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ApplicationofFisherLinearDiscriminant(FLD)inDataClassification1、FLDalgorithmoverviewFisherLinearDiscriminant(FLD)isaclassicalalgorithmforpatternrecognition.ItwasintroducedbyBelhumeurin1996inthefieldofpatternrecognitionandartificialintelligence.Thebasicideaofsexualdiscriminantanalysisistoprojecthigh-dimensionalsamplepatternsintotheoptimaldiscriminantvectorspacetoachievetheeffectofextractingtheclassificationinformationandcompressingthefeature'sspatialdimension.Afterprojection,thesamplemodelisguaranteedtohavethelargestinterclassspaceinthenewsubspace.Thedistanceandtheminimumintra-classdistance,ie,thepattern,havethebestseparabilityinthisspace.Therefore,itisaneffectivefeatureextractionmethod.Usingthismethod,theinter-classscattermatrixofthesampleafterprojectioncanbemaximized,andtheintra-classscattermatrixcanbeminimizedatthesametime.Thatistosay,itcanguaranteethatthemodelsampleafterprojectionhasthesmallestintra-classdistanceandthelargestinter-classdistanceinthenewspace,thatis,thepatternhasthebestseparabilityinthespace.2、Formuladerivation,thealgorithmformalizeddescription.Algorithmfunction:dimensionalityreductionandclassification2、Formuladerivation,thealgorithmformalizeddescription.Generallyspeaking,agooddirectioncanalwaysbefoundwhichisveryeasytoseparatesamplesfromthelinesprojected,seekingthebestprojectionvectorW*isFisheralgorithmtodo，thespecificmethodisasfollows：1）Calculatingthemeanvectorofallkindsofsamplesmi；NiisthenumberofsamplesofthewiclassiwXiiXNm1i=1,2(1)i=1,2(2)iwXiimXmXiS(3)21SSSw2）TheintraclasdispersionmatrixSiandthetotalintraclassscattermatrixSwarecaculated2、Formuladerivation,thealgorithmformalizeddescription.3）calculateaninterclassdiscretenessmatrixforsamplesSb4）FindthebestprojectionvectorW*(4)2121mmmmSbWSWWSWWJwbfisherWehopethataftertheprojection,thesamplesareasseparateaspossibleintheone-dimensionalspace,thatis,thedifferencebetweenthemeanofthetwotypesofsamplesisbettertolarge.wealsohopethatthelessthebettertheintraclassflickeris,soweintroducetheFisheridentificationexpressionhere:21~~mm(5)2、Formuladerivation,thealgorithmformalizeddescription.1WSWw(7)The,AfterjoiningLagrangemultiplier,guide1WSWWSWWCwb(8)022WSWSdWCdwb(9)WSWSwb(6)211mmSWwTheobtainthemaximum,theW*isWJfisherThederivationprocessisasfollows：By(4),ithastobe(11)wbmmWmmmmWS212121Thefinaleigenvalueformulaisderived.(12)WmmSWSSwwbw2111Duetowthesizeofanytimesdoesnotaffecttheresults,soitcanbeanunknownconstanttobothsidesand,obtained:w(13)211mmSWwTheresultsafterthederivationaremultipliedby，obtain1wS(10)WWSSbw12、Formuladerivation,thealgorithmformalizeddescription.XWy6）Computingthesegmentationthresholdy0ontheprojectionspace(14)7）ForagivenX,caculatedtheprojectionyontheW*XWy*5）Projectionofallsamplesinthetrainingset.212211~~NNmNmNy0(15)(16)8）Classificationaccordingtodecisionrules21wXyywXyyoo(17)3、MatlabinstanceoperationOpentheprogram'zhucx.m',the'lda_touyin.m‘and'local_roc_calculate.m'isthesubroutine，Theexecutionprogramhasthefollowingresults：Fig.1scatterplottoclassifydatafeaturesThedataisatwo-class2-dimensionalfeaturematrix,eachofwhichis2×100Fig.2TheprojectionvectorsobtainedbytheFisheralgorithmprojectionthetwodimensionalfeaturesintoastraightlineandbecomeonedimensionalfeature,preparingforthelatterclassificationFig.3Inthelaststep,wegetonedimensionalfeatureasascatterplot,whichcanclearlyseetheseparationoftwotypesofdata.Fig.4ClassificationresultFig.5ROCcurveAUC=0.9996ROC（ReceiverOperatingCharacteristic）CurvesandAUC(AreaUnderCurve)areoftenusedtoevaluateabinaryclassifier.Thecloserthecurveistotheupperleft,theclassificationeffectisbetter.Fig.6ProbabilitydensitydistributiondiagramFig.7ProbabilitydensitydistributioncurveTheresultsshowthattheFisherlineardiscriminantalgorithmachievesgoodresultsintheclassificationoftwoclassificationdata.Theaccuracyoftheclassificationobtainedbytheseexperimentsisover95%.Referencesandotherlinks：[1]ROC和AUC介绍以及如何计算AUC：[2]线性判别分析(LinearDiscriminantAnalysis,LDA）算法分析：[3]线性判别分析（LinearDiscriminantAnalysis）（一）: