresentationofJapanese

consonantsal!4!30%-probabaityellipsoidof

-126-

/d/

3.80

/n/

aPi

64

00

0.2

0

0.2

64

00

0.2

0

-0.2

-0.4

1

20channel

250

5

←

500 1K 2K4KHz

Fie.7.6Directioncosinesofam*,am2,Op,andOp,

-127-

工nFig.7.5wealsoshowthe90%-probabilityellipsoid

(whereAp

れi1/n)‘1N-I

Z= X2p (a) (7.12)

(a)isthenumbersuchthatEq｡(2.7)ho:Ldswhenp=3,

(加0.1)correspondingtotheresidual

R(3×3)=(AI(λA･)‾1)･R(A･(AA写1) (7.13)

byiijklm(1)(3)゛hichisorthogonalproiectionof20-dir!ensional

vector■^iiklmontheabovethree-dimensionalsubspace,whereR(20×20)

istheresidualvarianceasinEq.(5.4)andn=abcde.

(SeeAppendixHandSection2.2.2.)

Wecanseeverticesofthisprismseparatedsignificantlyfrom

eachotherascomparedwiththesizeoftheresidualellipsoidR/n.

°P1,

Vesignifieddirectioncosinesofeigenvectors "mi,"m,.

°P2-inFig.7.6.Thepatternsarenevercontrarytothe

traditionalspectralknowledgeonmannerandplaceofarticulation.

7.6Conclusion

Weanalyzednineconsonants,andinvestigatedrelationsoftheir

relativedistributﾆioninthree-dﾆimensionalspace.Theseconsonants

wereplacedatphonemeC20fCiViC2V2Words(a:L:Lofthetotalcombi-

nationsof.Ci,C2=/p,t,k,b,d,g,ra,n,o/,andVl,V2=/a,e,o/).To

beginwithweextracted,fromspeechspectra(20-dimensionalspectra),

componentswhichrepresentindependentlyofco-articulation,then

selectedtheboundaryofC2-V2asasectionatwhichtheseconsonants

areevenlyseparated.VJhenprojectedorthogonallythenineconsonants

onthree-dl皿ensionalsubspacewhichisgeneratedbythreedirections

andpromotesseparationofconsonantsin･viewpointofmannerandplace

-128-

ofarticulation,theytooktheshapenearlyofatriangularprismin

thespace.Itisguaranteedbymultivariatestatisticalanalysisthat

theseverticesaresignificantlydifferentfromeachother｡

Spatialexpressionofconsonantsfromphysicalandanalytical

point二sofviewbasedonactualspeechseems,forthefirsttime,to

betriedinthisstudy,thoughspatialexpressionofvowelshasbeen

investigatedfromtﾆheearlystage.Spatialexpressionofother

phonemesisabouttobeinvestigated.

-129-

ﾄ即項欠

CHAPTER8

Conclusion

Weconsideredthefrequencycomponentsofspeechspectrumas

componentsofmulti-dimensionalvector;performedmultivariate

statisticalanalysis;andunderstoodvariouscharacteristicsof

speechsoundsmainlyco-articulationsanddifferencesofspeakers

quantitatively･

-131-

Startﾆingwiththeconclusion･,wewilldescribethecharacteris-

ticsofthemethodsofanalysisintroducedinthisstudy,andnew

proposalsonthemethodsofanalysis.

(1)Asithasbecomepossibletoanalyzedirectlyspeechspectraby

introducingmu:Ltivariatestatisticalanalysis,analysisofconsonants

hasalsobecomeaseasyasthatofvowels.Onlyvowelscanbeanalyz-

edbyusingthetraditionalformantfrequencies.Themethodadopted

inthisstudyisalsosuperiorInthefactthatspectrumisfavorable

forasimp:Lerexpressionofphysicalinformationofspeechsoundto

themethodusingformantfrequencies.

(2)Particularlymultivariateanalysisofvarianceamongmultivariate

statisticalanalyseshastendedtobeundervaluedaboutpractical

applicationsindespiteofthemathematicalresultsofitstheory.

InthisStﾆudy,however,thismethodofanalysiswasextremelyuseful･

工tmeansthatthepracticalvalueofthemethodofmultivariate

analysisofvariancewas,forthefirsttime,confirmedhere.

(3)Twomatters,intheoryaswellasinapplication,werefoundto

contﾆributetomultivariateanalysisofvariance;

(i)Notionofdirection,besidesnotionofamountindicated

generallybytheanalysisofvariance,wasnecessarytounderstand

theeffectsofrespectivefactorsofanalysisofvariance,andone

ofitstechniquesofexpressionwasdescribed｡

(ii)Amodelofthe'Divided-type'"analysisofvariancewas

proposed.Thisisanup-to-datemodelinwhichananalysisof

variancefortwo-factordesignwasdevelopedbydividingoneoftwo

factorsintomorethantwoportﾆions.Thismodelwasintroducedto

doresearchonpatternsofsometypesofco-articulations,andformu-

lated｡Itwasfurthertriedtoexpressspatial:1-ytheresultofthe

analysis.

(4)Wefoundsometypesof]Limitinthemethodoftheprincipal-

componentanalysis.

Wewilldescribetheresultsonspeechscienceacquiredinthis

study.Themostfavorablepointthroughouttﾆheseresu:Ltﾆsisthat

theywereinvestigatedquantitativelyonsufficientlystatistical

grounds.｡

TheresultsofanalyzingVCV-typewordsutteredbyfivemale

adultsareasf0110WS:

(1)Atthestationarypartofnasalconsonants,theeffectof

speaker-factoristhelargestamongfourfactﾆorS:

(2)Atﾆtﾆhestationarypartofthevowel,theeffectofvowel-factor

isthelargestamongfourfactors.Theinfluenceofthenasalし

consonantﾆcontiguoustothevowelissmallerthanthatofspeaker:

(3)Theeffectofspeakeris,generally,considerab:Lyinfluentﾆial,

v/hilethatofconsonantisnotﾆsoInfluentia:L.Putthedirections

ofthemainaxes,seeninthedistributionsofthethreefactﾆorsof

vowel-effect,consonant-effectandspeaker-effectﾆ,meetnear:1-yat

rightangleswitheachother:く

(4)Intensivecorrelationisfound･betweenvowelandspeaker-factor:

(5)Formantinclinesto"vocalicfactor,"andhasnotsufficient

informationonanyfactorotherthanvowel-factorascomparedwiththe

caseofspectrumやI･･,●･●｡｡●

-132-

Theresults,obtainedfromanalysisofCVCV-syllablewordson

co-articulation,arethen;

(6)Themaineffectofthephonemewhichweremarked1S,

0fcourse,thelargestamongalleffectsatitﾆSstationarypart｡

Theinteractionbetweentheremarkedphonemeandthephoneme

whichwasjustcontiguoustoitwaslargerthanthemaineffectof

othernon-contiguousphonemesthoughitwas,ofcourse,smallerthan

maineffectsoftwocontiguousphonemesthemselves:

(7)Theabove(6)signifiesthemainpartsofthetotalvariance･

Therefore,whenwetakeacertainphoneme,wemayhave

totakeintoaccountoftﾆheinfluencesmaineffectsofjust

precedingandf01:Lowingphonemes,butnothavetoconsidermaineffects

ofphonemesfartherthanthem.IfweIntendtoconsiderl･thelattereffects,

wehavetoconsidertheinfluencetwo-factorinteraction一specificto

thecombinationoftheremarkedphonemeandjustprecedingone(o｢

followingone｣priortoconsideringthemaineffectsoffarther

phonemes.

(8)l･leanwhile,thetwo-factorintﾆeractionisnotsosmall,but,

toneglectitinanalysisofco-articulationdoesnotyielda

significantdifferenceintheresultsofthemaineffect,because

thedirectionofdlstributﾆionofthetwo-factorinteractﾆion1S

differentfromthatofthemaineffect.

FromtheresultofanalyzingcvandVV-syllablewords;

(9)Thewayofinfluenceofprecedingphonemeonfollowingvowelis

considerab:lyandsignificantlydifferentduetowhetherpreceding

phonemeisavoweloraconsonant.

(10)Themaineffectofprecedingvowelintheregionoffollowing

vowelislargerthanthatofprecedingconsonant,butmakesnogreat

difference.

Astotrial0fspatialrepresentationofconsonants:

(11)Thespectra,whichrepresentnineconsonants/p,t,k,b,d,g,m,n,o

-133-

/jweredefinedattheboundarybetweenfollowingvowel.Nine

consonantsof20-dimensionalvectors,whichareprojectedorthogonal-

1yonthree-dimensiona1subspace(thatiscreatedbythethreedirec-

tionswhichpromoteseparationinmannerandplaceofarticulation),

producenearlyatriangularprism.Althoughspatialrepresentation

ofvowelshasearlybeentried,physicalandanalyticexpressions

ofconsonantsbasedonactualspeecharesupposedtobe,forthefirst

time,triedhere.へ

Abovearetheprincipalresultsobtainedinthisstudy.

Problems

speakers

-134-

(1)Evaluationoftheinteractionsbetweenspeechsoundsand

(19).

(2)Spatialrepresentationofa11Japanesephonemes:

,stﾆi11lefthereafter,areinvestigatedbythepresentwriter.

Wehopeinconclusionthattheresultsofthisstﾆudywillbe

helpfultothedevelopingstudyof"speechscience"asfundamental

data｡

Wealsohopethatthemethodsofmultivariatestatistical

ana:Lysisconfirmedanddevelopedinthisstudy,willbefully

utilizedInmanyotherresearchfieldsuponthisopportunity･

ACKNOI-TLEDGMENTS

IwishtoexpressmyheartygratitudetoProfessorToshiyuki

Sakaiforhisguidance,constantﾆsupportandencouragementduringthe

courseofthisstudy｡

lalsowishtothankAssistantProfessorShujiDoshita(Tokyo

InstituteofTechonology),AssistantProfessorYasuhisaNiimi(Kyoto

Univ.of工ndustrialArtsandTextileFibers)forgivingmemuchneeded

andinformativehelp｡

IamindebtedtoAssistantProfessorMakotoNagao,Assistant

ProfessorHidenosukeNishio,AssistantProfessorShigeharuSugita,

Dr.KeniiOtani,Mr.TakeoKanade,Ilr.SiniiTomita,Mr.Shigeyoshi

Kitazawa,Mr.IlasatoshiKuboandmembersofProfessorSakai'sResearch

Groupforhelpfuldiscussion,co-operationandvariousconveniences.

-135-

1罫項欠

REFERENCES

(1)D,J.ProadandR.H.Fertig,"Formant-FrequencyTrajectories

inSelectedCVC-SyllableNuclei'″,J.Acoust.Soc.Amer.,47,

6,p.:L572(1970)･

(2)M.G.Kendall,"ACourseinMultivariateAnalysis",Charles

Griffin,London(1957)･

(3)H.L.Seal,"MultivariateStatisticalAnalysisforBiologists

(Reprintedwithminorcorrections)",p.97,Methuen,London

(1964and1968).

(4)T.Okuno,H.Kume,T.FagaandT.Yoshizawa,''Multivariate

StatisticalIlethods",p.A,Nikka-giren,Tokyo(1971).

(5)T,U.Anderson,"AnI:ntroductlontoMultivariateStatistical

Analysis≒JohnWiley&Sons,NewYork(:L958).

(6)D.F.Morrison,"llultivariateStatisticall･!ethods",

IIcGraw-Hill,NewYork(1967)･

(7)II.SiotaniandT.Asano,"MultivariateAnalysisTheory",

p.55,Kyoritsu,Tokyo(1967).

(8)II.SiotaniandT.Asano,"MultﾆivariateAnalysisTheory",

p.104,Kyoritsu,Tokyo(1967).

(9)M.SiotaniandT.Asano,"llultivariateAnalysisTheory",

p.60,Kyoritsu,Tokyo(1967)･

(10)K.Ito,':TheoryofMultivarlateStatisticalAnalysis",

p.162,Baihu-kan,Tokyo(1969)･

(11)W.Klein,R.PlompandL.C.X･7.Pols,"VowelSpectra,Vowel

Spaces,andVowelIdentification",J.Acoustﾆ.Soc.Amer.,

48,4,p.999(1970)･

(12)M.NakatsuiandJ.Suzuki,"FormantFrequencyExtractionby

InverseFilteringandMomentCalculationandItsEvaluation

usingSyntheticSpeechSound",J.Acoust.Soc.Japan,26,5,

p･.211(Jun.1970).

(13)S.Kltazawa,"AQuantitativeApproachtoCo-Articulationby

FormantﾆFrequencies'",Graduationthesisforthebachelar

-137-

.degreeofKyotoUniversity(1971).

(14)M.SiotaniandT.Asano,"MultivariateAnalysisTheory",

p.21,Kyoritsu,Tokyo(1967)･

(15)II.Kubo,"IlultivariateAnalysisofcvcvsyllables',

GraduationthesisforthebachelardegreeofKyotﾆoUniversity

(1972).づ

(16)Y.Kato,K.OchiaiandS.Azami,"SpeechSynthesisbyRule

SupplementarilyUsingNatualSpeechSegments'゜,The6th

工nternationalCongressonAcoustics,Tokyo(1968).

(17)J.L.Flanagan,"SpeechAnalysisSynthesisandPerception(

Second,expandededition)",p.19,Springer-Verlag,Berlin

(1965and1972)･

(18)S.Siegel･,"NonparametricStatisticsfortheBehavioral

Sciences'',p･,47･,McGraw-Hill･,NewYork(1956)･

(19)K.TabataandT.Sakai,"EvaluationofSpeaker-Factorinvcv

Utterances",RecordofJointConventionoftﾆheAcous.Soc.of

Japan(May1973),(Tobelectured).

OTHERREFERENCES

- 一 一

- 一 一

- 一 一

- - -

S.S.Vilks,"l･lathematicalStﾆatistics'JohnUiley&Sons,

NewYork(1962).

W.J.DixonandF.J.llassey,Jr･,''IntroductiontoStatistical

Analysis'･,I'icGraw-Hill,NewYork(1951,1957and1969).

T.OkunoandT.Haga,"DesignofExperiment≒Eaihu-kan,

Tokyo(1969)･

K.TakeuchiandH.Yanai,"Foundationofl･:ultivariateAnalysis",

Tpyo-keizai-shlnpoh,Tokyo(1972).

-138-

- ● - 四

- - -

- 一 一

- 一 一

- 一 一

- 一 一

- 一 一

一 一 -

THEAUTHOR゛SPAPERSRELATEDTOTHISTHESIS

K.TabataandT.Sakai,"MultivariateAnalysisofvcv

Syllables',RecordofJointConventionoftheAcous.Soc.of

Japan(Nov.1971).

T.SakaiandK.Tabata,"MultivariatestatisticalAnalysisof

vcvSy1:Lables',TechnicalReportoftheProfessionalGroupon

AutﾆomatonandInformationTheoryofthe工nstituteofElectronics

andCommunicationEngineersofJapan(Jan.1972).

T.SakaiandK.Tabata,"MultivariateStﾆatﾆiStﾆicalAnalysisof

cvcvSyllables':,RecordofJointConventionoftheAcous.Soc.

0fJapan(May1972)･

T.SakaiandK.Tabata,'"Three-DimensionalRepresentation

ofJapanesePhonemes:!,RecordofJointConventionofthe

Acous.Soc｡0fJapan(May1972).

T.SakaiandK,Tabata,"Three-DimensionalRepresentation

ofJapanesePhonemes'‘,TechnicalReportoftheProfessional

GrouponSpeechofAcous.Soc.ofJapan(July1972).

T.SakaiandK.Tabata,"llultivariateStatisticalAnalysisof

cvandvvUtterances'･,RecordofJointConventionofthe

Acous.Soc.0fJapan(Oct.:L972).

T.SakaiandK.Tabata,"MultivariateStatisticalAnalysisof

vcvSyllables'",Jour,oftheInstituteofElectronics

andCommunicationEngineersofJapan56-D,p.63(Jan.1973)

K.TabataandT.Sakai,"EvaluationofSpeaker-Factorinvcv

Uttﾆerances",RecordofJointConventionoftheAcous.Soc.of

Japan,(May1973)べ(To‘belectured).

-139-

14a項欠

AppendixA

TheAsymptoticDistﾆributionoftheLikelihood

RatﾆioTestCriterioninMultivariatﾆeAnalysis

ofVarianceforFour-FactorDesignWitﾆhSingle

Observation

Letusconsidertheasymptoticdistributionofthelikelihoodratio

testcriterionindicatedbyEq°(4°7).Arrange゛ijkl(the11°iberof

whichisn°a･b･c･d)ofEq,(4,2)inacertainorder,andassumea-th

,一一IIvectortobe｡%(a=l,..･,n),andx゛(pxn)=(.X;1,‥･,･χn).Thenlet

F'(゛l゛○゜[fic.]･G゛(bXn)゜[Sjoc'i･H'(cXn)゜[hk(Z)･andM'(dXn)

゜[1111α]9i-jhereﾀ‥

fμ

一

一

1,

0,

if心e{゛りklIi=1.･･･,b

otherwise.

kI}i

一

一 1,…,c;1=1,…,d}

andKjot>･^kQ!'･“liαaresimilarlydefined‘

1!eanwhile,ifwedefinethatl;(1xn)=(1,‥･,1),BI゛(Pxa)=

((ぢ,‥･,0!'),B2'(pXb)゜(βし‥･･凡')･Bj'CpXc)゜(r;･‥･･

ぺ)andB'(pXd)=(≪,',‥･,≪d),itispossiblet.0alterFq･

(4.2)to

e(x)=!,μ十FB1十GB2十HB3十MBu

=(>nFGHM)(μ'Bi'E2'B3'E^')″≡ZB

whereCdenotes"expectedvalue";Z°(lnFGroi)thedesignmatrixof

theexperiment;Bunknownparametermatrix.FromEq.(4.4),the゛'

densityfunctﾆionofxis

｣享..一讐'.1-1(2TT)21Aﾄ2e°ip{--trΛ(X-ZB)'(X-ZB)卜

wheretrindicatesthetraceofmatrix.Fromthisdensity,wewill

findRandAthemaχimumlikelihoodestimatesofBandΛ,respec-

-141-

tively.Bsatisfies

Z'ZB=Z'X,

(SeeEq.(2.28))

(A.I)

butsinceZ゛Zhasnotitsinversematﾆrix,weobtainBbycomparing

thebothsidesof(A,l)undertheconditionofEq.(4.3),thatis,

Jn'B,°09･｀?9ln'B^°O.

八μ

'χ.●゛●9

§

1･

゜(尽' 3°'゜3

a

al)゜(

゜X;1'゜'1‾'X"'゜'

1

9

°゜'yza°｀‘;‾‘x'゜‘'゜1).“゜yB4

/

:‘.･

Therefore,themaximumlikelihoodestimateofΛis

(A.2)

J=(χ-ZB)･(χ､-ZB)=(χ-lj_F礼一G92-臨3い瓜)･

･(X-1丿-FEi-GB,-HB3-mu)

abcd=Σ

i=1

ΣΣ

j二1k=1

Σ

l=1

(Zり.)

(A.4)

(゛ijkl‾゛i…‾゛:.:..-X..k.-X゜¨｢+ろ゛¨¨｣(A°3)

whichcoincideswiththeresidualRitselfofEq.(4.5)･

Subsequently,considerthehypothesisthatthereisnoeffect

offactorA

Ha:(zlこ･･.こ(za=0,thatis,Bi=0

《公《《《Letμ,B2,B3,Bit,andΛbethemaximum:Likelihoodestimatesﾆofμ

B2,B3,BいandΛundert:hehypothesisK^,

then,

;

=゛･‥･,Bz'=(x.i.

:

-x…

'

.,･･',X･b.

:

-z…

:)

《｡,｡R心1余http://www...

nA=(X-l

n

μ-GBo-HB5-MB4)(χ-lnμ-GBo-HB3

…,§4

而4)

abcd

°ぶ1j=ik=1ぶ1(゜x°ijkl―X.j..~JC..k°‾ぷ...1+2x....)'(A:ijki-AT.j,,-ぷ･･k-~ぷ"｀1

aabcd

-t-2*....)=bcdぶ1(凪'゛゜-JC....)'(‘゛i¨゜-X)+ぷlj?FIぶ1ぷ](jfijklXi°¨

-142-

areallidempotentmatrices,

‾z･j･.‾z･･k.‾x‥･1+ろx･･‥)･(zijkl‾xi‥.‾‘x;･j･●‾z‥k.‾z‥･1ｷろz･･･.)

=Q1-4-R

Thus,thelikelihoodratiocritﾆerionforthehypothesisHais

|謡1㈹

"|nA|1ﾏh+p-l

(SeeF-q.(2.37))

(A.5)

(A.6)

Thedistributionofl,･underthehypothesis(A.4)isobtainedfrom

Cochran'stﾆheorem.

LetY=χ-ZB=χ-1,1μ二FBi-GB,-FB3-IIBii,andbreakdownY'Y

asfollowsこ

Y゛Y=Y゛(ヰ)Y十Y'(百一斗)Y十Y'(万一脂Y十Y'

(Jy一子)Y十YI(響一寧)Y十Y'dn一昔-ご一首

)Y

一

一

-

-

一

一

Y'WoY十Y'WiY十Y'W2Y+Y'W3Y十Y'W4Y十Y'WsY

VO十V1十V2十V3+V4十V5.

(ln:n-dimensionalunitmatrix.)

I<7herenxnmatrices

forexample,as

!･T09･゜･9W5

-143-

'.bed

こ‾牛

゜Wi

Theseareeasilyverifiedifwepayattentiontofollowingrelations.

F゛F°bcdlg,I'F°bedla゛;ヽ3Fla'゜IngF'G°cdla'lb″'

Therefore,takingthetraceofmatricesinordertoobtainthe

ranksofeachmatrix,

aΣJ

｡Σ心

晨 n

一一n

-

- -bed

tr(W2)=b-1,trCW,)=c-1.tr(W4)=d-:L',

tr(W5)=n-a-b-c-d+3

-1=a-1

(A.7)

areobtained.Andsincen=(1)十(a-1)十(b-1)十(c-1)十(d-1)十(n-a-b-c-

d+3),Vo,･･･>v5‘areindependentofeachother,andcanbeexpressed

tyCochran'stheoremasfollows:fi,･I･

Vi°j=1

"j≪j “j(lxp)~N(O,A),(i=0,1,...,5)

VJhere,りistherankofW;showninEq.(A.7),and"jareassumedto

bedistributedindependentlyofeachother.(X但enて戸p,V.is

distributedaccordingtoWishartdistributionU(Λ,P,てi).)

Now,keepingattentiont01a゛BI°'b゛B2°Ic'B3°Id･b.°09

substituteX-ZBforYinVj,andexpressViasthefunctionof

･^ijkl.Then

Vi゛bedjl(Xi...-x....-cCi)'(゜Xi..゜-x....-rfi)

ParticularlywhenHa:1?1°Oj;strue!Vi°Qi

Ontheotherhand,

abc

V5°ΣΣΣi=1j=1k=1

dΣμ

(゛りkl‾.゛゜i...-x･j･.-゛‥k.-.x°‥･1ｷﾞ叙….)

'(xijkl‾xi･.I.‾Z･j･,X..Ir.X…l+ろX….)゜R(A.8)

V5=Rhold･inomatterwhatBI~B41!laybe,andR~W(Λ,p,T5)when

●●･.ﾐ●.･●;･I･r■･

T5°n-a-b-C-d+31p.

-144-

Therefore,themomentof11JofEq.(A.6)isaccuratelyobtained

sothatﾆwecanlearnitsasymptoticdistributionbyBoχ.(9)

Namely,ﾘ=-{nぺ2-(p十ill+l)/2}loRwisdistributedasaChi-

squaredvariatﾆ;ewithP兌1degreesoffreedomﾄaS十thesampleSizentends

toinfinity.VJhereli=a-l.£2=b十c十d-2,n-a-b-C-d+3≧p.

-145-

Appendix･B

TheIntersectionProducedbytheEllipsoidandthe

StraightlineorthePlane

Thepointofintersection,producedbytheellipsoidZA‾IZ'=1

andthestra･ightlinekCthroughtheorigin(krepresentﾆSarbitrary

realnumber),isexpressedby±c//ca"''c'.(Itisobtainedbysub-

StﾆitutingkCfor.X°inZA‾IZI°1.)

Asthedistancebetweentheoriginandthepointofintersection

is

(B.:L)

thisexpressioncoincideswithV^whenCistheeigenvectorof

AC'=λC'.(c=入A‾IC゛;CC゛=λCA‾IC″;λ=ccVCA'Ic.)

Theintersection,drawnbytheellipsoidandtheplanedeter-

minedbyCiandC2,willbeobtainedbyconnectingthepointsof

intersectionswhichareproducedbytheellipsoidandarbitrary

straightlinesontheplanewhichpassthroughtﾆheorigin.

SinceanarbitrarystraightlineisexpressedbykiCi十k2C2upon

choosingkiandk2arbitrarily.thedistancebetweenthepointof

intersection(madebythestraightline)andtheoriginisobtained

bysubstitutingklCI十koCpforCinEq.(B.l).

-146-

AppendixC

VarianceofFactorA

LeteA(゛りkl)betheexpectedvalueof゛りkl(゜μ十αj十βj+71

十δI+61jkl)underthehypothesistﾆhatHA:α1°¨'゛αa°Oistrue.

Then,fromEq.(A.5)inAppendixA,themaximumlikelihoodestimate

ofeA(刷jkl)is

a

A(zijkl),゜j゛βけ;けs,

=z….十(X.j..-X....)十(x..k.一心‥.)

十(x...1-ふ‥●)

人余《余Sinceμ‾μ-0,βj‾βjきO゛rk‾rkきoandδ1-≪lso,

(C.1)

thedifferencebetweensa°pievectoi:･^ijkland4(゛jjkl)isapproxi-

matelyequaltoαi十りiklasfollows:

《｡《

刷jkl‾eA(゛ijkl)゛(μ‾μ)十(zi十(βj‾βj)

十(rk-Fk)十(δl-21)十りjkl

~a.十陥kl

(C.2)

Therefore,theabovedifferencemaybeconsideredtobethedeviation

onlyduetofactorA(Ai,...,Aa).

(Ofcourse,ifhypothesisI--Aisactuallytrue(itmeansofj=0),

刷jkl‾ら.(りikl)゛njklandthisdifferenceistheresidualitself.)

Hencethematriχofsumsofsquaresandcrossproductsonthebasis

ofthedeviation(C.2)

abcd会4乱見ぶ1叫kl呵【りjk】))(りjkl‾eA(りjkl))

ﾍﾀﾞ

1

ﾑﾑ,t<りjkl-゛¨¨‾(゛lj°゜‾゛....)-(゛¨k‘‾゛¨¨)

‾(゛…l‾゛….))･(りjkl-゛･...-(゛･j･.‾゛….)-(X･･k.‾゛….)‾(z…1‾x･･‥).)

-147-

=bcdvO£:i‥.-x‥‥)゛(xi‥.-x‥‥)

十

万

j

ﾀﾞﾑA(xijkl‾xi‘..-X.j°.-X゜゜k°¨づ¨'けろx¨゜゜)

ﾆQI･十R

'(J:ijkl‾xi"'‾:xj‘j¨‾X°‘k'‾゜X;●●●1+5*･･●●)

canbeconsideredtorepresentthevarianceoffactorA

(C.3)

Moreover,letAbethemaximumlikelihoodestimateofthe

covariancematrixAunderthehypothesis＼

fromEq.(A.5)inAppend工XA

《nA°Qi十R,

whoseexpressioniscoincidentwithEq.(C.3)

-148-

:α1°‥.゜αa.Then,

ApoendiχD

TheVectorSpaceNormalizedbytheResidualR

Supposethefollowingnonsingulartransformation(notalways

orthogonaltransformation).

.x･→i=χ(R/n51 (D.:L)

!･ThereRistheresidualgivenbyEq.(4.5),nisthesamplesize

(n°abed),and.randiarevectorsintheoriginalandnewspaces,

respective:Ly･

Then,十Qに'き1(zi‥.-i‥‥)1(,x;i‥.-z‥‥)(QlgivenbyEq.(4.5))

istransformedto

こそj゛‾

とﾀ](Xj...-X….)'(ij…‾i….)

a-!-

゜‾yE:

1(R/11)2(゛j...゛｀'｀゜)゛(゛i'∵‾゛゜...)(R/n)

lal

°R‾7{EI

(ら‥‾゛‥‥)'(゛i…-χ….;}R"2

=R‾九1R唾

and

!R(pxp)4

n‥､

Similarly,:.

悩ﾉRj'仙･

and

∠―(Qi+R)→十(61+ii)∠μ

(QI+l)す,

i[ま[Qi+R)]X=X[Iけ(QI+n)Iﾄﾞを]‾li'･

=i(R/n)2 t'÷(Qi+R)]‾1

=z(十(Q1十R))'1z'

･

(R/n)23c'

-149-

■

2

(D.2)

(D.3)

(D.4)

(D.5)

AppendixE

TheKolmogorov-SmirnovOne-SampleTest(18)

TheKolmogoでov-Smirnovone-sampletestisatestof:goodnessof

fit.Thatis,itisconcernedwiththedegreeofagreementbetween

thedistributionofasetofsamplevalues(observedscores)andsome

specifiedtheoreticaldistribution.Itdetermineswhetherthescores

inthesamp1ecanreasonablybethoughttohavecomefromapopu:Lation

havingthet:heoreticaldistﾆribution.｡

Briefly,thetestinvolvesspecifyingthecumulativefrequency

distributionwhichwou:Ldoccurunderthetheoreticaldistﾆribution

andcomnaringthatﾆwiththeobservedcumulativefrequencydistribution.

Thetheoreticaldistributionrepresentswhatwouldbeexpectedunder

H0｡

LetFO(X)=acompletﾆelyspecifiedcumulatﾆivefrequencydistribu-

tionfunction,thetheoreticalcumulativedistributionunderKo.

Thatis,foranyvalueofX,thevalueofFo(X)istheproportionof

casesexpectedtohavescoresequaltoorlessthanχ｡

AndletSN(X)゜theobservedcumulativefrequencydistribution

ofarandomsampleofNobservations.1･Tiere,χisanypossiblescore,

SN(X)=k/N,wherek=thenumberofobservationsequaltoorless

thanχ｡

Nowunderthenu:LIhypothesisthatthesamplehasbeendrawn

fromthespecifiedtheoreticaldistribution,itisexpectedtﾆhatfor

every゛8:Lueofx･SN(X)Sho゛1dbefairlyclosetoFO(X).Thatis,

underHowewouldexpectthedifferencesbetweenSvj(X)andFo(X)to

besmallandwithinthelimitsofrandomerrors.TheKolmogorov-

Smirnovtestfocusesonthelargestofthedeviations.Thelargest

valueofFO(X)-SN(X)iScalledthemaximumdeviation,D;

D°maximum1Fo(X)‾SN(X)|

-150-

(E.I)

ThesamplingdistributionofDunderHoisknown.TheTableE.1

givesDa゛tﾆhecritﾆicalvalueofDwhenNisover35.1fNisover

35,onedeterminesDa゛thecriticalvaluesofDbythedivisions

indicatedinTableE.I.白Foreχample,supposearesearcheruses

N=43cases･andsetsa=･.05.Table(E.l)showsthatﾆanyDequalto

orgreatertﾆhanDa°ﾐFy捗‘willbesignificant.ThatiS゛anyD,as

definedbyformula(E.I)･whichisequa:Lto01°greatei°thanD(Z°

譜=.207willbesignificantatthe05level(two-tailedtest)

TableE.1TableofB^(criticalvalueofD)intheKolmogorov-Smirnovone-sampletest.

Sample

size(N)

a.LevelofsignigicanceforD=max Fo(X)-SN(X)

0.20 0.ﾆL5 0.10 0.05 0.01

Over35 1.07

-＼｢N‾

1.14

一汗

1.22可N‾

1.36

ｸ‾1.63可『

-151-

Appendix,F

TheAsymptoticDistributionoftheLikelihood

RatioTestCriterioninHultivariateAnalysis

ofVarianceforFour-FactorDesignwithRepeated

MeasurementsinWhichA11InteractionsAreDisregarded

Letus･considertheasymptot:icdistributionofthelikelihoodratio

testcriterionindicatedbyEq.(5.17).InEq.(5.12),arrange.x;りklni

(thenumberofwhichisn=abcde)inacertainorder,andassumea-th

vectortobeX£^(a=l,...,n)andχ'(pxn)゜(Xi',..･>･*■nz･Thenlet

F'(aXn)=[fiα],G'(bxn)=[g,μ],H'(cXn)=[hkα],andII'(dXn)

゜[ma],where.'尚｀

fi(Z=

]

.,

0,

if-≫^tfI^ijklmlj=1,…,b;k=i,…,c;1=1,…,d;ni―1,…,e}

otherwise,

andgjαﾀhkα9n＼＼aaresimilarlydefined･

Meanwhi:Le,ifwedefinethatU(1xn)=(1,‥･,1),Bで(pxa)=((z11

‥･,αa'),B2'(pXb)゜(βら･･･,βb‘),Bろ'(pXc)=(r;,‥･.r')

andB4'(pXd)゜(δll9･･･,δd'),itispossib:letoalterEq.(5.:L2)

to

e(x)=lnμ十FB1+GB2十HB3十MBi,

=(lnFGml)(μ'Bi'Bo'Ba゛B4')'sZB

where6denotesI:expectedvalue";Z=(InFGHII)thedesignmatrixof

theexperiment;Bunknownparametermatrix.FromEq.(5.1A),the

densityfunctionofxis

.巴

(27T)2--

|Λ12exp{ 一手tΓΛ-1(X - ZB)1(X-ZB)},

wheretrindicatesthetraceofmatrix.Fromthisdensity,wewill

findBandK-themaximumlike:LihoodestimatesofB上andA,

-152-

respectively･Bsatisfies.

z゛ZB=Z'X,

(SeeEq.(2.28).)

(F.I)

butsinceZ゛Zhasnotitsinverseir.atrix,weobtain%bycomparing

thebothsidesof(F.I)undertheconditionofEq.(5.13),thatis,

'n'Bi°0,‥･1･n'B,°O.

g=z‥‥.,沁'=(a11,…,もI)=(a･1…:-z….:....Xa...:-.χ･….:)

....B4(F●2)

Therefore,themaχimumlikelihoodestimateofAis

∧∧八∧∧A∧s岨=(X-ZB)゛(X-ZB)=(X-lnμ-FBi-GB2-HB,-l!B4)

べX-lj-F礼一･G礼一池3-M礼)

abcde°ぶ:

lj脊

I

Jj

11

召

ln

F

I(Xりklm~-'^i･...-X･j･･X.,J{..X,,･,.+5･X)

゜(゛ijklm-゛i･…-j;･j･･.-゛･･k･.-゛…l.4'3゛････.)(F.3)

whichcoincideswiththeresidualR.itselfofEq.(5.15).

Subsequently,considerthehypothesisthatthereIsnoeffect

offactorA

HA

Letμ,

:(ZI°‥.゜αa°0,thatis,B,°0 (F.4)

B29B3,Bit,andAbethemax:imumlikelihoodestimatesof

μ9B2ﾀB39BいandAunderthehypothesisl!A≫then.

;l°z･･…,n2'゜(z･1･..―X･･･.:,‥･,z･b.i:‾z･･･.:).¨･,n4･

nA=(X-lnβ一品2-H礼一MB4)'(χ-lnβ-GB2-HB3-MB4)=

(ぶり

dΣけ

｡Σり1

bΣ･F

aΣJ

････.)･(zijklm‾.x;･j･･.‾X

Σ

1m=1

･k..-AT...I.

-153-

-i-2x)=bcde｡Σ

' ド

abcde

1

(‘･V|･■･■a^a･■･゜) 'iXi...゜‾'X°)ﾅぶ

1j=1k=1

ぶ

lrT

y

l(£りk1“1

‾Zi･･‥‾Z･j･･.‾Z･･k‥‾X･･･|.+ろ.X;･‥‥)I(a;ijklm‾.x;i･･‥‾ぷ･j･..-X..k･･

一X..小斗ろぶ)゜Qi+R-

(F.5)

Thus,thelikelihoodratiocriterionforthehypothesisKais

w=-いに|旦1

い||Oi+r|.

(SeeEq.(2.37))

(F.6)

ThedistributionofZ∂underthehypothesis(F.4)isobtainedfron

Cochran'stheorem.､･

LetY=χ-ZB=χ-lnμ-FBi-GB,-KB3-MB4､andbreakdowny'Y

asfo1:Lows:

十Y'(

)Y十Y゛(FF曹

-bcde

in≪n-

)Y十Y゛(GG'In'n

ΞΥ'WoY+Y'WiY+Y'W2Y十YIW3Y+Y゛W4Y十■Y'W5Y

ΞVO+V1十V2十V3十V4十V5.

(ln:n-dlmensionalunitmatrix.)

WherenXnmatrices

forexample,as

x･712=(⊇?y

)Y十YI〔Ji;‾1戸〕Y

一響'÷ろ|五に|)Yabcen

WO゛゜..,V･areallidempotentmatrices.

-

'n'n°Wi

,n

Iχ

Theseareeasilyverifiedifwepayattentﾆiontofollowingrelations.

F'F=bcdela,≫nF=bcdela,Fla=ln,F'G=cdelalb･

-154-

Therefore,takingthetraceofmatricesinordertoobtainthe

ranksofeachmatﾆrix,

tr(Wo)=1,tr(Wi)=jUヱj

l‰2‾そJji‾i♪a-1,tr(W2)=b一丿L,trCWa)=c-1,tr(W1,)=d-1,

trCWs)=n-a-b-c-d+3 (F.7)

areobtained.Andsincen=(1)+(a-1)十(b-1)十(c-1)十(d-1)+(n-a-b-c-d･

+3),Vo,･･･>V5canbeexpressedbyCochran'stheoremasfollows:

r.

Vi°Σ

に1"j"j' Uj(lxp)~N(O,A),(i=0,l,..5)

1･Jhere,T:istherankofWishowninEq.(F.T),and≪Jareassumedto

bedistributedindependentlyofeachother.(WhenT:>p,V:isdls-

tributedaccordingtoWishartdistributﾆionW(A,p,Ti).)

NOWﾀkeepingattentiont0･a'Bl°lb'B2°≫c'B3°ld'B4°07

substituteX-ZBforYinVi,andexpressViasthefunctionof

Zりklm°Then

V1=bcdeE(゛i‥‥-･x°…･-a.y(X.….-゛･….-CC,).1=1

Particularlywhen!lA:B1=Oistrue,Vi=Q1.

Ontheotherhand,

V5=

abcΣΣΣ

eΣ

dΣ

i=1i=1k=11=1m=1(xijklm‾゛i….‾゛･j･･.‾゛･･k‥‾゛…1.+3X..…)l

'(゛ijklm‾j:i….‾゛･j･･.‾゛･･k･.‾X…1.+3X…･･)=旦.

V5=RholdsnomatﾆterwhatBI~B4maybe,andR~W(A,p,て5)when__

T5=n-a-b-c-d+3≧p.

Therefore,themomentofw0fEq.(F.6)isaccuratelyobtained

sothatwecanlearnitsasymptoticdistributoinbyBoχ.(9)･

Namelv,ﾘ=-{n-G-(p+らり)/2}1o肪iSdistributedasaChi-

squaredvariatewithpj!･1degreesof<^reerlomasthesamplesizentendsJ

toinfinity.VJhere£1=a-l,£2=b十c+d-2,n-a-b-c-d+3≧p.

-155-

AppendixG

TheAsymptoticDistributionoftheLikelihoodRatioTestCriterioninIlultivariateAnalysisofVarianceforDivided-TypeTwo-･FactﾆorDesignwithRepeatedlﾆeasurements

Lとtusconsidertheasymptﾆoticdistributionofthelikelihood

ratiocriterionindicatedbyEa.(6.12),

'|■･･Arrange゛りk(thenumberofwhichisn=abr)ofE(!.(6.ね)inthe

●j･-lfollowingorder,inwhichletぷαbethea-thvector:

Z(Z°*ijk,

where a=(i-l)br十(j-l)r+k,

(1!i!a,1≦j≦b,1≦k≦r,anda=1,...,n)

Similarly,arranger4'inthefollowing:order:

rq°咄),(叫1副･瓦'7j゛)(a.゛1111a)･

whereq=.(i-l)b+j(q=!,..,ab).

LetG{(axn)=[g池],G2'(axn)=[g?11,1!･(bxn]=H;十H2;

Hi'(bxn)゜[h?ﾙ],H,'(bxn)゜[h(ﾙ)'L'゛(8b)q1)祀[ﾍﾟﾙ],and

L,'(abxn)=[11ﾕ.1,

where

巡=

瑞

1

0

1

0

if1≦i≦ai,andifj(1≦j≦b)andk(1≦k≦r)exist

suchthata=(i-l)br4-(j-l)r十kholds,

otherwise,

ifai+l<i<a,andifj(1≦j≦b)andk(1≦k≦r)exist

suchthata=(i-l)br十(J-l)r十kholds,

otherwise.

-156-

hμ=

関戸h

(1)

qα

口がＩ

一一

一一

1

0

1

0

1

0

1

0

ifi(1≦i≦a-i)andk(1≦k≦r)existsuchthat

af=(i-l)br十(j-l)r十kholds,

otherwise,

ifi(ai+1≦i≦a)andk(1≦k≦r)existsuchthat

α=(i-l)br十(j-l)r十kholds

otherwise,

If1≦q≦a^b,andifk(1≦k≦r)･existssuchthat

a:=(q-l)r十kholds,

otherwise.

ifalb+11qiab,andifk(1≦k≦r)existssuch

thatcc=(q-l)r十kholds,

otherwise.

Meanwhile,wedefinethatX'(pxn)゜(zl゛,‥･9xnl)s

1:n(11)(11):n-d:imensiona:Lunitmatrix･ln'(1)(11)゜(1,..‘μ)ﾊﾟl[ﾌ]z(1'"｀)s

on(1xn)゜(09°..,0),Jali(1xn)゜(1いbc゛032bc)゛J327(1xn)

゜(Oalbc'la2bj),andTq,(axab)=[(Γal)iq],T(axab)゜[(Γ32)iq]'

Γbl(b)(3b)゜[(rbl)jq],r(bxab)゜[(Γb2)jq]^where

(Γal)iq°

1

0

if1≦i≦ai≫andifj(1≦j≦h)existssuch

thatq=(i-1)b+jholds,

otherwise,

-157-

≪..o

E

1

｢IJ

O:0

(Γa2)iq°

(Γbl

)jq --

(rb2)jq゛

1

0

1

0

1

0

ifal+1≦i<a,ar!difj(l≦j≦b)exists

sucht!latq=(i-l)b十jholds,

otherwise,

if1≦q≦“lb3andif

that･q=ｲi-1)b+j

otherwise.

i(1≦i≦a)existssuch

holds,

ifallbf1≦q≦ab,andifi(1≦i≦a)exists

suchthatq=(i¬1)b六十jholds,

otherwise.

Besides,definethatAi'(pxa)゜(α(1)'...･(Zjl)I･l(i112)･A2'(pxa)゜

(肱1ﾀ屯伊1(2ぺ'りαj2)')9T'(pxb)=(β1い‥,βb'),Ti'(pxb)゜(il(1)1

‥'･lb(1)')5Tz'Cpxb)゜(11(2)≒゛゜'･ib(2)')9Ci'(pxab)゜(rl(1)≒"゛9ral(ぴ･

略12b),C2i(p)(8b)゜(皿11b,(9)'･‥゜･rj2)I)･(ai+a2=a).

ThenitispossibletoalterEq.(6.U)to

e(x)゜lnμ十Jalμ(1)十Ja2μ(2)十GiAi十G2A2+HT十HiTi十H2T2

十LiCi十L2C2ヽ

゜(inJaiJa2GiGjHH1H2L1L2)(μ｀μ(1)゛μ(2)'Ai'A2'T'Ti'T2゛Cl'C2')'

ΞZB,(G.1)

where6denotes"expectedvalue":Z°('nJa-|Ja2GiG2HHiH2LiL2)

thedesignmatrixoftheexperiment;B°(μ゛μ(1)゛μ(2)'Ai'A2'T'Ti'T2'

Ci'Cz')'unknownparametermatrix.Eq･(6.6)correspondstoEq.(G.2):

alμ(1)十a2μ(2)゜0,

(la;･Oa2)AI°o･(Oal>lap)A2°o'

-158-

Andalso

Ib

Γal

T=0

C1=

】b'Ti°091b'T2°0,31T1+32T2=0,

Γ゛12C2°ΓblCI°ΓbpCa°0

(la^b'9Oa2b)Ci°(031b9132bl)C2°0

holds.

FromEq.(6.3),thedensityfunctionofxis

(2TT)‾苧|川弓exp{一子tΓΛ'■"(X-ZB)'(X-ZB)},

(G.2)

(G.3)

wheretrindicatesthetraceofmatriχ.Fromthisdensity,wewill

findBand/?

respectively･

Bsatisfies

In'n

'n'H

Λμ

=n

一一

themaximumlikelihoodestimatesofBandA,

茫と=z‥jt)-;x;･･･9

-159-

(G.4)

(G.5)

Z'ZB=z'x,

(SeeEq.(2.28).)

butsincez'zhasnotitsinversematrix,weobtainBbycomparing

thebothsideof(G.4)undertheconditionofEq.(G.2)andEq.(G.3).

゜.χr･.･5

&i(1)゜:zi‥‾.x;‥.(tへ

ら(t‰(z･.(t)_゛…(1))-(゛･j.-゛‥.)･

βj=･･ﾚｰJC････

'it"^lj･-Vi..X.j.十'x°‥'(t≒

(t=l,2).

Theseareverifiedifwepayattentiontofollowingrelations:

'lnIJal゛fllbr'lnIGI゛b｢(1311'032)

arlu'･11.:Hi°¥Ib".In'H2°32｢Ib''ln'Li°r(lalbl'Oa2b)

1n'L2ﾆr(0°ibllI)'Ja;J31'゛a-ibr)J311J32ﾆO'J゛111Qiﾆb｢(1311'032)゛

Ja;G2ﾆ0,J'H°air1b'゛Jal'HIﾆalrlbl'J31'H2ﾆO'J311L1ﾆ｢(laib''0a2b)ﾀ

J311L2ﾆO'Ja^^ao°aobc.Ja2'Gi゛O゛J32'G2°b｢｢031｣32')'

Ja21Hﾆa2｢lb'゛Ja2IHFO'J321H2ﾆ゜2｢V.Ja2'Li°o'

J321L2ﾆ｢(Oaib'1a2b')

G1゛G1=br10

Ｌ

Gi゛Hi=Gi'H,

G2゛G2=br

00

Ｌ

G11

100 G11G21=O･,Gi'H=r[

;lj

1b''

H2°09Gi'Li°rr11,ﾀGjLo'°09

I

?

2

]

s

Ga'H=r[;jj]

1b'゛G21H1°O゛

Go'Rz°Gz'H,G2'Li°O゛G2'L2゛ΓΓ32゛

H'H°ar工b>H'Hi°alrlbﾀH'Hz°a2rlbﾀHi゛H1°a.rib

Hi'H2=0,H'L,=HiLi=rrK,H2'Li=0,Hz'Hz=a2rlb,

i七2°H2'L2°lΓΓb2'WL2ぷ

'Li'Li°lr[乱lbO]

･,--………『0

Li'La°0゛L2゛L2゛r[;1

2b

]

Therefore,themaximumlikelihoodestimateofAis

(G.6)

J°(X-ZB)

'(X-ZB)゜(X-1nA-Jil

l

β(1)

‾J･12

μ(2)

-GiAi-G2A2･-HT-Hifi-H2T2-LiCi-LoC2)"

(X-Inβ‾Jal

β(1)‾Ja2β

(2)

‾G1

λ

l‾G2

λ

2‾

�

-HiTi

-H212-LiCi―L2C2)

一一

bΣJbΣJ

･j.J

r●ぶ

1(ぶりk‾りj゛)'(ぶりk‾りj°)

｢ぶ

1(‘x'ijk‾りr)'(ぶりk‾りj゛)

abr

゛ぶlj

lﾐF

I

JF

I(zりk‾りj.)'(Xりk‾りj°) (G.7)

whichcoincideswiththeresidualRitselfofEq.(6.10)･

Subsequently,considerthehypothesisthatthereisnoeffect

-160-

(G.10)

1

offactorA(1)一A,(2)

Ko:μ(1)=μ(2)゜0 (G.8)

Letμ,Ai,A2,T,Ti,T2,Ci,C2andAbethemaximumlikelihood

estimatesofμ9Ai,A29T,Ti,T2,Ci,C2,andΛunderthehypothesis

Ho,then'

ぶ=

ll(叱

≪t≫a7

x･‥j?)゜zi‥_jp(t)

(゛゛j°(t)‾゛‥'(t))‾(゛･j･~゛...)

゜X.j.X...,Tjj-Xjj.―X-^.,-X..(t)十V,(t)

(t=l,2). (G.9)

nΛ=(X-ZB)゛(X-ZB)=(X-lnμ-GiAi-G2A2-HT-HiT,

-

H2T2-LiCi-L2C2)゛･(X-lnμ-GiAi-G2A2-HT-HiTi

,-H2T2 -LiCi-L2C2)

゜χ

j

y

l

ぶ{(範jl)‾゛...)+(り‾゛り゜)白(゛'¨(1)‾゛゜¨)

I

ﾊﾞ勺k-*ii°)}几乱もﾕ{(゛…(2)_゛…)゛(i印う町川'

りﾘ‥.(2)‾孔･.)十(ぷりk‾xり.)}

=br'lat(゛･･.(t)-゛･‥)'(゛‥.(t)-z...)

1=1

十Aχk

ｼりjk‾乱ｺﾞ(xりk-*ii.)

-

-

ql十R

-161-

Thus,thelikelihoodratiocriterionforthehypothesicHois

|｡則|引w=一一こｰ-'-

ln別|(h+則

(G.ll)

｡(SeeEq.(2.37))

●●●瀞●'●Thedistributionofz･･underthehypothesis(G.8)isobtainedfrom

Cochran'stheorem.

Let Y°X-ZB°X‾lnμ‾Jal μ(1)-J

a2μ(2)‾GIAI‾G2A2

- HT-HiTi-H2T2-LiCi-L2C2,

andbreakdownY'Yasfollows:,

Y'Y°Y゛(二万is-)Y十Y゛(ln一三ぺ這)Y

=Y'(-il≒い一一)ｸy十YI(j計J!L十

十Y゛(

十YI(

+

air

GiGi'-

br

ヰ -

l

匯)Y

-づ柚ﾀ!')Y゛い(jVﾝj‾べjぶ;)↓

Inl;1一一

Ja2Ja21可こ

)Y十YI(

-162-

HiHi'+

H2H2'HH゛

a2｢

LiLi'GiG'i

GoGj.-br-

a｢

H2H2'-

a^r

)Y

HH゛

-a｢

_JajJai-

‘a-ibr

一一HiHi'十j≒jlﾕｰ)Y十Y'(≒-り2'-

WherenXnmatrices-w..w,X'>,---,We

ces,forexample,as

a2｢

ΞΥ'WoY十Y'WiY十YIW2(14十Y゛xjjY十Y゛F3Y

十Y'WuY十Y'WJI)Y十YI匈2)Y十Y'W6Y

≡vo十VI十V2(1)十V2(2)十V3十V4十V5(1)十V5哨十V6.(G.n)

areallidempotentmatri-

●

-々斗ぎ!ﾕﾂﾞﾀﾞ=?1_GI/-:^i_£3i_'=waibrbraibr2

Theseareverifiedifweattentiontothe!relationsofEq.(G.6)and

■･･･.･'･■,■り¶WII¶が倆,･■¶-･･4r一一j■--･●'･･

followingrelatﾆions.

LI

｢乙al

.

E｣

゜Li

i

引10

{Ｌ

IG

100

000yo

じ[卜~Ｌ

LG

I

a2

゜G1,

{――}

0

｢I-~｡｣

T『

゜L2,

l

G2･i

I'1ぺ゜G

ﾑ

L√几)yHI'LI

Q]

゜Jai

GI

CL｣

゜･Jai゛Hilb°Ja!゛

L匹;=GいLがいり,L

ぐづ

=≒

G2

[7:1]

゜J32'H2'b=Ja2゛

HIb=In,HH'=(Hi十H2)(HI十H,)'=HiHi'十H2H2',etc.(G.12)

Therefore,takingthetraceofmこtricesinordertoobtainthe

ranksofeachmatriχ.

tr(Wo)=1,tr(Wi)=1,゛(匈1))゜1n1.(1)>2器

=ぺU牛一1=ai-1,tr(Wj2))=a2-1,tr(W3)=b-1,

tr(W4)=b-].,tir(WJ1))=alb-ai-b十1,tr(W5(2))゜32b'a2

- b+1 tr(U6)=n-ab (G.13)

areobtained.Andsincen･(1)十(1)十(ai-l)十(a2-1)十(b-1)十(b-1)十(aib-ai

-b+1)十(a2b-a2-b+1)十(n-ab),

｀J｡,Vいvr)9¨¨,V5canbeexpressedbyCochiran'stheoremasfollows:

163

岬=Ｉい≪j'"j Uj(ixp)二~N(0･A)･

雪

(

じ･･･,6J(G.14)

where,7r7istherankofVI■shovヽ7ninEq.(G.13),and≪jareassumedto

bedistributedindependentlyofeachother.‘(WhenぐをP,V?is

distributedaccordinetoUishartdistributionU(Λ,p,T?)).)

Now,keepingattentiontoEq.(G.2),substitutex-ZBfor

YinVi,andexpressViasthefunctionofzりk Then

Visbr

jal(.x;!?.-x...-μ(t))I(x!?.-X...-μ(t))

1二1'

ParticularlywhenB.:μ(1)ごμ(2)゜Oistrue･9Vi°Qi

Ontheotherhand.

V6=

bΣ

aΣ E(Xijk-X･'.)I(zijk‾'x;'¨)゜RμIJﾆ1k=1

(G.15)

(G.16)

V6=RholdsnomatterwhatBmaybe.andR~U(Λ･PjTe)whenT6°n-ab

≧P●ﾉｰ,･

Therefore,themomentofwofEq･(G.ll)isaccuratelyobtained

sothatwecanlearnitsasymptoticdistributionbyEox.(9)

Namely,V=-{n一見2-(p十見1十:L)/2}10glむ･isdistributedas-

Chi-squared･variatewithP見1degreesoffreedomasthesamplesizen

tendstoinfinity,where£i°1,聡十見2°ab,n-ab≧p･

-164-

4

/

●'

1 1 1

1

♂ ･

AppendixH

MatrixofSumsofSquaresandCrossProductsinSubspace.

LetB(pxq)betheorthogonalprojectionmatrixofp-dimensional

spaceonitsq-dimenslnalsubspace.

X(1Xp)→X(1Xq)=xB (H.I)

And,letamatrixQbesuchamatrlχofsumsofsquaresandcross

productﾆSasshowninEq.(2.15).

Q(pxp)=2(xi-z.)'(xにX.),X.=!11Xi

i=lni=1

ThenQistransformedbytheaboveprojectionasfollows.

where

6(9･･9)=j(ii-i･.)･(Xi-X.)

1=1

=.ZB'(Xi-x.)'(Xi-z.)Bt=1

=B゛n-z.)'(xi-x.))Bt=1

°B'QB,

WZ

1

-n

n

Σ

i=1

W*i

(H.2)

(H.3)

NowletusconsidertheEq.(7.13).

SinceRisamatrixofsumsofsquaresandcrossproducts,and

B=A゛(AAI)‾1fromEq.(7.11),thentheresidualRinthesubspace1S

equalto

R=(A゛(AAI)‾1)゛R(A'(AA°)‾1)

-165-

(H.4)