Genetic learning of accurate and compact fuzzy rule based systems based on the 2-tuples linguistic representation q

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1 Internatonal Journal of Approxmate Reasonng 44 (2007) Genetc learnng of accurate and compact fuzzy rule based systems based on the 2-tuples lngustc representaton q Rafael Alcalá a, *, Jesús Alcalá-Fdez a, Francsco Herrera a, José Otero b a Department of Computer Scence and Artfcal Intellgence, Unversty of Granada, Granada, Span b Department of Computer Scence, Unversty of Ovedo, Campus de Vesques, Gjón, Span Receved 19 July 2005; receved n revsed form 12 January 2006; accepted 6 February 2006 Avalable onlne 24 July 2006 Abstract One of the problems that focus the research n the lngustc fuzzy modelng area s the trade-off between nterpretablty and accuracy. To deal wth ths problem, dfferent approaches can be found n the lterature. Recently, a new lngustc rule representaton model was presented to perform a genetc lateral tunng of membershp functons. It s based on the lngustc 2-tuples representaton that allows the lateral dsplacement of a label consderng an unque parameter. Ths way to work nvolves a reducton of the search space that eases the dervaton of optmal models and therefore, mproves the mentoned trade-off. Based on the 2-tuples rule representaton, ths work proposes a new method to obtan lngustc fuzzy systems by means of an evolutonary learnng of the data base apror(number of labels and lateral dsplacements) and a smple rule generaton method to quckly learn the assocated rule base. Snce ths rule generaton method s run from each data base defnton generated by the evolutonary algorthm, ts selecton s an mportant aspect. In ths work, we also propose two new ad hoc data-drven rule generaton methods, analyzng the nfluence of them and other rule generaton methods n the proposed learnng approach. The developed algorthms wll be tested consderng two dfferent real-world problems. Ó 2006 Elsever Inc. All rghts reserved. q Supported by the Spansh Mnstry of Scence and Technology under Projects TIC C05-01 and TIN C * Correspondng author. E-mal addresses: alcala@decsa.ugr.es (R. Alcalá), jalcala@decsa.ugr.es (J. Alcalá-Fdez), herrera@decsa. ugr.es (F. Herrera), jotero@ls.unov.es (J. Otero) X/$ - see front matter Ó 2006 Elsever Inc. All rghts reserved. do: /j.jar

2 46 R. Alcalá et al. / Internat. J. Approx. Reason. 44 (2007) Keywords: Fuzzy rule-based systems; Lngustc 2-tuples representaton; Learnng; Interpretablty accuracy trade-off; Genetc algorthms 1. Introducton One of the problems assocated to lngustc fuzzy modelng (FM), modelng of systems buldng a lngustc model clearly nterpretable by human bengs, s ts lack of accuracy when modelng some complex systems. It s due to the nflexblty of the concept of lngustc varable, whch mposes hard restrctons to the fuzzy rule structure [1]. Ths drawback leads lngustc FM to sometmes move away from the desred trade-off between nterpretablty and accuracy, thus losng the usefulness of the model fnally obtaned. Many dfferent possbltes to mprove the accuracy of lngustc FM whle preservng ts ntrnsc nterpretablty have been consdered n the specalzed lterature [2,3]. These approaches try to nduce a better cooperaton among the rules by actng on one or two dfferent model components: the data base (DB) contanng the parameters of the lngustc parttons and the rule base (RB) contanng the set of rules. An effcent way to do that s to obtan the whole knowledge base (KB) RB and DB by learnng the DB a pror [4 10],.e., consderng a process that learns the DB and wraps a smple method to derve a set of rules for each DB defnton. Most of the works based on these knds of learnng use genetc algorthms (GAs) for the learnng of the DB parameters. In fact, the automatc defnton of fuzzy systems can be consdered as an optmzaton or search process and nowadays, evolutonary algorthms, partcularly GAs, are consdered as the more known and used global search technque. Moreover, the genetc codng that they use allow them to nclude pror knowledge to lead the search up. For ths reason, evolutonary algorthms have been successfully appled to learn fuzzy systems n the last years, gvng way to the appearance of the so called genetc fuzzy systems [11,12]. On the other hand, to ease the genetc optmzaton of the DB parameters a new lngustc rule representaton model was presented n [13]. It s based on the lngustc 2-tuples representaton [14] that allows the lateral dsplacement of a label consderng an unque parameter. Ths way to work nvolves a reducton of the search space that eases the dervaton of optmal models. In ths work, we propose a new method to obtan whole KBs by means of an evolutonary learnng of the DB a pror that s based on the lngustc 2-tuples rule representaton [14]. Ths method conssts of an evolutonary process that learns the optmal number of labels per varable and the lateral dsplacement of such labels. For each DB defnton generated by the evolutonary algorthm, a quck rule generaton process s run to obtan the RB. Addtonally, n order to mprove the generalzaton ablty of the models so obtaned we propose a new nference system consderng non-covered nput examples. Ths way to work, makes the selecton of the rule generaton process become an mportant aspect. A prelmnary study of the proposed technque was presented n [15] consderng the Wang and Mendel s (WM) algorthm [16] as a frst approach for the rule dervaton. To perform a better study, we also propose two new ad hoc data-drven rule generaton methods n ths contrbuton, analyzng ther nfluence n the proposed KB learnng technque. Furthermore, these methods wll be analyzed by solvng two realworld problems from both, the accuracy and the nterpretablty pont of vew.

3 R. Alcalá et al. / Internat. J. Approx. Reason. 44 (2007) Ths contrbuton s arranged as follows. The next secton descrbes the lngustc rule representaton model based on the lngustc 2-tuples and proposes the new nference system. Secton 3 ntroduces the learnng scheme consdered n ths work and proposes the new evolutonary learnng algorthm to obtan whole KBs. Secton 4 presents two new ad hoc data-drven rule generaton methods and explans how they can be ntegrated n the proposed evolutonary algorthm. Secton 5 shows an expermental study consderng two dfferent real-world problems. Fnally, Secton 6 ponts out some concludng remarks. 2. Rule representaton based on the lngustc 2-tuples In [13], a new model of tunng of fuzzy rule-based systems (FRBSs) was proposed consderng the lngustc 2-tuples representaton scheme ntroduced n [14], whch allows the lateral dsplacement of the support of a label and mantans a good nterpretablty assocated to the obtaned lngustc FRBSs. Ths tunng proposal also ntroduces a new model for rule representaton based on the concept of symbolc translaton [14] (the lateral dsplacement of a label). Respect to the classcal tunng [11,17 23], usually consderng three parameters n the case of trangular membershp functons (MFs), ths way to work nvolves a reducton of the search space that eases the dervaton of optmal models, preservng the orgnal shape of the MFs. The followng subsectons present the concept of symbolc translaton, the lngustc 2- tuples rule representaton and the new nference system proposed n ths work to consder non-covered nput examples The symbolc translaton of a label The symbolc translaton of a lngustc term s a number wthn the nterval [ 0.5, 0.5) that expresses the doman of a label when t s movng between ts two lateral labels. Let us consder a set of labels S representng a fuzzy partton. Formally, we have the par, ðs ; a Þ; s 2 S; a 2½ 0:5; 0:5Þ: Fg. 1 depcts the symbolc translaton of a label represented by the par (S 2, 0.3), consderng a set S wth fve lngustc terms represented by ther ordnal values ({0, 1,2,3,4}). Actually, the symbolc translaton of a label nvolves the lateral dsplacement of the MF that represents such label. As an example, Fg. 2 shows the lateral dsplacement of the label M. The MF of the new label y 2 s located between S and M, beng stll closer to M S 0 S 1 S 2 S 3 S (S,-0.3) 2 Fg. 1. Symbolc translaton of a label.

4 48 R. Alcalá et al. / Internat. J. Approx. Reason. 44 (2007) ES α = -0.5 VS S y 2 M L VL EL ES VS S y 2 M L VL EL Fg. 2. Lateral dsplacement of the lngustc label M consderng the set of labels S = {ES,VS,S,M,L,VL,EL} Rule representaton In [14], both the lngustc 2-tuples representaton model and the needed elements for lngustc nformaton comparson and aggregaton are presented and appled to the decson makng framework. In the context of the FRBSs, we are gong to see ts use n the lngustc rule representaton. In the next we present ths approach consderng a smple control problem. Let us consder a control problem wth two nput varables, one output varable and a DB defned from experts determnng the MFs for the followng labels: Error; rerror!fn; Z; Pg; Power!fL; M; Hg: Based on ths DB defnton, an example of classcal rule and lngustc 2-tuples represented rule s: Classcal rule, If error s Zero and $Error s postve then Power s hgh. Rule wth 2-tuples representaton, If error s (Zero,0.3) and $Error s (Postve, 0.2) then Power s (Hgh, 0.1). In [13], two dfferent rule representaton approaches were proposed, a global approach and a local approach. In our partcular case, the learnng s appled to the level of lngustc parttons (global approach). In ths way, the par (X,label) takes the same a value n all the rules where t s consdered,.e., a global collecton of 2-tuples s consdered by all the fuzzy rules. For example, X s (Hgh, 0.3) wll present the same value for those rules n whch the par X s Hgh was ntally consdered. The man achevement s that, snce the three parameters usually consdered per label [11,17 23] are reduced to only one symbolc translaton parameter, ths proposal decreases the learnng problem complexty easng ndeed the dervaton of optmal models. Other mportant ssue s that, the learnng of the dsplacement parameters keeps the orgnal shape of the MFs (n our case trangular and symmetrcal). In ths way, from the parameters a appled to each label, we could obtan the equvalent trangular MFs, by whch a FRBS based on lngustc 2-tuples could be represented as a classcal Mamdan FRBS [24,25] A new fuzzy nference system Once the 2-tuples represented model s transformed to ts equvalent classcal Mamdan FRBS (obtanng the dsplaced MFs from the learned 2-tuples), a classcal fuzzy reasonng

5 R. Alcalá et al. / Internat. J. Approx. Reason. 44 (2007) could be consdered. In our case, the fuzzy reasonng method s the mnmum t-norm playng the role of the mplcaton and conjunctve operators, and the center of gravty weghted by the matchng strategy actng as defuzzfcaton operator [26] (FITA scheme). However, snce we are searchng for models wth the smallest possble number of rules (compact lngustc models) and the support of the fnal MFs comprsng that rules can be dsplaced, there could be non-covered zones n the nput space. Takng nto account that the learnng algorthm s based by error measures, ths fact should not be a problem (noncovered tranng data usually provokes hgh errors n the system and fnally they would be covered). However, a good behavor of the obtaned model s not ensured for the non-covered test data (.e., the generalzaton of the fnal lngustc model could not be good for uncovered nputs). In ths way, to consder non-covered nput data for the system output computaton, the followng mechansm s appled when non-covered ponts are found: (1) The nearest rule to the non-covered pont s dentfed (normalzed eucldean dstance to the vertex of the labels). The non-covered coordnates of the pont are set to the value of the vertex of the correspondng label. (2) The second nearest rule s dentfed. Then, f the consequent labels of both rules present overlappng to some degree, we only nfer wth the nearest rule snce t wll be the most representatve n a subspace that does not present strong changes n the output doman. (3) In other case, the fnal FRBS output should be obtaned by nterpolaton of both rules, snce strong changes are detected n ths subspace output doman. To do that, the coordnates of the pont that are ntally covered are dsplaced towards the second rule, ensurng a mnmum coverng degree of the nearest rule (nearng these coordnates to the correspondng label extreme at the 10% of the support sze). As an example, let e be a coordnate of the non-covered pont e that s ntally covered by the correspondng label of the nearest rule fa 1st ; b 1st ; c 1st g (left extreme, vertex and rght extreme). And let fa 2nd ; b 2nd ; c 2nd g be the defnton ponts of the correspondng label of the second nearest rule. Then, the new value e 0 s computed as follows: 8 >< a 1st þðc 1st a 1st Þ0:1; If b 2nd < b 1st ; e 0 ¼ c 1st ðc 1st a 1st Þ0:1; If b 2nd > b 1st ; >: e ; If b 2nd ¼ b 1st : (4) Fnally, we nfer wth the new nput values consderng the whole RB. 3. Evolutonary algorthm for learnng of the knowledge base Ths secton presents the learnng scheme and the specfc evolutonary algorthm proposed n ths work to obtan whole KBs based on the lngustc 2-tuples rule representaton KB dervaton by learnng the DB a pror As sad, an effcent way to generate the whole KB of a FRBS conssts of obtanng the DB and the RB separately, based on the DB learnng a pror [4 10]. Ths way to work allows us to learn the most adequate context [5,8] for each fuzzy partton, whch s

6 50 R. Alcalá et al. / Internat. J. Approx. Reason. 44 (2007) Learnng Process GA Rule Learnng Process Ad-hoc Evaluaton Module (DB+RB) DB RB Fg. 3. Learnng scheme of the KB. necessary n dfferent contextual stuatons (dfferent applcatons) and for dfferent fuzzy rule extracton models. Although dfferent optmzaton technques could be consdered for the learnng of the DB parameters a pror, n ths work, we consder an evolutonary algorthm for ths task. In ths way, the learnng scheme consdered for the learnng of whole KBs s comprsed of two man components (see Fg. 3): An evolutonary process to learn the DB, whch allows to defne: The number of labels for each lngustc varable. The lateral dsplacements of such labels. A quck ad hoc data-drven method to derve a RB from each DB defnton generated by the evolutonary process. In ths way, the cooperatve acton of both components allows to fnally obtan the whole defnton of the KB (DB and RB). The smple WM algorthm [16] wll be consdered for ths task as a frst approach Evolutonary algorthm (the CHC approach) Evolutonary algorthms n general and, GAs n partcular, has been wdely used to derve FRBSs. In ths work, we wll consder the use of a specfc GA to desgn the proposed learnng method, the CHC [27] algorthm. The CHC algorthm s a GA that presents a good trade-off between exploraton and explotaton, beng a good choce n problems wth complex search spaces. Ths genetc model makes use of a mechansm of selecton of populatons. M parents and ther correspondng offsprng are put together to select the best M ndvduals to take part of the next populaton (wth M beng the populaton sze). To provoke dversty n the populaton the CHC approach makes use of an ncest preventon mechansm and a restartng approach, nstead of the well-known mutaton operator. Ths ncest preventon mechansm s consdered n order to apply the crossover operator,.e., two parents are crossed f ther dstance (consderng an adequate metrc) dvded by two s over a predetermned threshold, L. Ths threshold value s ntalzed as the maxmum possble dstance between two ndvduals dvded by four. Followng the orgnal CHC scheme, L s decremented by one when there s no new ndvduals n the populaton n one generaton. Furthermore, the algorthm restarts the populaton when L s below zero.

7 R. Alcalá et al. / Internat. J. Approx. Reason. 44 (2007) Intalze populaton and THRESHOLD Crossover of M parents Evaluaton of the New ndvduals KB DB WM (RB) no Selecton of the best M ndvduals Restart the populaton and THRESHOLD THRESHOLD < 0.0 yes If NO new ndvduals, decrement THRESHOLD Fg. 4. Scheme of the algorthm consderng the CHC approach. Consderng the learnng scheme proposed n the prevous subsecton, the CHC algorthm have to defne both, the granularty of the lngustc parttons and the lateral dsplacements of the nvolved labels. A global scheme of the proposed algorthm consderng the CHC approach s shown n Fg. 4. In the followng, the components needed to desgn ths process are explaned. They are: DB codfcaton, chromosome evaluaton, ntal gene pool, crossover operator (together wth the consdered ncest preventon) and restartng approach DB codfcaton A double codng scheme (C = C 1 + C 2 ) to represent both parts, granularty and translaton parameters, s consdered: Number of labels (C 1 ): Ths part s a vector of nteger numbers wth sze N (beng N the number of system varables). The possble numbers of labels depend on the problem beng solved and are establshed by the system expert for each varable (usually the set {3,...,9} for the N varables): C 1 ¼ðL 1 ;...; L N Þ: Lateral dsplacements (C 2 ): Ths part s a vector of real numbers wth sze N * 9(N varables wth a maxmum of nne lngustc labels per varable) n whch the dsplacements of the dfferent labels are coded for each varable. Of course, f a chromosome does not have the maxmum number of labels n one of the varables, the space reserved for the values of these labels s gnored n the evaluaton process. In ths way, the C 2 part has the followng structure (where each gene s the tunng value of the correspondng label): C 2 ¼ a 1 1 ;...; a1 L 1;...; an 1 ;...; an L : N 3.4. Chromosome evaluaton As sad, to evaluate a determned chromosome we wll apply the well-known rule generaton method of Wang and Mendel [16] on the DB coded by such chromosome. To decode ths DB, strong fuzzy parttons are defned consderng the granularty values of C 1. After that, each MF s dsplaced to ts new poston consderng the dsplacement

8 52 R. Alcalá et al. / Internat. J. Approx. Reason. 44 (2007) values of C 2. Once the whole KB s obtaned and usng the nference system presented n Secton 2.3, the mean square error (MSE) s computed and the followng functon s mnmzed: F C ¼ w 1 MSE þ w 2 NR; where, NR s the number of rules of the obtaned KB (to penalze a large number of rules), w 1 = 1 and w 2 s computed from the MSE and the number of rules of the KB generated from a DB consderng the maxmum number of labels (usually 9) and wthout consderng the dsplacement parameters, w 2 ¼ a ðmse max-lab =NR max-lab Þ wth a beng a weghtng percentage gven by the system expert that determnes the tradeoff between accuracy and complexty. Values hgher than 1.0 search for lngustc models wth few rules, and values lower than 1.0 search for lngustc models wth hgh accuracy. A good neutral choce s for example 1.0 (good accuracy and not too many rules) Intal gene pool The ntal populaton wll be comprsed of two dfferent parts (wth the same number of chromosomes): In the frst part, each chromosome has the same random number of labels for all the system varables, settng all the translaton parameters to zero. In the second part, the only change s that each varable could have a dfferent number of labels. Snce CHC has no mutaton operator, the translaton parameters reman unchanged and the most promsng number of labels s obtaned for each lngustc varable. The algorthm works n ths way untl the frst restartng s reached Crossover operator Two dfferent crossover operators are consdered dependng on the two parent s scope to obtan two offsprng: When the parents encode dfferent granularty levels n any varable, a crossover pont s randomly generated n C 1 and the classcal crossover operator s appled on ths pont n both parts, C 1 and C 2 (exploraton). When both parents have the same granularty level per varable, an operator based on the concept of envronments (the offsprng are generated around one parent) s appled only on the C 2 part (explotaton). These knds of operators present a good cooperaton when they are ntroduced wthn evolutonary models forcng the convergence by pressure on the offsprng (as the case of CHC). Partcularly, we consder the Parent Centrc BLX (PCBLX) operator [28], whch s based on the BLX-a. Fg. 5 depcts the behavor of these knds of operators.

9 R. Alcalá et al. / Internat. J. Approx. Reason. 44 (2007) Fg. 5. Scheme of the behavor of the BLX and PCBLX operators. The PCBLX s descrbed as follows. Let us assume that X =(x 1 x n )andy =(y 1 y n ), ðx ; y 2½a ; b ŠR; ¼ 1;...; nþ, are two real-coded chromosomes that are gong to be crossed. The PCBLX operator generates the two followng offsprng: O 1 =(o 11 o 1n ), where o 1 s a randomly (unformly) chosen number from the nterval ½l 1 ; u1 Š, wth l1 ¼ maxfa ; x I g, u 1 ¼ mnfb ; x þ I g,andi = jx y j. O 2 =(o 21 o 2n ), where o 2 s a randomly (unformly) chosen number from the nterval ½l 2 ; u2 Š, wth l2 ¼ maxfa ; y I g and u 2 ¼ mnfb ; y þ I g. On the other hand, the ncest preventon mechansm wll be only consdered n order to apply the PCBLX operator. In our case, two parents are crossed f ther hammng dstance dvded by 2 s over a predetermned threshold, L. Snce we consder a real codng scheme (the C 2 part s gong to be crossed), we have to transform each gene consderng a Gray Code (bnary code) wth a fxed number of bts per gene (BITSGENE), that s determned by the system expert. In ths way, the threshold value s ntalzed as: L ¼ð#GenesC 2 BITSGENEÞ=4:0: Followng the orgnal CHC scheme, L s decremented by one when there are no new ndvduals n the next generaton. In order to avod very slow convergence, n our case, L wll be also decremented by one when no mprovement s acheved respect to the best chromosome of the prevous generaton Restartng approach Snce no mutaton s performed, to get away from local optma a restartng mechansm s consdered [27] when the threshold value L s lower than zero. In ths case, all the chromosomes set up ther C 1 parts to that of the best global soluton, beng the parameters of ther C 2 parts generated at random wthn the nterval [ 0.5,0.5). Moreover, f the best global soluton had any change from the last restartng pont, ths s ncluded n the populaton (the explotaton only contnues whle there s convergence). Ths operaton mode was ntally proposed by the CHC authors as a possblty to mprove the algorthm performance when t s appled to solve some knds of problems [27]. 4. Two new ad hoc data-drven rule generaton methods and ther ntegraton n the evolutonary learnng of the DB a pror As sad, the selecton of the method consdered for rule generaton n the learnng of the DB a pror becomes an mportant aspect. Ths method should allow to the learnng

10 54 R. Alcalá et al. / Internat. J. Approx. Reason. 44 (2007) process to obtan accurate and, at the same tme, compact KBs. Furthermore, snce ths method s run each tme a DB s evaluated, ts computaton tme must be as short as possble. In ths secton, we dscuss about the knds of methods that could favor ths behavor, proposng two new ad hoc data-drven methods specfcally desgned for ths task. We can dstngush between two man possbltes to select ths method: (1) The frst possblty s the use of advanced methods to obtan rules wth the best accuracy. In [29], the authors analyzed dfferent ad hoc data-drven methods to propose a new approach called mxed method (MM) that presents a better approxmaton ablty. It s based on the combnaton of a method guded by examples (the WM [16] algorthm) and a method guded by fuzzy grd (the nput space orented strategy, ISS [30]), and conssts of addng rules to the lngustc model obtaned by WM n the fuzzy nput subspaces that havng examples do not stll have a rule. Although at frst, ths approach could seem a good choce, the use of these knds of advanced methods wthn our learnng approach presents some mportant drawbacks that should be taken nto account. On the one hand, the computatonal tme needed by these methods s hgher than that of smpler methods. Moreover, the accuracy mprovement obtaned by a more sophstcated approach s often acheved by ncreasng the fnal number of rules (less nterpretable models). On the other hand, some studes [4,5,8] have shown that the system performance s much more senstve to the learnng of the DB than to the composton of the RB. In ths way, t s not clear that the dervaton of a more elaborated RB favors the learnng of better DB defntons respect to other smpler RBs, snce the RBs obtaned could askew the learnng of optmal DBs. (2) The second possblty s the use of smpler and faster algorthms that favors the learnng of the MFs. These knds of methods quckly obtan a small set of basc rules based on the examples wth the best coverng degree n each fuzzy subspace. Therefore, the qualty of the obtaned rules drectly depends on a successful DB defnton to well cover the examples that better represent the system behavor. Ths way to work leads the DB learnng a pror to obtan more optmal DBs and smpler RBs,.e., more accurate and compact KBs. Furthermore, the dervaton of smpler models s a way to reduce the overfttng, whch eases the dervaton of models also presentng a good generalzaton ablty [31]. For these reasons, a basc and smple algorthm as WM performs so well when t s ntegrated n a method based on the a pror learnng of the DB [4,5]. Snce our man am s the learnng of accurate but also compact FRBSs and the computatonal tme s also an mportant factor, we wll focus our attenton on methods fttng wth the second possblty,.e., smple methods that favors and gude the learnng of the MFs. An example of these knds of methods s the WM algorthm, consdered n the prevous secton as a frst approach to derve the RB. In the followng subsectons, we propose two new smple ad hoc data-drven methods that allow the dervaton of smpler models mantanng the same or a smlar accuracy. They are based on the selecton of more general consequents consderng a group of the best covered examples and not only the one wth the best coverng degree. The use of more general consequents also mproves the generalzaton ablty of the models so obtaned, reducng the effect of nose ponts. In any case, for the experments and wth comparatve purposes, we wll also consder the MM algorthm by drectly replacng the WM algorthm n the method proposed n Secton 3.

11 R. Alcalá et al. / Internat. J. Approx. Reason. 44 (2007) Rule generaton method based on averaged outputs (AV algorthm) Ths method tres to obtan more general consequents by means of a weghted average of the output of the examples matchng the rule antecedents to a certan degree. The use of an averaged output decrements the nfluence of nose ponts. The method s based on the exstence of a predefned DB and a set of nput output tranng data E ={e 1,...,e l,..., e m } wth e l ¼ðx l 1 ;...; xl N 1 ; yl Þ, l 2 {1,...,m}, m beng the data set sze, and N 1 beng the number of nput varables. The RB s generated by means of the followng steps: Intally the RB s empty. For each example e l n E: (1) Generate the rule antecedent wth the labels best coverng the nput data ðx l 1 ;...; xl N 1 Þ. (2) If there s not a rule wth the same antecedent n the RB: (a) Select the examples wth a matchng degree 1 hgher than d, where d 2 [0.5,1] s a value provded by the system expert. If no examples can be selected, select all the examples covered to some degree. (b) Calculate the mean of the outputs of the selected examples weghted by ther matchng degrees, M. (c) Generate the rule consequent wth the label best coverng M. (d) Add the obtaned rule to the RB. The d parameter determnes how general or specfc are the consequents obtaned respect to the covered examples. Snce t depends on the problem beng solved, the granularty and the MFs postons, ths parameter should be obtaned together wth the DB n the evolutonary process. At the end of ths secton we explan how the proposed methods are ncluded n the evolutonary DB learnng a pror. Ths method s a bt slower than the WM algorthm snce for each rule antecedent, the matchng of all the examples must be computed Rule generaton method based on modal consequents (MO algorthm) Ths method tres to obtan more general consequents obtanng the modal labels of those proposed by the examples matchng the rule antecedents to a certan degree. Snce nose ponts usually appear wth a small frequency, these knds of ponts would not be consdered to compute the output. Ths method s also based on the exstence of a predefned DB and a set E of nput output tranng data. Ths algorthm conssts of the followng steps: Intally the RB s empty. For each example e l n E: (1) Generate the rule antecedent wth the labels best coverng the nput data ðx l 1 ;...; xl N 1 Þ. (2) If there s not a rule wth the same antecedent n the RB: 1 Usng the mnmum t-norm as conjunctve operator on the obtaned antecedent.

12 56 R. Alcalá et al. / Internat. J. Approx. Reason. 44 (2007) (a) Select the examples wth a matchng degree 1 hgher than d, where d 2 [0.5,1] s a value provded by the system expert. (b) If any example has been selected: Calculate the label best coverng the output of each selected example, countng the number of tmes that each output label s obtaned. Generate the rule consequent wth the modal output label,.e., the output label more tmes obtaned. Else: Generate the rule consequent exactly as WM (that of the rule obtaned by the example wth the hghest coverng degree on the N varables). (c) Add the obtaned rule to the RB. As n the case of the prevous method, the d parameter s obtaned together wth the DB wthn the evolutonary process. Ths method can be mplemented exactly as the WM algorthm but countng the frequency of the consequents proposed and fnally selectng the modal labels. Therefore, t s faster than the AV algorthm and very smlar to the WM algorthm Integraton of the proposed methods n the evolutonary learnng of KBs To consder these algorthms wthn the proposed approach for the DB learnng a pror, the WM algorthm s drectly replaced by the AV or the MO algorthms and the d parameter should be obtaned together wth the DB. In ths way, the method proposed n Secton 3 must nclude the learnng of the d parameter. In the followng, we wll only explan the needed changes respect to ths algorthm: Codng scheme The codng scheme s modfed by addng the new d parameter that wll be consdered to obtan the RB: C ¼ C 1 þ C 2 þ d: Intal gene pool It works n the same way, but settng the d parameters at random n [0.5,1]. Crossover Consderng the crossover operator presented n Secton 3, when the C 1 part s crossed, the d parameter s generated at random n [0.5,1]. When only the C 2 part s crossed, the PCBLX s also appled on the d parameters. Restartng approach As n the orgnal algorthm but settng up the d parameters at random n [0.5,1]. If the best global soluton had any change from the last restartng pont, ths s ncluded n the populaton consderng the d part. The d parameter s only needed to be consdered n the rule generaton process, but once the learnng process ends and a fnal 2-tuple represented KB s obtaned, ths parameter s no more needed. 5. Expermental study To evaluate the goodness of the proposed algorthms (DB learnng a pror consderng WM, AV or MO algorthms), two real-world electrcal energy dstrbuton problems [32] of dfferent complextes are consdered:

13 Estmatng the length of low voltage lnes n rural nucle. Ths problem wth only two nput varables nvolves a small search space (small complexty). However, t s stll an nterestng problem snce the system s strongly nonlnear and the avalable data s lmted to a low number of examples presentng nose. All of these drawbacks make the modelng surface complcated ndeed and, n ths case, produce a strong overfttng of the obtaned models. Estmatng the mantenance costs of medum voltage lnes n a town. Ths problem conssts of four nput varables and the avalable data set s comprsed of a representatve number of well dstrbuted examples. In ths case, the learnng methods are expected to obtan a consderable number of rules. Therefore, ths problem nvolves a larger search space (hgh complexty). To correctly solve both problems s a hard task snce, n general, methods presentng a good approxmaton ablty do not show a good generalzaton n real problems (smlar to the frst problem), snce these knds of methods can easly overft the obtaned models. In ths way, the proposed methods present a good approxmaton ablty (specally n the second problem) and at the same tme a good generalzaton ablty (specally n the frst problem). In the followng subsectons these problems are ntroduced and solved to analyze the behavor of the proposed methods Expermental set-up R. Alcalá et al. / Internat. J. Approx. Reason. 44 (2007) A bref descrpton of the studed methods s presented n the next three paragraphs (Table 1 summarzes the man characterstcs of these methods): The proposed methods are named as GLD-WM, GLD-AV and GLD-MO (presented n Sectons 3 and 4.3 respectvely). The GLD-MM s consdered for comparson purposes drectly replacng the WM algorthm by the MM method [29] n GLD-WM. The WM [16], COR [33] (wth Best Worst Ant System) and MM [29] algorthms are consdered as a smple and two advanced rule generaton methods to quckly obtan Table 1 Methods consdered for the expermental study Ref., Year Method Type of learnng [16], 1992 WM AHDD method [29], 2004 MM Improved AHDD method based on WM [33], 2005 COR Cooperatve rules by usng the BWAS algorthm [13], 2004 WM + GL Global lateral tunng from WM Methods consderng DB learnng a pror [4], 2001 Gr-MF Gr. + MF. parameters + RB by WM [5], 2001 GA-WM Gr. + Scalng factors + Domans + RB by WM [8], 2004 GA-COR Gr. + Scalng factors + Domans + RB by COR Proposed algorthms GLD-MM Gr. + Global lateral parameters + RB by MM GLD-WM Gr. + Global lateral parameters + RB by WM GLD-AV Gr. + Global lateral parameters + RB by AV GLD-MO Gr. + Global lateral parameters + RB by MO AHDD: Ad hoc data-drven BWAS: Best Worst Ant System Gr.: Granularty.

14 58 R. Alcalá et al. / Internat. J. Approx. Reason. 44 (2007) RBs from a predefned DB. We also show the results of the WM + GL tunng method [13] based on the lngustc 2-tuples representaton. All of these methods wll be consdered as a reference snce the proposed algorthms are based on some of them. The ntal lngustc parttons for these methods are comprsed by fve lngustc terms wth unformly dstrbuted trangular MFs gvng meanng to them. On the other hand, three methods to obtan a complete KB (DB learnng a pror) are consdered for comparsons. The frst one, Gr-WM [4], learns the granularty (number of labels) of the fuzzy parttons and the MFs parameters (ther three defnton ponts). GA-WM [5] and GA-COR [8] learn the granularty, scalng factors and the domans (.e., the varable doman or workng range to perform the fuzzy parttonng) for each system varable. These methods respectvely obtan the correspondng RB by means of the WM and COR algorthms. To develop the dfferent experments we consder a 5-folder cross-valdaton model,.e., fve random parttons of data 2 wth a 20%, and the combnaton of four of them (80%) as tranng and the remanng one as test. For each one of the fve data parttons, the studed methods have been run sx tmes, showng for each problem the averaged results of a total of 30 runs. Moreover, a t-test (wth 95% confdence) was appled n order to ascertan f dfferences n the performance of the proposed approaches are sgnfcant. Fnally, the followng values have been consdered for the parameters of each method: 3 50 ndvduals, 50,000 evaluatons, 30 bts per gene for the Gray codfcaton and the set {3,...,9} as possble numbers of labels n all the system varables; 0.6 and 0.2 as crossover and mutaton probabltes n the case of the Gr-MF, GA-WM and GA-COR algorthms; snce the GA-COR algorthm spends too much tme to derve the RB, the authors propose the use of only 2000 evaluatons n both problems. The a factor for the ftness functon of the GLD methods was set to 1 n both problems. Nevertheless, to obtan models wth dfferent levels of accuracy and smplcty, n the second problem (problem wth more varables and rules) we also prove wth a = Estmatng the length of low voltage lnes Ths problem conssts of relatng the length of the low voltage lne of a certan vllage (as output varable) wth the followng two nput varables: the radus of the vllage and the number of users n the vllage. A complete descrpton of ths problem can be found n [32]. To learn the dfferent system models, we are provded wth the measured lne length, the number of nhabtants and the mean dstance from the center of the town to the three furthest clents n a sample of 495 rural nucle. Fve parttons 2 consderng an 80% (396) n tranng and a 20% (99) n test are consdered for the experments. The exstng dependency of the two nput varables wth the output varable n the tranng and test data sets of one of the fve parttons s shown n Fg. 6 (notce that they present strong non-lneartes). 2 These data sets are avalable at: 3 Wth these values we have tred to ease the comparsons selectng standard common parameters that work well n most cases nstead of searchng very specfc values for each method. Moreover, we have set a large number of evaluatons n order to allow the compared algorthms to acheve an approprate convergence. No sgnfcant changes were acheved by ncreasng that number of evaluatons.

15 R. Alcalá et al. / Internat. J. Approx. Reason. 44 (2007) X 1 X 2 X 1 X Y 4000 Y 4000 Y 4000 Y (a) Tranng data (b) Test data Fg. 6. (a) (X 1,Y) and (X 2,Y) dependency n the tranng data; (b) (X 1,Y) and (X 2,Y) dependency n the test data. Table 2 Results obtaned n the lne length estmaton problem wth parameter a = 1 for the ftness functon Method #R MSE tra r tra t-test MSE tst r tst t-test h:m:s WM :00:00.01 MM :00:00.2 COR :00:04 WM + GL :01:03 Gr-MF :01:31 GA-WM :01:24 GA-COR q :37:49 GLD-MM :01:47 GLD-WM = 00:01:26 GLD-AV = 00:01:54 GLD-MO q 00:01:25 The results obtaned n ths problem by the analyzed methods are shown n Table 2, where #R stands for the number of rules, MSE tra and MSE tst respectvely for the averaged error obtaned over the tranng and test data, r for the standard devaton, h:m:s for the averaged tme of one run n an Intel Centrno (1.73 GHz, 512 MB of RAM) and where t-test represents the followng nformaton: q represents the best averaged result. + means that the best result has better behavor than the one n the correspondng row. = denotes that the results are statstcally equal accordng to the t-test. Analyzng the results presented n Table 2 we can pont out the followng conclusons: Although the GLD-based methods do not obtan the best tranng errors, the trade-off between approxmaton and generalzaton s pretty good n a problem wth nose and poor example data. Takng nto account ths fact and the hgh test errors of the remanng methods, we could state that the remanng methods overfts whle the GLD-based methods really learns the system behavor. Furthermore, GLD-WM, GLD-AV and GLD-MO obtan the models wth the least number of rules. Respect to the use of the more advanced MM method, t slghtly mproves the tranng error of WM at the cost of addng much more rules. Fortunately, the GLD approach favors the dervaton of more smple models, although GLD-MM stll presents more

16 60 R. Alcalá et al. / Internat. J. Approx. Reason. 44 (2007) rules and less generalzaton ablty than the remanng GLD-based methods. Therefore, n ths problem, the use of ths algorthm does not nvolves an advantage. We can see how the computatonal tme of the rule dervaton methods affects to the DB learnng a pror. The most clear case s that of the COR algorthm, multplyng the tme of WM per 400 and forcng the GA-COR to spend more than two hours to reach 2000 evaluatons. In the followng problem, ths fact wll be even more clear. Fg. 7 depcts one of the 30 KBs obtaned by GLD-MO n ths problem. Ths fgure shows how small varatons n the MFs lead to mportant mprovements n the behavor of the obtaned FRBSs. In ths way, the two nput varables respectvely present three and four labels whose MFs are more or less unformly dstrbuted, whch makes easy to fnd ther correspondng meanngs for an expert. The output varable presents fve labels that are balanced to the left, representng a hgher concentraton of examples wth small outputs (see Fg. 6). However, snce they are agan more or less well dstrbuted to the left and to the rght of the mddle label, we can stll easly name these labels Estmatng the mantenance costs of medum voltage lnes Ths problem conssts of relatng the mantenance costs of the medum voltage lne of a certan town (as output varable) wth the followng four nput varables: sum of the lengths of all streets n the town, total area of the town, area that s occuped by buldngs, and energy supply to the town. A complete descrpton of ths problem can be found n [32]. In ths case, we wll deal wth estmatons of mnmum mantenance costs based on a model of the optmal electrcal network for a town n a sample of 1059 towns. Fve parttons 2 consderng an 80% (847) n tranng and a 20% (212) n test are consdered for the experments. The results obtaned n ths problem by the analyzed methods are shown n Table 3 (these knds of table was descrbed n the prevous subsecton). Analyzng the results presented n Table 3 we can stress the followng facts: X1 X2 Y l1'=(l1,0.2) l2'=(l2,0.1) l3'=(l3,0.1) l1 l2 l3 l1'=(l1,-0.3) l2'=(l2,-0.3) l3'=(l3,-0.2) l4'=(l4,0.0) l1 l2 l3 l4 l1'=(l1,-0.5) l2'=(l2,-0.1) l3'=(l3,-0.2) l4'=(l4,-0.4) l5'=(l5,0.5) l1 l2 l3 l4 l5 l6 #R: 7 X1 X2 Y l1' l1' l1' l1' l2' l2' l1' l3' l3' l1' l4' l4' Labelng the fnal MFs: X1 Y l1' = Small l2' = Medum l3' = Large X2 l1' = Small l2' = Moderately Small l3' = Moderately Large l4' = Large MSE-tra: MSE-tst: X1 X2 Y l2' l2' l3' l2' l3' l5' l3' l3' l3' l1' = Very Small l2' = Small l3' = Moderately Small l4' = Medum l5' = Large Fg. 7. DB wth/wthout lateral dsplacements (black/gray), RB and dsplacements of a model obtaned by GLD + MO (the unused labels were removed from ths fgure).

17 R. Alcalá et al. / Internat. J. Approx. Reason. 44 (2007) Table 3 Results obtaned n the mantenance costs estmaton problem Method #R MSE tra r tra t-test MSE tst r tst t-test h:m:s WM :00:00.02 MM :00:00.4 COR :01:00 WM + GL :08:15 Gr-MF :07:53 GA-WM :10:26 GA-COR :45:41 Proposed methods wth parameter a = 1 n the ftness functon GLD-MM q q 14:19:02 GLD-WM = = 00:09:10 GLD-AV :22:06 GLD-MO = = 00:09:20 Proposed methods wth parameter a = 3 n the ftness functon GLD-MM :41:02 GLD-WM :09:23 GLD-AV :13:32 GLD-MO :07:55 In ths problem, the drawbacks of the use of more advanced rule dervaton methods are even more obvous. In ths case, the MM method presents sgnfcant mprovements n tranng and test respect to WM at the cost of obtanng an excessve number of rules and ncreasng the computatonal tme. Ths makes the GLD-MM method to take more than 14/12 h to obtan a model wth more than two/one hundred rules and wthout sgnfcant mprovements respect to the use of more smple models. On the other hand, although we do not consder the COR algorthm n our methods, a second analyss could be done about ts use for the DB learnng a pror, GA-COR. The man problem of COR s the long computatonal tme t takes to obtan a RB (approxmately 3000 tmes more than WM), whch makes the GA-COR algorthm to take more than one day to reach a total of 2000 evaluatons. The man achevement, of ths method respect to ts homologous, GA-WM, s the dervaton of a lngustc model wth less number of rules. It s due to the rule smplfcaton performed by COR durng the RB learnng, whch results n lngustc models wth too few rules and therefore, wth no much better accuracy. The GLD-based methods proposed n ths work show an mportant reducton of the mean squared error over the tranng and test data n a problem wth a large search space. It s due to the use of the lngustc 2-tuple representaton that reduces the search space respect to the classcal learnng of MFs, easng the dervaton of more optmal models. We must take nto account that the Gr-MF method theoretcally could obtan at least the same results than GLD-WM, snce Gr-MF learns the three defnton ponts of the MFs, beng a generalzaton of GLD-WM. GLD-AV and GLD-MO performs so well when we search for smpler models wth a smlar accuracy to those obtaned by GLD-WM or GLD-MM. Furthermore, the lngustc models so obtaned are nterpretable n a hgh level snce the orgnal shapes of the ntal MFs are mantaned. In ths way, we can hghlght the GLD-MO method because of the low number of rules, the errors and the computatonal tmes obtaned n both problems.

18 62 R. Alcalá et al. / Internat. J. Approx. Reason. 44 (2007) X1 X2 l2'=(l2,-0.5) l3'=(l3,-0.2) l4'=(l4,0.0) l1 l2 l3 l4 l1'=(l1,0.0) l2'=(l2,0.3) l3'=(l3,0.1) l1 l2 l3 l1'=(l1,0.1) l2'=(l2,0.1) l3'=(l3,0.2) l4'=(l4,0.4) l5'=(l5,0.4) l6'=(l6,0.3) #R: 33 MSE - tra:12805 MSE - tst:12381 X1 X2 X3 X4 Y X1 X2 X3 X4 l2' l1' l1' l1' l1' l3' l2' l2' l3' l2' l1' l1' l2' l2' l3' l2' l3' l1' l2' l1' l2' l1' l2' l3' l2' l3' l2' l2' l1' l2' l2' l3' l3' l2' l3' l3' l3' l1' l1' l1' l1' l3' l2' l4' l2' l3' l1' l1' l2' l2' l3' l2' l4' l3' l3' l1' l2' l1' l2' l3' l2' l5' l2' l3' l1' l2' l2' l3' l3' l2' l5' l3' l3' l1' l2' l3' l4' l4' l2' l2' l1' l3' l2' l2' l1' l2' l4' l2' l2' l2' l3' l2' l2' l2' l3' l4' l2' l2' l3' Y l4' l3' l4' l5' l5' l6' l6' l7' l3' l3' l4' X1 X2 X3 X4 l4' l2' l2' l4' l4' l2' l3' l1' l4' l2' l3' l2' l4' l2' l3' l3' l4' l2' l3' l4' l4' l2' l4' l2' l4' l2' l4' l3' l4' l3' l4' l2' l4' l3' l4' l3' l4' l3' l6' l2' l4' l3' l6' l3' Y l6' l3' l4' l5' l6' l5' l7' l5' l6' l6' l7' X3 X4 Y l1 l2 l3 l4 l5 l6 l1'=(l1,-0.2) l2'=(l2,0.1) l3'=(l3,0.1) l4'=(l4,0.4) l1 l2 l3 l4 l1'=(l1,0.1) l2'=(l2,0.1) l4'=(l4,0.1) l3'=(l3,0.1) l5'=(l5,-0.2)l6'=(l6,0.2) l7'=(l7,0.1) l1 l2 l3 l4 l5 l6 l7 Labelng the fnal MFs: X1 X4 l2' = Small l3' = Medum l4' = Large X2 l1' = Small l2' = Medum l3' = Large X3 l1' = Very Small l2' = Small l3' = Moderately Small l4' = Moderately Large l5' = Large l6' = Very Large l1' = Small l2' = Moderately Small l3' = Moderately Large l4' = Large Y l1' = Very Small l2' = Small l3' = Moderately Small l4' = Medum l5' = Moderately Large l6' = Large l7' = Very Large Fg. 8. DB wth/wthout lateral dsplacements (black/gray), RB and dsplacements of a model obtaned by GLD + MO (the unused labels were removed from ths fgure). Fg. 8 presents the KB obtaned by GLD-MO from one of the 30 runs performed n ths problem wth a = 3. Analyzng ths lngustc model, we can observe a smlar DB confguraton to that obtaned n the prevous problem. The MFs are more or less well dstrbuted whch allows us to easly gve a meanng to the correspondng labels. 6. Conclusons Ths work presents a new method for learnng KBs by means of an a pror evolutonary learnng of the DB (granularty and translaton parameters) that uses the lngustc 2-tuples rule representaton model and a new nference system. Furthermore, two new ad hoc data-drven rule generaton methods have been proposed to analyze the nfluence of them and other rule generaton methods n the proposed learnng approach. In the followng, we present our conclusons and further works: The used learnng scheme together wth the 2-tuples rule representaton model and the new nference system allows an mportant reducton of the search space that eases the dervaton of more precse and compact lngustc models. The use of smple rule dervaton methods searchng for basc rules better coverng the example data, favors the learnng of a better DB and the dervaton of RBs wth a smaller number of rules. Snce the DB learnng has more nfluence n the system behavor than the RB composton, these knds of methods also eases the dervaton of more precse and compact models.

19 R. Alcalá et al. / Internat. J. Approx. Reason. 44 (2007) Moreover, snce a global approach s consdered and the shapes of the ntal MFs are preserved, the nterpretablty of the obtaned models s mantaned to a hgh level respect to the classcal learnng of fuzzy systems. The use of dfferent a values to penalze the number of rules n the second problem has demonstrated the exstence of optmal models wth dfferent levels of accuracy and smplcty. An nterestng further work could be the use of multobjectve genetc algorthms to obtan the pareto front wth these solutons. In ths way, we could easly select a soluton wth the desred accuracy nterpretablty trade-off consderng two man objectves, the tranng error and the number of rules. References [1] A. Bastan, How to handle the flexblty of lngustc varables wth applcatons, Internatonal Journal of Uncertanty, Fuzzness and Knowledge-Based Systems 3 (4) (1994) [2] R. Alcalá, J. Alcalá-Fdez, J. Casllas, O. Cordón, F. Herrera, Hybrd learnng models to get the nterpretablty accuracy trade-off n fuzzy modelng, Soft Computng 10 (9) (2006) [3] J. Casllas, O. Cordón, F. Herrera, L. Magdalena (Eds.), Accuracy Improvements n Lngustc Fuzzy Modelng, Sprnger-Verlag, [4] O. Cordón, F. Herrera, P. Vllar, Generatng the knowledge base of a fuzzy rule-based system by the genetc learnng of the data base, IEEE Trans. Fuzzy Syst. 9 (4) (2001) [5] O. Cordón, F. Herrera, L. Magdalena, P. Vllar, A genetc learnng process for the scalng factors, granularty and contexts of the fuzzy rule-based system data base, Informaton Scences 136 (2001) [6] B. Flpc, D. Jurcc, A genetc algorthm to support learnng fuzzy control rules from examples, n: F. Herrera, J.L. Verdegay (Eds.), Genetc Algorthms and Soft Computng, Physca-Verlag, 1996, pp [7] D. Smon, Sum normal optmzaton of fuzzy membershp functons, Internatonal Journal of Uncertanty, Fuzzness and Knowledge-Based Systems 10 (4) (2002) [8] J. Casllas, O. Cordón, F. Herrera, P. Vllar, A hybrd learnng process for the knowledge base of a fuzzy rule-based system, n: Proceedngs of the 2004 Internatonal Conference on Informaton Processng and Management of Uncertanty n Knowledge-Based Systems, Peruga, Italy, 2004, vol. 3, pp [9] W. Pedrycz, Assocatons and rules n data mnng: a lnk analyss, Internatonal Journal of Intellgent Systems 19 (7) (2004) [10] Y. Teng, W. Wang, Constructng a user-frendly ga-based fuzzy system drectly from numercal data, IEEE Transactons on Systems, Man, and Cybernetcs B 34 (5) (2004) [11] O. Cordón, F. Herrera, F. Hoffmann, L. Magdalena, GENETIC FUZZY SYSTEMS. Evolutonary tunng and learnng of fuzzy knowledge bases, Advances n Fuzzy Systems Applcatons and Theory, vol. 19, World Scentfc, [12] O. Cordón, F. Gomde, F. Herrera, F. Hoffmann, L. Magdalena, Ten years of genetc fuzzy systems: current framework and new trends, Fuzzy Sets and Systems 41 (1) (2004) [13] R. Alcalá, F. Herrera, Genetc tunng on fuzzy systems based on the lngustc 2-tuples representaton, n: Proceedngs of the 2004 IEEE Internatonal Conference on Fuzzy Systems, Budapest, Hungary, 2004, vol. 1, pp [14] F. Herrera, L. Martínez, A 2-tuple fuzzy lngustc representaton model for computng wth words, IEEE Transactons on Fuzzy Systems 8 (6) (2000) [15] R. Alcalá, J. Alcalá-Fdez, F. Herrera, J. Otero, A new genetc fuzzy system based on lngustc 2-tuples to learn knowledge bases, n: Proceedngs of the Frst Internatonal Workshop on Genetc Fuzzy Systems (GFS 2005), Granada, Span, 2005, pp [16] L. Wang, J. Mendel, Generatng fuzzy rules by learnng from examples, IEEE Transactons on Systems, Man, and Cybernetcs 22 (6) (1992) [17] O. Cordón, F. Herrera, A three-stage evolutonary process for learnng descrptve and approxmatve fuzzy logc controller knowledge bases from examples, Internatonal Journal of Approxmate Reasonng 17 (4) (1997)