2-chapter_4-Somedetectioncriterions

1UESTCSomedetectioncriterionsChapter4SingleSampleDetectionofBinaryHypotheses2UESTC4.1Chapterhighlights•HypothesistestingandtheMAPcriterion.•ExtensiontothegeneralBayescriterion.•Minimaxcriterion•Neyman-Personcriterion.3UESTC4.2Hypothesistesting&MAPcriterionConsidertwopossiblevaluess0ands1.Wewishtodecidebetweenthesetwoalternativeswhenweobserveameasurementcorruptedbynoise.H0:thevalues0beingpresentH1:thevalues1beingpresent,0,1(4.1)iysni00)(HP11)(HPTheaprioriprobabilities:4UESTC01(|)(|)(4.2)PHyPHy10(|)(|)(4.3)PHyPHy000(|)()(|)(4.4)()pyHPHPHypy=MAPcriterion(MaximumAPosteriori):Ifwedecides0arepresent,andconverselywedecides1ifUsingBayes’theorem,wecanwriteLikelihoodratio5UESTC,0,1(4.1)iysni0(|)pyH1(|)pyHLikelihoodfunction:ConditionalPDF(Likelihoodfunction:)000000(|)()(|)()(|)()()(|)()()PHyPyPyHPHPHypydypyHdyPH==Similarly,Inordertocalculatetheaposterioriprobabilities,itisnecessarytoknowtheaprioriprobabilities.Let6UESTC111(|)()(|)(4.6)()pyHPHPHypy00)(HP11)(HP7UESTCD0:theeventassociatedwithdecisionH0D1:theeventassociatedwithdecisionH1101100(|)(|)(4.10)DDpyHpyH11001010(|)(4.11)(|)()DDDMAPDpyHpyHLyLikelihoodratio8UESTC4.2.1Maximum-Likelihoodcase•Iftwohypothesesareequallylikelyapriori,then011,12MAPMaximum-likelihood(ML)criterionwithathresholdItisoftenusedindetectionproblemswhenthereisnoknowledgeoftheaprioriprobabilities.1ML9UESTCExample4.1Considertheexamplewhereoccurringwithaprioriprobabilitiesand.Wewillassumethatbislargerthanzero.Ifthenoisepdfisazero-meanGaussianrandomvariablewithvariance,01,sbsb00.810.22221()exp()(4.13)22nnpn10UESTC2121()(|)exp()(4.14)22nybpyH2021()(|)exp()(4.15)22nybpyH014(4.16)MAPThetwohypothesesconditionalpdfThethreshold11UESTC22222()()2()exp[]exp()(4.17)22ybybybLy22exp()4(4.18)yb22ln4(4.20)yb()ln()(4.19)yLyislog-likelihoodratio)(yWedecideD1ifWedecideD1,if12UESTC102ln4(4.21)2DDybIf,then5.010ln0MLThen,theMLcriteriondecision100DDy13UESTC14UESTCExample4.2ManyopticalcommunicationssystemscanbemodeledbaseduponadiscretePoissonprobabilitydescription.kisthenumberofphotonswhicharriveinaunitoftime.000(|),0,1,2...(4.23)!kaapkHekk111(|),0,1,2...(4.24)!kaapkHekk15UESTCHere,itisassumedbothhypothesestobeequallylikely,and.As,wedecideD1if10aa011()0()()(4.25)aakaLkea1MAPML0110()[lnln]0(4.26)kaakaa1010(4.27)lnlnaakaa16UESTC101,()(4.29)0,yRyyR011()(4.30)MAPyRLy0()(4.31)MAPyRLyAdecisionrulecanbedefinedasIntermsofMAPcriterion,4.2.2DecisionRule17UESTCItfollowsthattheMAPdecisionruleis01011,(4.32)0,MAPyLyyyLy18UESTC4.2.3TypesoferrorsPij:theprobabilityofdecidingDiwhenhypothesisHjiscorrect.R0R119UESTC4.2.3Typesoferrors(|)(4.33)iijjRPpyHdy010101(4.34)EPPPPij:theprobabilityofdecidingDiwhenhypothesisHjiscorrect.iscalledtheprobabilityofa“miss”01PmP11PiscalledtheprobabilityofdetectiondP10PiscalledtheprobabilityoffalsealarmfP20UESTCExample4.32`00222`01222102`22112`21()exp()221()exp()221()exp()221()exp()(4.35)22ybPdyybPdyybPdyybPdyInexample4.1,ifwedefinealog-likelihoodthresholdas,then21UESTC22UESTC202()(4.36)zterfzedtTheerrorfunctionisdefinedas22()1()(4.37)tzerfczedterfzandcomplementaryerrorfunctionisdefinedas23UESTC000110111[1()]221[1()]221()221()(4.38)22bPerfbPerfbPerfcbPerfcMakingtheappropriatechangesofvariables,wecanwrite24UESTC221,ln42()(4.39)0,ln42MAPybyyb01222lnlnln4(4.40)222MAPbbbInexample4.1,theMAPdecisionrulewhereand1{:}Ryy0{:}Ryy25UESTC000110110.9905260.0491220.0094740.950878(4.41)PPPPFor21,0.25,b26UESTCExample4.4(不讲)1010(4.42)lnlnaakaaInexample4.2,theintegerisdefinedaskwhereindicatesthelargestintegerlessthanorequaltotheinteriorexpression.Thedecisionregion0{:}Rkkk1{:}Rkkk27UESTC01010000101011011111!!!(4.43)!kkakkkakkakkkakkaPekaPekaPekaPekThefourprobabilitiesare28UESTC1001(4.44)1PP1101(4.45)1PP000110110.9196890.2381030.0803010.761897(4.46)PPPP1,3()(4.47)0,2MAPkkkIf,then011,4aa2kDecisionrule29UESTCExample4.5（不讲）/201,0(|)(4.48)20,0yeypyHy/211,0(|)(4.49)40,0yyeypyHy()(4.50)2yLyUnderthehypothesis,conditionalpdf0HUnderthehypothesis,conditionalpdf1H30UESTC/2110210.3679(4.51)2yPedyeForequallylikelyaprioriprobability,thedecisioncriterionotherwiseDandyifDdecide012forthisexample10P31UESTC/20(|)1,0(4.52)yFyHey2ln(4.53)yu102ˆ(4.54)numberofsampleswithyPtotalnumberofsamplesValuesoftherandomvariableycanbegeneratedfromisuniformlydistributedininterval(0,1)uEstimate10ˆP32UESTC33UESTC4.3Bayescriterion(4.55)000010100010111111[][]rPCPCPCPC1101(4.56)1PP1000(4.57)1PP（TheBayescriterionintroducestheconceptofcost）Letbethecostofdecidingwheniscorrect.ijCiDjHTheBayesdecisionistominimize.rAccordingtoAveragerisk34UESTC01011101000001011101()()(4.58)rCCCCPCCP0000(|)(4.59)RPpyHdy0011(|)(4.60)RPpyHdyFurthermore0010111101111010000{()(|)()(|)}(4.61)RrCCCCpyHCCpyHdy