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@126.comFCMCFCMWFCMIrisWFCMFCMGermanFCMFCM(filterapproach)(wrapperapproach)[1]JoshuaZhexueHuang[2]k-meansC2.1mmAAA×××L21C{}nXXXS,,,ˆ21L=mimiiiAAAxxxX×××∈=LL2121),,,(),,2,1(niL=iiXjX()2112),(),(⎟⎠⎞⎜⎝⎛⋅=∑=mkjkikkkjixxdwXXD(1)),(jkikkxxdkk≠==,1,0{),(kjkikjkikkxxxxd−=),(∑==mkkmw1),,2,1(0mkmwkL=≤≤12.2k-meansJoshuaZhexueHuang[2]k-meansW-k-MEANSTYPEALGORITHMS,()∑∑∑===⋅⋅=niCjmkjkikkqkijvxdwuWVUP1112),(),,((2)UVW2.3FCM[2]()∑∑∑===⋅⋅=niCjmkjkikkqkrijCvxdwuWVUE1112),(),,((3)Subjectto10≤≤iju,11=∑=Cjiju,Cj,,2,1L=,ni,,2,1L=.),,2,1(0,1mkmwmwkmkkL=≤≤=∑=.iju),,2,1(niXiL=j),,2,1(CjVjL=),,,(21jmjjjvvvVL=j),,,(21m=2.3.1U,VW()∑∑∑===⋅⋅=niCjmkjkikkqkrijCvxdwuWVUE1112)ˆ,(ˆ),ˆ,ˆ((4)),,2,1(0,1mkmwmwkmkkL=≤≤=∑=L())()ˆ,(ˆ),(11112∑∑∑∑====−⋅+⋅⋅=mkkniCjmkjkikkqkrijmwvxdwuWLααL0()0)ˆ,(ˆ)(1121=+⋅⋅⋅=∂∂∑∑==−αniCjjkikkrijqkkvxduwqwL=−=∂∂∑=mkkmwLα),,2,1(mkL=()0)ˆ,(ˆ)(1121=+⋅⋅⋅∑∑==−αniCjjkikkrijqkvxduwq()11112)ˆ,(ˆ−==⎟⎟⎟⎟⎟⎠⎞⎜⎜⎜⎜⎜⎝⎛⋅⋅−=∑∑qniCjjkikkrijkvxduqwα01=−∑=mkkmw()mvxduqmkqniCjjkikkrij=⎟⎟⎟⎟⎟⎠⎞⎜⎜⎜⎜⎜⎝⎛⋅⋅−∑∑∑=−==111112)ˆ,(ˆα()()∑∑∑=−==−⎟⎟⎟⎟⎟⎠⎞⎜⎜⎜⎜⎜⎝⎛⋅⋅=−mkqniCjjkikkrijqvxduqm11111211)ˆ,(ˆ1α()()∑∑∑∑∑=−====⎟⎟⎟⎟⎟⎠⎞⎜⎜⎜⎜⎜⎝⎛⋅⋅=mlqniCjjlillrijniCjjkikkrijkvxduvxdumw111112112)ˆ,(ˆ)ˆ,(ˆ2.3.2W,VU()∑∑∑===⎟⎠⎞⎜⎝⎛⋅⋅=niCjmkjkikkqkrijCvxdwuWVUE1112)ˆ,(ˆ)ˆ,ˆ,((5)Subjectto10≤≤iju,11=∑=Cjiju,Cj,,2,1L=,ni,,2,1L=.L()∑∑∑∑∑=====⎟⎟⎠⎞⎜⎜⎝⎛−⋅+⋅⋅=niCjijiniCjmkjkikkqkrijuvxdwuUL111112)1()ˆ,(ˆ),(ααL0()0)ˆ,(ˆ)(121=+⋅⋅⋅=∂∂∑=−imkjkikkqkrijijvxdwuruLα011=−=∂∂∑=CjijiuLαni,,2,1L==,ni,,2,1L=()0)ˆ,(ˆ)(121=+⋅⋅⋅∑=−imkjkikkqkrijvxdwurα()1112)ˆ,(ˆ−=⎟⎟⎟⎟⎠⎞⎜⎜⎜⎜⎝⎛⋅⋅−=∑rmkjkikkqkiijvxdwruα011=−∑=Cjiju()1)ˆ,(ˆ11112=⎟⎟⎟⎟⎠⎞⎜⎜⎜⎜⎝⎛⋅⋅−∑∑=−=Cjrmkjkikkqkivxdwrα()()∑∑=−=−⎟⎟⎟⎟⎠⎞⎜⎜⎜⎜⎝⎛⋅⋅=−Cjrmkjkikkqkrivxdwr1111211)ˆ,(ˆ11α()()∑∑∑=−=−=⎟⎟⎟⎟⎠⎞⎜⎜⎜⎜⎝⎛⋅⎟⎟⎟⎟⎠⎞⎜⎜⎜⎜⎝⎛⋅=Clrmklkikkqkrmkjkikkqkijvxdwvxdwu111121112)ˆ,(ˆ1)ˆ,(ˆ1()()∑∑∑=−==⎟⎟⎟⎟⎠⎞⎜⎜⎜⎜⎝⎛⋅⋅=Clrmklkikkqkmkjkikkqkijvxdwvxdwu1111212)ˆ,(ˆ)ˆ,(ˆ1.2.3.3W,UV()∑∑∑===⋅⋅=niCjmkjkikkqkrijCvxdwuWVUE1112),(ˆˆ)ˆ,,ˆ((6)k()∑∑∑===−⋅⋅=niCjmkjkikqkrijCvxwuWVUE1112)(ˆˆ)ˆ,,ˆ(()0)()2(ˆ)ˆ(1=−⋅−⋅⋅=∂∂∑=nijkikqkrijjkCvxwuvE()()∑∑==⋅⋅⋅=niqkrijniikqkrijjkwuxwuv11ˆ)ˆ(ˆ)ˆ(()∑∑==⋅=nirijniikrijjkuxuv11)ˆ()ˆ(.kk-type(){}()∑∑∑===∈⋅⋅=niCjmkjkikkqkrijkvjkvxdwuvjk1112),(ˆˆminarg2.4FCM2.3FCMStep1.ε)0(V{)0(,),0(),0(21CVVVL}Tm),,,(21L=(1)0=t;r2q2;Step2.0)0(=−jiVX)0(iju1()()∑∑∑=−==⎟⎟⎟⎟⎠⎞⎜⎜⎜⎜⎝⎛⋅⋅=Clrmklkikkqkmkjkikkqkijvxdwvxdwu1111212))0(,()0())0(,()0(1)0(Cj,,2,1L=ni,,2,1L=.()CnijuU×=)0()0(;Step3.()∑∑==⋅=+nirijniikrijjktuxtutv11))(())(()1(Cj,,2,1L=,mk,,2,1L=)(nmCOk-type(){}()∑∑∑===∈⋅⋅=+niCjmkjkikkqkrijkvjktvxdtwtutvjk1112))(,()(ˆ)(ˆminarg)1(Step4.()()∑∑∑=−==⎟⎟⎟⎟⎠⎞⎜⎜⎜⎜⎝⎛+⋅+⋅=+Clrmklkikkqkmkjkikkqkijtvxdtwtvxdtwtu1111212))1(,()())1(,()(1)1(ni,,2,1L=Cj,,2,1L=.)(2nmCOStep51+=ttTm),,,(21L=()()∑∑∑∑∑=−====⎟⎟⎟⎟⎟⎠⎞⎜⎜⎜⎜⎜⎝⎛⋅⋅=mlqniCjjlillrijniCjjkikkrijktvxdtutvxdtumtw111112112))(,()())(,()()(mk,,2,1L=)(2CnmOStep5.max{)()1(tutuijij−+}ε;Step3.FCM3.1FCMFCMBezdek[3]1973()∑∑==−⋅=niCjjirijCVXuE112)((7)(7)iju),,2,1(niXiL=j),,2,1(CjVjL=11=∑=CjijuFCMrFCMStep1.ε)0(V{)0(,),0(),0(21CVVVL}0=t;r2)0(=−jiVX)0(iju1∑=−−⎟⎟⎠⎞⎜⎜⎝⎛−⎟⎟⎟⎠⎞⎜⎜⎜⎝⎛−=CkkirjiijrVXVXu1211211)0(1)0(1)0(Cj,,2,1L=ni,,2,1L=.()CnijtutU×=)()(;Step3.()∑∑==⋅=+nirijniirijjtuXtutV11)()()1(Cj,,2,1L=.(8)Step4.∑=−−⎟⎟⎠⎞⎜⎜⎝⎛+−⎟⎟⎟⎠⎞⎜⎜⎜⎝⎛+−=+CkkirjiijrtVXtVXtu1211211)1(1)1(1)1(Cj,,2,1L=ni,,2,1L=.Step5.max{)()1(tutuijij−+}ε;1+=ttStep3..)(nmCtOnCsmk-means()∑∑==⋅=+nirijniikrijjktuxtutv11)()()1(()),(),(11∑∑==⋅=nlrljnllkrljikkjkikkuxuxdvxd()()∑∑∑∑====−⋅=⋅−nllkikrljnlrljnlrljnllkrljikxxuuuxux1111)(1()∑∑==−⋅=nllkikrljnlrljjkikkxxuuvxd11)(1),(()∑∑==−⋅=nlrljnllkikrljjkikkuxxuvxd11)(),()(jkikxx−jkikjkikjkikxxxxxx≠==−,1,0{)(FCM∑=−−⎟⎟⎠⎞⎜⎜⎝⎛−⎟⎟⎟⎠⎞⎜⎜⎜⎝⎛−=CkkirjiijrtVXtVXtu1211211)(1)(1)(()()∑∑∑=−==⎟⎟⎟⎟⎠⎞⎜⎜⎜⎜⎝⎛⋅⋅=Clrmklkikkkmkjkikkkijtvxdwtvxdwtu1111212))(,())(,(1)(()∑∑==−⋅=nllkikrljnlrljjkikkxxuuvxd11)(1),(∑∑∑∑∑∑∑=−======⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠⎞⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝⎛⎟⎟⎟⎟⎟⎠⎞⎜⎜⎜⎜⎜⎝⎛⎟⎟⎟⎟⎠⎞⎜⎜⎜⎜⎝⎛−⋅⋅⎟⎟⎟⎟⎟⎠⎞⎜⎜⎜⎜⎜⎝⎛⎟⎟⎟⎟⎠⎞⎜⎜⎜⎜⎝⎛−⋅⋅=+Clrmknsskikrslnsrslkmknsskikrsjnsrsjkijxxtutuwxxtutuwtu11112111211))()(()(1))()(()(11)1((9)3.2FCMStep1.{}nXXXS,,,21L=C)}0(,),0(),0({)0(21CVVVVL=ε0=t;1FCMr2Step2.))0(,(jiVXD0)0(iju1∑=−⎟⎟⎠⎞⎜⎜⎝⎛=ClrlijiijVXDVXDu11122))0(,())0(,(1)0(()()∑∑∑=−==⎟⎟⎟⎟⎠⎞⎜⎜⎜⎜⎝⎛⋅⋅=Clrmklkikkkmkjkikkkijvxdwvxdwu1111212))0(,())0(,(1)0(ni,,2,1L=Cj,,2,1L=.()CnijuU×=)0()0(;Step3.∑∑∑∑∑∑∑=−======⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠⎞⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝⎛⎟⎟⎟⎟⎟⎠⎞⎜⎜⎜⎜⎜⎝⎛⎟⎟⎟⎟⎠⎞⎜⎜⎜⎜⎝⎛−⋅⋅⎟⎟⎟⎟⎟⎠⎞⎜⎜⎜⎜⎜⎝⎛⎟⎟⎟⎟⎠⎞⎜⎜⎜⎜⎝⎛−⋅⋅=+Clrmknsskikrslnsrslkmknsskikrsjnsrsjkijxxtutuwxxtutuwtu11112111211))()(()(1))()(()(11)1(ni,,2,1L=Cj,,2,1L=.{)()1(tutuijij−+}εStep5;1+=ttStep3.Step5.ijCjuk,,2,1maxargL==)(kclusterXi∈ni,,2,1L=.)(22tmCnOnCtm3.3FCM3.2FCM()∑∑==−⋅=nllkikrljnlrljjkikkxxuuvxd11)(1),(FCM()∑∑∑∑∑∑∑=−===−===⎟⎟⎟⎟⎟⎠⎞⎜⎜⎜⎜⎜⎝⎛⎟⎟⎟⎟⎟⎠⎞⎜⎜⎜⎜⎜⎝⎛⎟⎟⎟⎟⎠⎞⎜⎜⎜⎜⎝⎛−⋅⋅⎟⎟⎟⎟⎟⎠⎞⎜⎜⎜⎜⎜⎝⎛⎟⎟⎟⎟⎟⎠⎞⎜⎜⎜⎜⎜⎝⎛⎟⎟⎟⎟⎠⎞⎜⎜⎜⎜⎝⎛−⋅⋅=+Clrmknsskikrslnsrslqkrmknsskikrsjnsr