Pattern Recognition and Machine Learning

11.1.I'rindpall.:"l11pon~nt Anal~·.i. 567

FIIIUr. 1 :1:.5 An",>gi",,1~mpIeIromlI>e011·_digilsdata...ttOll"1herwithitsPeAre<:onstnxlions

oblair...:!by'e1aio"li!Xl,Ifj)<incipal~n1S10<variousval,," 01 ,If.As,IIincreason

!tiere<:onst,uctiOfI~smoreao::urateandwoukl~portee!when.-IfK D~

28 x 28 ~."-1.

whereweha"emademoeoftherelation

x= L" (x'",)u;

,-,

(12.21)

AI''''''''/;'\'A

Seer/on9.1

whichfollow.fromthecompletene"ofthe{u,I, Thi.represent.acontpre"ioo

"fthedata>ct. Ilttau>eforeachdatapoimweha,..repla«dthe V·dimensiooal

"o<:lorx"Wilhan,I[.din>en,ional"o<:torhavingcomponem,(x~'" _X'",).11Ie

'mailerthe"alueofM.thegreaterthedegreeofcomp.-e",ion. Example.ofPeA

,""on't""tioosofdatapointsforthedigitsdatasetareshowninFigure12.5

Anolherapplicationofpriocipalcompcmenlanaly,i.i'todatapre-processing.

Inthi'case,lhegoalisnO!dimensionalityredUC1ionbutratherthetmn,formmionof

a dataselinor<k'tostanda'lli'.eeenainofilSpmpenies.Thiscanbein'portanlin

allowing.ubsequentpallem,""ognitionalgorithm. 10 beappliedsuccessfully 10 the

data>ct.Typically.il isdonewilentheoriginal"ariable.aremea,uredin"arioosdif.

ferentunil'or!la"esignificantlydifTerent,'ariabilil}'.ForinstanceintheOldFaithful

datasel.thetimebetv.-eeneruption.i.typicanyanorderofmagni1Udegreaterthan

lhedUrali"" of.nerupt;,,". WhenW'eappliedthe".nlCansalgorill"" 10 thi<data

set,".-efirstmadeasepar.telinearre-sealingoftheindividual"anable'socbthm

each"ariablehadzeromeanandunit"ariance.llUsisknownasslllNlardiv·.,gthe

dota.andthecO\'anancematrixforlhe'lando,di/,eddalahascomponents

(12,22)

where<1,isthe,'anaoceof:c,.Thisi<knownasthe(",,,el,,,;,,,,matri.'oftheoriginal

dotaandha'thepropenythaiif t""orompooent,X;andx,ofthedataareperfee1ly

correl.ted.thenAi_ I.•ndiftheya.-euocorrelated.thenAi_O.

11",,'1""',usingPeAwecanmakeaIt>Of'esubst.mialnonnalizat;ooofthedata

togi\'Cit zeromeanandunitco'·ariance.sothatdifferent"anablesbecomederorre-

late<l Todothis.wefirst""riletheei8Cn"cclorequation(12,17)intheform

su=UL (12.23)