说明案例origin7 0软件删g03acc

g03–MultivariateMethods

g03acc

NAGCLibraryManual

nagmvcanonvar

[NP3275/5/pdf]

3.g03acc.

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3.g03acc.

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nagmvcanonvar(g03acc)

Purpose

nagmvcanonvar(g03acc)performsacanonicalvariate(canonicaldescrimination)ysis.

Specification

#include<nag.h>#include<nagg03.h>

voidnag_mv_canon_var(Nag_Weightstypeweight,Integern,Integerm,doublex[],Integertdx,Integerisx[],Integernx,Integering[],Integerng,doublewt[],Integernig[],doublecvm[],Integertdcvm,

doublee[],Integertde,Integer*ncv,doublecvx[],Integertdcvx,doubletol,Integer*irankx,NagError*fail)

Description

P

Letasampleofnobservationsonnxvariablesinadatamatrixcomefromnggroupswithn1,n2,...,nngobservationsineachgroup,ni=n.Canonicalvariateysisfindsthelinearcombinationofthenxvariablesthat izestheratioofbetween-grouptowithin-groupvariation.Thevariablesformed,thecanonicalvariatescanthenbeusedtodiscriminatebetween

groups.

Thecanonicalvariatescanbecalculatedfromtheeigenvectorsofthewithin-groupsumsofsquaresandcross-productsmatrix.However,nagmvcanonvarcalculatesthecanonicalvariatesbymeansofasingularvalue position(SVD)ofamatrixV.Letthedatamatrixwithvariable(column)meanssubtractedbeX,andletitsrankbek;thenthekby(ng−1)matrixVisgivenby:

X

—

V=QTQg,whereQgisannby(ng1)orthogonalmatrixthatdefinesthegroupsandQXisthefirstkrowsoftheorthogonalmatrixQeitherfromtheQRpositionofX:

X=QR

ifXisoffullcolumnrank,i.e.,k=nx,elsefromtheSVDofX:

X=QDPT.

LettheSVDofVbe:

g

V=Ux∆UT

thenthenon-zeroelementsofthediagonalmatrix∆,δi,fori=1,2,...,l,arethelcanonicalcorrelationsassociatedwiththelcanonicalvariates,wherel=min(k,ng).

i

Theeigenvalues,λ2,ofthewithin-groupsumsofsquaresmatrixaregivenby:

δ

i

2

λ2= .

i 1 2

—δi

i

i

andthevalueofπi=λ2/Pλ2givestheproportionofvariationex inedbytheithcanonical

variate.Thevaluesoftheπi’sgiveanindicationastohowmanycanonicalvariatesareneededtoadequaydescribethedata,i.e.,thedimensionalityoftheproblem.

j

Totestforasignificantdimensionalitygreaterthanitheχ2statistic:

2

(n−1−ng−1(k−ng))

X

l

j=i+1

log(1+λ2)

− −−

canbeused.Thisisasymptoticallydistributedasaχ2distributionwith(ki)(ng1i)degreesofdom.Ifthetestfori=hisnotsignificant,thentheremainingtestsfori>hshouldbeignored.

TheloadingsforthecanonicalvariatesarecalculatedfromthematrixUx.Thismatrixisscaledsothatthecanonicalvariateshaveunitwithingroupvariance.

Inadditiontothecanonicalvariatesloadingsthemeansforeachcanonicalvariatearecalculatedforeachgroup.

Weightscanbeusedwiththeysis,inwhichcasetheweightedmeansaresubtractedfromeachcolumnandtheneachrowisscaledbyanamount√wi,wherewiistheweightfortheith

observation(row).

Parameters

weight

Input:indicatesthetypeofweightstobeusedintheysis.Ifweight=NagNoWeights,thennoweightsareused.

Ifweight=NagWeightsfreq,thentheweightsaretreatedasfrequenciesandtheeffectivenumberofobservationsisthesumoftheweights.

Ifweight=NagWeightsvar,thentheweightsaretreatedasbeinginverselyproportionaltothevarianceoftheobservationsandtheeffectivenumberofobservationsisthenumberofobservationswithnon-zeroweights.

Constraint:weight=NagNoWeights,NagWeightsfreqorNagWeightsvar.

n

Input:thenumberofobservations,n.Constraint:n≥nx+ng.

m

Input:thetotalnumberofvariables,m.Constraint:m≥nx.

x[n][tdx]

− −

Input:x[i 1][j 1]mustcontaintheithobservationforthejthvariable,fori=1,2,...,n;

j=1,2,...,m.

tdx

Input:thelastdimensionofthearrayxasdeclaredinthecallingprogram.Constraint:tdx≥m.

isx[m]

Input:isx[j−1]indicateswhetherornotthejthvariableistobeincludedinthe ysis.

—

Ifisx[j 1]>0,thenthevariablecontainedinthejthcolumnofxisincludedinthecanonicalvariate ysis,forj=1,2,...,m.

Constraint:isx[j−1]>0fornxvaluesofj.

nx

ing[n]

ng

Input:thenumberofvariablesintheysis,nx.Constraint:nx≥1.

—

Input:ing[i 1]indicateswhichgrouptheithobservationisin,fori=1,2,...,n.Theeffectivenumberofgroupsisthenumberofgroupswithnon-zeromembership.

Constraint:1≤ing[i−1]≤ng,fori=1,2,...,n.

Input:Thenumberofgroups,ng.Constraint:ng≥2.

wt[n]

Input:ifweight=NagWeightsfreqorNagWeightsvarthentheelementsofwtmustcontaintheweightstobeusedintheysis.

Ifwt[i−1]=0.0thentheithobservationisnotincludedinthe ysis.Constraints:

P

wt[i−1]≥0.0,fori=1,2,...,n,

ni=1

wt[i−1]≥nx+effectivenumberofgroups.

Note:Ifweight=NagNoWeightsthenwtisnotreferencedandmaybesettothenullpointer

NULL,i.e(double*)0.

nig[ng]

Output:nig[j−1]givesthenumberofobservationsingroupj,forj=1,2,...,ng.

cvm[ng][tdcvm]

− −

Output:cvm[i 1][j 1]containsthemeanofthejthcanonicalvariatefortheithgroup,fori=1,2,...,ng;j=1,2,...,l;theremainingcolumns,ifany,areusedasworkspace.

tdcvm

Input:thelastdimensionofthearraycvmasdeclaredinthecallingprogram.Constraint:tdcvm≥nx.

e[min(nx,ng-1)][tde]

Output:thestatisticsofthecanonicalvariateysis.

e[i−1][0],thecanonicalcorrelations,δi,fori=1,2,...,l.

i

e[i−1][1],theeigenvaluesofthewithin-groupsumofsquaresmatrix,λ2,fori=1,2,...,l.e[i−1][2],theproportionofvariationex inedbytheithcanonicalvariate,fori=1,2,...,l.e[i−1][3],theχ2statisticfortheithcanonicalvariate,fori=1,2,...,l.

—

e[i−1][4],thedegreesofdomforχ2statisticfortheithcanonicalvariate,fori=1,2,...,l.e[i1][5],thesignificancelevelfortheχ2statisticfortheithcanonicalvariate,fori=1,2,...,l.

tde

ncv

Input:thelastdimensionofthearrayeasdeclaredinthecallingprogram.Constraint:tde≥6.

—

Output:thenumberofcanonicalvariates,l.Thiswillbetheminimumofng 1andtherankofx.

cvx[nx][tdcvx]

− −

Output:thecanonicalvariaoadings.cvx[i1][j1]containstheloadingcoefficientfortheithvariableonthejthcanonicalvariate,fori=1,2,...,nx;j=1,2,...,l;theremainingcolumns,ifany,areusedasworkspace.

tdcvx

tol

Input:thelastdimensionofthearraycvxasdeclaredinthecallingprogram.Constraint:tdcvx≥ng−1.

Input:thevalueoftolisusedtodecideifthevariablesareoffullrankand,ifnot,whatistherankofthevariables.Thesmallerthevalueoftolthestricterthecriterionforselectingthesingularvalue position.Ifanon-negativevalueoftollessthanmachineprecisionisentered,thenthesquarerootofmachineprecisionisusedinstead.

Constraint:tol≥0.0.

irankx

Output:therankofthedependentvariables.

Ifthevariablesareoffullrankthenirankx=nx.

×

Ifthevariablesarenotoffullrankthenirankxisanestimateoftherankofthedependentvariables.irankxiscalculatedasthenumberofsingularvaluesgreaterthantol(largestsingularvalue).

NAGCLibraryManual

nagmvcanonvar

g03–MultivariateMethods

g03acc

3.g03acc.

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3.g03acc.

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fail

TheNAGerrorparameter,seetheEssentialIntroductiontotheNAGCLibrary.

ErrorIndicationsandWarnings

NEBADPARAM

Onentry,parameterweighthadanillegalvalue.

NEINTARGLT

Onentry,nxmustnotbelessthan1:nx=hvaluei.Onentry,ngmustnotbelessthan2:ng=hvaluei.Onentry,tdemustnotbelessthan6:tde=hvaluei.

NEREALARGLT

Onentry,tolmustnotbelessthan0.0:tol=hvaluei.

NE2INTARGLT

Onentry,m=hvalueiwhilenx=hvaluei.Theseparametersmustsatisfym≥nx.

Onentry,tdx=hvalueiwhilem=hvaluei.Theseparametersmustsatisfytdx≥m.

Onentry,tdcvx=hvalueiwhileng=hvaluei.Theseparametersmustsatisfytdcvx≥ng−1.Onentry,tdcvm=hvalueiwhilenx=hvaluei.Theseparametersmustsatisfytdcvm≥nx.

NE3INTARGCONS

Onentry,n=hvaluei,nx=hvalueiandng=hvaluei.Theseparametersmustsatisfyn≥nx+ng.

NEINTARRINT

Onentry,ing[hvaluei]=hvaluei,ng=hvaluei.Constraint:1≤ing[i−1]≤ng,i=1,2,...,n.

NEWTARGS

ThewtarrayargumentmustnotbeNULLwhentheweightargumentindicatesweights.

NENEGWEIGHEMENT

h i h i

Onentry,wt[value]=value.

Constraint:Whenreferenced,allelementsofwtmustbenon-negative.

NEVARINCLINDICATED

h i

h i

Thenumberofvariables,nxintheysis=value,whilenumberofvariablesincludedintheysisviaarrayisx=value.

Constraint:thesetwonumbersmustbethesame.

NESVDNOTCONV

Thesingularvalue positionhasfailedtoconverge.Thisisanunlikelyerrorexit.

NECANONCORR1

Acanonicalcorrelationisequaltoone.

Thiswillhappenifthevariablesprovideanexactindicationastowhichgroupeveryobservationisallocated.

NEGROUPS

Eithertheeffectivenumberofgroupsislessthantwoortheeffectivenumberofgroupsplusthenumberofvariables,nxisgreaterthanthetheeffectivenumberofobservations.

NERANKZERO

Therankofthevariablesiszero.

Thiswillhappenifallthevariablesareconstants.

NEALLOCFAIL

Memoryallocationfailed.

NEINTERNALERROR

Aninternalerrorhasoccurredinthisfunction.Checkthefunctioncallandanyarraysizes.IfthecalliscorrectthenpleaseconsultNAGforassistance.

FurtherComments

Accuracy

Asthecomputationinvolvestheuseoforthogonalmatricesandasingularvalue positionratherthanthetraditionalcomputingofasumofsquaresmatrixandtheuseofaneigenvalue

position,nagmvcanonvarshouldbelessaffectedbyillconditionedproblems.

References

ChatfieldCandCollinsAJ(1980)IntroductiontoMultivariateysisChapmanandHall.GnanadesikanR(1977)MethodsforStatisticalDataysisofMultivariateObservationsWiley.HrlingS(1985)Thesingularvalue positioninmultivariatestatisticsSIGNUM20(3)

2–25.

KendallMGandStuartA(1979)TheAdvancedTheoryofStatistics(3Volumes)Griffin(4thEdition).

SeeAlso

None.

Example

Asampleofnineobservations,eachconsistingofthreevariablesplusgroupindicator,isreadin.Therearethreegroups.Anunweightedcanonicalvariateysisisperformedandtheresultsprinted.

ProgramText

/*nag_mv_canon_var(g03acc)ExampleProgram.

*

Copyright1998NumericalAlgorithmsGroup.

*

Mark5,1998.

*/

#include<nag.h>#include<stdio.h>#include<nag_stdlib.h>#include<nagg03.h>

#defineNMAX9

#defineMMAX3

#defineTDE6

main()

{

doublee[MMAX][6];doublex[NMAX][MMAX];doublewt[NMAX];

doublecvm[MMAX][MMAX],tol,cvx[MMAX][MMAX];

Integeri,j,m,n;

Integerng;

Integernx;

Integering[NMAX],nig[MMAX],ncv;Integerirx,isx[2*MMAX];

Integertdx=MMAX,tdc=MMAX,tde=TDE;

charwtchar[2];Nag_Weightstypeweight;

NAGCLibraryManual

nagmvcanonvar

g03–MultivariateMethods

g03acc

3.g03acc.

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3.g03acc.

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Vprintf("g03accExampleProgramResults\n\n");

/* Skipheadingindatafile*/Vscanf("%*[^\n]");

Vscanf("%ld",&n);

Vscanf("%ld",&m);

Vscanf("%ld",&nx);

Vscanf("%ld",&ng);Vscanf("%s",wtchar);

if(n<=NMAX&&m<=MMAX)

{

if(*wtchar==’W’||*wtchar==’V’)

{

for(i=0;i<n;++i)

{

for(j=0;j<m;++j)Vscanf("%lf",&x[i][j]);

Vscanf("%lf",&wt[i]);

Vscanf("%ld",&ing[i]);

}

if(*wtchar==’W’)

weight=Nag_Weightsfreq;else

weight=Nag_Weightsvar;

}

else

{

for(i=0;i<n;++i)

{

for(j=0;j<m;++j)Vscanf("%lf",&x[i][j]);

Vscanf("%ld",&ing[i]);

}

weight=Nag_NoWeights;

}

for(j=0;j<m;++j)Vscanf("%ld",&isx[j]);

tol=1e-6;

g03acc(weight,n,m,(dou