Fuzzy Membership Function Based on Structural Information of Data Fang-Zhi ZHU a, Hui-Ru WANG b, Zhi-Jian ZHOU c,*

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1 06 Intenatona Confeence on Sevce Scence, Technoogy and Engneeng (SSTE 06 ISBN: Fuzzy Membeshp Functon Based on Stuctua Infomaton of Data Fang-Zh ZHU a, Hu-Ru WANG b, Zh-Jan ZHOU c,* Coege of Scence, Chna Agcutua Unvesty, Bejng 00083, Chna a zfzckh@sna.cn, b wanghuu59@63.com, c zhjanzh@63.com *Coespondng autho Keywods: Fuzzy Membeshp, Stuctua Infomaton, Fuzzy Suppot Vecto Machne. Abstact. Membeshp functon based on affnty among sampes consdes not ony the dstance between sampes and ts cente, but aso the eatonshp among sampes. Howeve t gnoes the stuctua dstbuton of sampes n dffeent casses and the nfuences of between-cass dstance on membeshp. Theefoe, ths pape poposed an mpoved membeshp functon, named fuzzy membeshp functon based on stuctue nfomaton of data, whch ncopoates stuctua dstbuton and between-cass dstance nto the cacuaton of membeshp. The man dea s to fnd the optma hype-sphee fo the postve and negatve sampes espectvey by suppot vecto data descpton method and gve a educton ato fo sampes out of sphee accodng to the between-cass dstance. Numeca expements demonstate that the poposed membeshp functon s consstent wth the pactca appcaton, and t can sgnfcanty mpove the cassfcaton pefomance of fuzzy suppot vecto machne. Intoducton In ode to sove the pobem that suppot vecto machne (SVM s senstve to noses and outes, fuzzy suppot vecto machne (FSVM was poposed by Ln et a. whch gves a fuzzy membeshp to each sampe []. FSVM descbes the mpotance of sampes on the cassfcaton wth membeshp and educes the nfuence of outes fo cassfcaton hype-pane though educng the membeshp of outes. The choce of membeshp functon often decdes the cassfcaton pefomance of FSVM. Thus the cacuaton of membeshp s a key step n FSVM. Thee ae some desgns of membeshp functon. Ln et a. poposed membeshp functon wth dstance between the sampes and the cente whch satsfes that the sampe cose to the cente pont, the bgge of the coespondng membeshp [,]. An et a. constucts membeshp functon by sampe densty whch s defned as the numbe of suoundng sampes [3,]. Zhang et a. fnds an optma hype-sphee whch can suound amost a sampes though the eatonshp between sampe and othe sampes, and desgns dffeent membeshp functon fo sampes accodng to the eatve poston of hype-sphee [5,6]. Howeve these membeshp functons gnoe the nfuence of sampes stuctue dstbuton chaactestcs and the between-cass dstance on membeshp. Fuzzy membeshp functon based on stuctua nfomaton of data (S-FM s poposed n ths pape to sove ths pobem. Fsty, we obtan hype-sphee contanng postve and negatve sampes espectvey by suppot vecto data descpton (SVDD method [7] accodng to the stuctue dstbuton of sampes, and then espectvey gve a cetan popoton to educe membeshp 75

2 of sampes whch s outsde of the hype-sphee accodng to between-cass dstance of two casses. Ths poposed membeshp functon can make fu use of the stuctue nfomaton of the data, dstngush the outes and suppot vecto, and s moe sutabe fo the FSVM mode. Po Wok Fuzzy Suppot Vecto Machne Fuzzy suppot vecto machne fnds a cassfcaton hype-pane by mnmzng the weght eos of sampes and maxmzng the cassfcaton ntevas to cassfy sampes. Concetey, gven tanng sett = {( x, y, s, L, ( x, y, s }, whee s s a fuzzy membeshp of sampe x n satsfedδ s, =, L,, δ > 0 ; ϕ( s a mappng fom to the featue space, then the optmzaton pobem of FSVM can be expessed as: mn w, b, ξ = T. ( w ϕ( x b s t y w ξ 0, =, L, C s ξ ξ WheeC s a paamete. FSVM ntoduces the Lagange functon, and obtans ts sadde pont by the pata devaton of the vaabes, so the fom of dua pobem of FSVM s the foow: max α j j j = = j= = α s. t α y = 0 0 α s C, =, L, α α y y K( x, x We can obtan the hype-pane though sovng ths quadatc pogammng pobem. Fuzzy Membeshp Based on Affnty among Sampes Fuzzy membeshp based on affnty among sampes (A-FM uses SVDD method to fnd the smaest voume of hype-sphee that can suound a sampes, and defnes the membeshp functon by dstance between sampes and the cente. Its mode s [3]: d ( x 0.6 ( 0., d( x d ( x s = 0. (, d( x > d ( x whee d( x = x α, R andα ae the adus and cente of hype-sphee espectvey. An mpoved A-FM s gven n Lteatue [8], named IA-FM, and the mode s: s = ( x d x x > 0. ( 0.6, d( 0.6 (, d ( d ( x ( ( (3 ( 76

3 Fuzzy Membeshp Functon Based on Stuctua Infomaton of Data Athough A-FM takes nto account the compact of sampes, t st gnoes some aspects, as shown n Fg. and Fg.. In Fg., two ed sampes and eatvey cose to the cente of optma hype-sphee, theefoe, the fuzzy membeshp cacuated by A-FM method ae age. Howeve, accodng to the sampes dstbuton of postve and negatve casses, sampe and sampe ed n the edge of postve and negatve casses espectvey ae maybe outes, thus the membeshp shoud be smae, whch s nconsstent wth pevous esut. In Fg. (a and Fg. (b, sampes n two casses have the same wthn-cass poston, whe the between-cass dstance s totay dffeent. The membeshp of sampes ed n the edge, just as sampe n Fg. (a shoud age than that n Fg. (b, because the between-cass dstance n Fg. (a s age than that n Fg. (b. But, accodng to the A-FM method, t has the same membeshp n Fg. (a and Fg. (b, whch contadcts wth the ea stuaton. (a (b Fgue. Hype-sphee Poston Based on Wthn-cass Scatte Method. Fgue. Compason of Dffeent Between-cass Dstance. Note: whee back hoow ccuas stand fo postve sampes wth vetca dstbuton; back pus sgns stand fo negatve sampes wth hozonta dstbuton. Theefoe, the stuctua nfomaton of sampes, such as stuctua dstbuton and between-cass dstance shoud be consdeed n the cacuaton of fuzzy membeshp. On the bass of IA-FM method, we cacuate the adus of optma hype-sphee n accodance wth the sampes dstbuton of postve and negatve casses espectvey, meanwhe, consde the nfuence of between-cass dstance on membeshp and then popose a fuzzy membeshp functon based on stuctua nfomaton of data (S-FM. The cacuaton fomua of S-FM s 0. ( 0.6, d ( x f y =, s = 0.6 (, d ( x > and γ 0.6 (, d ( x > and < γ γ d ( x 3 0. ( 0.6, d ( x f y =, s = 0.6 (, d ( x > and γ 0.6 (, d ( x > and < γ γ (5 77

4 whee,, 3, s the paametes to adjust the tend of fuzzy membeshp of sampes nsde and outsde hype-sphee wth 0 <, 3 <,, > ;, ae the adus of postve and negatve optma hypesphee espectvey; paamete γ efects the between-cass dstance of two casses wth γ >,and we take γ = n ths pape; d ( x stands fo the dstance fom postve sampes to the cente of postve cass; d ( x stands fo the dstance fom postve sampes to the cente of negatve cass; d ( x stands fo the dstance fom negatve sampes to the cente of negatve cass; ( d x stands fo the dstance fom negatve sampes to the cente of postve cass. Its eatonshp s shown n Fg. 3. Fgue 3. Dffeent Dstance Meanng n Fomua (5. Fom fomua (5, the sampe ed nsde the optma hype-sphee cose to the cente, the bgge of the coespondng fuzzy membeshp and ts mnmum vaue s 0.6; the smae of between-cass dstance of the sampes ed outsde the optma hype-sphee, the smae of the fuzzy membeshp. Theefoe, S-FM not ony deceases the membeshp of outes but aso ensues the fuzzy membeshp of suppot vectos s not too sma. Expements In ode to vefy the vadaton of poposed S-FM method, we espectvey conduct expements on atfca datasets and benchmak datasets n FSVM mode to compae wth A-FM and IA-FM method. We take fve-coss-vadaton and gd seach to fnd optma paamete and ecod testng accuacy. A expements ae caed out n the kene space wth Gauss kene usng FSVM mode. The ange of Gauss kene paamete s {,, L,9,0} C =, 3, L,7,8 ; =, the paamete n FSVM s chosen wth { } the paametes n IA-FM ae set by = 0., = 0. A benchmak datasets ae deved fom the UCI standad database[7].the atfca datasets ae constucted by Matab whch satsfed two-dmensona noma dstbuton. Atfca Dataset contans 00 sampes wth 00 sampes fo evey categoy. Fo postve sampes, ts mean s [-,] and covaance matx s [0.75,0;0,6]; fo negatve sampes, ts mean s [,0] and covaance matx s [5,0;0,0.75]. In atfca Datasets, each sampe x = x, x s a andom two-dmensona aay satsfed spheca dstbuton, and the ( cacuaton fomua s x = R cos θ, x = R snθ, whee θ satsfes the unfom dstbuton of [0, π ], paamete R satsfes N(0, and N(0, of postve and negatve sampes espectvey. In ode to see t ceay, we daw Fg.. Fom Fg. (a, we get that postve 78

5 sampes satsfy vetca dstbuton and negatve sampes satsfy Hozonta dstbuton n Datasets. Sampes n Datestes satsfy spheca dstbuton n Fg. (b (a Dataset Fgue. The Dstbuton of Atfca Datasets (b Dataset Note: whee bue * stand fo postve sampes, ed stand fo negatve sampes. The expementa esuts s shown n Tabe. Fom Tabe, We can get the foow concusons. Compaed wth A-FM and IA-FM, the poposed S-FM s moe sutabe fo FSVM so that t has a hghe cassfcaton accuacy, especay n Pma-Indan dataset. Because ou method compehensvey consdes the stuctua dstbuton featues wth the nfuence of between-cass dstance on fuzzy membeshp. It dstngshs outes wth noma sampes, outes wth suppot vectos to a age degee and ensues that the membeshp of the suppot vecto s not too sma. Theefoe ou method of S-FM s n accodance wth actua chaactestcs of data, and moe sutabe fo the FSVM mode. Tabe. Expementa Resuts of Thee Fuzzy Membeshp Cacuaton Methods usng FSVM. Datasets (,, 3, functon C Accuacy(% Tme(s Bupa A-FM ± IA-FM ± (0.,8,0., S-FM ± Pma-Indan A-FM ± IA-FM ± (0.,,0.,8 S-FM ± Wdbc A-FM ± IA-FM ± (0.,,0.,8 S-FM ± Wne A-FM ± IA-FM ± (0.,,0.,8 S-FM ± Dataset A-FM ± IA-FM ± (0.,,0.,8 S-FM ± Dataset A-FM ± IA-FM ±. 0.8 (0.,,0.,8 S-FM ±

6 Concuson In ths pape, we poposed a fuzzy membeshp functon based on stuctua nfomaton of data. It consdes stuctua dstbuton of dffeent casses, and cacuates membeshp of postve and negatve sampe espectvey. Meanwhe, t consdes the nfuence degee of between-cass dstance on the fuzzy membeshp accodng to the sze of eatve dstance ato fom sampes to the cente of two casses. Theefoe, t educes the nfuence of outes on the cassfcaton esut and ensues that the membeshp of suppot vecto w not be educed too fast. Thus outes and noma sampes ae bette to be dstngushed. Expements on atfca and benchmak datasets show that the poposed method s effectve, and t can pay a bette oe n the FSVM mode. Thee ae many paametes n the mode, and the nfuence of the paametes on the cassfcaton esuts can be futhe studed. Acknowedgement Ths wok s suppoted by Chnese Unvestes Scentfc Fund(No.06LX00. Refeences [] Ln C F, Wang S. Fuzzy suppot vecto machnes.[j]. IEEE Tans Neua Netwoks, 00, 3(3: []Jang X, Y Z, Lv J C. Fuzzy SVM wth a new fuzzy membeshp functon[j]. Neua Computng & Appcatons, 006, 5(3-: [3] An J, Wang Z, Ma Z. Fuzzy suppot vecto machne based n densty [J]. Jouna of Tanjn Unvesty: Natua scence and Engneeng Technoogy, 00, 37(6:5-58. DOI:0.3969/j.ssn []L L, Zhou M, Lu Y. Fuzzy Suppot Vecto Machne Based on Densty wth Dua Membeshp [J]. Compute Technoogy and Deveopment, 009, 9(: -6. [5] Zhang X, Xao X, Xu G. Fuzzy suppot vecto machne based on affnty among sampes [J]. Jouna of Softwae, 006, 7(5: [6] An W, Lang M. Fuzzy suppot vecto machne based on wthn-cass scatte fo cassfcaton pobems wth outes o noses[j]. Neuocomputng, 03, 0(8:0 0. [7] Tax D M J, Dun R P W. Suppot vecto data descpton[j]. Machne Leanng, 003, 5(:5-66. [8] Xe L. The key technooges of fuzzy suppot vecto machne [D]. Habn Insttute of Technoogy, 0. [9] 80