TAKAGI-SUGENO REASONING PROCEDURE FOR PATTERN RECOGNITION

Transcription

1 TAKAGI-SUGEO REASOIG PROCEDURE FOR PATTER RECOGITIO Ramundas Jasnecus, Vytautas Petrauskas, Radle Krusnskene 2 Kaunas Unersty f Technlgy, Studentu A, Kaunas, Lthuana, ramundas.jasnecus@ktu.lt, ytautas.etrauskas@ktu.lt 2 Lthuanan Academy f Physcal Educatn, Deartment f Infrmatcs and Languages, Srt 6, Kaunas, Lthuana; r.krusnskene@lkka.lt Abstract. The rblem f attern recgntn s well knwn as well as the reasnng rcedure f Takag-Sugen (T-S), but u tll nw bth f them hae nt been used tgether t descrbe slutns f the same fuzzy rblems. Therefre, the aer analyses the attern recgntn rcess as Takag-Sugen reasnng rcedure and defnes (T-S) rules weghts, slng secal lnear rgrammng rblem, whch s cnstructed accrdng t exerts fuzzy knwledge. Such an arach fr attern recgntn rblem has been used fr the frst tme. A ractcal, cmrehensble and usable thery enablng racttners t buld adequate hardware/sftware tls fr attern recgntn s resented n the artcle. Keywrds: attern recgntn, Takag-Sugen reasnng, fuzzy rule, lnear rgrammng rblem. Intrductn The attern recgntn rblem s ld enugh, ery well-knwn and erfectly descrbed n scentfc lterature. The wrk f R. O. Duda and thers [] may be descrbed as ne f the hghly recmmended texts cncernng ths tc and ncludng an extremely gd lst f references. In general, a frmulatn f ths rblem tself mles a retty gd amunt f fuzzness. Accrdng t [2], attern recgntn s defned as a search fr structure n data, whch s erfrmed n three stes: data acqustn, the extractn f attern features frm data by the reductn f dmensnalty and the mang f extracted features t attern classes. The man rblem, related wth the ssue, s cnnected wth technques amng t ce wth the rblem f fuzzness f exert knwledge. There are three man gals f ths artcle.. Cnsderng the fuzzness f the descrtn f attern recgntn rblem n general, t use fr the frst tme Takag-Sugen (T-S) fuzzy reasnng rcedure, whch s knwn and successfully used n ther alcatns. 2. T rse t aly slutns f secally frmulated lnear rgrammng rblem (LPP) fr calculatn f weghts n (T-S) reasnng rcedure nstead f usng tme cnsumng and rly determned neural-tye tranng rcesses. It s cnsdered that frmulatn f LPP tself, usng exert s knwledge, s mre adequate t the fuzzy descrtn f the attern recgntn rblem. 3. T resent a ractcal, cmrehensble and usable thery enablng racttners t buld adequate hardware/sftware tls fr attern recgntn. Successful mlementatns f the rsed thery n dfferent felds wll be summarzed and resented n the next artcle f the authrs. The nference f rdnary fuzzy systems s based n: ) a derng erbal (lngustc) r arametrc cnsequents by re-rcessng lsts f fuzzy rules that cntan erbal r arametrc antecedents lnked by certan fuzzy lgc eratns, and 2) a defuzzfcatn rcess, alyng sme cmstnal rule r frmulae [2], [3]. The tyes f rules are: IF x s A AD y s B THE z s C (fr Mamdan fuzzy mdels) IF x s A AD y s B THE z = F(x, y) (fr T S fuzzy mdels) () Defuzzfcatn rcedures fr the tw cases mentned abe can be descrbed as reasnng n the bass f a set f cnsequents C usng the centre f graty (CG) r a mean f maxmum (MM) methds fr Mamdan tye mdels, and a fuzzy mean (FM) methd as reasnng by ealuatn f all results z ncluded and rcessed accrdng t the certan frmula Φ(z) fr T-S mdels. 2 Pattern Recgntn Based n T-S Reasnng Prcedure A attern f a class s cnsdered (as a hyscal r abstract structure f class s bjects) descrbed by a set f dstncte features as descrbed n [] and [2]. The attern recgntn rblem can be frmulated as fllws. It s cnsdered n the artcle that =, 2, S classes f bjects and each class has ts wn attern. Each bject s descrbed by n =, 2, j, features. After the extractn f a feature as well as the rcedures f measurement and nrmalzatn as descrbed n [4], the -th feature f an bject that belngs t

2 the -th class (.e. crresndng t the -th attern) can be reresented by a real number α, whch exresses a degree f ntensty f ths artcular feature, and the bject s reresented by a ectr-rw α = α, α 2, α,, α. If there are l =, 2, k, L bjects and ther deendence t L class (.e they rgnate frm the -th attern) s knwn n adance, class s reresented by a set f ectrs l α, l., The man task f the attern recgntn rcedure nles the deelment f T-S rules and defuzzfcatn nstruments. Rules and nstruments must be defned alyng all aalable nfrmatn abut atterns, whch s stred n sets f l α, l and exerts exerence that s resented n a erbal frm and was, cllected when wrkng wth features f bjects and atterns. Such reasnng allws cnstructng an nstrument f attern recgntn rdng a ssblty t assgn any unknwn but rerly descrbed bject x r t any f ssble classes (and atterns). An accuracy f the assgnment deends n the nstrument s decsn makng effcency t rcess fuzzy nfrmatn. Frequently, better reasnng results are acheed when features f bjects are nrmalzed and centred [4], l l l l l [5]. Then the bject s reresented as a ectr-rw α = ( α, α 2, α, α ) the cmnents f whch are calculated as fllws: l l l α = α α (2) j= Therefre, T-S reasnng rcedure can be exressed usng Eq (). It means that a set f ste rules ([6]) fr T-S attern recgntn rcedure cnssts f a lst f statements: IF < degree f certanty that feature wth ntensty x belngs t the attern s K > THE < z + = K x > RECOMMEDED It means: + + IF < μ ( x ) = K > THE z = K x,, (3) Smlarly, a set f negate rules ([6]) fr T-S attern recgntn rcedure cnssts f a lst f statements: IF < degree f certanty that feature wth ntensty x belngs t any ther attern excet s -K > THE < z - = - K x > OT RECOMMEDED Ths means: ( x ) = ( K ) > THE z = K x, j IF < μ (4) Accrdng t the cncet f hyer-nference ([6]) and When the unknwn bject deendence functn: x r, + ( x ) = { μ ( x ), ( μ ( x )} μ max (5) + z = z z = K x,,. (6) s cnsdered, ts degree f belngng t class can be ealuated by a Φ ( x ) = x K, Ths defuzzfcatn methd fr T-S rcedure n a ectr ntatn s exressed as: r r T Φ ( x ) = x K,, (8) where T stands fr a transstn f a certanty ectr r K = K, K 2,, K,, K (9) A blck dagram, reresentng a fnal decsn makng act f fuzzy attern recgntn based n T-S reasnng rcedure, s shwn n Fgure. (7)

3 Fgure. Decsn makng act fr T-S reasnng. In ractce, there s n ssblty t frmulate erbally understandable lsts f rules fr sets resented by Eq. (3) and (4). Usually neural-tye tranng rcedures based n gradent methds are used []. An ntentn t ad cmarately unredctable and ery clumsy neural-tye tranng rcedures fr the determnatn f certanty ectrs K, n T-S reasnng rcedure leads t a frmulatn f an bjecte functn maxmzatn rblem, subjected t a set f cnstrans and cnstructed accrdng t fuzzy nfrmatn. In the next sectn f ths aer a secal lnear rgrammng rblem (LPP) s rsed t fnd tmal certanty ectrs K, fr (3)-(6). LPP rblem s frmulated accrdng t exerts fuzzy nfrmatn whch enables t defne rule weghts n T-S reasnng based attern recgntn rcedure. 3 Lnear Prgrammng Prblem fr Determnatn f T-S Prcedure Rule Weghts A structure f Eq. (7) mles a smle (lnear) frm f an bjecte functn t be maxmzed as well as lnearty f cnstrants. Accrdng t the descrtn f attern recgntn rblem resented n sectn 2 f ths aer, nfrmatn abut the atterns s stred n the sets f l α,. In ste f ts fuzzness, the frmulatn f a rblem fr the determnatn f certanty ectrs K, fr T-S reasnng rcedure can be cnstructed k α, whch reresents class and s central, s randmly selected and crdnates f as fllws. One bject k certanty ectr K, are fund n rder the deendence functn Φ ( α ) wuld be maxmzed: k k Φ ( ) = α K max under the fllwng cnstrans: α (0) l l r k k α α K γ α K, l () α K κ K, r, l. (2) Eq. () s tghtly cnnected wth the cncet f ste smlartes wthn the class and extracts them usng the set f ste rules, as t was descrbed n sectn 2. Eq. (2) reflects dssmlartes between certan classes (n ths artcular case class ) and all ther classes (r negate smlartes ) and extracts thse dssmlartes by the set f negate rules. Otmal alues f γ and κ are recmmended frm the nteral [0-], and γ > κ [4]. Partcular alues deend n the rr knwledge (r guess) cncernng the structure (nternal cnnectns and dsersn f atterns features) f classes (r atterns). The dfference determnes γ κ exerts fuzzy assumtn cncernng a ssble structure f classes under nestgatn and must be chsen n the bass f sme exerence [4]. Ceffcents γ and κ allw cntrllng fuzzy leel f thse smlartes and dssmlartes

4 As t can be easly ntced the rblem belngs t the class f LPP where nequaltes () and (2) need addtnal cnstrans: 0 K A,, (3) where A s any ractcally cnenent real number. A slutn f Eq. (0)-(3) fr the class cnssts f the btaned alue f k max Φ ( α ) = Φ max, r (4) K = ( K, K 2,, K,, K ). The rcedure must be reeated fr all classes. In ths way a set f S slutns wll be btaned and the recgntn rcedure must be erfrmed accrdng t the Fgure 3 takng nt accunt the need f fulfllng a cndtn f rrtnalty: c Φmax = L = c Φ max = L = cs Φs max = B, (5) where c are real numbers. Eq. (5) lays a rle f nrmalzatn and t means that the fuzzy term ery smlar must be ealuated by the same number B whateer attern s cnsdered. As t was delered befre, T-S reasnng rcedure resents degrees f certanty, ndcatng that the descrtn f an unknwn bject x r belngs t attern (see Fgure ). If there s a need t use ths r Φ ( x ) nfrmatn as a recmmendatn fr a decsn maker, the alue u = must be determned. max x Practcally an nteral fr u s [0-] wth sme zne f accuracy as t s shwn n the Fgure 2, where the degree μ s resented. f certanty x r Fgure 2. Recmmended certanty that x r belngs t When u s n the zne, secal addtnal nestgatns f the bject s rertes are strngly recmmended befre the fnal decsn s taken. The entre classfcatn rcess s shwn n Fgure 3 and reresents the rcess f attern recgntn cmbned wth decsn makng recmmendatns

5 Fgure 3. T-S hyer-nference and the rcedure f decsn makng recmmendatns. 4 Fnal remarks In the analytcal art f ths aer a new arach t the rblem f fuzzy attern recgntn was resented. Frst f all the rule-based fuzzy nference, cncernng the measure f atterns smlarty, was ntrduced, then a descrtn f the rcess f attern recgntn was based n T-S reasnng rcedure, enhanced by secfc decsn makng recmmendatns. The weghts f rules n T-S rcedure were defned, slng secal LPP, cnstructed accrdng t the fuzzy nfrmatn resented r sused n adance. A ractcal, cmrehensble and usable thery, enablng racttners t buld adequate hardware/sftware tls fr attern recgntn s resented n the artcle. Successful mlementatns f the rsed thery n dfferent felds wll be summarzed and resented n the next artcle f the authrs f ths aer. 5 Acknwledgements The authrs f the artcle want t exress ther grattude fr the stmulatng deas t carry ut ths nestgatn by rf. Chrstan Brgelt frm the Eurean Centre fr Sft Cmutng (n Meres, Astura, San). Ths research was n art surted by COST rgramme and fnancal fundng receed frm Lthuanan Agency fr Internatnal Scence and Technlgy Prgrammes. References [] Rchard O. Duda, Peter E. Hart, Dad G. Strk, Pattern Classfcatn, Jhn Wley & Sns, 200, [2] Amt Knar, Cmutatnal Intellgence (Prncles, Technques and Alcatns), Srnger, 2005, [3] Ksk B., Fuzzy Engneerng Prentce Hall, [4] Jasnecus R., Parallel Sace-Tme Cmutng Structures (n Russan), Mkslas, Vlnus, [5] Jasnečus R., Petrauskas V., Fuzzy exert mas: the new arach // WCCI 2008 Prceedngs: 2008 IEEE Wrld Cngress n Cmutatnal Intellgence, June -6, 2008, Hng Kng : 2008 IEEE Internatnal Cnference n Fuzzy Systems IEEE Internatnal Jnt Cnference n eural etwrks IEEE Cngress n Elutnary Cmutatn. Pscataway: IEEE, ISB [6] Kendl H., Kncker R., ewels F., Tw-way fuzzy cntrllers based n hyernference and nference flter, // Prceedngs Secnd Wrld Autmatn Cngress, l. 4, Intellgent Autmatn and Cntrl, TSI Press, Mnteller