Computational Issue of Fuzzy Rule-based System

Transcription

1 IJCSNS Itertol Jourl of Computer Scece d Networ Securty, VOL.6 No.A, Februry 6 Computtol Issue of Fuzzy Rule-bsed System Chushe L Deprtmet of Computer Scece d Iformto Egeerg, Ntol Uversty of T 33, Sec., Shu-L St., T, 7, Tw, ROC. Summry A ovtve soft-computg system s proposed ths pper to ssuge the prdox of curse of dmesolty (COD d yet to preserve the property of completeess. Although the supreme merts of fuzzy ferece system (FIS re ts smplcty, uderstdblty of fuzzy rules d model-free pproch, t suffers wth the problem of COD. The COD problem c be hppeed esly f the umber of ether put vrbles or prttos of ech put uverse creses gretly. Ths drwbc of COD cuses serous stuto especlly for hrdwre relzto tht computtol resources wll be suced exhustvely. A evet-trggerg bsed euro-fuzzy system (NFS s preseted to llevte the COD problem d to reduce wstg computtol resources. The proposed soft-computg system c sve the computtol resources effectvely wthout mssg the property of completeess. The put to the proposed NFS s cosdered s evet. Becuse the comg put evet H( decdes the posto the put spce roud whch fuzzy sets wth membershp degree beyod threshold re detected d re used to costruct the fuzzy rules closely to the H(, the fuzzy rules volved the owledge bse c be very compct. The evet-trggerg bsed owledge bse s effectve, whch there re o redudt fuzzy rules d oly the eeded rules re the proposed system for the H(. oreover, the proposed softcomputg system, whose structure s tme-vryg d s depedet o the comg put evet, possesses the property of evet-trcg structure. The owledge bse of the proposed NFS s trggered off by the evet, d oly few rules re fred loclly roud the evet. It s sutble for lrge-scle system operto. A exmple s demostrted for the proposed pproch. ey words: Fuzzy ferece system (FIS, eurl-fuzzy system (NFS, curse of dmesolty (COD, COD-completeess prdox, softcomputg system.. Itroducto Fuzzy ferece systems [8][9] hve bee powerful tool for rel-world pplctos such s utomtc cotrol, dt clssfcto, decso lyss, expert systems, robotcs, ptter recogto, d my others. Amog the pplctos the most frutful reserch re s cotrol systems [][][3], whch expertse, egeerg experece d udgmet c be tegrted to the desg of owledge bse for the fuzzy ferece system (FIS [7][][][]. The supreme merts of FIS [8][9] re ts smplcty, uderstdblty of fuzzy rules, d model-free pproch by whch plt s vewed s blc box, whose output s observble to the FIS, servg s cotroller. The prttog of the put spce s crtcl to the desg of FIS o mtter wht pplcto purpose. There re three m types of prttog commoly used the desg of FIS, whch re grd-type, tree-type, d cluster-type. For most types of prttog for FIS such s grd-type d tree-type, the problem of curse of dmesolty (COD [5] rses tht the mout of fuzzy rules creses expoetlly f put vrbles d fuzzy prttos of ech put uverse re cresed. For the cluster-type prttog [6], the COD problem s depedet o the mout of clusters the put spce of FIS for ech cluster correspods to fuzzy rule. Although the COD problem s ot tht serous wth the prttos of cluster-type, t s compromse md the feess of prtto, the completeess of the rule bse d the sze of rule bse FIS. I other words, the fewer the fuzzy rules wth the cluster-type prttog, the corser the prttos d the worse the completeess to cover the put spce of the FIS, d vce vers. For the tree-type prttog, t usully does ot correspod to good lgustc megs to uderstd, d t s more complex th the grd-type prttog topologcl dstrbuto of fuzzy regos d more membershp fuctos my be eeded. For desg smplcty, the grdtype prttog of the put spce for FIS s most used. The grd-type of fuzzy prtto possesses the property of prtto completeess d megful lgustc descrpto d uderstdg, but suffers wth the COD problem. The COD problem of FIS wll cuse drwbcs hrdwre relzto d mplemetto such s FPGA, ASIC or DSP-processor. It my be terestg to th bout the prdox betwee the COD problem d the completeess property of owledge bse, the so-clled COD-Completeess Prdox, tht FIS possesses excellet property of completeess for the rule bse to cover the put spce d yet vods the COD problem to speed up the ecessry clculto of fuzzy ferece [][7] d to sve computtol resource. A evet-trggerg bsed (or clled evet-bsed euro-fuzzy system s proposed the uscrpt receved Jury 5, 6. uscrpt revsed Februry 5, 6.

2 IJCSNS Itertol Jourl of Computer Scece d Networ Securty, VOL.6 No.A, Februry 6 pper to overcome the COD problem d to preserve the completeess property for the system to cover the etre the put spce. The euro-fuzzy system (NFS [][][8] s relzed by fusg both FIS d eurl et [5], exhbtg excellet resog d lerg bltes to cope wth complex d ll-defed systems. The rule bse of the evet-trggerg bsed NFS s fcttously lgorthmed, but s ot relly frmly setup. Whe evet s hppeed tht put to the NFS s guged, evettrggerg bsed rule bse s vsulzed, whch oly few fuzzy rules closely d loclly relted to the evet re relly costructed. The evet-trggerg bsed rule bse s oly smll frcto compred to the etre rule bse of the NFS, d t becomes the rel rule bse wth the fuzzy rules closely relted to the evet. Becuse evet to the NFS s vred wth tme, the evet-bsed rule bse of the NFS s vred wth the evet d tme. I other words, the fuzzy rules the evet-bsed rule bse lwys eep good trc wth the evet d the evet-bsed rule bse possesses tme-vryg structure. Wth the tme-vryg structure of the evet-trggerg bsed rule bse, the proposed NFS c both overcome the COD prdox d preserve the completeess of the NFS. The pper s orgzed s follows. A covetol fuzzy ferece system s overvewed Secto. The evettrggerg bsed euro-fuzzy system s proposed Secto 3. A exmple demostrto s gve Secto to llustrte the proposed soft computg system. Flly, dscusso d cocluso re gve Sectos 5 d 6, respectvely.. themtcl Descrpto of Fuzzy Iferece System The supreme merts of fuzzy ferece system (FIS re ts smplcty, uderstdblty of fuzzy rules, d expertse-oreted pproch. A FIS c use expertse, egeerg experece d udgmet to the desg of the owledge bse. I FIS, there re my methods to decde the umber of rules, whch s determed by put spce prttog. Amog the types of prttog, the grd-type prttog s frequetly used becuse of ts completeess, smplcty, d good lgustc megs for uderstdg. Although the grd prttog s smple d complete desg of the owledge bse of FIS, the problem of curse of dmesolty s lwys occur wth the prttog for FIS s dscussed the prevous secto. I ths secto the put-output behvor of FIS wth grd-type prttog for put spce s specfed. Phlosophy of fuzzy prttog for FIS s relted wth the cocept of dvde-d-coquer for ll h Fg.. Iput spce prttog of grd-type fuzzy ferece system. stutos of put to the FIS such tht every put codto to the FIS c be rected by the FIS. Ech prtto the put spce correspods to tecedet of fuzzy rule, d the cosequet descrbes the recto behvor for the fuzzy rego. Prttog of the put spce comes out wth umber of fuzzy regos, terpreted s the costructo of the correspodg fuzzy rules. A typcl grd prttog two-dmesol put spce of FIS s show Fg., whch fuzzy regos re overlpped wth trset boudres d re led up such tht the desg of fuzzy sets for the fuzzy regos s smple d esy to uderstd wth few megful lgustc terms. The put spce s covered everywhere wth the fuzzy grd prttos so tht the property of completeess s lwys stsfed for the FIS. Suppose tht there re crsp put vrbles to FIS d they re the bse vrbles h (,,,,. The put vrbles re collected together to form put crsp vector H(, H h h h h Assocted wth ech crsp vrble h (, there s correspodg lgustc vrble x. Let X deote the set of lgustc vrbles, tht s X(x, x,, x. Ech uverse of dscourse of ech lgustc elemet of X c be prttoed to severl regos tht overlp ech other. Ad ech prtto s lbeled wth lgustc term, such s postve lrge, postve, or egtve. Thus the lgustc vrbles hve correspodg lgustc term sets, T,,,,. I ech term set, there s collecto of lgustc vlues. The crdltes, c,,,,, for the put lgustc vrbles re collected to form the crdlty vector C gve s follows. (

3 IJCSNS Itertol Jourl of Computer Scece d Networ Securty, VOL.6 No.A, Februry 6 3 c c C c The umber of f-the rules the owledge bse for FIS wth grd prttog of ts put spce s determed s follows. Π c (3 The fuzzy rules the pper use the method of Tg d Sugeo [6] tht the cosequet s ler combto of the compoets of H(. A fuzzy f-the rule s gve s Rule : IF ( x s s ( h d x s s ( h d d x s s ( h THEN σ + h + h +... h ( ( (,,, +, t ( where s ( h s the fuzzy set for the -th lgustc put vrble the -th fuzzy rule, for,,, d,,,, d h ( s the -th crsp put t tme t to the FIS. Ech rule of the FIS my hve multple outputs σ (,,,,Q. The coeffcets,, l,,,,, l,,,,,,,,q, wll become pproprte vlues v ether desg or lerg [3][][9]. Let the followg dt types be defed. ϕ,, L, h (,, L, A H h O Q, Q, L Q,, h, S ( H( ( s ( h, s( h,, s ( h (5 The costt ϕ c be set to uty f costt terms the cosequets of fuzzy rules re volved the ferece process, otherwse t s set to zero. Let the rule ctos σ (,,,,Q, from the -th rule be collected together to form the rule cto vector, gve s σ ( σ ( Σ A H (. (6 σ Q Wth the dt types defed Eqs.(5 d (6, fuzzy fthe rule c be expressed compct form, gve s IF ( X s S ( H(, THEN Σ A H (7 for,,,. Let μ (H( be the set of membershp degrees of the crsp put vrbles the -th fuzzy rule, gve s μ ( H( ( μ( h, μ( h,..., μ ( h (8 where μ ( h s the membershp degree evluted wth h ( t tme t for,,,, the -th fuzzy rule. The frg stregth β ( of the -th rule s obted wth β ( (μ (H(, (9 where (μ (H( s the fuzzy-d operto over ll elemets the set μ (H(. Usully the fuzzy-d operto s clculted usg t-orm opertor. The fuzzy ferece results, z (,,,,Q, re obted by combg ll dvdul fred rule ctos, gve s β, +, h z ( β d λ, +, h β λ, ( β for,,,q d,,,. All ormlzed frg stregths λ (,,,,, re collected together to form the ormlzed frg stregth vector λ(, gve s λ β λ β λ( ( β λ β Let ll rule cto vectors defed Eq. (6 be collected together to hve the followg mtrx. Σ([Σ ( Σ ( Σ (] (3

4 IJCSNS Itertol Jourl of Computer Scece d Networ Securty, VOL.6 No.A, Februry 6 whch s clled the rule cto mtrx for the FIS. The output of the FIS t tme t s expressed s the product of the rule cto mtrx d the ormlzed frg stregth vector, gve s z( z( Z( Σ λ(. ( zq Altertvely, the FIS output Z( c be expressed explctly terms of the crsp puts, gve s β Z( [ Σ ( t Σ ( t Σ ( t ] β β β ( μ ( H [ A H A H L A H ] ( μ ( H ( μ ( H ( μ ( H (5 By the equto bove, the relto betwee the put vector H( d the output vector Z( for the FIS s estblshed, whch s hghly oler mppg fucto. Becuse of the hghly oler mppg betwee put d output, the FIS s esly ble to hdle wth oler rel-world problems. Note tht the mthemtcl dervto of the put-output relto gve bove for FIS s for the grd-type FIS. It c lso be ppled to the other types of FIS, wth some modfcto. Although the FIS s wth excellet oler mppg blty, t suffers wth the problem of curse of dmesolty, especlly for the grd-type prttog of put spce. IF the system scle of FIS or the crdltes of the term sets T (x,,,,, get lrger d lrger, the COD problem for the FIS gets worse. For stce, grd-type FIS wth 3 put vrbles d lgustc vlues for ech term set wll get 3 fuzzy f-the rules, whch s uusully lrge for the gve system scle d crdltes of the FIS. 3. Evet-Trggerg Bsed Soft-Computg System I the secto evet-trggerg bsed euro-fuzzy system (NFS s proposed to overcome the prdox of dmesolty curse for fuzzy ferece system (FIS. The pproch s bsed o the de tht put to the NFS s regrded s evet to trgger off the owledge bse of the NFS. The put evet fres o the fuzzy rules loclly roud the evet so tht oly few rules re trggered t ech tme. Ths de my be performed wth the dstrbuted structure of eurl etwor. Not le the structure of FIS whose owledge bse s estblshed frmly, the owledge bse of the proposed NFS s pseudo-set up, whose structure s tme-vryg d s depedet o the comg put evet. I the proposed pproch, o mtter wht sze of the owledge bse of FIS, t s trggered off by evet d oly few rules re fred loclly roud the evet. I such wy, the COD problem c be overcome. I the followg, the proposed euro-fuzzy pproch s specfed detl. As put evet s occurred d guged, the evet s mpped to the correspodg uverses of dscourse so tht the fuzzy sets roud the evet re fred. For ech of the lgustc vrbles, there s lgustc term set. Ech lgustc vlue c be defed wth fuzzy set. Let the membershp fucto set for the -th put lgustc vrble be deoted by μ,( h μ (h ( μ,( h (6 μ ( (, c h t for,,,, where μ, (h ( s the -th membershp fucto of fuzzy set for the -th lgustc put vrble. The membershp fucto sets of put vrbles re collected together to form the membershp fucto bss set of ll lgustc vrbles, gve s μ(h({ μ (h (, μ (h (,, μ (h (} (7 The membershp fucto bss set provdes wth ll ecessry formto eeded to set up FIS. Whe put evet s occurred d guged, the membershp degrees re clculted for fuzzy sets ll put uverse of dscourse. Tht s, the membershp degrees of ll fuzzy sets re processed for tht put evet t ths momet ll uverses of dscourse. Note tht up to ths pot the owledge bse s ot set up d the rules re ot prtcpted these clcultos yet. For the put evet t the momet, the fuzzy sets correspodg to the membershp degrees beyod threshold re detected d the qulfed to prtcpte the structure setup of owledge bse d ferece process of the FIS. I other words, there s o rel fuzzy rules setup utl the put evet s occurred d mesured. Ths s the m de of the proposed pproch to del wth the COD problem. I such wy, the structure of the proposed FIS c be very compct, d fuzzy rules volved ferece process re oly ty frcto to the orgl owledge bse of covetol setup. A detector s used to detect the fuzzy sets to whch the membershp degrees re beyod threshold ε for comg put vector H( ech correspodg

5 IJCSNS Itertol Jourl of Computer Scece d Networ Securty, VOL.6 No.A, Februry 6 5 uverse of dscourse U for,,,. H( s vewed s evet occurred t tme t. I ech put uverse, these fuzzy sets detected wth membershp degree greter th ε re qulfed to prtcpte the setup of the FIS. Assume tht the formto of the qulfed fuzzy sets by the detector for the -th lgustc vrble s cluded the vector N (, gve s,, N (8, c ' for,,,, where c deotes the crdlty of the N (, tht s the umber of qulfed fuzzy sets detected the -th put uverse, d, s the -th elemet of N ( whose vlue s the correspodg sequetl umber of fuzzy set the uverse. Note tht the crdlty c s depedet o the -th compoet of the put vector H( t tme t. The crdltes c,,,,, re collected together to form the crdlty vector C, gve s C' c c c ' (9 whch s clled the evet-trggerg crdlty bsed vector of fuzzy sets for the proposed pproch. For proposed pproch, the umber of fuzzy rules t tme t the grd-type owledge bse s determed s ' c ( Π To determe the fuzzy rules trggered by comg evet t tme t, the procedure of rule costructo for rule s specfed s follows. The rule costructor s defe s, γ G, γ (, γ for,,,, where, γ s from oe of the compoets of N ( for,,,. Note tht ech N ( provdes oly oe elemet to the G. The γ,,,,, re clculted usg the followg equtos. c q ' γ +, γ c, c ',,..., q q c ' + θ, θ < c, c ' γ θ +,,,,...,. ( For,,,, ech G wll correspod to fuzzy rule wth rule umber deoted s, whch s clculted usg the followg equtos. τ c +,, γ, γ c. τ τ c +,,-,-,,., γ τ. (3 By the proposed evet-trggerg pproch, the fuzzy ferece results, z (,,,,Q, whe the evet H( s occurred, re obted s follows. ' R ( R ( β, +, h z ( t ' R ( ( β d t λ, +, h R ( λ R ( β t, (5 R ( β for,,,q d,,,. All ormlzed frg stregths λ (,,,,, re collected together to form the ormlzed frg stregth vector λ H (, gve s λ β H λ β λ ' β ' ' λ β (6 The rule ctos t d σ,,,,,,,q c be collected together to form the rule cto vector, gve s

6 6 IJCSNS Itertol Jourl of Computer Scece d Networ Securty, VOL.6 No.A, Februry 6 Σ ( σ σ σ Q A H ( (7 where A dσ ( re defed usg Eqs.(5 d (6. All rule cto vectors Σ,,,,, re collected together to hve the followg mtrx Σ H ( t [Σ ( Σ ( Σ (] (8 The output of the proposed FIS t tme t s expressed s the product of the rule cto mtrx Σ H ( t d the ormlzed frg stregth vector λ H (, gve s z( z( H H H Z Σ λ. (9 zq Altertvely, the FIS output Z( c be expressed explctly terms of the crsp puts, gve s Z H ([ Σ R ( ( t Σ R ( ( t Σ ( t ] ' β R β R β β ( ( ( t ( μ ( H( t [ A H( A H( L A H( ] ( μ ( H( ( μ ( H( ( μ ( H( (3 Note tht, compred Eq.(3 to Eq.(5, the dmesos of the mtrx Σ H ( t d the ormlzed frg stregth vector λ H ( re much less th those of Σ ( t d λ(, becuse the s much smller th. Ths dctes tht much computtol resource s sved d tht the sze of the owledge bse of the proposed system s much less th tht of the covetol FIS. The owledge bs of the proposed system s relzed roud the put evet H(, whose sze d structure both re compct d best-ft for the comg evet H(. Neuro-Fuzzy Structure to Implemet the Proposed Evet-trggerg Bsed NFS The lyered structure of euro-fuzzy system s sutble to mplemet the proposed de of evet-trggerg bsed FIS becuse tlly there s o rule costructed Fg.. Evet-trggerg bsed euro-fuzzy system (NFS wth two puts. physclly, d the lyered structure provdes excellet specfcto for how the proposed system s relzed. There re sx lyers used the eruo-fuzzy system show Fg.. The explto for ech lyer of the proposed evet-trggerg bsed NFS s gve s follows. Lyer : The odes the lyer receve the compoets of the evet put vector H( d the drectly sed them to the correspodg odes lyer. A ler fucto s used s ctvto fucto. The et puts d ode outputs re gve s follows. h O f ( for,,,, where f ( dctes the ctvto fucto for the -th ode of lyer, d O the output of the -th ode. h ( t h ( t the et put, Lyer : The fuzzy mtchg process s performed the lyer. The odes the lyer receve correspodg outputs of odes from lyer to clculte the membershp degrees of the fuzzy sets the put spce for the comg put evet H(. The ode outputs re membershp degrees, s gve Eqs.(6 d (7. The vlutos of the membershp degrees greter th threshold ε re detected for the qulfed fuzzy sets to prtcpte the costructo of the proposed euro-fuzzy system, s specfed Eq.(8. The evet-trggerg bsed crdlty vector, s show Eq.(9, for the qulfed fuzzy sets re to be used to the evet-trggerg bsed owledge bse. Oly the odes of qulfed fuzzy sets re llowed to sed the ode outputs to ext lyer. The et puts d ode outputs of the lyer re gve s follows Z (t

7 IJCSNS Itertol Jourl of Computer Scece d Networ Securty, VOL.6 No.A, Februry 6 7 O O ( f for,,, d c q q c + s( d s(,,,c, where f ( s the ctvto fucto for the -th ode of lyer d s membershp fucto. Lyer : The costructo of the evet-trggerg bsed owledge bse s begu the lyer. Bsed o the qulfed fuzzy sets detected ech uverse, the fuzzy rules re costructed. Ech put lgustc vrble provdes oly oe qulfed fuzzy set to rule formto, by whch the tecedet of the rule s set up. The process of rules creto s permutto of the qulfed fuzzy sets, s show Eqs. ( to (3. The ode outputs re frg stregths of fuzzy rules. The et puts d ode outputs of the lyer re gve s follows.,,..., (,,, O f ( (,,,,...,, for,,,, where, s the -th put from lyer to the -th odes of lyer d s t-orm opertor to perform fuzzy-d operto. Lyer 3: The ormlzed frg stregths of the evet-bsed fuzzy rules re performed the lyer, s show Eqs. (5 d (6. The umber of rules volved the fuzzy ferece s depedet o the comg put evet H(, s show Eq.(. The et puts d ode outputs of the lyer re gve s follows O, O,..., 3 l ( O ' 3 3 Ol O l f l ( l ' O for l,,,. Lyer : The ormlzed cosequets of the evet-bsed fuzzy rules re performed the lyer. The Tg d Sugeo method s used to qutfy fuzzy cotrol rule, whose cosequet s polyoml fucto of the compoets of the comg put evet H(. Defuzzfcto process s ot ecessry. Ech ode ths lyer receves output from correspodg ode lyer 3 d the outputs of ll odes lyer. The odes ths lyer must be cpble of represetg -put-q-output fuzzy rules, s gve Eq.(. To cope wth Q outputs, Q recepto sub-fuctos wth Q correspodg ctvto sub-fuctos re tegrted to odes the lyer, clled super odes. Ech sub-et-put of super ode gve below s set of two terms, ormlzed frg stregth d ler combto of h (,,,,. ([ w + w h ], for,,, d,,,q, where s the sub-et-put of the -th ctvto sub-fucto the -th super ode of lyer, w the coecto stregth from the -th ode of lyer to the -th super ode of lyer for the -th recepto sub-fucto, d w weght for extr put to the -th recepto sub-fucto. Compred wth the coeffcets of the cosequet, Eq.(, the weghts re gve by w,. A product fucto s used s ctvto fucto for the super odes. The ode sub-output from the -th ctvto sub-fucto of the -th super ode s gve by O f ( 3 O [ w + 3 O w h ] for,,, d,,,q. Lyer 5: The outputs of the evet-trggerg bsed NFS re summrzed the lyer, s show Eq. (3. The etputs d outputs of odes the lyer re gve s follows. 5 ' O O f ( for,,,q.. Exmple Demostrto for the Proposed Soft-Computg System I ths secto, exmple s used to demostrte how the proposed evet-trggerg bsed NFS fuctos. A twoput-oe-output evet-trggerg bsed NFS s gve for demostrto purpose, lthough the proposed NFS c be exteded to be -put-q-output system. Assume tht the two put uverses of dscourse re prttoed to sx d fve tervls overlppg ech other, respectvely. The correspodg crdlty vector for the two put lgustc vrbles s deoted s follows. c 6 C. 5 c

8 8 IJCSNS Itertol Jourl of Computer Scece d Networ Securty, VOL.6 No.A, Februry 6 Note tht for covetol FIS the umber of fuzzy rules wll be thrty the owledge bse, but for the proposed system the rules eeded wll be reduced sgfctly. Wth the defto Eq.(6, suppose tht the membershp fucto sets of the two put vrbles for the comg evet H([h ( h (] T [.89.37] T t tme t, show Fg. 3, re gve s follows. d.8. μ ( h μ ( h As defed Eqs.(8 d (9, the qulfed fuzzy sets detected wth the threshold ε. d the evet-trggerg crdlty vector C ( re gve s follows., N (,, 5, N,, 3 d c C '. c ' Wht ths mes s tht oly fuzzy rules re volved costructo of the grd-type NFS, c c ', d tht the secod d thrd fuzzy sets the frst put uverse d the fourth d ffth fuzzy sets the secod put uverse re qulfed. Up to ths pot, the clculto correspods to the operto of lyer of the evet-trggerg bsed NFS. Wth the defto of rule costructor G defed Eqs.( to (, the four rule costructors re clculted s follows., G,,, G,, 3 3, 5 G,, d, 5 G., 3 These rule costructors defe the tecedets of the four evet-trggerg bsed fuzzy rules, s show Fg.. The correspodg ode umbers lyer re h h.37 h.89 H( h.37 h.89 Fg. 3. Clculto of membershp degree sets for the two put uverse of dscourse whe the comg evet H([.89.37] T. h.37 h h.89 Fg.. Four evet-trggerg bsed fuzzy rules for the proposed softcomputg system. determed usg Eq.(3 d they re clculted s follows. 7, 8, 3, d 3 whch re the umbers of evet-trggerg bsed rules whe the evet H( s putted t tme t. The four fuzzy rules re correspodg to odes 7, 8,, d 3 lyer. If the m opertor s used for t-orm operto, the the outputs of the four odes, s the frg stregths of the rules, re clculted s β 7(m(.76,.8.76, β 8(m(.76,.3.3 β (m(.3,.8.3, d β 3(m(.3, h h.89 H( h.37 3 h

9 IJCSNS Itertol Jourl of Computer Scece d Networ Securty, VOL.6 No.A, Februry 6 9 The ormlzed frg stregth vector defed Eq.(6 λ H ( s obted s follows. 7 λ λ 8 H λ λ λ 3 ( λ t λ 3 λ λ 7 β.5 8 β.973 β.53 β 3 β. 53 Ths clculto s correspodg to the operto of lyer 3 of the NFS. The outputs of odes 7, 8,, d 3 re terpreted s the ormlzed frg stregths for the evettrggerg bsed fuzzy rules. Assume tht the coeffcets of the cosequets of the four rules, defed Eq.(5, re gve s follows [ 3], A [ ] 7 A [ ] A [ ] d A [ ] [ 5 6], [7 8 9], [ ], Wth Eqs.(7 d (9, the correspodg four rule ctos re expressed s follows. Σ 7 ( A 7 H ([ 3] 6.89, Σ 8 ( A 8 H ([ 5 6] 8.89, Σ ( A H ([7 8 9] 9.689, d Σ 3 ( A 3 H ([ ] The outputs of the four rules re correspodg to the operto of lyer of the NFS. Ech rule output s clculted by ts correspodg ode lyer, whch ormlzed frg stregth from lyer 3 d rule cto from ts et put formto re multpled together. The four rule outputs re obted s follows. u 7 (Σ 7 (λ 7 ( , u 8 (Σ 8 (λ 8 ( , u (Σ (λ (.53.9, d u 3 (Σ 3 (λ 3 ( The output of evet-trggerg bsed NFS, defed Eq.(3, s clculted s follows. 7 λ Z H ([Σ 7 ( Σ 8 ( Σ ( Σ 3 8 (] λ λ 3 λ.5 [ ] Ths clculto for the output of the proposed two-putoe-output soft-computg system s summrzed lyer 5 of the NFS. The evet-trggerg bsed NFS of the exmple demostrto s show Fg., wth whch the de of the proposed evet-trggerg bsed soft-computg system s llustrted very clerly. As oe c observe Fg., the rules costructed for the proposed NFS re closely roud the comg put evet H(. The owledge bse of the evet-trggerg bsed NFS s much smller th tht of the correspodg covetol grd-type FIS, d s lwys relted to the evet H(. 5. Dscussos I ths secto, severl propertes for the proposed softcomputg system re dscussed, whch re the reducto of computtol resource, the mxml sze of owledge bse, the evet-trcg d compct structure of the system, the preservto of completeess property, the sutblty for lrge-scle system operto, d the overcomg of curse of dmesolty. These propertes re couplg together. As dscussed the secto of exmple demostrto, the proposed soft-computg system eeds much less computtol resource th the correspodg covetol FIS. I the exmple gve, the covetol FIS eeds the computto opertos of 9 ddtos, 5 multplctos, 6 dvsos, d comprsos to fsh the fuzzy ferece, whle the proposed softcomputg system wth rules fred eeds oly 3 ddtos, multplctos, dvsos, d 3 comprsos. ore comprsos of computto opertos re summrzed Tbles d. The sze of owledge bse of the proposed softcomputg system s depedet o both the comg put evet H( d the overlppg of fuzzy sets. The overlppg of fuzzy sets put uverse my decde

10 3 IJCSNS Itertol Jourl of Computer Scece d Networ Securty, VOL.6 No.A, Februry 6 the mxmum umber of qulfed fuzzy sets for tht correspodg lgustc vrble to prtcpte costructo of owledge bse, whle the comg evet H( decdes the posto t whch membershp degrees re clculted the correspodg put uverse. For grd type prttog of the put spce, the umber of evettrggerg bsed fuzzy rules re determed usg Eq.(. Becuse the comg put evet H( decdes the posto the put spce roud whch fuzzy sets wth membershp degree beyod threshold re detected d re used to costruct the fuzzy rules closely to the H(, the fuzzy rules volved the owledge bse c be very compct. I other words, the evet-trggerg bsed owledge bse s effectve, whch there re o redudt fuzzy rules d oly the eeded rules re the proposed system for the H(. oreover, the proposed soft-computg system possesses the property of evettrcg structure. Ths to the propertes specfed, the proposed soft-computg system s sutble for lrge-scle system operto. The property of completeess for the proposed softcomputg system s lwys preserved s log s the correspodg covetol FIS possesses the property tht the prttos cover everywhere ll put uverses d ll put codtos re cred wth. The property of completeess s heretly depedet o the prttos of put spce. The proposed soft-computg system preserves the excellet property of completeess, d yet vods the curse of dmesolty, for the owledge bse s compct d s costructed roud the evet H( oly. The proposed soft-computg system hs the structure of evet-trcg owledge bse d tme-vryg property. Tble : Comprso of computto for the proposed soft-computg system d the correspodg covetol FIS ( puts d outpu. Fuzzy Rules Operto FIS 3 Proposed evet-trggerg bsed NFS (c d c (c d c (c d c Comprso Addto ultplcto Dvso 6 8 * System wth put crdltes of fuzzy sets c 6 d c 5. Tble : Comprso of computto for the proposed soft-computg system d the correspodg covetol FIS (3 puts d outpu. Fuzzy Rules Operto FIS 5 Proposed evet-trggerg bsed NFS 8 (c, c d c 3 (c, c d c 3 (c, c d c 3 Comprso 5 6 Addto ultplcto 5 6 Dvso * System wth put crdltes of fuzzy sets c 6, c 5 d c Coclusos A evet-trggerg bsed soft computg system s proposed the pper to llevte the prdox of the problem of curse of dmesolty (COD d to preserve the completeess property. The proposed soft computg system hs bee specfed detl. A exmple s gve for demostrto of the proposed pproch. The proposed soft-computg system possesses the propertes of the reducto of computtol resource, the evet-trcg d compct structure of owledge bse, the preservto of completeess property, the sutblty for lrge-scle system operto, d llevto of COD prdox, lthough these propertes re couplg together. Acowledgmets The uthor s grteful to s Y.C. Che for the help the preprto of the pper. Refereces [] Debrup Chrborty d Nhl R. Pl, Itegrted feture lyss d fuzzy rule-bsed system detfcto euro-fuzzy prdgm, IEEE Trsctos o Systems,, d Cyberetcs, vol. 3, o.3, pp. 39-, Jue. [] J-S R. Jg, ANFIS: dptve-etwor-bsed fuzzy ferece system, IEEE Trsctos o Systems, d Cyberetcs, vol. 3, o.3, pp , y/jue 993. [3] J-S R. Jg, Self-lerg fuzzy cotrollers bsed o temporl bc propgto, IEEE Trsctos o Systems,, d Cyberetcs, vol. 3, o.5, pp. 7-73, September 99. [] J-S R. Jg d C. -T. Su, Fuctol equvlece betwee rdl bss fucto etwors d fuzzy ferece systems, IEEE Trsctos o Neurl Networs, vol., o., pp , Jury 993.

11 IJCSNS Itertol Jourl of Computer Scece d Networ Securty, VOL.6 No.A, Februry 6 3 [5] J-S R. Jg, C. -T. Su d E. zut, Neuro-Fuzzy d Soft-Computg, Pretce Hll, pp , 997. [6] Ch-Feg Jug d Ch-Teg L, A o-le selfcostructg eurl fuzzy ferece etwor d ts pplctos, IEEE Trsctos o Fuzzy Systems, vol.6, o., pp. -3, Februry 998. [7] N.. sbov d Qu Sog, DENFIS: dymc evolvg eurl-fuzzy ferece system d ts pplcto for tme-seres predcto, IEEE Trsctos o Fuzzy Systems, vol., o., pp. 5, Apr.,. [8] C. C. Lee, Fuzzy logc cotrol systems: fuzzy logc cotroller Prt I, IEEE Trsctos o Systems, d Cyberetcs, vol., o., pp. 8, r./apr., 99. [9] C. C. Lee, Fuzzy logc cotrol systems: fuzzy logc cotroller Prt II, IEEE Trsctos o Systems, d Cyberetcs, vol., o., pp. 9 35, r./apr., 99. [] Chushe L d R. Premer, Fuzzy cotrol of uow multple-put-multple-output plts, Fuzzy Sets d Systems, vol., o., pp. 5 67, 999. [] Chushe L d Chu-Y Lee, Fuzzy moto cotrol of d uto-wrehousg cre system, IEEE Trsctos o Idustrl Electrocs, vol.8, o.5, pp.983-9, October. [] Chushe L d R. Premer, Self-lerg geerl purpose PID cotroller, Jourl of the Frl Isttute, vol. 33B, o., pp , 997. [3] Chushe L d Chu-Y Lee, Soft computg system for moto cotrol, IEEE Itellget Trsportto Systems Coferece Proceedgs, Old (CA, USA, pp , August. [] C. T. L d C.S.G. Lee, Neurl-etwor-bsed fuzzy logc cotrol d decso system, IEEE Trscto o Computers, vol., o., pp , Dec., 99. [5] R. P. Lppm, A troducto to computg wth eurl ets, IEEE ASSP gze, pp. -, 987. [6] T. Tg d. Sugeo, Fuzzy detfcto of systems d ts pplcto to modelg d cotrol, IEEE Trsctos o Systems,, d Cyberetcs, vol. SC-5, o., pp. 6-3, Jury/Februry 985. [7] L. X. Wg, Fuzzy systems re uversl pproxmtors, IEEE Itertol Coferece o Fuzzy Systems, pp , r., 99. [8] Shq Wu d eg Joo Er, Dymc fuzzy eurl etwors- ovel pproch to fucto pproxmto, IEEE Trsctos o Systems,, d Cyberetcs, vol. 3,o., pp , Aprl. [9] Jee-Shg Wg; C.S.G. Lee, Self-dptve euro-fuzzy ferece systems for clssfcto pplctos, IEEE Trsctos o Fuzzy Systems, vol., o.6, pp. 79 8, Dec.,. [] L. A. Zdeh, Fuzzy sets, Iformto d Cotrol, vol. 8, pp , 965. [] L. A. Zdeh, Fuzzy lgorthm, Iformto d Cotrol, vol., pp. 9-, 968. [] L. A. Zdeh, Fuzzy logc d pproxmte resog, Sytheses, vol. 3, pp. 7-8, 975. Chushe L receved the Ph.D. degree Electrcl Egeerg d Computer Scece 996 from Uversty of Illos t Chcgo, USA. He s curretly wth Deprtmet of Computer Scece d Iformto Egeerg, Ntol Uversty of T, Tw. He ws wth the Deprtmet of Electrcl Egeerg, Chg Gug Uversty, To-Yu, Tw. He receved the Reserch Awrd from the Ntol Scece Coucl, Tw, 999 d the Reserch Awrd of Excellet Techer from Chg Gug Uversty, Tw,. He hs bee the frst uthor of more th 6 techcl ppers. He s bogrphzed rqus Who swho Scece d Egeerg, edce d Helthcre, d the World. He hs served s referee of pper revew for the ourls of IEEE Trsctos o Idustrl Electrocs, IEEE Trsctos o Systems, d Cyberetcs-Prt B: Cyberetc, IEEE Trsctos o Fuzzy Systems, Fuzzy Sets d Systems, d other tertol ourls. Hs curret reserch terests clude soft computg, computtol tellgece, euro-fuzzy systems, ptter recogto, tellget sgl processg, tellget systems d cotrol, mche lerg, d chp/processor-bsed rel-tme systems.