A New ANFIS Model based on Multi-Input Hamacher T-norm and Subtract Clustering

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1 Sed Orders for Reprs o reprs@behmscece.e The Ope Cyberecs & Sysemcs Jourl, 04, 8, Ope ccess New NFIS Model bsed o Mul-Ipu Hmcher T-orm Subrc Cluserg Feg-Y Zhg Zh-Go Lo * Deprme of Mgeme, Gugx Uversy of Scece Techology, Luzhou, Gugx, Ch bsrc: Ths pper proposed ovel dpve euro-fuzzy ferece sysem (NFIS, whch combes subrc cluserg, employs dpve Hmcher T-orm mproves he predco bly of NFIS. The expresso of mul-pu Hmcher T-orm s relve feher hs bee orglly gve, whch suppors he opero of he proposed sysem. Emprcl sudy hs esfed h he proposed model overweghs erly work he spec of bechmrk Box-Jeks dse my provde prccl wy o mesure he mporce of ech rule. Keywords: NFIS, hmcher T-orm, subrc cluserg, T-orm.. INTRODUCTION Tkg, Sugeo Kg hve esblshed wh s clled he Tkg-Sugeo-Kg (TSK mehod [5-7]. Ths eurlework-bsed fuzzy resog scheme s cpble of lerg he membershp fuco of he IF pr deermg he mou of corol he THEN pr of he ferece rules. Wh s more, s cely sued o mhemcl lyss usully works well wh opmzo dpve echques. Subsequely, my mproved lgorhms exesos were developed for he TSK model. I prculr, he dpve euro-fuzzy ferece sysem (NFIS s mpor pproch o mpleme he TSK fuzzy sysem, whch hs bee pu forwrd by Jg 993 [3]. However mog ll erly work, oly Ilds e l. hve replced he lgebrc produc T-orm wh oher fuzzy T-orms o hle erseco opero []. Bu hey hve o expled he reso why hey selec he oe. Besdes, he cse of pu dmeso cresg, he umber of rules wll crese wh he pu dmeso expoelly, whch evbly leds he coveol NFIS srucure dmeso o dsser. I order o mprove he ole ccess speed of NFIS T- S rules for complex sysem, vrous cluserg lgorhms hve bee used o cosruc ew muldmesol srucure of NFIS, whch combed mechsm of T-S fuzzy ferece cluserg lgorhm from he perspecve of kowledge dscover.i hs pper, we selec Hmcher T- orm o ckle he erseco opero for wo resos:. lgebrc produc T-orm s used wdely NFIS, whe equls o, he Hmcher T-orm s cully lgebrc produc T-orm, whch mes s o coflcg wh he regulr NFIS.. Hmcher produc T-orm, cluserg of fuzzy produc T-orms, dffers depedg o. So, o selec he mos suble fuzzy T-orm by chgg he umbercl prmeer s dvsble. Subrc cluserg, whch could ob he mou vlue of cluserg ceer, ws used o deerme he If pr of ech rule, for s wde pplco NFIS ceer deermo. The res of he pper s orgzed s follows. Seco provdes some ecessry bckgroud formo, he proposed sysem s essel erferece re dscussed Seco 3. Seco 4 preses he smulo resuls for bechmrk Box-Jeks dse. Flly, he summry of hs pper s gve Seco 5.. BCKGROUND I hs seco, he bsc heory of NFIS model ormlzo mehod whch hs bee used hs experme wll be roduced... dpve Nework Bsed Fuzzy Iferece Sysem (NFIS Boh rfcl eurl ework fuzzy logc re used NFIS rchecure. NFIS cosss of f-he rules couples of pu-oupu. For NFIS rg, lerg lgorhms of eurl ework re lso used. To smplfy he explos, he fuzzy ferece sysem uder cosdero s ssumed o hve wo pus (x y oe oupu (f. For regulr NFIS model, ypcl rule se wh bsc fuzzy fhe rules c be expressed s f x s y s B, he f p x q y r ( where p s ler oupu prmeers. The NFIS rchecure wh wo pus oe oupu re s show Fg. (. Ths rchecure s formed by fve lyers e f-he rules: Lyer-: Every ode hs lyer s squre ode wh ode fuco X/4 04 Behm Ope

These prmeers hs lyer re referred o s premse prmeers. Lyer-: Every ode hs lyer mulples he comg sgls seds he produc ou. For sce, O,( μ ( x μb ( y,,, Ech ode oupu represes he frg sregh of rule.

2 830 The Ope Cyberecs & Sysemcs Jourl, 04, Volume 8 Zhg Lo Fg. (. The srucure of regulr NFIS. O, μ ( x, O,3 μb ( y,, where x y re pus o ode, B re lgusc lbels for pus. I oher words, O, s he membershp fuco of B. Usully μ (x μ B (y re chose o be bell-shped wh mxmum equllg o mmum equllg o 0, such s μ ( x exp( (( x /( c (3 where, c s he prmeer se. These prmeers hs lyer re referred o s premse prmeers. Lyer-: Every ode hs lyer mulples he comg sgls seds he produc ou. For sce, O,( μ ( x μb ( y,,, Ech ode oupu represes he frg sregh of rule. Lyer-3: Every ode hs lyer clcules he ro of he rule s frg sregh o he sum of ll rule??s frg sreghs: O w~ w /( w w w,,,,4 (5 3, 9 Lyer-4: Every ode hs lyer s squre ode wh ode fuco O w~ f w~ ( p x p x,,,,4 (6 4,,, where w s he oupu of lyer 3 p,, p,, p,3 s he prmeer se. Prmeers hs lyer wll be referred o s coseque prmeers. Lyer-5: The sgle ode hs lyer compues he overll oupu s he summo of ll comg sgls: O 5, w~ f w f w ( (4 (7.. Hmcher T-orm Hmcher T-orm s kd of T-orms wh prmeer, ssfes boudry codos,commuvy, ssocvy mooocy. The prmeer of Hmcher T-orm s lso moooous, s expresso s gve below: xy T ( x, y (8 ( ( x y xy where > 0. Especlly, whe, Hmcher T-orm equls o lgebrc produc T-orm. I s esy o recogse h lgebrc produc T-orm s specl Hmcher T-orm whch hs cos prmeer. However, employg cos prmeer s o lwys ppropre. For y rule, here mus be correspodg prmeer sued for. I s wse o use bckpropgo lgorhm o deerme he correspodg. 3. PROPOSED SYSTEM O The oupu of lyer-,( refers o he resul of erseco opero bewee μ (x μ B (y, whch mes he membershp degree h x belogs o x belogs o B. I s commo o use lgebrc produc T-orm " " o del wh he membershp degree erseco opero, bu s s well-kow h lgebrc produc T-orm s o proper y suo. Wh (8 shows s h, lgebrc produc T-orm s specl Hmcher T-orm whose prmeer s cos o. So modfyg he prmeer o su o he d prs s megful wy o overcome he dlemm. I s o esy o deerme he vlue of h should be served Hmcher T-orm o hle erseco opero. Ilds e l. hve red o use oher cos o ob beer performce bu o ll lwys resuled good suo []. I s good soluo o mke NFIS o dp-

New NFIS Model bsed o Mul-Ipu Hmcher T-orm The Ope Cyberecs & Sysemcs Jourl, 04, Volume 8 83 Fg. (. The srucure of NFIS combed wh subrc cluserg. vely selec s ow for ech rule.

Bck-propgo lgorhm could be doped he process of deermg he prmeer of ech rule, bu hs mehod eeds o ob T (x, y x T (x, y whch s he grde of T ( x, y. 3.

3 New NFIS Model bsed o Mul-Ipu Hmcher T-orm The Ope Cyberecs & Sysemcs Jourl, 04, Volume 8 83 Fg. (. The srucure of NFIS combed wh subrc cluserg. vely selec s ow for ech rule. If NFIS could selec for ech rule respecvely,ccordg o he rg d prs, s more lkely o f o he performce curve close o he here lw. Bck-propgo lgorhm could be doped he process of deermg he prmeer of ech rule, bu hs mehod eeds o ob T (x, y x T (x, y whch s he grde of T ( x, y. 3.. Mul-pu Hmcher T-orm T Clculg (x, y T (x, y s esy, NFIS x my hve more h wo pus how o clcule her grdes s rel problem. More eo should be pd o how o clcule her grde wh more h pus. Now he defo of mul-pu Hmcher T-orm s gve below. T ( s mul-pu Hmcher T-orm o whch hs elemes, where {,,, } N,,0. T ( T ( T (,. Especlly, whe, T ( T (,. The defo gve bove s recursve defo, oher word he meg of upper lyer s correspodg o he lower oe he lowes s clrfed. To express clerly, useful ool ( hs bee used. The defo feures of re gve below: ( c c c 3 c, where N, c,,c {,,},c c N,. Especlly, 0 (. For exmple, Corollry Whe, ( ( ( The proof s gve ppedx. ( 4,k. Corollry Whe, ( ( The proof s gve ppedx. Corollry 3 k ( ( \ k k \,,,,,, } { k k, where,k N The proof s gve ppedx. Oe evde feure of mul-pu Hmcher T-orm s he mooocy wh respec o. fer cocse proof, he feure he expresso of T ( re cofrmed below: Proposo T ( s decresg wh respec o. Especlly, whe [, ] 0, T ( s srcly decresg wh respec o. The proof s gve ppedx B. Proposo T ( ( ( ( ( ( The proof s gve ppedx B Proposo 3 T ( ( R,where Q Q ( ( ( ( R ( ( [( ( ] ( ( ( The proof s gve ppedx B.

4 83 The Ope Cyberecs & Sysemcs Jourl, 04, Volume 8 Zhg Lo 3.. Hmcher T-orm Subrc Cluserg bsed NFIS The proposed model dffers from regulr NFIS wo pos:. I mkes Hmcher prmeer vrble dpve by dopg bck-propgo lgorhm, eeds o clcule he grde wh respec o ech prmeer pu.. I s combed wh subrc cluserg employ o deerme he mou vlue of ech rule. s s gve bove, he grde of Hmcher T-orm s prmeer pus hve bee cheved by Proposo 3 Proposo 4. Dffere from he regulr NFIS, he oupu of lyer- for proposed model whch hs 3 pus s gve below: O T,3 (3( k ( 3 (9 where 3 {μ (x,μ B ( y,μ Ck (z}. Ech rule hs sme poso regulr NFIS, becuse her hve bee uformly se o, whch hmmers he sysem o fd he mos sgfc rule dpvely. However, proposed model s ech rule wh dffere he ed c ly he foudo for mesurg he mporce of self. Boh he IF pr he THEN pr correle o he he prcple of updg s o mmze he error, whch gurees h upded w s hrmoous o he sysem. s he wegh of h rule decresg wh respec o ccordg o Proposo Equo(9. I mes h he less w leds o bgger, he bgger w ~, so he h rule plys more mpor role proposed model. I ddo, hs model volves he feld h he ohers hve ever ouched upo. Ths feld s ched o he mproveme fuzzy resog, could be combed wh he mproveme boh fuzzy resog oher process, becuse provdes ew mehodology for hlg erseco opero. Wh he vrble dpve prmeer, he predco bly of proposed model my be mproved; he prmeer s modfed ccordg o he grde so s o f o he here lw. Emprcl sudy wll be gve ex seco, whch proves h he proposed oe overweghs he regulr NFIS. 4. EXPERIMENT Fmous Box-Jeks dse s he bechmrk dse o vlde he performce of fg mehod. The Box-Jeks dse represes he CO cocero s oupu, y(, erms of pu gs flow re, u(, form combuso process of mehe r mxrue.los of erly work hs bee doe o fg Box-Jeks dse.mog hem 7 pu-ype hs bee wdely used:( u(k-4,y(k-; (B u(k-3, y(k-; (C u(k-3, u(k-4, y(k-; (D u(k, u(k-, y(k-, y(k-; (E u(k-, u(k-, y(k-, y(k-; (F u(k, u(k-, u(k-, y(k-, y(k-, y(k-3; (G u(k-, u(k-, u(k-3, y(k-, y(k-, y(k- 3. Whole experme ws uderoke he evrome of Mlb The resuls of proposed model erly work re lsed Tble, where we c fd h he proposed model hs ousg performce. CONCLUSION Hmcher T-orm s oe of he mos fluel T-orms. I hs pper, we vesge he fesbly of pplyg N- FIS mplemeed wh Hmcher T-orm. Whle employg bechmrk Box-Jeks dse, he proposed mehods hve more compeve performce predco ccurcy compred o erly work. There re wo m dvge of proposed model: o he oe h, s he exeo of NFIS fuzzy resog, mkes possble o mprove whe be mplemeed wh oher mproveme fuzzfco, defuzzfco eve rg mehod oher opml lgorhm such s G PSO; o he oher h, provdes very vl prmeer o fer he mporce of ech rule, bu he orml form of ferrg mesurg hs o bee proposed. The sudy of ll bove expecos s progress. Tble. The resuls of ech model. Model Ipu-ype MSE PMDE [4] F 0.47 PMG [4] F TS-GMDH [9] G 0.99 TS-GMDH [9] G 0.97 TS-GMDH3 [9] G Mes [3] B 0.39 Yue [8] Yue [8] B 0.40 Yue [8] C Yue [8] F Yue [8] G Proposed Model F Proposed Model G CONFLICT OF INTEREST The uhors cofrm h hs rcle coe hs o coflc of eres. CKNOWLEDGEMENTS The uhors would lke o hk for he suppor by ovo Proec of Gugx Grdue Educo (YCSZ0403, Proec of Ousg Youg Techers' Trg Hgher Educo Isuos of Gugx gr of Phlosophy Socl Scece Fud of Gugx (3BGL009.

5 New NFIS Model bsed o Mul-Ipu Hmcher T-orm The Ope Cyberecs & Sysemcs Jourl, 04, Volume 8 PPENDIX. PROOF OF COROLLRY Proof of Corollry Whe 0 <, c,,c {,,,,, },c c ( c c c c 3 ( c c c c ( c c c c c,,c {,,,,, },c c c,,c {,,,,, },c c 3 ssume whe, proposo s rgh oo. So, T ( Proof of Corollry Whe, ( 3 ( 3 ( Proof of Corollry 3 Whe 0 <, ( ( [( ( ] ( ( ( \ ( ( \ 0 ( Q ( ( ( \ 0 Whch complees he proof. ( ( PPENDIX B. PROOF OF PROPOSITION Especlly, whe,, 0, T ( s srcly decresg wh respec o. The proposo s cofrmed. ssume whe -,he proposo s rgh oo, he whe, T ( T (T (, T (T (,. ( ( ( ( ( f (, f ( ( ( ( ( ( Proof of Proposo The proof s gve below: Le Q ( ( ( (, so T ( (. Q, T ( ( ( ( ( ( ( ( ( ( [( ( ] [( ( ] ( ( Whch complees he proof. 0, T ( s srcly decresg wh respec o [( ( ] ( ( ( ( ( T (T (, T (. Especlly, whe [0, ] [( ( ] ( Q ( ( ( ( ( Q Proof of Proposo, [0, ] <, whe, T ( T (, T (, T (. Whe ( Q ( ( ( ( ( Q Q ( ( ( ( ( 0 Whch complees he proof. (. Q ( T ( Q ( ( ( ( ( Q Q Whch complees he proof. ( Q The proposo s rgh. 3 ( \ ( \ ( ( ( ( ( ( 833 [( ( ] ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( (

6 834 The Ope Cyberecs & Sysemcs Jourl, 04, Volume 8 Zhg Lo Q So, T (. Q I cocluso, whe N, ( T ( ( ( ( ( Whch complees he proof. Proof of Proposo 3 Q ( ( ( ( ( ( ( ( (( ( ( R ( T ( Q ( R, Q ( ( ( Whch complees he proof. REFERENCES [] L.S. Ilds, S. Sprls S. Tchos,008. pplco of fuzzy T-orms owrds ew rfcl Neurl Neworks evluo frmework: cse from wood dusry, Iformo Sceces, 78, [] J.S. R, 993. NFIS: dpve-ework- bsed fuzzy ferece sysems, IEEE Trs. Sys., M, Cyber., 3 (3, [3] Mes,., Romero, S. Moreo, F.J., 008. Sysemc Mehodology o Ob Fuzzy Model Usg dpve Neuro Fuzzy Iferece Sysem. pplco for Geerg Model for Gs- Furce Problem, Dsrbued Compug rfcl Iellgece, DOI: 0.007/ _53. [4] Ozer S. Zorlu H., 0. Idefco of bler sysems usg dfferel evolu-o lgorhm, Sdh-cdemy Pro-ceedgs Egeerg Sceces, 36(3, 8-9. DOI: 0.007/s [5] M. Sugeo, G. T. Kg, 988. Srucure defco of fuzzy model, Fuzzy Ses Sys., 8(, [6] M. Sugeo, K. Tk, 99. Successve defco of fuzzy model s pplcos o predco of complex sysem, Fuzzy Ses Sys., 4(3, [7] T. Tkg, I. Hysh, 99. NN-drve fuzzy resog, I. J. pprox. Reso., 5(3, 9-. [8] Yue, J.H., Lu, J.Z., Lu X.J. T, W., 006. Idefco of Noler Sysem Bsed o NFIS wh Subrcve Cluserg,Proceedgs of he 6h World Cogress o Iellge Corol uomo,, DOI: 0.09/WCIC [9] Zhu, B., He C.-Z., Lssb P. L X.Y., 0. GMDH-bsed fuzzy modelg pproch for cosrucg TS-model, Fuzzy Ses Sysems, 89, 9-9. DOI: 0.06/.fss Receved: Sepember 6, 04 Revsed: December 3, 04 cceped: December 3, 04 Zhg Lo; Lcesee Behm Ope. Ths s ope ccess rcle lcesed uder he erms of he Creve Commos rbuo No-Commercl Lcese (hp://crevecommos.org/- lceses/by-c/3.0/ whch perms uresrced, o-commercl use, dsrbuo reproduco y medum, provded he work s properly ced.

Interval Estimation. Consider a random variable X with a mean of X. Let X be distributed as X X

Interval Estimation. Consider a random variable X with a mean of X. Let X be distributed as X X ECON 37: Ecoomercs Hypohess Tesg Iervl Esmo Wh we hve doe so fr s o udersd how we c ob esmors of ecoomcs reloshp we wsh o sudy. The queso s how comforble re we wh our esmors? We frs exme how o produce