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1、BayesDecisionTheoryMinimum-Error-RateClassificationClassifiers,DiscriminantFunctionsandDecisionSurfacesTheNormalDensity0CSE555:SrihariMinimum-Error-RateClassification•ActionsaredecisionsonclassesIfactionαistakenandthetruestateofnatureisωijthen:decisioniscorrectifi=jandinerrorifi≠jSeekadecisio
2、nrulethatminimizestheprobabilityoferrorwhichistheerrorrate1CSE555:SrihariMinimumErrorRateClassifierDerivation•zero-onelossfunction:⎧0i=jλ(αi,ωj)=⎨i,j=1,...,c⎩1i≠j•Therefore,theconditionalriskis:j=cR(αi
3、x)=∑λ(αi
4、ωj)P(ωj
5、x)j=1=∑P(ωj
6、x)=1−P(ωi
7、x)j≠1Theriskcorrespondingtothislossfunctionistheaver
8、ageprobabilityerror”•MinimizetheriskrequiresmaximizeP(ω
9、x)i(sinceR(α
10、x)=1–P(ω
11、x))ForMinimumerrorrateiiDecideωifP(ω
12、x)>P(ω
13、x)∀j≠iiij2CSE555:SrihariLikelihoodRatioClassification•Regionsofdecisionandzero-onelossfunction,λ−λP(ω)P(x
14、ω)122221Let.=θthendecideωif:>θλ1λλ−λP(ω)P(x
15、ω)211112•Ifλisthezero
16、-onelossfunctionwhichmeans:⎛01⎞λ=⎜⎟⎜⎟⎝10⎠P(ω)2thenθ==θλaP(ω)1⎛02⎞2P(ω)2ifλ=⎜⎟thenθ==θ⎜⎟λb⎝10⎠P(ω1)•LikelihoodRatiop(x/ω1)/p(x/ω2).•Ifweuseazero-onelossfunctiondecisionboundariesaredeterminedbythresholdθ.a•Iflossfunctionpenalizesmiscategorizingω2asωmorethanconversewegetlarger1Class-conditional
17、pdfsCSE555:SriharithresholdθbandhenceR1becomessmaller3Classifiers,DiscriminantFunctionsandDecisionSurfaces•ManymethodsofrepresentingpatternclassifiersSetofdiscriminantfunctionsg(x),i=1,…,ciClassifierassignsfeaturextoclassωifg(x)>g(x)∀j≠iiijClassifierisamachinethatcomputescdiscriminantfunction
18、sFunctionalstructureofageneralstatisticalpatternClassifierwithdinputsandcdiscriminantfunctionsgi(x)4CSE555:SrihariFormsofDiscriminantFunctions•Letg(x)=-R(α
19、x)ii(max.discriminantcorrespondstomin.risk!)•Fortheminimumerrorrate,wetakeg(x)=P(ω
20、x)(max.discriminationcorrespondstomax.posterior!)iig(x
21、)≡P(x
22、ω)P(ω)iiig(x)=lnP(x
23、ω)+lnP(ω)iii5CSE555:SrihariDecisionRegion•Featurespacedividedintocdecisionregionsifg(x)>g(x)∀j≠ithenxisinRiji2-D,two-categoryclassifierwithGaussianpdfsDecisionBoundary=twohyperbolasHencedecisionregionR2isnotsimplycon
