APPROXIMATE FUZZY REASONING BASED ON INTERPOLATION IN THE VAGUE ENVIRONMENT OF THE FUZZY RULEBASE AS A PRACTICAL ALTERNATIVE OF THE CLASSICAL CRI

Transcription

1 Kovác, Sz., Kóczy, L.T.: Approxmate Fuzzy Reaonng Baed on Interpolaton n the Vague Envronment of the Fuzzy Rulebae a a Practcal Alternatve of the Clacal CRI, Proceedng of the 7 th Internatonal Fuzzy Sytem Aocaton World Congre, pp.44-49, Prague, Czech Republc, (997). Draft veron. APPROXIMATE FUZZY REASONING BASED ON INTERPOLATION IN THE VAGUE ENVIRONMENT OF THE FUZZY RULEBASE AS A PRACTICAL ALTERNATIVE OF THE CLASSICAL CRI Szlvezter Kovác Computer Centre, Unverty of Mkolc Mkolc-Egyetemváro, Mkolc, H-355, Hungary e-mal: zkzlv@gold.un-mkolc.hu Lázló T. Kóczy Department of Telecommuncaton an Telematc, Techncal Unverty of Budapet Sztoczek u.2, Budapet, H-, Hungary Abtract Ung the concept of vague envronment decrbed by calng functon [2] ntead of the lngutc term of the fuzzy partton gve a mple way for fuzzy approxmate reaonng. In mot of the practcal applcaton, the fuzzy partton (ued a prmary et of the fuzzy rulebae) can be decrbed by vague envronment (baed on the mlarty or ndtnguhablty of the element [2]). Comparng a decrpton of a unvere gven by a fuzzy partton to the way of ung the concept of vague envronment we can ay, that the lngutc term of the fuzzy partton are pont n the vague envronment, whle the hape of the fuzzy et decrbed by the calng functon. Th cae the prmary fuzzy et of the antecedent and the conequent part of the fuzzy rule are pont n ther vague envronment, o the fuzzy rule themelve are pont n ther vague envronment too (n the vague envronment of the fuzzy rulebae). It mean, that the queton of approxmate fuzzy reaonng can be reduced to the problem of nterpolaton of the rule pont n the vague envronment of the fuzzy rulebae relaton [4,5]. In other word, ung the concept of vague envronment, n mot of the practcal cae we can buld approxmate fuzzy reaonng method mple enough to be a good alternatve of the clacal Compotonal Rule of Inference method n practcal applcaton. In th paper two method of approxmate fuzzy reaonng baed on nterpolaton n the vague envronment of the fuzzy rulebae, and a comparon of thee method to the clacal CRI wll be ntroduced.. Connecton between mlarty of fuzzy et and vague dtance of pont n a vague envronment The concept of vague envronment baed on the mlarty or ndtnguhablty of the element. Two value n the vague envronment are ε-dtnguhable f ther dtance grater then ε. The dtance n vague envronment are weghted dtance. The weghtng factor or functon called calng functon (factor) [2]. Two value n the vague envronment X are ε-dtnguhable f ε > δ ( x x ) ( ), 2 x2 where δ ( x x ) x x dx, 2 the vague dtance of the value x, x 2 and (x) the calng functon on X. For fndng connecton between fuzzy et and a vague envronment we can ntroduce the memberhp functon µ A ( x) a a level of mlarty a to x, a the degree to whch x ndtnguhable to a [2]. The α-cut of the fuzzy et µ A ( x) the et whch contan the element that are ( α)-ndtnguhable from a (ee fg..): δ ( ab, ) α b µ A( x) mn { δ ( ab, ), } mn ( x) dx, a Fg.. The α-cut of µ A ( x) contan the element that are ( α)-ndtnguhable from a It very eay to reale (ee fg..), that th cae the vague dtance of pont a and b (δ (, ab) ) bacally the Dcontency Meaure (S D ) of the fuzzy et A and B (where B a ngleton): S D ( x) x X f δ ( ab, ) [ 0, ] up µ δ ( ab, ) A B

2 where A B the mn t-norm, µ A B( x) mn µ A( x), µ B( x) x X. [ ] It mean, that we can calculate the dcontency meaure between member fuzzy et of a fuzzy partton and a ngleton, a vague dtance of pont n the vague envronment of the fuzzy partton. The man dfference between the dcontency meaure and the vague dtance, that the vague dtance a crp value n range of [0, ], whle the dcontency meaure lmted to [0,]. That why they are ueful n nterpolate reaonng wth nuffcent evdence. So f t poble to decrbe all the fuzzy partton of the prmary fuzzy et (the antecedent and conequent unvere) of our fuzzy rulebae, and the obervaton a ngleton, we can calculate the extended dcontency meaure of the antecedent prmary fuzzy et of the rulebae and the obervaton, and the extended dcontency meaure of the conequent prmary fuzzy et and the conequence (we are lookng for) a vague dtance of pont n the antecedent and conequent vague unvere. 2. Generatng vague envronment from the fuzzy partton of the lngutc term of the fuzzy rule The vague envronment decrbed by t calng functon. For generatng a vague envronment we have to fnd an approprate calng functon, whch decrbe the hape of all the term n the fuzzy partton. The method propoed by Klawonn [2], for choong the calng functon (x), gve an exact decrpton of the fuzzy term after ther recontructon from the calng functon: dµ x ( ) µ '( x) dx So we alway fnd a calng functon, f we have only one fuzzy et n the fuzzy partton. Uually the fuzzy partton contan more than one fuzzy et, o th method requre ome retrcton for the memberhp functon of the term [2]: dµ x ( ) µ '( x) dx { } ext mn µ ( x), µ ( x) >0 µ ' ( x) µ ' ( x) ff j j, j I Generally the above condton not fulfllng, o the queton how to decrbe all fuzzy et of the fuzzy partton wth one unveral calng functon. For th reaon we propoe to ue the approxmate calng functon. 3. The approxmate calng functon The approxmate calng functon an approxmaton of the calng functon decrbe the term of the fuzzy partton eparately. Suppong that the fuzzy term are trangle, each fuzzy term can be charactered by two contant calng functon, the calng factor of the left and the rght lope of the trangle. So a trangle haped fuzzy term can be charactered by three value (by a trple), by the value of the left and the rght calng factor and the value of t core pont (e.g.fg.2.). Fg.2. Two trangle haped fuzzy et charactered by two trple, by the left and the rght calng factor and the value of the core pont For generatng the approxmate calng functon we ugget to adopt the followng non-lnear functon for nterpolatng the neghbourng calng factor: kw w ( d ) L k w k ( d ) ( x x ) R L, ( x) kw w ( d ) k w w k ( d ) ( x x ) R L <, x [ x, x ), [, n ] where L R w,, n, [ ] w R

3 [ ], d x -x,, n (x) the approxmate calng functon, x the core of the th term of the approxmated L fuzzy partton,, R are the left and rght de calng factor of the th trangle haped term of the approxmated fuzzy partton, k contant factor of entvty for neghbourng calng factor dfference, n the number of the term n the approxmated fuzzy partton The above functon ha the followng ueful properte: If the neghbourng calng factor are equal, (x) lnear [ ) ( x) R L R, x x, x L If one of the neghbourng calng factor nfnte e.g. R (the rght de of the th term crp) then R L and fnte, x [ x, x) x x ( x) 0 otherw e mlarly L and R fnte, x [ x, x ) ( x) x x 0 otherw e Fg.3. Approxmate calng functon generated by the propoed non-lnear functon (k), and the orgnal fuzzy partton (A,B) a th calng functon decrbe t (A,B ) 4. Calculatng the concluon by approxmatng the vague pont of the rulebae If the vague envronment of a fuzzy partton (the calng functon or the approxmate calng functon) ext, the member et of the fuzzy partton can be charactered by pont n the vague envronment. (In our cae the pont are characterng the core of the term, whle the hape of the memberhp functon are decrbed by the calng functon.) If all the vague envronment of the antecedent and conequent unvere of the fuzzy rulebae are ext, all the prmary fuzzy et (lngutc term) ued n the fuzzy rulebae can be charactered by pont n ther vague envronment. So the fuzzy rule (buld on the prmary fuzzy et) can be charactered by pont n the vague envronment of the fuzzy rulebae too. Th cae the approxmate fuzzy reaonng can be handled a a clacal nterpolaton tak. Applyng the concept of vague envronment (the dtance of pont are weghted dtance), any nterpolaton, extrapolaton or regreon method can be adapted very mply for approxmate fuzzy reaonng. For example we can ue the Lagrange nterpolaton. The orgnal formula the followng: ( x-x2) ( x-x3) ( x-xn) ( x -x2) ( x -x3) ( x -xn) ( x-x) ( x-x3) ( x-xn) y 2 ( x2 -x) ( x2 -x3) ( x2 -xn) ( x-x) ( x-x2) ( x-xn-) yn ( x -x ) ( x -x ) ( x -x ) Yx ( ) n n 2 n n- y where Y(x) the Lagrange nterpolaton of the n pont (x,y ), (x 2,y 2 ),, (x n,y n ). Ung the concept of vague dtance n the cae of one dmenonal antecedent unvere: dt( y 0,y) ( 2 ) ( 3 ) ( r ) dt( y 0,b ) ( ) ( ) ( ) 2 3 r ( ) ( 3 ) ( r ) dt( y 0, b ) ( ) ( ) ( ) r 2 dt( a, x) dt( a 2, x) dt( a r, x) dt( y 0, br ) ( ) ( ) ( ) dt a,x dt a,x dt a,x dt a,a dt a,a dt a,a dt a, x dt a, x dt a, x dt a,a dt a,a dt a,a dt a, a dt a, a dt a,a where r 2 r r r a l ( ) ( ) dt a k, al X x d x a k y 0 the frt element of the unvere Y: y 0 y y Y (a Y a one dmenonal unvere)

4 Another example a an adaptaton of a mple ratonal nterpolaton the followng: where w k ( ) dt y, y 0 r k ( dt( xa, k )) p k ( ) w dt y, b k r a weghtng factor nverely proportonal to the vague dtance of the obervaton and the k th rule antecedent, x m dt ( ak, x) dt( x, ak) X ( x) dx a b k ( ) ( ) dt y 0, bk Y y dy, y0 X the th calng functon of the m dmenonal antecedent unvere, Y the calng functon of the one dmenonal conequent unvere, x the multdmenonal crp obervaton, a k are the core of the multdmenonal fuzzy rule antecedent A k, b k are the core of the one dmenonal fuzzy rule conequent B k, R A B are the fuzzy rule, p the entvty of the weghtng factor for dtant rule, y 0 the frt element of the one dmenonal unvere (Y: y 0 y y Y), y the one dmenonal concluon we are lookng for. Comparng the two propoed nterpolaton method we can etablh the followng: Havng only two rule, between the two rule antecedent the two method gve the ame concluon (e.g.fg.4.). Becaue of the abolute antecedent dtance of the propoed ratonal nterpolaton functon, the approxmate reaonng method baed on th functon can be ued n cae of multdmenonal antecedent unvere too w k 0 k, k 2, Fg.4. Interpolaton of fuzzy rule (R :A B ) n the approxmated vague envronment of the fuzzy rulebae, ung the propoed ratonal nterpolaton (p) and the adopted Lagrange nterpolaton 5. Comparng the crp concluon generated by approxmate reaonng n the vague envronment of the fuzzy rulebae to the crp concluon generated by the clacal Compotonal Rule of Inference For comparng the crp concluon generated by the propoed approxmate reaonng method to the clacal Compotonal Rule of Inference (CRI), we are choong a a repreentatve one, the mn-max. compotonal rule of nference and the centre of gravty defuzzfcaton method. Comparng the crp concluon of the propoed approxmate fuzzy reaonng and the clacal CRI (fg.5.), the mot trkng dfference, that the control functon of the approxmate fuzzy reaonng alway ft the pont of the fuzzy rule. (Th a property of the

5 nterpolaton functon ued for the approxmate reaonng.) Whle, the control functon of the CRI uually not ft thee pont. In practcal ene t mean that, f an obervaton ht a rule antecedent exactly, than the concluon generated by approxmate fuzzy reaonng wll be equal to the conequent part of the ame fuzzy rule. The next dfference the man reaon of the approxmate fuzzy reaonng method for nuffcent evdence. The approxmate fuzzy reaonng method gve concluon for all the obervaton of the antecedent unvere, even f the fuzzy rulebae not complete, whle the CRI gve no concluon f there are no overlappng between the obervaton and at leat one of the rule antecedent. (Fg.5.) The lat mentoned dfference a knd of phloophcal queton. The wde rule conequent ha more nfluence to the defuzzfed crp concluon of the CRI, becaue of the wde conequent are more heavy n the fuzzy concluon (ung the centre of gravty defuzzfcaton). Whle ung the method baed on approxmaton n the vague envronment of the fuzzy rulebae, the tuaton the oppote. (Fg.5.) The dea we ued, that the rate of the dtance of the obervaton and the rule antecedent mut be equal to the rate of the dtance between the concluon and the correpondng rule conequent, ha a pecal property n cae of ung the concept of vague envronment for calculatng the dtance of fuzzy et. If a rule conequent narrower than the other, the calng functon hgher there, the urroundng vague envronment more dene. It mean maller dtance n the conequent unvere. So the narrow rule conequent domnatng the wder one. In other word t mean, that ung the CRI, n the crp concluon thoe rule ha domnance, whoe conequent part more global (more mprece, fuzzy, wder ), n pte of the approxmate reaonng method, where thoe rule ha the domnance, whoe conequent are more prece (more crp, narrower ). Bacally th a queton of the mportance of the vaguene n the rule conequence. Whch rule need more attendance, thoe, whoe conequence are more global, or thoe, whoe conequence are more prece. Fg.5. Interpolaton of two fuzzy rule (R :A B ) n the approxmated vague envronment of the fuzzy rulebae, ung the propoed ratonal nterpolaton (p) and the mn-max. CRI wth the centre of gravty defuzzfcaton

6 Concluon Ung the concept of vague envronment n mot of the practcal cae we can bult approxmate fuzzy reaonng method mple enough to be a good alternatve of the clacal Compotonal Rule of Inference method n practcal applcaton. The advantage (compared to CRI) of the method propoed n th paper are the followng: - the computatonal effort needed for the concluon can be reduced by reducng the number of the fuzzy rule (the unmportant fllng rule can be elmnated) - the propoed method gve concluon n cae of nuffcent evdence (pare fuzzy rulebae) too - ung the propoed approxmate fuzzy reaonng method, f crp concluon needed, t can be fetched drectly from the vague concluon (there are no addtonal defuzzfcaton tep needed) The vague concluon calculated by the propoed approxmate fuzzy reaonng method bacally one pont. For tranformng th pont to a fuzzy concluon, we have to examne the conequence unvere. Suppong that the term n the fuzzy partton of the conequence unvere decrbe all the man properte of the conequence unvere and the calng functon approxmated from th term proper, we can calculate the memberhp functon of the fuzzy concluon a a level of mlarty to the vague concluon n the vague envronment of the conequence unvere. [4] Kovác, Sz., Kóczy, L.T.: Fuzzy Rule Interpolaton n Vague Envronment, Proceedng of the 3rd. European Congre on Intellgent Technque and Soft Computng, pp.95-98, Aachen, Germany, (995). [5] Kovác, Sz.: New Apect of Interpolatve Reaonng, Proceedng of the 6th. Internatonal Conference on Informaton Proceng and Management of Uncertanty n Knowledge-Baed Sytem, pp , Granada, Span, (996). Reference [] Turken, I.B., Zhong, Z.: An Approxmate Analogcal Reaonng Schema Baed on Smlarty Meaure and Interval-valued Fuzzy Set, Fuzzy Set and Sytem, vol.34, pp , (990). [2] Klawonn, F.: Fuzzy Set and Vague Envronment, Fuzzy Set and Sytem, 66, pp207-22, (994). [3] Kóczy, L. T., Hrota, K.: Interpolatve reaonng wth nuffcent evdence n pare fuzzy rule bae, Informaton Scence 7, pp 69-20, (992).