Pattern Recognition and Machine Learning

566 12.COl\'TINUOliSLATf;I\'T\'ARIAIILES

Figure12.3 Themean~'"xaklogwith!heII"'tlou'PCAe;gerrvecl<)rllUl,. ..'" lorthe011-.....

cligitsdataset.t<>getl'lerwith!hecorrespondi~~.

;ntheOIigi",,1D-<limensionalspace.wecanrepresentthoeigenw:cto<sasimago<of

thosamesilOas,1>0datapoi",,_11,.firstIh'e.ig.n,'occOfS.alongwichtl>ocorre-

sponding.igen,'slue,.are<IIo"'ninFigure12,3,AplO!ofll>ocompletespect"'muf

oigo",·alue,.sone<!intodecreasingorder.is showninFigure12.4{ai.Thedi'tortion

measureJaSSQCiatedwilhchoo<inga particularvalueofM isgi.'enbythosum

oftheeig.n",luesfromM +Iupto 0 andisptO!tedfordifferent,'aluo<of.\1in

Figure12,4(b).

If "'e<utlslitut.(12,12)and(12.13)into(12.10).wecanwritetheI'CAappro~­

imationtoa data"eel'"x~i"thefonn

M

"

'-

~ L{x~",)u,+

I:

(xl'u,)u, (12.19)

.-. ._M+l

M

• x+L(X~U,-XTU,)U;

(12.20)

• ,

, to' ,, 10'

, ,

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, "-

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", ~ ~ ; 0 ", ~ ~

,., ,., "

FIIIUre12,4 (a)PIolat!heeJoI;nv.loo.".,etrumlortheoff·1inedigitsdataset (b)P10t 01 !hesumatthe

<:liscarded."".Ioos,which"'l'fesoots!hes.um-ol·SQ"",esdistortlonJi<*~byprojecti<Xlthedataonto

apspaee'"dimensionalitvM.