Extending the Concept of Fuzzy Rule Interpolation with the Interpolation of Fuzziness

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1 WCCI 01 IEEE World Congress on Computatonal Intellgene June, 10-15, 01 - Brsbane, ustrala FUZZ IEEE Extendng the Conept of Fuzzy Rule Interpolaton wth the Interpolaton of Fuzzness Szlveszter Kovás Department of Informaton Tehnology Unversty of Mskol Mskol, Hungary szkovas@t.un-mskol.hu bstrat Fuzzy Rule Interpolaton (FRI) methods are not always sutable for desrbng hanges n the onluson fuzzness. For example, t s dffult to desrbe ases n whh the onluson for a rsp observaton must be fuzzy, or n whh an nrease n the fuzzness of an observaton yelds less fuzzness n the onluson. Ths problem s manly nherted from a lak of nformaton n the model, and orgnates from the defeny of the fuzzy rule representaton. In ths paper, a novel rule representaton onept alled double fuzzy dot s suggested, whh helps the elaboraton of FRI methods n order to be able to better handle the fuzzness of the onluson. Two smple examples of suh FRI methods are also brefly ntrodued n ths paper. Keywords- fuzzy rule nterpolaton, nterpolaton of the fuzzness, fuzzy funton, double fuzzy dot rule representaton. I. INTRODUCTION ommon problem wth exstng fuzzy reasonng methods based on fuzzy nterpolaton s that they have dffultes n expressng stuatons n whh the onluson to a rsp observaton s fuzzy, or n whh an nrease n the fuzzness of an observaton yelds less fuzzness n the onluson. The man reason for ths weakness n expressng the level of fuzzness of the onluson as a funton of the level of fuzzness of the observaton s manly nherted from the shortage of nformaton n suh models, whh orgnates from the defeny of the fuzzy dot rule representaton. s a soluton to ths problem, the paper proposes a novel rule representaton onept alled double fuzzy dot, whh helps the elaboraton of fuzzy rule nterpolaton (FRI) methods able to better handle the fuzzness of the onluson. Two smple examples of FRI methods based on the suggested double fuzzy dot rule representaton are also ntrodued brefly n ths paper. II. FUZZY FUNCTIONS, INTERPOLTION OF FUZZY DT There are two ommon ways to defne fuzzy funtons. The frst s the lassal desrptve way, where the fuzzy funton s vewed as a speal fuzzy relaton. The seond s the delaratve way, whh onsders the fuzzy funton as a mappng between two fuzzy spaes. In the desrptve vew the fuzzy rules represent relatons between rule anteedents and onsequent fuzzy sets. In ths ase, the entre rule-base s also onsdered to be a fuzzy relaton of the anteedent and onsequent unverses, bult upon the dsjunton of the fuzzy rule relatons. The lassal ompostonal rule of nferene [3] also follows the desrptve vew, as the onluson fuzzy set s ganed as a omposton of the fuzzy observaton and the fuzzy rule-base relaton. In the rest of the paper we wll onsder the fuzzy funton form the delaratve vew, as a mappng between two fuzzy spaes. pplyng the notaton ntrodued n [1], [] by Perfleva et al. the fuzzy funton as a mappng an be defned n the followng way: Let L X and L Y are fuzzy spaes on X and Y, respetvely. X Y orrespondene f : L L s a fuzzy funton f for X every, B L, = B mples f ( ) = f( B). = B ff x X x = B x., ( ) ( ) X Y fuzzy funton f : L L s extensonal f for every X, B L, ( B) ( f( ) f( B) ), where s a smlarty L X, L. Y relaton defned on In ths ase, the fuzzy nterpolaton of fuzzy data an be onsdered as an extenson of a fuzzy funton: Let fuzzy data be gven by a set of pars {(, B ) = 1,..., n}, where L X, B L Y, = 1,..., n, and 1,..., n are dfferent wth respet to =. Moreover, let f : B be a fuzzy funton on the doman = { 1,..., n }, and P { : D L Y }, where D L X, s a hosen subset of fuzzy funtons. The problem s to fnd P (P s a set of nterpolatng funtons) suh that the nterpolaton ondton ( ) = B, = 1,..., n s fulflled. In ths ase, s alled an nterpolatng fuzzy funton for fuzzy data {(, B ) = 1,..., n}. In other words, the nterpolatng fuzzy funton s an extenson of a fuzzy funton f on the doman D. III. FUZZY RULE INTERPOLTION METHODS Turnng the notaton of fuzzy funton and fuzzy nterpolaton to the form of fuzzy rules and fuzzy rule nterpolaton, n the delaratve vew of the fuzzy funton, the U.S. Government work not proteted by U.S. opyrght 1106

2 n fuzzy rules of the fuzzy rule-base represent the rules as fuzzy data {(, B ) = 1,..., n}, alled fuzzy dots. In ths ase, the way of fuzzy reasonng based on fuzzy nterpolaton an be understood as a searh for a sutable nterpolatng fuzzy funton ( ) = B, = 1,..., n. Based on the nterpolatng fuzzy funton, the onluson y L Y for observaton x L X s smply ganed as y = (x). There are numerous methods of fuzzy nterpolaton whh have appeared n the lterature n the last 0 years. One of the frst FRI tehnques was publshed by Kózy and Hrota [7]. It s usually referred as the KH method. It s applable to onvex and normal fuzzy (CNF) sets. It determnes the onluson by ts α-uts n suh a way that the rato of dstanes between the onluson and the onsequents should be dental wth the ones between the observaton and the anteedents for all mportant α-uts. It s shown n e.g. [8], [9] that the onluson of the KH method s not always dretly nterpretable as a fuzzy set. Ths drawbak motvated many alternatve solutons. modfaton was proposed by Vass, Kalmár and Kózy [10] (VKK method), where the onluson s omputed based on the dstane of the entre ponts and the wdths of the α-uts, nstead of lower and upper dstanes. The VKK method dereases the applablty lmt of the KH method, but does not elmnate t ompletely. The tehnque annot be appled f any of the anteedent sets s sngleton (the wdth of the anteedent s support must be nonzero). In spte of the dsadvantages, KH s popular beause ts smplty nfers advantageous omplexty propertes. It was generalzed n several ways. mong them the stablzed KH nterpolator s emerged, as t s proved to hold the unversal approxmaton property [11], [1]. Ths method takes nto aount all flakng rules of an observaton n the alulaton of the onluson n extent to the nverse of the dstane of anteedents and observaton. The unversal approxmaton property holds f the dstane funton s rased to the power of the nput s dmenson. nother modfaton of KH s the modfed alphaut based nterpolaton (MCI) method [13], whh allevates ompletely the abnormalty problem. MCI s man dea s the followng: t transforms fuzzy sets of the nput and output unverses to suh a spae where abnormalty s exluded, then omputes the onluson there, whh s fnally transformed bak to the orgnal spae. MCI uses vetor representaton of fuzzy sets and orgnally applable to CNF sets [14]. MCI s one of the pratally applable FRI methods [15], sne t preserves advantageous omputatonal and approxmate nature of KH, whle t exludes ts abnormalty. n applaton orented aspet of the FRI, the low omputatonal and resoure demand s also emergng n the onept of FIVE (Fuzzy Interpolaton based on Vague Envronment) [16]. nother fuzzy nterpolaton tehnque was proposed by Kózy et al. [17]. It s alled onservaton of relatve fuzzness (CRF) method, whh noton means that the left (rght) fuzzness of the approxmated onluson n proporton to the flankng fuzzness of the neghborng onsequent should be the same as the (left) rght fuzzness of the observaton n proporton to the flankng fuzzness of the neghborng anteedent. The tehnque s applable to CNF sets. n mproved fuzzy nterpolaton tehnque for multdmensonal nput spaes (IMUL) was proposed n [18], and desrbed n detals n [15]. IMUL apples a ombnaton of CRF and MCI methods, and mxes advantages of both. The ore of the onluson s determned by MCI method, whle ts flanks by CRF. The man advantages of ths method are ts applablty for multdmensonal problems and ts relatve smplty. In [] a novel FRI dea able to handle hghly overlappng rule anteedents by nonlnear mappng s appears. There are a lot of works done n the dreton fuzzy nterpolaton based on fuzzy relaton vew of fuzzy rules by Dubos, Prade et al. [3]. The suggested nterpolaton by onvex ompleton method [4] s based on the extenson of nterval nterpolaton to fuzzy nterpolaton. Coneptually dfferent approahes were proposed by Bouhon-Meuner et al. [5], [6]. The suggested analogybased nterpolaton frst nterpolates the referene pont poston of the onluson, then onstruts the shape of onluson fuzzy sets based on the smlarty (dstngushablty) relaton of the rule anteedents and observaton wth respet to the smlarty of the orrespondng rule onsequents and the onluson. smlar two step method dea s appearng n the work of Barany et al. [4]. They extend the onept to be able to handle arbtrary shaped fuzzy sets. The suggested two step General Methodology (GM) frst generate an nterpolated ntermedate rule n the referene pont poston of the observaton, then n the seond step a sngle rule reasonng method (revson funton) s appled to determne the fnal fuzzy onluson based on the smlarty of the fuzzy observaton and an nterpolated observaton. Pratal applaton of the GM appears n the work of Johanyák et al. [7]. The suggested Polar Cut nterpolaton solves the task of the sngle rule reasonng step by smlarty of arbtrary shaped fuzzy sets defned based on polar uts. n extenson of the GM s also appearng n the work of Shen et al. [8]. The suggested sale and move transformaton an extend the orgnal method to extrapolaton. The ode of many of the above mentoned FRI methods are freely avalable as a part of the MTLB FRI Toolbox [19], together wth a sparse fuzzy model dentfaton toolbox [0] s avalable at [1]. IV. INTERPOLTION OF THE FUZZINESS It an be easly stated, that many of the above mentoned methods have dffultes when the hange of the fuzzness of the onluson n the funton of the fuzzness of the observaton have to be delberately defned. E.g. for a rsp observaton the onluson have to be fuzzy, or f nreasng the fuzzness of the observaton have to lead to dereasng the fuzzness of the onluson. The problem s straghtforward n the ase f the FRI method s defned for a sngle dmensonal anteedent unverse and two flankng anteedent rules to the observaton, lke the orgnal KH method, the orgnal GM, or the analogy-based nterpolaton. The freedom of ndependently handlng the fuzzness s mssng from the defnton of the methods. On the other hand t s also a lak of nformaton. There s no way for expressng the hange of the fuzzness n the poston of the observaton havng two neghbourng fuzzy rules only. 1107

3 In ase of FRI methods able to handle multple rules from the rule base (usually defned as an extenson of the orgnal method) by takng more rules nto the onsderaton the hange of the fuzzness an be expressed e.g. by overlappng rules havng dfferent fuzzness on the anteedent and onsequent sde. Therefore the man reason of the dffultes n expressng the level of fuzzness of the onluson n the funton of the level of fuzzness of the observaton s manly nherted from the shortage of nformaton orgnates from the defeny of the fuzzy rule representaton. To dsmss the problem of delberately defnng the hange of the fuzzness of the onluson n the funton of the fuzzness of the observaton n most of the FRI methods there are two ommon preoneptons hold about the nterpolaton of the fuzzness: In ase of rsp observaton, the onluson must be rsp. If the level of fuzzness n the observaton s nreasng, the level of the fuzzness n the onluson must be also nrease. The nterpolatng fuzzy funton usually hosen to be sutable for smoothly desrbng a sparse fuzzy rule-base, a rule-base, whh not fully overs the anteedent unverse, and usually ontans only a few fuzzy rules,.e. fuzzy dots, {(, B ) = 1,..., n}, where n s usually relatvely small. The nterpolatng fuzzy funton s usually takng less are about how the fuzzness s hangng n the lose neghborhood of a gven rule. It s also a lak of nformaton,.e. n a sparse rulebase t s also unommon to have overlappng rules ( j =0, j 1,..., n). Hene n the neghborhood of a gven rule the only exstng nformaton about the hange of the fuzzness s the fuzzness relaton of the anteedent and onsequent fuzzy sets of the rule, whh s a onstant n a rule represented as a fuzzy dot. To somehow handle the problem of havng observatons wth dfferent fuzzness than the fuzzness of the rule anteedent, most of the fuzzy nterpolaton methods follow some gudelnes alled axomat approah of the fuzzy nterpolaton. omprehensve overvew of the axomat approah of the fuzzy nterpolaton an be found n []. The most ommon gudelne for handlng the fuzzness relaton of the observaton and onluson n the axomat approah of the fuzzy nterpolaton funton s the Monotonty ondton. In [5] t s the ondton I and defned for the sngle dmensonal anteedent ase, n [6] t s alled Property 6. and extended for multdmensonal anteedents n the followng way: Monotonty [6]: Let *, + are two observatons suh that * s more spef n all dmenson than +, that s for + j =1,...,n, * + j j, then ( * j) ( j ),.e. the same relaton should be held for the two onlusons. In other words the monotonty ondtons mean that we have no nformaton about the rate or lmt the hange of the fuzzness of the onluson wth respet to the hange of the fuzzness of the observaton, only that they have to hange n the same dreton (nreasng or dereasng). The monotonty property s orgnally ntrodued n [5], where t had no partular reason save that the author proves that the method proposed n that paper satsfy ths property. To have some gudelne about the extreme ases of the fuzzness, some fuzzy rule nterpolaton methods also holds the property, that n ase of a rsp observaton, t has to have rsp onluson. But ths ondton s usually not appearng among the axomat approah of the fuzzy nterpolaton, exept n [6] as Property 9. : The fuzzness of the onluson: mappng of a FRI s allowed to generate sngleton onluson n any of the next two ases: a) Whenever the observaton * s sngleton, then the onluson ( * ) should be sngleton as well. b) Whenever all B, where denotes ndes of rules that ontrbutes to the alulaton of onluson ( * ) and the observaton * are sngleton, then ( * ) should also be sngleton. Both the axomat approahes of the fuzzy nterpolaton,.e. the monotonty property and fuzzness of the onluson property are gvng some gudelne for handlng the fuzzness of the onluson n ase of lak of nformaton about the hange of the fuzzness. But they are reatng unneessary restrtons for the reasonng f the nterpolaton of the fuzzness an be delberately handled. V. EXPRESSING THE FUZZINESS CHNGE To be able to express the hange of the fuzzness n a fuzzy rule, the fuzzy dot representaton of the fuzzy rules has to be extended by the ablty of handlng the fuzzness hange n the loaton of the fuzzy dot. In ths paper for ths purpose a smple way s suggested. The onept of the sngle fuzzy dot rule representaton has to be extended to two overlappng fuzzy rules, to the double fuzzy dot rule representaton. Ths ase a sngle fuzzy rule s bult upon two fuzzy rules, where n both the overlappng rules the anteedent and onsequent fuzzy sets have the same referene pont, but they an have dfferent fuzzness. For smplfyng the explanaton and vsualzaton of the man onept, n the further part of the paper we restrt the shape of the fuzzy sets to normal soseles trangular shaped membershp funtons (nludng the sngleton). Ths restrton s a loss of generalty, but serves for the better demonstraton purposes, as the goal of ths paper s to ntrodue the man double fuzzy dot onept wthout solvng the omplex and general problem of arbtrary shaped fuzzy sets. In ase of normal soseles trangular shaped fuzzy sets the referene pont s a sngle pont, the ore: µ ore( ) = 1 Ths ase the th rule of the double fuzzy dot rule representaton has the followng form: p, q p, q B, 1108

4 p p whh s bult upon two overlappng fuzzy rules: B q q p p q q and B, where ( ) = B and ( ) = B, where n both the overlappng rules the anteedent and the onsequent p q fuzzy sets have the same referene pont: ore ( ) = ore( ), p q ore ( B ) = ore( B ), = 1,..., n, but they an have dfferent fuzzness (see e.g. on Fg. 1. and Fg..). For demonstratve purposes n ths paper to be the frst example the extenson of the fuzzy sngle rule reasonng (the revson funton n GM [4]) was hosen. Ths ase the sngle rule s a double fuzzy dot rule p, q p, q p p q q B, where ( ) =B, ( ) =B, and the =. onluson s ganed by the nterpolatng funton: ( *) B* The nterpolatng funton ould be defned based on the lose X, =, lose Y ( B, B) = 1: loseness relaton ( ) 1 The loser the observaton * to one of the element of the double fuzzy dot anteedent j, the loser s the onluson B * to the orrespondng onsequent B j : X j j ( ) lose ( B B ), Y * lose *,. Wthout more detaled alulatons, the onlusons of some relevant double fuzzy dot rule onfguratons are ntrodued on Fg Fg.. Fgure 1. Double fuzzy dot rule representaton when the dreton of the fuzzness hange s remanng the same. Fgure 3. Sngle Rule Reasonng Double fuzzy dot rule representaton when p, B p are fuzzy and q, B q are sngletons. Fgure. Double fuzzy dot rule representaton when the dreton of the fuzzness hange s reversng. Please note, that n some of the exstng FRI methods the double fuzzy dot rule representaton n the rule base means ontradtve rules. (Rules havng same (.e. hghly overlappng) anteedents and dfferent onsequents. VI. EXMPLE: SINGLE RULE RESONING Most of the exstng FRI methods followng the delaratve way of the fuzzy funton nterpolaton ould be extended wth the apablty of nterpolaton of the fuzzness by ntrodung the double fuzzy dot rule representaton. Fgure 4. Sngle Rule Reasonng Double fuzzy dot rule representaton when p, B p, B q are fuzzy and q s sngleton. 1109

5 Fgure 5. Sngle Rule Reasonng Double fuzzy dot rule representaton when p, B p and q, B q are fuzzy. Fgure 8. Sngle Rule Reasonng Double fuzzy dot rule representaton when the dreton of the fuzzness hange s reversng and q, B p are sngletons. Fgure 6. Sngle Rule Reasonng Double fuzzy dot rule representaton when p, B p, q are fuzzy and B q s sngleton. VII. EXMPLE: DOUBLE-LINER FUZZY INTERPOLTION For havng a more omplex example, n the followngs, the smplfed verson of the sngle dmensonal double-lnear fuzzy nterpolaton method s extended by the ablty of nterpolaton of the fuzzness. The smplfaton of the example s a restrton of the orgnal method from handlng trapezodal to trangular shaped fuzzy sets only. Ths method has a very smple way for vsualzaton of the fuzzy nterpolaton and hene sutable for the demonstraton purposes of ths paper. The double-lnear fuzzy nterpolaton method was orgnally ntrodued by Detynek et al. n [9]. The method was defned for sngle dmensonal anteedent unverse only, where all fuzzy sets n the rule-base are normal and onvex trapezodal shaped fuzzy sets defned on the unverse of R real numbers. The rule-base onssts of two rules 1 B and 1 B, where, are fuzzy sets of 1 L X, X R, and B,B 1 are fuzzy sets of L Y, Y R. The rule-base s sparse n the sense that the support sets of 1 and are dsjont and 1, where s a preedene relaton. The preedene relaton s also exsts among B,B. 1 Modfyng the equatons of [9] aordng to trangular shaped fuzzy sets, B, havng a trangular observaton * we get the followng lnear equaton for the B * onluson (see Fg. 9 for notaton): ( ) = α α (1) * ( ) B = α α () B* B1 + 1 Fgure 7. Sngle Rule Reasonng Double fuzzy dot rule representaton when the dreton of the fuzzness hange s reversng. 1110

6 * α = (3) 1 = (4) * * 1 B* B1 + B 1 1 where B * s the ore of the onluson. unertantes of the observaton *L, to the newly nterpolated rule anteedent el must be the same as the rate of the unertantes of the newly nterpolated rule onsequent B el and the onluson B *L we are lookng for (see e.g. on Fg. 10): B L L = * * el B Please note, that n onept the double-lnear fuzzy nterpolaton method s also very smlar to the two step nterpolaton methodology ntrodued n [4] (see also on Fg. 11). el (7) Fgure 9. Notaton for the 1, and * trangular shaped rule anteeden fuzzy sets and observaton (the notaton s the same for the onsequent sets and onluson B 1, B and B *) For the left el, B el (and rght el, B er ) unertantes of the nterpolated rule e B e we get smlar lnear equatons (see Fg. 10 for notaton): ( ) L = α α (5) el 1L + 1 ( ) B L = α α (6) BeL B1L + 1 Fgure 11. Double-lnear nterpolaton of two fuzzy rules 1 B 1, B, the nterpolated rule e B e and the onluson * B * To extend the double-lnear fuzzy nterpolaton method to be able to handle the nterpolaton of the fuzzness by the double fuzzy dot rule representaton s very smple. The man onept s remanng the same. There are no hanges n the use of Eq. (1) (6), exept the nterpolated fuzzy rule must be alulated separately for the p B p and the q B q rulebases. s a result we get the double fuzzy dot representaton of the nterpolated fuzzy rule pq e B pq e (see e.g. on Fg. 13). For alulatng the left B *L (and rght B *R ) unertantes of the onluson we an follow the onept of the orgnal method (Eq. (7)), save ths ase we have to keep the relatve unertantes of the observaton to the two newly reated rules,.e. the rate of the unertantes of the observaton *L, to the two newly nterpolated rule anteedents p el and q el must be the same as the rate of the unertantes of the two newly nterpolated rule onsequents B p el and B q el and the onluson B *L we are lookng for (see e.g. on Fg. 1, Fg.13 and Fg.14): Fgure 10. Calulatng the left el, B el unertantes of the nterpolated rule e B e and the onluson * B * The left B *L (and rght B *R ) unertantes of the onluson an be alulated by keepng the relatve unertantes of the observaton to the newly reated rule,.e. the rate of the B p el * L q * L * L = B p q el+ p el el el q el q el B p el (8) 1111

7 Fgure 14. The nterpolated rule p,q e B p,q e and the onluson * B * n ase of double fuzzy dot rule representaton. Fgure 1. Calulatng the left p,q el, B p,q el unertantes of the nterpolated rule p,q e B p,q e and the onluson * B * n ase of double fuzzy dot rule representaton. VIII. CONCLUSION The double fuzzy dot rule representaton proposed n ths paper ntrodues a novel approah for extendng FRI methods n ways that are able to nterpolate arbtrary hanges of fuzzness n the onluson. The nterpolated fuzzness adds a new dmenson for amelorated fuzzy funton defntons based on FRI methods. CKNOWLEDGMENT Ths researh was supported by the Hungaran Natonal Sentf Researh Fund grant no: OTK K Ths researh was arred out as part of the TMOP-4..1.B- 10//KONV projet wth support by the European Unon, o-fnaned by the European Soal Fund. Fgure 13. Double-lnear nterpolaton of two fuzzy rules p,q 1 B p,q 1, p,q B p,q and the nterpolated rule p,q e B p,q e n ase of double fuzzy dot rule representaton. REFERENCES [1] I. Perfleva, Fuzzy funton as an approxmate soluton to a system of fuzzy relaton equatons, Fuzzy Sets and Systems 147, 004, pp [] I. Perfleva, et al., Interpolaton of fuzzy data: nalytal approah and overvew, Fuzzy Sets and Systems, 010, do: /j.fss [3] E. H. Mamdan and S. sslan, n experment n lngust synthess wth a fuzzy log ontroller, Int. J. of Man Mahne Studes, (7), 1975, pp [4] P. Barany, L. T. Kózy, and T. D. Gedeon, Generalzed Conept for Fuzzy Rule Interpolaton, IEEE Trans. on Fuzzy Systems, vol. 1, No. 6, 004, pp [5] S. Jene, Interpolatng and extrapolatng fuzzy quanttes revsted an axomat approah, Soft Comput., vol. 5., 001, pp [6] D. Tkk, Z. C. Johanyák, Sz. Kovás, K. W. Wong, Fuzzy Rule Interpolaton and Extrapolaton Tehnques: Crtera and Evaluaton Gudelnes, Journal of dvaned Computatonal Intellgene and Intellgent Informats, ISSN , Vol.15, No.3, 011, pp [7] L. T. Kózy and K. Hrota, Rule nterpolaton by α-level sets n fuzzy approxmate reasonng, In J. BUSEFL, utomne, UR-CNRS. Vol. 46. Toulouse, Frane, 1991, pp [8] L. T. Kózy and Sz. Kovás, On the preservaton of the onvexty and peewse lnearty n lnear fuzzy rule nterpolaton, Tokyo Inst. Tehnol., Yokohama, Japan, Teh. Rep. TR 93-94/40, LIFE Char Fuzzy Theory, [9] L. T. Kózy, and Sz. Kovás, Shape of the Fuzzy Conluson Generated by Lnear Interpolaton n Trapezodal Fuzzy Rule Bases, n Proeedngs of the nd European Congress on Intellgent Tehnques and Soft Computng, ahen, 1994, pp

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