OPTIMIZATION OF FUZZY RULE BASES USING CONTINUOUS ANT COLONY SYSTEM

Transcription

1 Proceedng of the Frst Internatonal Conference on Modelng, Smulaton and Appled Optmzaton, Sharah, U.A.E. February -3, 005 OPTIMIZATION OF FUZZY RULE BASES USING CONTINUOUS ANT COLONY SYSTEM Had Nobahar Sharf Unversty of Technology Department of Aerospace Engneerng P.O.Box , Tehran, Iran. Sed H. Pourtadoust Sharf Unversty of Technology Department of Aerospace Engneerng P.O.Box , Tehran, Iran ABSTRACT The well-nown Ant Colony Optmzaton method s appled to the problem of tunng the parameters of a Taag-Sugeno fuzzy rule base. The latest developed Contnuous Ant Colony System algorthm s used to fnd the global soluton of the contnuous optmzaton problem. The method s systematcally tested for the approxmaton of one analytcal test functon and the performance of the proposed scheme s compared wth the Standard Genetc Algorthm.. INTRODUCTION The concept of Fuzzy Logc (FL) was proposed by Professor Lotf Zadeh n 965, at frst as a way of processng data by allowng partal set membershp rather than crsp membershp. Soon after, t was proven to be an excellent choce for many control system applcatons snce t mmcs human control logc. Today, fuzzy rule bases are wdely used to express the nowledge of operators or experts n varous domans such as decson, control, modelng, dentfcaton and so on [,]. However there are some lmtatons to ther development, the most mportant of whch s: the nowledge does not always completely exst and the manual tunng of all the base parameters taes tme. The lac of portablty of the rule bases when the dmensons of the control system change, maes the later dffculty stll more serous. To coop wth these problems, the learnng methods have been ntroduced. The frst attempt was made by Procy and Mamdan n 979, wth a "self tunng controller" [3]. The gradent descent method was used by Taag and Sugeno n 985 as a learnng tool for fuzzy modelng and dentfcaton [4]. It was used by Nomura, et al. n 99 as a self tunng method for fuzzy control [5]. The gradent descent method s approprate for smple problems and real tme learnng, snce t s fast. But t may be trapped nto local mnma. Also the calculaton of the gradents depends on the shape of membershp functons employed, the operators used for fuzzy nferences as well as the selected cost functon. So, there s a need for more general and flexble methods capable to search wthn wde soluton spaces and ntellgent enough to avod local mnma. These methods must not be lned to a gven type of applcaton and they must be applcable to several types of fuzzy rule bases and cost functons. In 998, Sarry and Guely appled the well-nown Genetc Algorthm to the problem of tunng the parameters of a Taag- Sugeno fuzzy rule base [6]. In ths artcle, we have ntegrated ths parametrc learnng problem wth our latest developed Contnuous Ant Colony System (CACS) [7], whch s based on the well-nown Ant Colony Optmzaton (ACO) [8,9]. ACO has been nspred from the ants' pheromone tral followng behavor. It can fnd the global mnmum of the problem to optmze, and t can handle any cost functon.. THE TAKAGI-SUGENO FUZZY RULE BASE Taag-Sugeno (TS) rules dffer from Mamdan rules n that ther outputs are not defned by membershp functons but by nonfuzzy analytcal functons (usually constants). Ths feature should permt to express complcated nowledge wth small number of rules. Unfortunately, TS rules are less ntutve and t s more dffcult to translate human expertse nto such rules as compared wth Mamdan rules. Therefore TS rules have been mostly used tll now for modelng purposes and buldng adaptve controllers [6]. Ther wder applcaton s becomng more common, as effectve learnng technques are devsed. In ths paper, a Taag-Sugeno fuzzy rule base was used to approxmate an m nputs analytcal functon f(x). The rule base has m nputs, one output, and s composed of n rules. The output of the fuzzy rule base s denoted by y(x). Each rule R can be wrtten as R : f ( x s g ) and...( x then y = f ( x,..., x ) m s g ) and...( x m s g where x s the -th nput, g the membershp functon of the -th rule defned on x, the output of rule R s y, and analytcal functon of the output defned for the -th rule. m ) f s the ICMSA0/05-

2 Proceedng of the Frst Internatonal Conference on Modelng, Smulaton and Appled Optmzaton, Sharah, U.A.E. February -3, 005 In ths paper, rulesr wth constant outputs w are consdered, so that y = w. But the method s applcable to other types of output functons as well. Fgure. Trangular symmetrc membershp functon The membershp functons g are defned as follows (normed trangular symmetrc, shown n Fg. ): x g = 0 a / b f x a otherwse The and operator s the multplcaton, so that µ = =, m b () g ( x ) () where µ s the actvaton degree of the rule R. The output of the rule base s calculated usng the center of gravty operator: µ w =, n f µ y x = = n 0 ( ), µ =, n 0 otherwse Let us defne the quadratc error E: E = p samp ( y( x p ) f ( x p )) p= where x p s the p-th tranng sample and p samp s the number of samples. The purpose of learnng s to mnmze the quadratc error E. mnmzaton of E s performed by tunng the a, b and w parameters. Optmzng the membershp functon parameters s a complex problem for the followng reasons:. The cost functon s not dervable everywhere.. µ b a g ( x ) the cost functon s not contnuous everywhere (the case when membershp functons do not overlap).. numerous parameters have to be optmzed (for the case consdered n ths paper, there s 60 parameter to optmze) x (3) (4) 3. ANT COLONY OPTIMIZATION Ant algorthms were nspred by the observaton of the real ant colones. An mportant and nterestng behavor of ant colones s ther foragng behavor, and n partcular, how ants can fnd the shortest path wthout usng vsual cues. Whle walng from the food sources to the nest and vce versa, ants depost on the ground a chemcal substance called pheromone whch maes a pheromone tral. Ants use pheromone trals as a medum to communcate wth each other. They can smell pheromone and when they choose ther way, they tend to choose paths wth more pheromone. The pheromone tral allows the ants to fnd ther way bac to the food source or to the nest. Also, the other ants can use t to fnd the locaton of the food sources, whch are prevously found by ther nest mates. Ths pheromone tral followng behavor can converge to the shortest path, once employed by a colony of ants. It means that, when there are more paths from the nest to a food source, a colony of ants may be able to use the pheromone trals left by the ndvdual ants to dscover the shortest path from the nest to the food source and bac. Consder two dfferent paths from the nest to the food source wth dfferent lengths. Intally there s no pheromone on the two branches, so ants select them wth the same probablty. Snce the ants move at approxmately constant speed, at each nstant of tme the number of ants who have passed the shorter path s greater than the number of ants who have not. Therefore when the ants start ther return trp, more pheromone s present on the shorter path, ncreasng the probablty of choosng t. Returnng the ants through the shorter path refreshes t faster than the other one and compensates the pheromone evaporaton. Thus n ths way pheromone s accumulated on the shorter path and for the new ants who want to go to the food source, the probablty of choosng t, wll ncrease. Very soon all the ants wll be usng the shorter path. 3.. Ant Colony System Basc Features Ant Colony System (ACS) was one of the frst algorthms proposed based on ACO. It was a dscrete algorthm, and at frst t was appled to the well-nown Travelng Salesman Problem (TSP) [8], whch s a dscrete optmzaton problem. In ths part we wll shortly revew the basc dea of ACS. Then n the subsequent part, the contnuous verson of ACS wll be presented. Consder a set of ctes. TSP s defned as the problem of fndng a mnmal cost closed tour that vsts all ctes and each cty only once. In a graph representaton, the ctes are the nodes and the connecton lnes between them are the edges. Each edge s assocated wth a cost measure, whch determnes the dstance or cost of travel between two ctes. Ant colony system uses a graph representaton, whch s the same as that defned for TSP. In addton to the cost measure, each edge has also a desrablty measure, called pheromone ntensty, updated at run tme by the ants. ICMSA0/05-

3 Proceedng of the Frst Internatonal Conference on Modelng, Smulaton and Appled Optmzaton, Sharah, U.A.E. February -3, 005 Ant colony system wors as follows: Each ant generates a complete tour by choosng the ctes accordng to a probablstc state transton rule. Ants prefer to move to ctes, whch are connected by short edges wth a hgh amount of pheromone, whle n some nstances, ther selecton may be random. Every tme an ant n one cty has to choose another cty to move to, t samples a random number, q n [0,]. If q becomes less than a gven q 0, then the destnaton cty s chosen by explotaton. It means that the one connected by the edge wth the most rato of pheromone ntensty to dstance, s chosen. Otherwse a cty s chosen by exploraton. In ths case the one connected by the edge wth the most rato of pheromone ntensty to dstance, has the most chance to be chosen, but all other ctes have also ther chances to be chosen proportonal to ther rato of pheromone ntensty to dstance. Whle constructng a tour, ants also modfy the amount of pheromone on the vsted edges by applyng a local updatng rule. It concurrently smulates the evaporaton of the prevous pheromone and the accumulaton of the new pheromone deposted by the ants when they are buldng ther solutons. Once all the ants have completed ther tours, the amount of pheromone s modfed agan, by applyng a global updatng rule. Agan a part of pheromone evaporates and all edges that belong to the globally best tour, receve addtonal pheromone conversely proportonal to ther length. 3.. Contnuous Ant Colony System Algorthm A contnuous optmzaton problem s defned as the problem of fndng the absolute mnmum of a postve non-zero contnuous cost functon f(x), wthn a gven nterval [a,b], whch the mnmum occurs at a pont x s. In general f can be a mult-varable functon, defned on a subset of a, b ], =,..., n. [ n R delmted by n ntervals The latest developed Contnuous Ant Colony System (CACS) has all the maor characterstcs of ACS, but certanly n a contnuous frame. These are a pheromone dstrbuton over the search space whch models the desrablty of dfferent regons for the ants, a state transton rule wth both exploraton and explotaton strateges, and a pheromone updatng rule whch concurrently smulates pheromone accumulaton and pheromone evaporaton Contnuous Pheromone Model Although pheromone dstrbuton has been frst modeled over dscrete sets, le the edges of the travelng salesman problem, n the case of real ants, pheromone deposton occurs over a contnuous space. Consder a food source, whch s surrounded by several ants. The ants aggregaton around the food source causes the most pheromone ntensty to occur at the food source poston. Then ncreasng the dstance of a sample pont from the food source wll decrease ts pheromone ntensty. CACS models ths varaton of pheromone ntensty, n the form of a normal dstrbuton functon: ( x xmn ) σ τ( x ) = e (5) Where x mn s the best pont n the nterval [a,b] whch has been found from the begnnng of the tral and σ can be nterpreted as an ndex of the ants aggregaton around the current mnmum. Note that τ has been modeled as a Probablty Dstrbuton Functon (PDF) whch determnes the probablty of choosng each pont x wthn the nterval [a,b] State Transton Rule In CACS, pheromone ntensty s modeled usng a normal PDF, the center of whch s the last best global soluton and ts varance depends on the aggregaton of the promsng areas around the best one. So t contans explotaton behavor. In the other hand, a normal PDF permts all ponts of the search space to be chosen, ether close to or far from the current soluton. So t also contans exploraton behavor. It means that ants can use a random generator wth a normal PDF as the state transton rule to choose the next pont to move to Pheromone Update Ants choose ther destnatons through the probablstc strategy of equaton (5). At the frst teraton, there sn't any nowledge about the mnmum pont and the ants choose ther destnatons only by exploraton. It means that they must use a hgh value of σ (assocated wth an arbtrary x ) to approxmately model a unform dstrbuton postons. Durng each teraton, pheromone dstrbuton over the search space wll be updated usng the acqured nowledge of the evaluated ponts by the ants. Ths process gradually ncreases the explotaton behavor of the algorthm, whle ts exploraton behavor wll decrease. Pheromone updatng can be stated as follows: The value of obectve functon s evaluated for the new selected ponts by the ants. Then, the best pont found from the begnnng of the tral s assgned to x. Also the value of σ s updated based on the evaluated ponts durng the last teraton and the aggregaton of those ponts around x. To satsfy smultaneously the ftness and aggregaton crtera, a concept of weghted varance s defned as follows: ( x xmn ) = f f mn (6) σ =, for all n whch f f mn f f = mn where s the number of ants. Ths strategy means that the center of regon dscovered durng the subsequent teraton s the last best pont and the narrowness of ts wdth s dependent on the aggregaton of the other compettors around the best one. The closer the better solutons get (durng the last teraton) to the best one, the smaller σ s assgned to the next teraton. Durng each teraton, the heght of pheromone dstrbuton functon ncreases wth respect to the prevous teraton and ts narrowness decreases. So ths strategy concurrently smulates pheromone accumulaton over the promsng regons and mn mn mn ICMSA0/05-3

4 Proceedng of the Frst Internatonal Conference on Modelng, Smulaton and Appled Optmzaton, Sharah, U.A.E. February -3, 005 pheromone evaporaton from the others, whch are the two maor characterstcs of ACS pheromone updatng rule. procedure CACS() ntalze() whle(termnaton_crteron_not_satsfed) move_ants_to_new_locatons() pheromone_update() end whle procedure ntalze() ntally guess the best pont usng a unform random generator set the ntal values of weghted varances (large enough) procedure move_ants_to_new_locatons() for =, for =,n choose the new x() for the th ant usng a normal random generator end for end for procedure pheromone_update update the globally best pont for =,n update the value of sgma_ end for Fgure. The CACS scheme n pseudo-code, s the number of ants, n s the number of varables to optmze 3.3. Algorthmc Model of CACS The pseudo-code of CACS scheme s shown n Fg.. A hgh level descrpton of CACS desgned to mnmze a general multvarable contnuous cost functon f x,..., x ), can be summarzed as follows: ( n Step : choose randomly the ntally guessed mnmum pont x,..., x over the search space and calculate the value of the ( n) mn functon f at ths pont, namely f mn PDF over the nterval a, ]. [ b. For each x use a unform Step : Set the ntal value of weghted varance for each pheromone ntensty dstrbuton functon: σ = ( b a ), 3 ( =,..., n ). It wll be large enough to approxmately generate unformly dstrbuted ntal values of a, b ]. [ x wthn the nterval Step 3: Send ants to ponts ( x,..., ), =,...,. To generate these random locatons, a random generator wth normal PDF s x n utlzed for each x, where ts mean and varance are ( x ) mn and σ respectvely. Note that f a generated x s outsde the gven nterval [ a, b ], t s dscarded and a new x s regenerated agan. Step 4: Evaluate f, at each dscovered pont, namely Compare these values wth the current mnmum value determne the updated f mn and ts assocated (,..., xn ) mn f,..., f. f and mn x. Step 5: f a stoppng crteron s satsfed (for the case whch s examned n ths paper, the algorthm stops after evaluatons) then stop, else update the weghted varance parameter σ for each varable x usng equaton (7) and go bac to step 3. σ = f f = mn [( x ) f f = mn ( x ) mn 4. EXPERIMENTS The same problem as [6] s consdered to evaluate the learnng method and to compare the results wth those of GA. The problem s approxmatng the followng analytcal functon wth a fuzzy rule base. 9x + 3x + / 3 x [0,/ 3] f ( x) = x / 3 x [/ 3,/ ] (8) 0.03/( (4.5x 3.85) ) x [0,/ 3] It presents the followng propertes (Fg. 3):. A dscontnuty of the dervatve at x=/3. A dscontnuty at x=/. A pea wth a hgh curvature zone around x=0.85 As [6], a Taag-Sugeno fuzzy rule base s consdered wth 0 rules. Here the learnng process means to fnd the optmal values of the a, b and w ( =,..., 0). Therefore the number of unnowns s 60. The optmzaton problem s to mnmze the quadratc error E, whch s calculated wth a set of testng samples. The test samples are numerous enough and regularly dstrbuted n the nterval on whch the analytcal functon s defned. To mae the results comparable wth those of [6], 00 sample ponts were taen and a lmtaton of evaluatons of the quadratc error E, per run, was mposed. A typcal result s shown n Fg. 3, obtaned after evaluatons of E, usng 0 ants. The best value of E was E = It s clear from ths fgure that f (x) s well optmum approxmated, but that the obtaned membershp functons seem rather dffcult to nterpret n a lngustc way. The correspondng values of a, b and w are lsted n table. ] (7) ICMSA0/05-4

5 Proceedng of the Frst Internatonal Conference on Modelng, Smulaton and Appled Optmzaton, Sharah, U.A.E. February -3, columns report the average values, whle the numbers n the parenthess are the standard devatons. The notatons used n the subsequent tables, are defned as follows: f(x) Fuzzy System Output Analytcal Functon E best : Mnmum value of E obtaned over all evaluated ponts by the ants (It allows to evaluate the convergence qualty) Eval ( Ebest ) : Number of evaluatons at whch E best was obtaned Eval ( Eave /0) : Number of evaluatons at whch the average error s below tenth of ts ntal value at the frst teraton (It allows to evaluate the convergence speed at the begnnng) Fgure 3. Typcal approxmaton obtaned and the assocated membershp functons. Table. Typcal Soluton for a, b and a x e e e-00.4e e e e e e e-00.54e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e-00.44e e e e e e e e e e e e e e e e e-00 b w w Table. Averaged results vs. the number of ants E Eval E ) Eval E /0) best (0.0678) (0.0053) (0.008) (0.07) (5938) 6740 (65) 7385 (9048) 5530 (43) 384 (03) 04 (4) 4738 (38) 4708 (8758) The presented results of Fg. 4, Fg. 5 and table, show that, the sze of the populaton of ants has a clear effect on the observed results. As mentoned n [7], CACS s not responsve enough for < 0. Small number of ants (e.g. = 5 ) yelds a fast convergence speed at the frst evaluatons, but at the same tme t may cause an unrpe convergence to a local mnmum (Fg. 4 and Eval E ) n table ). In the case of small number of ants the probablty of fallng nto a local mnmum ncreases because t s very probable that durng one teraton all ants choose the ponts near to the current mnmum. Ths n turn decreases the values of the weghted varances, whch magnfy the aggregaton of the later teraton ponts around the local mnmum. When the value of the weghted varances become several tmes smaller than the dstance to the nearest local mnmum, t s no longer possble to ump to a better local mnmum and the mnmzaton process stops. 0 0 No. of Ants = 5 No. of Ants = 0 No. of Ants = 0 No. of Ants = Parameter Study E mn 0 - CACS has only one parameter to set whch s the number of ants,. Accordng to [7], CACS needs at least 0 ants to propose relatvely good results. The varaton of E versus the number of evaluatons s shown n Fg. 4 and Fg. 5. The later one s concentrated over the frst 5000 evaluatons to observe more precsely the ntal behavor. Also table gves the results obtaned wth dfferent number of ants ( = 5,0, 0,50). All presented results are averaged over 0 dfferent runs. The mn Number of Evaluatons Fgure 4. Mnma found vs. number of evaluatons (the effect of the number of ants). ICMSA0/05-5

6 Proceedng of the Frst Internatonal Conference on Modelng, Smulaton and Appled Optmzaton, Sharah, U.A.E. February -3, CONCLUSIONS E mn Number of Evaluatons No. of Ants = 5 No. of Ants = 0 No. of Ants = 0 No. of Ants = 50 Fgure 5. Mnma found durng the frst 5000 evaluatons (the effect of the number of ants). The best results are observed for = 0 (see Fg. 4 and the obtaned E and Eval E /0) n table ). The convergence best rate decreases for greater number of ants (wth 0 ants, E 0.0, whle E < 0. 0 wth 0 ants). opt opt 4.. Comparson wth Standard Genetc Algorthm In order to compare the overall performance of CACS wth other algorthms, the results obtaned for = 0 are lsted n table 3, and have been compared wth the well-nown Genetc Algorthm of [6], whch utlzes the Standard Genetc Algorthm (SGA) wth 500 chromosomes, a mutaton rate of 0.005, a crossover rate 0.6, 8 bt for codng fuzzy parameters and 400 generatons. Table 3 shows that the best mnma found are at the same order, but a lttle better for SGA than CACS. At the same tme comparng Eval E ) and Eval E /0) between two methods, reveals a better convergence speed for CACS than SGA. A ey pont n CACS s ts smplcty. One can easly fnd that mplementaton and use of CACS s very smpler than SGA, whch s due to ts smpler structure and ts fewer control parameters. Table 3. Comparson of CACS and GA results A recently developed Contnuous Ant Colony System algorthm, based on Ant Colony Optmzaton, s appled to the problem of fndng optmum parameters of a fuzzy rule base. The proposed scheme s used to approxmate an analytcal test functon. The performance of CACS s compared wth the Standard Genetc Algorthm. The overall results are compatble. Although SGA can fnd the mnmum slghtly better, but t has poorer convergence rate compared wth CACS. One of the best advantages of the CACS scheme s ts smplcty, whch s manly due to ts smpler structure. Also t has fewer control parameters, whch maes the parameter settngs process easer than many other methods. 6. REFERENCES [] Wang, L. X., A Course n Fuzzy Systems and Control, Pearson Educaton POD, 997. [] Dranov, D. and Palm, R., Advances n Fuzzy Control, Hedelberg: Physca-Verlag, 998. [3] Procy, T. J. and Mamdan, E. H., "A lngustc selforganzng process controller," Automatca, IFAC, Vol. 5, 979, [4] Taag, T. and Sugeno, M., "Fuzzy dentfcaton of systems and ts applcaton to modelng and control," IEEE Trans. Systems, Man and Cybernetcs, Vol. 5, 985, 6-3. [5] Nomura, H., Hayash, I., Waam, N., "A Self Tunng Method of Fuzzy Control by Descent Method," Proceedngs of the Internatonal Fuzzy Systems Asscaton, IFSA9, Engneerng Vol., Bruxelles, 99, [6] Sarry, P. and Guely, F., "A Genetc Algorthm for Optmzng Taag-Sugeno Fuzzy Rule Bases," Fuzzy Sets and Systems, Vol. 99, 998, [7] Pourtadoust, S. H. and Nobahar, H., "An Extenson of Ant Colony System to contnuous optmzaton problems," Lecture Notes n Computer Scence, Vol. 37, 004, [8] Colorn, A., Dorgo, M. and Manezzo, V., "Dstrbuted optmzaton by ant colones," Proceedngs of the Frst European Conference on Artfcal Lfe, Elsever Scence Publsher, 99, [9] Dorgo M. and Gambardella, L. M., "Ant colony system: A cooperatve learnng approach to the travelng salesman problem," IEEE Transactons on Evolutonary Computaton, Vol., No., 997, Method CACS SGA E Eval E ) Eval E /0) best (0.0053) (0.003) 6740 (5938) (683) 04 (03) 000 (87) ICMSA0/05-6