The Research of Algorithm for Data Mining Based on Fuzzy Theory

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1 The Research of Agorhm for Daa Mnng Based on Fuzzy Theory Wang Amn, L Je, Schoo of Compuer and Informaon Engneerng Anyang Norma Unversy Anyang, Chna Shenyang Insue of Compung Technoogy Chnese Academy of Scences Shenyang, Chna wam508@6.com, acky@vp.sna.com Journa of Dga Informaon Managemen ABSTRACT: Daa Mnng s a new fed n daa processng research. Suppor Vecor Machne (SVM s one of he new mehods usng n daa mnng, whch has ganed grea appcabe success. However, here are s peny of maons n SVM. For exampe, SVM won work f s ranng se conans unceran nformaon. In order o sove he probem presened above, hs paper dscusses he consranng programmng of fuzzy chance and he characersc of fuzzy cassfcaon as we as s expresson mehods. The agorhm for cassfyng Suppor Vecor Machne s aso ncuded n hs paper. Caegores and Subec Descrpors: H..8 [Daabase Appcaons]: Daa Mnng; I..3 [Deducon and Theorem Provng]: Fuzzy Genera Terms: Daa Mnng, Fuzzy Theory, Mnng Agorhm Keywords: Daa Mnng, Fuzzy Programmng, Fuzzy Cassfcaon, Fuzzy Lnear Separabe, Agorhm Receved: 8 Apr 03, Revsed 6 June 03, Acceped June 03. Inroducon Wh he adven of he nformaon age, humanknd s faced wh more and more nformaon processng probems such as daa sorage, organzaon and searches. These probems graduay ncrease n compexy n he herarchy. Ther scaes of space acves are growng. I s faser n me scae and more exensve and far-reachng n consequences and mpcaons. Daa mnng s a new fed n daa processng. I s a process ha dgs ou varous modes, summary and expor vaues from he known daa ses. Daa mnng s an nerdscpnary ha nvoves daabase echnoogy, machne earnng, sascs, neura neworks, knowedge engneerng, and hgh-performance cacuaon and so on. I has been wdey apped n ndusry, agrcuure, busness, economcs, heah and many oher ndusres. Suppor Vecor Machne (SVM s one of he new mehods usng n daa mnng. I was proposed by Vapnk e a [] [] [3]. Usng SVM, he probem of cassfcaon and regresson can be dea wh beer. SVM has been a research focus n machne earnng and was apped n many feds successfuy. Bu here are many maons n SVM. For exampe, when he ranng ses of SVM conan unceran nformaon, SVM w be ncapabe. In 00, Chunfu Ln and Shengde Wang, he professors of Tawan Unversy pu forward Fuzzy Suppor Vecor Machne (FSVM mehod wh Shgeo Abe and Takuya Inoue, he professors of Kobe Unversy. They made some mprovemens o SVM, bu dd no bud FSVM on agorhm eve. Ther work acks of he research on SVM ha conans unceran nformaon. To sove he probem of SVM conanng unceran nformaon (fuzzy parameers n common condons, we dscuss he consranng programmng of fuzzy chance and he characersc of fuzzy cassfcaon as we as s expresson mehods. The agorhm for cassfyng Suppor Vecor Machne s aso ncuded n hs paper.. Premnary Knowedge The research of hs paper s carred on n he gven possbe space whch s menoned n [6]. Journa of Dga Informaon Managemen Voume Number 5 Ocober 03 37

2 . Fuzzy Chance-consraned Programmng Defnon. Suppose A s a fuzzy subse n unverse of dscourse U. If U = R (se of rea number and A s a reguar cosed convex fuzzy se, A s caed fuzzy number, wren a. Defnon. Suppose a s a fuzzy number. If a s membershp funcon: µ a (x = r r r 3, r R ( =,, 3, x r, r r r x r, x = r x r 3, r r r x r 3 3 a s caed ranguar fuzzy number, wren a = (r, r. Rea number r and r are caed ranguar fuzzy number a s cener, ef and rgh endpons. Cener s he man ocaon of ranguar fuzzy number. Rea number a can be expressed as a speca ranguar fuzzy number a = (a, a, a. Defnon.3 Suppose f : R R R s a bnary funcon n rea number fed. a, b are fuzzy numbers. The membershp funcon of fuzzy number c = f (a, b can be defned as: µ c (z = sup{ mn (µ a (x, µ b (x x, y R, z = f (x, y}, z R Theorem. a = (r, r and b = (,, 3 are ranguar fuzzy numbers. ρ s a rea number. Then: a b = (r, r 3 ; ρ a = (ρr, ρr, ρr, ρ 0 3. (ρr 3, ρr, ρr, ρ < 0 Proof: The concuson can be drecy deduced from Defnon.3. Theorem. Suppose a = (r, r s ranguar fuzzy numbers. For any gven confdence eve λ (0 < λ, Pos {a 0} λ }( λ r λr. Proof: If Pos {a 0} λ, here mus be an esabshed r beween r 0 and r. If r 0, hen r < r 0, so r λ r ( λ r λr 0. If,, r λ (r r due o r < r, r r λ.e ( λ r λr 0. On he conrary, here are wo cases when( λ r λr 0. When r 0, Pos {a 0} =.e. Pos {a 0} λ; when r > 0, r r < 0, r r r λ.e Pos {a 0} λ due o ( λ r λr 0. ( ( Defnon.4 Programmng mn f s.. Pos{ f (x, ξ f } β Pos{ g (x, ξ 0, =,,..., p} α s caed fuzzy chance-consraned programmng [8]. In he programmng, x s decson varabe. ξ, ξ are fuzzy parameer vecors. f (x, ξ s obecve funcon. g (x, ξ 0, =,,..., p s consran condons. α, β (0 < α, β are he gven confdence eves of consran condon and obecve funcon. Pos { } s he possby measure of fuzzy even { }. Common fuzzy chance-consraned programmng s mn f (x s.. Pos{ h (x ξ c 0} f } λ ( - f (x s he funcon of decson varabe x. I s an obecve funcon ha does no conan fuzzy parameers. h (x s he funcon of decson varabe x. ξ s fuzzy number. c s rea number. h (x ξ c 0 s he consran condon. (0 < λ s he gven confdence eve. In [8], a souon of fuzzy chance-consraned programmng under norma crcumsances was pu forward. I convers fuzzy chance-consraned programmng o ceary equvaen programmng. Bu does no avaabe o he fuzzy chance-consraned programmng descrbed as (-. In hs paper, we provde a new souon of fuzzy chanceconsraned programmng descrbed as (-. Theorem.3 If ξ n fuzzy chance-consraned programmng equaon (- s ranguar fuzzy number.e. ξ = (r, r, ceary equvaen programmng of equaon (- s: mn f (x ( - s..( λ [r h (x r 3 h (x] λr h (x c 0 h (x = h(x 0 h(x,, h 0, h(x < 0 (x = 0, h(x 0. h(x, h(x < 0 I s an ordnary programmng ha equas o fuzzy chanceconsraned programmng descrbed as (-. Proof: Frs, we prove he cear equvaen cass of Pos{h (x ξ c 0} λ s: ( λ [r h (x r 3 h (x] λr h (x c 0 From he cacuaon of ranguar fuzzy number, h (xξ c s s a ranguar fuzzy number. Se h (x = h(x 0 h(x, 0, h(x < 0 h (x = 0, h(x 0. h(x, h(x < 0, 38 Journa of Dga Informaon Managemen Voume Number 5 Ocober 03

3 Then h (x, h (x are nonnegave and h (x = h (x h (x. So h(xξ c = (h (x h (xξ c = h (xξ h (xξ c = (h (xr, h (xr, h (xr 3 (h (xr, h (xr, h (xr 3 c = (h (xr h (xr 3 c, h (xr h (xr c, h (xr 3 h (xr c = (h (xr h (xr 3 c, (h (xr h (xr c, h (xr 3 h (xr c = (h (xr h (xr 3 c, h(xr c, h (xr 3 h (xr c From Theorem., he cear equvaen cass of Pos {h (xξ c} λ (0 < λ s ( λ [r h (x r 3 h (x] λr h (x c 0 Because fuzzy chance-consraned programmng equaon (- and ordnary programmng equaon (- have he same obecve funcon, and her consran condons are equa, fuzzy chance-consraned programmng equaon (- s equa o ordnary programmng equaon (-.. Characerscs of Fuzzy Cassfcaon Frs, we gve he defnon of cassfcaon. Defnon.5 Fnd a rue for ranng se S = {(x, y,...,(x, y } (x R n, y s fuzzy number ha means s fuzzy caegores =,..., of fuzzy number accordng o he oupu of gven ranng pons o deduce he correspondng fuzzy number y (means fuzzy caegores of x of any mode x. We ca hs probem fuzzy cassfcaon. The negra fuzzy nformaon suded n hs paper has hree fuzzy characerscs: fuzzy posve cass (sampe pons posve degree of membershp are more han negave one, fuzzy negave cass (sampe pons negave degree of membershp are more han posve one and cener cass (sampe pons posve degree of membershp and negave one are equa. Suppose sampe pons posve or negave degrees of membershp are δ or δ, δ, δ [0.5, ]. For convenence, we use δ [, 0.5] [0.5, ], make δ = δ, δ = δ. So, hree fuzzy characerscs can be descrbed by a speca ranguar fuzzy number as foow: y = (r, r = δ δ, δ, δ 3δ (, 0.5 δ δ ( - 3 3δ (, δ, δ δ, δ 0.5 ( Exampe of fuzzy posve cass: sampe pons posve degree of membershp s 0.9, and s negave degree of membershp s 0.. Ths fuzzy characersc can be descrbed by ranguar fuzzy number a = (0.58, 0.8,.0. In hs equaon, 0.8 s he cener of a. ( Exampe of fuzzy negave cass: sampe pons negave degree of membershp s 0.8, and s posve degree of membershp s 0.. Ths fuzzy characersc can be descrbed by ranguar fuzzy number b = (., 0.6, 0.. In hs equaon, 0.6 s he cener of b. (3 Exampe of cener cass: sampe pons posve degree of membershp s 0.5, and s negave degree of membershp s 0.5. Ths fuzzy characersc can be descrbed by ranguar fuzzy number c = (, 0,. In hs equaon, 0 s he cener of c. 3. FSVM Suppose he ranng se s S = {(x, y,...,(x, y } (3-. x R n, y s a ranguar fuzzy number as equaon (-3, =,...,. (x, y ( =,..., s fuzzy ranng pons. S s a fuzzy ranng se. Defnon 3. In equaon (-3, f δ (0.5, ], he correspondng fuzzy ranng pons are fuzzy posve pons. If δ [, 0.5, he correspondng fuzzy ranng pons are fuzzy negave pons. For smpcy, we do no consder he suaon of δ = 0.5 or δ = 0.5. Because he ranguar fuzzy number y = (, 0, does no provde posve or negave cass nformaon n hs suaon. To facae our research, we reorder he fuzzy ranng pons n he se. Pu he fuzzy posve cass pons n fron, and negave ones on back. Thus, we ge he fuzzy ranng se: S = {(x, y,..., (x p, y p, (x p, y p,..., (x, y } (3 - In equaon (3-, (x, y are fuzzy posve cass pons =,..., p. (x, y are fuzzy negave cass pons = p,...,. Defnon 3. Suppose he fuzzy ranng se s descrbed as equaon (3-, for he gven confdence eve λ (0 < λ, f has w R n, b R o make: Pos {y ((w. (x b } λ, =,...,, (3-3 We ca ranng se (3- fuzzy near separabe n he confdence eve λ. Now, we aso ca he queson of fuzzy cassfcaon fuzzy near separabe under confdence eve. As we a know, he basc condon of fuzzy near separabe s he npus of fuzzy posve cass pons and negave ones are separaed by possby λ (0 < λ. If he oupus of fuzzy ranng pons are a or, he fuzzy ranng se w degenerae o ordnary ranng se. Now, he near separabe of fuzzy ranng se becomes he near separabe of ordnary one. The fuzzy near separabe of fuzzy ranng se s he exenson of ordnary ses near separabe. In fuzzy ranng se, f ony we choose an approprae confdence eve, he fuzzy ranng se s s near separabe, ahough Journa of Dga Informaon Managemen Voume Number 5 Ocober 03 39

4 some ranng pons wh smaer membershp are mscassfed n he se. For exampe, hs s shown as Fgure : Suppose here are hree fuzzy ranng pons. Every pon s npu s one-dmensona. The oupus of fuzzy ranng pons (x, y and (x 3, y 3 are deermnae y = and y 3 =. The membershp degree of he frs fuzzy ranng pon ha beongs o negave cass s δ. Suppose δ = 0.5, 0.6, 0.7. x 3 = x = x = 0 y 3 = y = δ Fgure. The shown of fuzzy near separabe ( When δ = 0.5, accordng o equaon (-3, he correspondng ranguar fuzzy number y = (.94, 0.0,.9. So, he fuzzy ranng se s: S = {(x, y, (x, y, (x 3, y 3 } Choose λ = 0.7. If use separang hyper-pane x = 0, (w. x b =. So Pos{(y (( w. x b } = 0.7 > 0.7 And Pos{(y 3 (( w. x 3 b } = > 0.7 Pos{(y 3 (( w. x 3 b } = > 0.7 So, fuzzy ranng se S s near separabe under confdence eve λ = 0.7. ( When δ = 0.6, he correspondng ranguar fuzzy number y = (.53, 0.,.3. Choose λ = 0.47, smary, fuzzy ranng se s near separabe under confdence eve λ = ( When δ = 0.7, he correspondng ranguar fuzzy number y = (.6, 0.4, Choose λ = 0.08, smary, fuzzy ranng se S s near separabe under confdence eve λ = The probem of fuzzy cassfcaon has hree ypes: fuzzy near separabe (a he fuzzy ranng pons mee he condon of defnon 3.; approxmae fuzzy near separabe (mos of he fuzzy ranng pons mee he condon of defnon 3.; fuzzy nonnear (mos of he fuzzy ranng pons do no mee he condon of defnon 3.. Theorem 3. Under confdence eve λ (0 < λ, he cear equvaence cass of equaon (3-3 s: (( λ r 3 ((w. x b, =,..., p (3-4 (( λ r ((w. x b, = p,..., Proof: for any =,...,, se (w, b = ((w. x b, (w, b = 0, (w. x b > 0 ((w. x b, (w. x b 0 (w, b = (w. x b, (w. x b > 0 0, (w. x b 0 Accordng o heorem.3, ge he cear equvaence cass of Pos{( y (( w. x 3 b, =,..., } = Pos{( y (( w. x b 0} =, =,..., } λ s: ( λ [ r (w, b r 3 (w, b] λr (w, b 0, = (3-5 Under confdence eve λ (0 < λ, equaon (3-5 can be wren: (( λ r 3 ((w. x b, =,..., p (( λ r ((w. x b, = p,..., Tha s o say, under confdence eve λ (0 < λ, he cear equvaence cass of equaon (3-3 s equaon (3-4. In equaon (3-4, se k =, =,..., p ( λ r3 =, = p,..., ( λ r Then (3-5 can be expressed as (w. x b k, =,..., p (w. x b, = p,..., (3-6 Defnon 3.3 Consder he fuzzy near separabe probem gven by fuzzy ranng se equaon (3-3. We ca wo parae hyper-panes (w. x b = k and (w. x b = supporng hyper-panes abou fuzzy ranng se (3-3. If: (w. x b k, =,..., p mn{((w. x b } = k =,...p (w. x b, = p,..., max {((w. x b } = = p,... (3-7 k ( =,..., p, ( =,..., p s shown as equaon (3-7. k = mn {k }, = max { }. =,...p = p,... Obvousy, he par of supporng hyper-panes abou fuzzy ranng se (3-3 s unque. Dsance beween he par of supporng hyper-panes (w. x b = k and (w. x b = are k. We ca w nerva (k > 0 and < 0 are consans. 4. Fuzzy Lnear Separabe FSVMS Now we consder he near separabe probem of fuzzy 330 Journa of Dga Informaon Managemen Voume Number 5 Ocober 03

5 ranng se descrbed as equaon (3-. Under confdence eve λ (0 < λ, he probem of fuzzy cassfcaon ransform no fuzzy chance-consran programmng queson wh decson varabe (w, b T : mn w w, b s. Pos {y (4 - ((w. x b } λ, =,..., In equaon (4-, y ( =,..., s he ranguar fuzzy number of ranng se descrbed as equaon (3-. Pos{ }s possby measure of fuzzy even{ }. Theorem 4. Under confdence eve λ (0 < λ, he cear equvaen programmng of unceran chance-consran programmng descrbed as equaon (4- s he foowng quadrac programmng: mn w w, b s. (( λ r λr ((w. x b, =,..., p 3 (( λ r ((w. x b, = p,..., (4 - Proof: From he concuson of Theorem 3., we can draw hs concuson. In [], we know he opma souon of programmng (4- exss. Now, we w sove he dua programmng of quadrac programmng descrbed as equaon (4-. Theorem 4. he dua programmng of quadrac programmng descrbed as equaon (4- s: mn (A B C β Σ w, b = = p s.. (( λ r 3 Σ (( λ r = 0 = = p (4-3 0, =,..., p 0, = p,..., In equaon (4-3: A = B = = Σp s = = = C = Σ Σ = p q = p L (w, b, β, α = w ( β s (( λ r 3 (( λ r s3 λr s (x. x s (( λ r 3 (( λ r (x. x α q (( λ r (( λ r q (x. (, s s he subscrp of fuzzy posve cass pons,, q s he subscrp of fuzzy negave cass pons, β = (β,...,β p T, α = (α p,..., α T, (β, α T are decson varabes. Proof: Frs, we nduc Lagrange funcon Σ = p = ((( λ r 3 ((w. x b (( λ r ((w. x b In equaon (4-4, β = (β,...,β p T R p, α = (α p,..., α T R p,, are Lagrange mupers. Accordng o he defnon of Wofe anhess, we sove he mnmum of Lagrange funcon abou w, b. From he condon of exreme vaue: Ge = = b L (w, b, β, α = 0, L (w, b, β, α = 0 w (( λ r 3 Σ = p Σ = p (( λ r = 0 (4-5 w = (( λ r 3 x (( λ r x (4-6 Subsue equaon (4-6 no Lagrange funcon descrbed as equaon (4-4, and use equaon (4-5, we can ge he dua programmng of quadrac programmng descrbed as equaon (4-: max( Σ α (A B C = Σ β, = p s..σ p (( λ r 3 Σ (( λ r = 0 = = p (4-7 0, =,..., p = p,..., In equaon (4-7: A = B = = Σp s = = = C = Σ Σ = p q = p β s (( λ r 3 (( λ r s3 λr s (x. x s (( λ r 3 (( λ r (x. x α q (( λ r (( λ r q (x. β = (β,...β p T, α = (α p,...α T, (β, α T are decson varabe Conver he obecve funcon of quadrac programmng descrbed as equaon (4-7 o mnmum; we can ge equaon (4-3. The programmng descrbed as equaon (4-3 s a convex quadrac programmng. We sove s opma souon (β, α T = (β,...β p,α p,...α T. Se: w = = (( λ r 3 x b = (( λ r s3 λr s ( Σ = p s {s β s > 0}. or = Σ = p (( λ r (x. x s b = (( λ r q λr q ( Σ = p = (( λ r (x. α (( λ r x (( λ r 3 (x. x s (( λ r 3 (x. Journa of Dga Informaon Managemen Voume Number 5 Ocober 03 33

6 q {q α q > 0}. or And se g (x = (w. x b. We can prove he opma cassfcaon funcon (he mehod menoned n [] s: f (x = sgn (g(x = sgn ((w. x b, x R n (4-8 The opma separang hyper-pane s (w. x b = 0 The membershp funcon of opma cassfcaon funcon s µ (u = ϕ (u, 0 < u ϕ ( ϕ (u, ϕ ( u < 0, u > ϕ ( u < ϕ ( (4-9 In equaon (4-9, ϕ (u s he regresson funcon (monoone ncreasng funcon abou u geng from ε suppor vecor regresson. The consrucon mehod of ε suppor vecor regresson s: (a consruc he ranng se of regresson {(0, 0.5, (g (x, δ,...,(g (x p, δ p (4-0 (b use he ranng se as equaon (4-0, choose approprae ε > 0, penay parameer C > 0, choose near kerne as kerne funcon, consruc ε suppor vecor regresson. Smary, ϕ (u s he regresson funcon (monoone decreasng funcon abou u geng from ε suppor vecor regresson. The consrucon mehod of ε suppor vecor regresson s: (b consruc he ranng se of regresson {(0, 0.5, (g (x p, δ p,...,(g (x, δ (4- (c use he ranng se as equaon (4-, choose he same ε and C wh above, choose near kerne as kerne funcon, consruc ε suppor vecor regresson. ϕ ( s he vaue of funcon ϕ (u s nverse funcon a. ϕ ( s he vaue of funcon ϕ (u s nverse funcon a. Gven an npu x of es pon, subsue no equaon (4-8 and (4-9, we can ge f ( x = (or and s degree of membershp µ (g ( x. Conver hem o ranguar fuzzy number y. I s he oupu of es pons ha can refec he fuzzy cassfcaon of es pons (x, y. Ths cassfcaon shows he es pon s membershp of posve and negave cass. Through above dscusson, we can ge he agorhm for cassfyng Suppor Vecor Machnes. Ths agorhm s fuzzy near separabe. ( Gven he ranng se of fuzzy near separabe queson as equaon (3-, choose an approprae confdence eve λ (0 < λ o consruc quadrac programmng as equaon (4-3. ( Sove quadrac programmng as equaon (4-3, ge he opma souon (β, α T = (β,...β p, α p,...α T (3 Compung w = p Σ = (( λ r 3 x Σ = p α (( λ r x ; choose β, s posve componen β s or α s posve componen α. Accordngy, compue or q b = (( λ r s3 λr s ( Σ = p = (( λ r (x. x s b = (( λ r q λr q ( Σ = p = (( λ r (x. (( λ r 3 (x. x s (( λ r 3 (x. (4 Consruc opma separang hyper-pane (w. x b = 0. Accordngy, ge he opma cassfcaon funcon as equaon (4-8. (5 Use he ranng se as equaon (4-0 and (4- separaey, consruc ε suppor vecor regresson (choose approprae ε, penay parameer C, choose near kerne as kerne funcon. Ge regresson funcon ϕ (u and ϕ (u. Accordngy, consruc he membershp funcon of opma cassfcaon funcon as equaon (4-9. Normay, here s ony a e β and α are no zero n he opma souon (β,...β p, α p,...α T of quadrac programmng as equaon (4-3. The correspondng fuzzy ranng pons npu x and x are caed unceran suppor vecors. So, he opma cassfcaon funcon can be expressed as: f (x = sgn {( Σ (( λ r 3 (x x (FSV. Σ α (( λ r (x. x b } (FSV In hs equaon, (FSV s he se composed by a he unceran suppor vecors n fuzzy posve cass pons. (FSV s he se composed by a he unceran suppor vecors n fuzzy negave cass pons. If he oupus of fuzzy ranng pons are a or, he fuzzy ranng se w degenerae o ordnary ranng se. Now, he near separabe of fuzzy ranng se becomes he near separabe of ordnary one. 5. Numerca Expermenaon The unceran Suppor Vecor Cassfcaon Machne s bu on he bass of cassc SVM and fuzzy mahemacs. 33 Journa of Dga Informaon Managemen Voume Number 5 Ocober 03

7 To expan he raonay of our agorhm, we gve specfc daa o do numerca expermenaon. Suppose he npu of 6 ranng pons and her membershp degrees of posve (or negave cass are: x = (, T, s membershp degree of posve cass s (s membershp degree of negave cass s 0; x = (.7, T, s membershp degree of posve cass s 0.95 (s membershp degree of negave cass s 0.05; x 3 = (.5, T, s membershp degree of posve cass s 0.8 (s membershp degree of negave cass s 0.; x 4 = (0, 0 T, s membershp degree of negave cass s (s membershp degree of posve cass s 0; x 5 = (0.8, 0.5 T, s membershp degree of negave cass s 0.85 (s membershp degree of posve cass s 0.5; x 6 = (, 0.5 T, s membershp degree of negave cass s 0.8 (s membershp degree of posve cass s 0.. Accordng o he fuzzy characerscs and her expressons n fuzzy cassfcaon menoned n., conver he membershp degrees o ranguar fuzzy numbers; ge he foowng fuzzy ranng se: S = {(x, y, (x, y, (x 3, y 3, (x 4, y 4, (x 5, y 5, (x 6, y 6 } The oupus of ranng pons are ranguar fuzzy numbers. y = = (,,, y = (0.755, 0.9,, y 3 = (0., 0.6,., y 4 = = (,, y 5 = (.05, 0.7, 0.34, y 6 = (., 0.6, 0., Choose confdence eve λ = 0.8, we can verfy he fuzzy near separabe of ranng se S from Defnon 3.4. So: The opma separang hyper-pane s [x] [x] 4 = 0 Shown as Fgure : Fgure 3.Numerca expermena ( The opma cassfcaon funcon s: f (x = sgn (g(x = sgn ( [x] [x] 4 We consruc he membershp funcon of opma cassfcaon funcon: Use S = {(0, 0.5, (4,, (3.4, 0.95, (, 0.8} as fuzzy ranng se, choose ε = 0., C = 0 and he near kerne, consruc suppor vecor regresson, we can ge regresson funconϕ (u = 0.u 0.6. The membershp funcon of opma cassfcaon funcon s 0.0 g (x 0.6, 0 < g (x 4 µ (g (x = 0. g (x 0.6, 3.64 g (x 0, g (x > 4 g (x < 3.64 Suppose here are es pons npu x 7 = (, T, x 8 = (, 0 T Pu hem no f (x and µ (g (x, we ge f (x 7 = (fuzzy posve cass, s membershp degree of posve cass s 0.8 and negave s 0., so y 7 = (0., 0.6,. (ranguar fuzzy number; f (x 8 = (fuzzy negave cass, s membershp degree of negave cass s 0.8 and posve s 0.8, so y 8 = (.08, 0.64, 0. (ranguar fuzzy number. Compare Fuzzy Suppor Vecor Machne wh cassc suppor vecor machne. Frs, suppose fuzzy ranng se s: S = {(0, 0, ((0.8, 0.5,, ((, 0.5,, ((,,, ((.7,,, ((.5,, } Then he oupu of ranng pons ((.5,, w change: = (,, (0., 0.6,. (., 0.6, 0. (,, = So he oher fuzzy ranng ses are: S = {((0, 0, ((0.8, 0.5,, ((, 0.5,, ((,,, ((.7,,, ((.5,,, (.4, 0.6,.6} S 3 = {((0, 0, ((0.8, 0.5,, ((, 0.5,, ((,,, ((.7,,, ((.5,,, (.4, 0.6,.6} S 4 = {((0, 0, ((0.8, 0.5,, ((, 0.5,, ((,,, ((.7,,, ((.5,, } Choose λ = 0.8, we can verfy ha fuzzy ranng ses S, S, S 3, S 4 are fuzzy near separabe accordng Defnon 3.4. So, on he bass of he agorhm we propose n hs paper, we arrve a he opma separang hyper-pane are 4 sragh nes: : [x ] [x ] = ; : [x ] [x ] =.4; 3 : [x ].93 [x ] =.4; 4 : [x ].93 [x ] =.76; Shown as Fgure 3, he oupu of fuzzy ranng pons change: (,, (0., 0.6,. (., 0.6, 0. (,, and he sragh ne move as: 3 4. Ths change shows ha he fuzzy ranng pons membershp degrees Journa of Dga Informaon Managemen Voume Number 5 Ocober

8 of negave cass are ncreased and posve one decreased. From hs, can be seen ha he resu concdes wh nuve udgmens. 6. Acknowedgmen Fgure. Numerca expermena ( The auhors wsh o hank Compuer and Informaon Engneerng Schoo of Anyang Norma Unversy. L 4 L 3 L L References [] Vapnk, V. N. (998. Sasca Learnng Theory, New York: Wey. [] Vapnk, V. N. (995 The Naure of Sasca Learnng Theory, New York: Sprnger-Verag. [3] Neo, C. e a. (000. Inroducon o Suppor Vecor Machnes. Cambrdge Unversy Press. [4] Sakawa, M., Nshzak, I., Uemura, Y. (000. Ineracve fuzzy programmng for mu-eve near programmng probems wh fuzzy parameers, Fuzzy Ses and Sysems, 09, p [5] Lu, B. D., Zhao, R. Q. (998. Sochasc Programmng and Fuzzy Programmng. Tsnghua Unversy Press. [6] Deng, N. Y.,Tan, Y. J. (004. New Mehod n Daa Mnng: Suppor Vecor Machnes. Scence Press. [7] Mra, S., Pa, S. K., Mra, P. (00. Daa mnng n sof compung Communcang framework survey. IEEE Trans. Neura Neworks, 3 ( 3-4. [8] Kou, G., Xanao, L., Peng, Y. (003. Mupe crera near programmng approach J. C. o daa mnng. Modes, agorhm desgns and sofware deveopmen, Opmzaon Mehods and Sofware 8 ( Auhor Bography Wang Amn, born n 957. Professor and maser supervsor of compuer scence a Anyang Norma Unversy, Chna. Hs maor research neress ncude negen compung and fuzzy conro. L Je, born n 978. Lecure of compuer scence a Anyang Norma Unversy, Chna. Hs maor research neress ncude rea-me sysem and negen conro. 334 Journa of Dga Informaon Managemen Voume Number 5 Ocober 03