Optimization of Multiple Input Single Output Fuzzy Membership Functions Using Clonal Selection Algorithm

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1 Optimizatio of Multiple Iput Sigle Output Fuzzy Membership Fuctios Usig Cloal Selectio Algorithm AYŞE MERVE ACILAR, AHMET ARSLAN Computer Egieerig Departmet Selcuk Uiversity Selcuk Uiversity, Eg.-Arch. Fac. Computer Eg Koya TURKIYE Abstract: - A cloal selectio algorithm (CLONALG) ispires from Cloal Selectio Priciple used to explai the basic features of a adaptive immue respose to a atigeic stimulus. I this study, a ew method is proposed for optimizatio of the Multiple Iput Sigle Output (MISO) fuzzy membership fuctios usig CLONALG. The most appropriate placemet of membership fuctios with respect to fuzzy variables ca be determied usig our method for a fuzzy system whose rules table ad shape of membership fuctios were give previously. Also, how the membership fuctios compute as a parameter optimizatio problem usig CLONALG is descried for MISO fuzzy system o a illustrative example. Key-Words: - Multiple Iput Sigle Output Fuzzy Membership Fuctios, Optimizatio, Cloal Selectio Algorithm. Itroductio Fuzzy logic is used to solve a lot of problems related wide rage of area because of providig flexible solutios. Desigig fuzzy system cotais fuzzy sets which are de-fied by rule table ad membership fuctios. Whe the fuzzy sets have bee established, how best to determie the membership fuctio is the first questio that has to be tackled. For most fuzzy logic cotrol problems, the membership fuctios are assumed to be liear ad usually triagular i shape. So, the issues to be determied are the parameters that defie the triagles. Because of this, the membership optimizatio problem ca be reduced to parameter optimizatio problem. These parameters are usually based o the cotrol egieer's experiece ad/or are geerated automatically. To improve behavior of this parameter optimizatio problem some methods such as geetic algorithms (GA), self-orgaizig feature maps (SOFM), tabu search (TS) etc. ca be used. GA was used by Karr [] i determiatio of membership fuctios. Karr applies GA to desig of fuzzy logic cotroller (FLC) for the cart pole problem. Meredith et al. [2] have applied GA to the fie tuig of membership fuctios i a FLC for a helicopter. Lee ad Takagi [3] also tackle the cart problem. They take a holistic approach by usig GA to desig the whole system. Cheg et al. [4] have chose the im-age threshold via miimizig the measure of fuzziess. For selectig the badwidth of fuzzy membership fuctios, they use peak locatios which are chose from the histogram usig the peak selectio criterio. Bağiş [5] presets a method for the determiatio of the membership fuctios based o the use of Tabu Search algorithm. Cerrada et al. [6] proposed a approach permits icorporate the temporal behavior of the sys-tem variables ito the fuzzy membership fuctios. Simo [7] employed H state estimatio theory for the membership fuctio parameter optimizatio. He made some modificatios o the H filter with additio of state costraits so that the resultig membership fuctios are sum ormal. Yag ad Bose [8] described a method for geeratig a fuzzy membership fuctios with usupervised learig usig self-orgaizig feature map. They applied this method to patter recogitio. I our previous work, a ew method was proposed to determie the membership fuctios of a sigle iput ad output fuzzy system whose shape was triagular [9]. The suggested method is relevat for ay membership fuctio whose mathematical model is kow. The method focused o fidig the optimum values of the parameter of this membership fuctio model. This problem solved usig a artificial immue system algorithm: CLONALG. This algorithm is ispired from Cloal Selectio Priciple used to explai the basic features of a adaptive immue respose to a atigeic stimulus. I this study, we optimized the multiple iputsigle output (MISO) fuzzy system whose rule base is complete usig the adaptatio of the method metioed above. This paper is orgaized as follow. I Sectio 2, CLONALG are explaied. I Sectio 3, how the MISO membership fuctios is optimized usig CLONALG o a illustrative example. The experimetal results ad the ISSN: ISBN:

2 discussio of them have bee give i Sectio 4. Fially, i Sectio 5, we preset our coclusios. 2 Cloal Selectio Priciple &CLONALG Cloal selectio is the theory used to explai the basic properties of a adaptive immue respose to a atigeic stimulus. It establish the idea that oly those cells capable of recogizig a atigeic stimulus will proliferate ad differetiate ito effectors cells, thus beig selected agaist those that do ot. Maily features of the cloal selectio priciple are affiity proportioal reproductio ad mutatio. The higher affiity, the higher umber of offsprig geerated. The mutatio suffered by each immue cell durig reproductio is iversely proportioal to the affiity of the cell receptor with the atige. The stadard geetic algorithm does t accout for these immue properties [0]. De Castro& Vo Zube proposed a Cloal selectio algorithm amed CLONALG, to fulfill these basic processes ivolved i cloal selectio priciple. CLONALG will be iitially proposed to perform machie-learig ad patter recogitio tasks, ad the adapted to be applied to optimizatio problems []. The algorithm of the CLONALG for the optimizatio task is give below.. Geerate j atibodies radomly. 2. Repeat a predetermied umber of times: a. Determie the affiity of each atibody (Ab). This affiity correspods to the evaluatio of the objective fuctio. b. Select the highest affiity atibodies. c. The selected atibodies will be cloed proportioally to their affiities, geeratig a repertory C of cloes: the higher affiity the higher umber of cloes ad vice versa. d. The cloes from C are subject to hypermutatio process iversely proportioal to their atigeic affiity. The higher affiity, the smaller mutatio, ad vice versa. e. Determie the affiity of the mutated cloes C. f. From this set C of cloes ad atibodies, select the j highest affiity cloes to compose the ew atibodies populatio. g. Replace the d lowest affiity atibodies by ew idividuals geerated at radom. 3. Ed repeat [2]. illustrative example. I our example we have a fuzzy system with two iputs ad oe output whose shapes are triagular ad each of them cosist of three membership fuctios. They are deoted x, x2 ad y, respectively. The rages of the variables x, x2 ad y are [0,7], [0,80] ad [0,50]. Each of them use low, medium, high liguistic terms. I this case, the liguistic rules are as follows: Rule : If x is low ad x2 is high the y is medium. Rule 2: If x is medium ad x2 is low the y is low. Rule 3: If x is medium ad x2 is medium the y is medium. Rule 4: If x is medium ad x2 is high the y is high. Rule 5: If x is high the y is high. Some referece values must be measured beforehad for this system. These are the output values obtaied correspodig to specific iput values. : If x= ad x2=80 the y=20. 2: If x=3 ad x2=20 the y=6. 3: If x=4 ad x2=50 the y=26. 4: If x=5 ad x2= 80 the y=36. 5: If x=7 ad x2= 60 the y=40. The membership fuctios of fuzzy system for iput ad output variables will be as show i Fig.. 3 Computatio of the MISO membership fuctios usig CLONALG I this sectio, we described how the membership fuctios compute as a parameter optimizatio problem usig CLONALG for MISO fuzzy system o a Fig.: A sample membership fuctios for the MISO fuzzy system Expected from a CLONALG is to fid the base legths of right triagles ad itersectio poits of triagles ISSN: ISBN:

3 correspodig to the referece data. Its secod goal is to obtai the output values correspodig to all the appropriate iput values amog defiite rages of system. If the base legth of each membership fuctio is represeted by 6-bit, the maximum value each base ca take is 2 7 =28. The domai itervals for iput ad output variables are [0, 7], [0, 80] ad [0, 50], respectively. The base values are reflected uder these values. This is formulated as follows: d X i = X + *( X X ) mi L () i (2 ) maxi mii L is the legth of related variable (i this case, it is Atibody- Ab ) i bits, d is the decimal value of this variable, X mi is the miimum value of regio to be trasformed, ad X max is the maximum value of this regio, the X i is the trasformed figure of that variable. The affiity fuctio of this procedure is calculated as follows: Affiity = MaxError TotalError (2) TotalError = ( y CLONALGi y i ) i = [( y y ) ( y y )] (3) MaxError = max ; CLONALGi mi max CLONALGi (4) 2 these two membership fuctios are ANDed, outputs are determied from lower grades of membership fuctios, but if two membership fuctios are ORed, outputs are determied from higher grades of membership fuctios. If there is more tha oe membership fuctio itersectig with these give iputs, the the formulatio which is give i Eq.5 is used for acquirig the output value. y * = ( yi )* yi μ( yi ) μ (5) This example is described i Fig. 2. The give values are show as xi ad x2i. xi has itersected with low ad medium ad x2i has itersected with medium ad high. Firstly, the grades of membership are determied for each membership fuctio from the Fig.2. The the rules which are triggered accordig to these values are detected. I our example first, third ad fourth rules are triggered. For the first rule µ low is 0.7 ad µ medium is 0.4. where y i is the output of i th referece iput; y CLONALGi obtaied by CLONALG, is output for i th referece iput, ad is the umber of iput-output data pairs give.. The aim is to approximate the TotalError to zero as close as possible. Be cause of that affiity fuctio is coverted to MaxError-TotalError, i this way; miimizatio process is also coverted to maximizatio process. I order to prevet affiity fuctio from gettig egative values, the maximum umber 5628 is used ad this umber is also maximum error at the same time. How this umber acquired has bee showed i the below. max[(y CLONALGi -y mi );(y max - y CLONALGi )] 2 Error max[(20-0);(50-20)] 2 = max[(6-0);(50-6)] 2 = max[(26-0);(50-26)] 2 = max[(36-0);(50-36)] 2 = max[(40-0);(50-40)] 2 = max[(y CLONALGi -y mi );(y max - y CLONALGi )] Let solve a example for uderstadig how the output value has bee computed regardig to give base values. The base values are take 2 for x ad 45 for x2. Firstly the membership fuctios must be foud for these variables. The, it is explored if two membership fuctios ca be put i a rule or ot. If they ca, the grades of membership fuctios are calculated ad aalyzed to determie if the rule cotais AND or OR. If Fig.2: Fidig the appropriate outputs of CLONALG for give values for the MISO fuzzy system We choose the µ medium =0.4 because the rule ivolves AND. The same process has repeated for third ad fourth rule. I the third rule µ medium =0.2 ad i the forth rule µ high =0.2 are take. The acquired outputs correspodig to rules are 8, 6 ad 23, respectively. We defuzzificate the output value usig equatio 5 ad obtaied 8.75 as result. ISSN: ISBN:

4 4 Experimetal Results I this sectio, CLONALG used to optimize the MISO fuzzy membership fuctios is implemeted by Matlab 7. R4. After 25 iteratio completed, the idividual whose affiity is maximum has bee accepted as optimal solutio. The affiity value accepted as the optimum solutio is ad its membership fuctio shape is depicted i Fig. 3. For showig the results were t obtaied by coicidetally, 20 differet groups were geerated. A group cosisted of 0 differet iitial populatios ad a populatio cosisted of 0 Ab idividuals. The algorithm was set to ru for 25 geeratios o each populatio i the groups, respectively. The the maximum affiities values of each group are depicted i the Fig Max. Affiity Values Group No Fig.5: The maximum affiity values of each groups Results are compared group by group. As show i the figure, at least oe Ab coverged to the optimum solutio i each group. Tha it was straightforward to say the CLONALG represeted a good performace while fidig the optimum base distace of a fuzzy membership fuctios. Fig.3: The membership fuctio shape of the optimal solutio for the MISO fuzzy system Affiity Geeratio No Fig.4: The affiity values of CLONALG accordig to geeratio umber of a populatio The affiity values of algorithms accordig to geeratio umber of a populatio are give i the Fig. 4. As see from Fig. 4, CLONALG coverged to optimum solutio quickly because of the cloig ad hyper mutatio processes. Also, it provided the stability of the affiity values. 5 Coclusio The most appropriate placemet of membership fuctios with respect to fuzzy variables ca be foud for a fuzzy system whose rules table ad shape of membership fuctios were give previously. There are o restrictios for shape of membership fuctios. They ca be geerally used but it ca be mathematical fuctio whose model is kow. The, the issues to be determied are the parameters that defie the model. Because of this, the membership optimizatio problem ca be reduced to parameter optimizatio problem. How the membership fuctios compute as a parameter optimizatio problem usig CLONALG is descried for Multiple Iput Sigle Output (MISO) fuzzy system i this work. CLONALG used to optimize the fuzzy membership fuctios was implemeted by Matlab 7. R4. Firstly, 20 differet groups were geerated for showig the results were t obtaied by coicidetally. A group cosisted of 0 differet iitial populatios ad a populatio cosisted of 0 Ab idividuals. The algorithms were set to ru for 25 geeratios o each populatio i the groups, respectively. The the acquired results are examied. It was see that, at least oe idividual Ab coverged to the optimum solutio i each ISSN: ISBN:

5 group. As well, it could be said that accordig to results, CLONALG coverged to optimum solutio quickly because of the ow cloig ad hyper mutatio mechaisms. Also, the stability of the affiity values was provided by the CLONALG algorithm. Tha it was straightforward to say the CLONALG represeted a good performace while fidig the optimum base distace of a fuzzy membership fuctios of a MISO fuzzy system. Issue o Artificial Immue Systems, Vol.6, No.3, 2002, pp [2]N. C. Cortes, D.T. Perez, C.A. Coello, Hadlig Costraits i Global Optimizatio Usig a Artificial Immue System, Lecture Notes i Computer Sciece Vol. 3627,Spriger-Verlag,Berli Heidelberg, 2005, pp Ackowledgemets: The authors ackowledge the support of this study provided by Selçuk Uiversity Scietific Research Projects. The authors have also thaked to TUBITAK for their support of this study. Refereces: []C.L.Karr, Desig of a Adaptive Fuzzy Cotroller Usig a Geetic Algorithm, Proc. of the 4th Itl. Cof. o Geetic Algorithms, 99. [2]D.L. Meredith, C.L. Karr, K. Krisha Kumar, The Use of Geetic Algorithms i The Desig of Fuzzy Logic Cotrollers, 3rd Workshop o Neural Network WNN'92, 992. [3]M.A. Lee, H. Takagi, Itegratig Desig Stages of Fuzzy Systems Usig Geetic Algorithms, 2d IEEE Itl. Cof. o Fuzzy Systems, 993. [4]H.D. Cheg, Y.M. Lui, Automatic Badwidth Selectio of Fuzzy Membership Fuctios, Iformatio Scieces, Vol.03, 997, pp.-27. [5]A. Bağiş, Determiig Fuzzy Membership Fuctios with Tabu Search a Applicatio to Cotrol, Fuzzy Sets ad Systems,Vol. 39, 2003, pp [6]M. Cerrada, J. Aguilar, E. Colia, A. Titli, Dyamical Membership Fuctios: a Approach for Adaptive Fuzzy Modelig, Fuzzy Sets ad Systems, Vol , pp [7]D. Simo, H Estimatio for Fuzzy Membership Fuctio Optimizatio, Iteratioal Joural of Approximate Reasoig, Vol.40, 2005, pp [8]Yag, N.K. Bose, Geeratig Fuzzy Membership Fuctio with Self-Orgaizig Feature Map, Patter Recogitio Letters, Vol.27, 2006, pp [9]A.M. Sakiroglu, A. Arsla, Cloal Selectio Priciple for Fuzzy Membership Fuctio Optimizatio, Lecture Notes i Computer Sciece,Vol.443, Spriger-Verlag,Berli Heidelberg, 2007,pp [0]L.N de Castro, J.I. Timmis, Artificial Immue Systems: A New Computatioal Itelligece Approach, Spriger -Verlag, Lodo, 2002, 357 pages. []L.N de Castro, J.V. Zube, Learig ad Optimizatio Usig Cloal Selectio Priciple, IEEE Trasactio o Evolutioary Computatio, Special ISSN: ISBN: