Pattern Recognition and Machine Learning

598 12.CONTINUOUSLATENTVAK1AHU':S

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FllIu.e12.21 ?lotottrleoillkYw<:lataWllisualiz.edusingPeAontheleftandGTMonItlengr,tFOftileGTM
model.each<latapoinIlsplollfldattilemeanotitsposM'k><dislributionin..tents;>ace,Tile"","ineantymlhe

GTM1TlOd8I._lhasepamlionbetwoonthegroupsofdatapointstobe.....n""".ckl.arfy,

Ch"l'l~fj

S~etioo/.4

Theno"liotarmappingisgi,'enbya linearregressionmodelthaIallow,forgeneral

IIO/llinearilywhilebeinga linearfuoctionoftileadapli'-eparameler<,NOIethaItilt

usuallimitationoflinearregressionmodelsarisingfromtheen"",ofdimen,iooalily

does 1101 ariseintheContr~1oflheGT~Isi""'ethe"\3nifoldgenerall)'ha<t,,'odi"ltn·

sionsirrespecti'-eofthedimensionalityofthedataspace, Acoo""!",,nceofIllese

11"0cooicesisthatthelikelihoodfunclioncanbee~pressedanalyticallyindosed

formandcanbeoptimilC<.!efficientlyo,ingtheEMalgorithm_TheresoltingGTM

model hIsa lwo-dimensionalnonlinearmanifold 10 tiledalasel.andbye"alualing

theposteriordistrilJ",lion(Werlatentspaceforthedatapoi"",theycanheprojectt<J

backtothelalent'JI'K'"for.'isualilalionpurposes,Figure12,21sl"""sa comparison

oftheoildata..,1"isualiredwilhlincarPeAandwilhlheIIO/lhnearGT~I,

TIltGTMcanbeseenasa probabilistic"'rsionofanearliernlOd<lcallMthe'''If

org"nidng""'p.orSOM(Kohonen.1982:Kobonen. (995).whichalsorepresents

a Iwo-dimensiooal IIO/llincarmanifoidasa regulararrayofdisc"'lepoints. The

SOM i'somewhat remin;""'ntofthe K·trlCan,algorithmin thatdatapointsare

a.,igr.edtonearbyProlOl)'j>C'-eclonthaIarelhensubsc<juenllyupdale<!. Initially.

lheproIOI)'jl('Saredistribuledatrandom,andduringthetrainingprocessthey'selr

organize'soastoaPl'ro~imalea smoolhmanifold. UnlikeK-mean'.how'e"e.. the

SOMisTIOIoptimizinganywell.ddine<!costfunction(Erwin..al..1992)making"

difficulttos."theparametersofthemodeland 10 assesscon'-ergence.Therei'also

noguaranteethatthe'",If-<>rganilalion'willtakeplace..thisisdepen""nl 00 the

choiceofappropriateparanlttcr"aloC'f,,,anyparticulardatasel.

ByOOfItrast,GTMoptimize,theloglikelihoodfunctioo,andtheresoltingmodel

define'a probabililyden,ityindma,pace, InfaeLilcorre,pondstoa con,m,incd

mi,tureofGaussian,inwhichthecomponent.',h.reaCOnlnlOn".riance.•ndthe

mean,arecon'trainedtolieona 'mooIhtw-o-diITlCn,iooaln1anifold. Thisproba-