Cerrada et al. [4] proposed an approach permits incorporate the temporal behavior of the system variables into the fuzzy membership functions. Simon [5] ...
Optimization of Fuzzy Membership Function Using Clonal Selection Ayşe Merve Şakiroğlu and Ahmet Arslan Selcuk University, Eng.-Arch. Fac. Computer Eng.42075-Konya, Turkey {msakiroglu,ahmetarslan}@selcuk.edu.tr
Abstract. A clonal selection algorithm (Clonalg) inspires from Clonal Selection Principle used to explain the basic features of an adaptive immune response to an antigenic stimulus. It takes place in various scientific applications and it can be also used to determine the membership functions in a fuzzy system. The aim of the study is to adjust the shape of membership functions and a novice aspect of the study is to determine the membership functions. Proposed method has been implemented using a developed Clonalg program for a single input and output fuzzy system. In the previous work [1], using genetic algorithm (GA) is proposed to it. In this study they are compared, too and it has been shown that using clonal selection algorithm is advantageous than using GA for finding optimum values of fuzzy membership functions.
1 Introduction Fuzzy logic is used to solve a lot of problems related wide range of area because of providing flexible solutions. Designing fuzzy system contains fuzzy sets which are defined by rule table and membership functions. When the fuzzy sets have been established, how best to determine the membership function is the first question that has to be tackled. For solving this problem, some methods such as genetic algorithms (GA), self-organizing feature maps (SOFM), tabu search (TS) etc. can be used. GA was used by Karr [10] in determination of membership functions. Karr applies GA to design of fuzzy logic controller (FLC) for the cart pole problem. Meredith et al. [12] have applied GA to the fine tuning of membership functions in a FLC for a helicopter. Lee and Takagi [11] also tackle the cart problem. They take a holistic approach by using GA to design the whole system. Cheng et al. [2] have chosen the image threshold via minimizing the measure of fuzziness. For selecting the bandwidth of fuzzy membership functions, they use peak locations which are chosen from the histogram using the peak selection criterion. Bağiş [3] presents a method for the determination of the membership functions based on the use of Tabu Search algorithm. Cerrada et al. [4] proposed an approach permits incorporate the temporal behavior of the system variables into the fuzzy membership functions. Simon [5] employed H∞ state estimation theory for the membership function parameter optimization. He made some modifications on the H∞ filter with addition of state constraints so that the resulting membership functions are sum normal. Yang and Bose [6] described a method B. Beliczynski et al. (Eds.): ICANNGA 2007, Part I, LNCS 4431, pp. 694–701, 2007. © Springer-Verlag Berlin Heidelberg 2007
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for generating a fuzzy membership functions with unsupervised learning using selforganizing feature map. They applied this method to pattern recognition. In the previous work, proposed a new method for determination of fuzzy logic membership functions implemented using a genetic algorithm program for single input and output system. Due to space limitation in this paper, how the membership functions compute using GA didn’t explained. Details of it can be found in the earlier work [1]. In this work the same method was performed using an artificial immune system algorithm: Clonalg. This algorithm is inspired from Clonal Selection Principle used to explain the basic features of an adaptive immune response to an antigenic stimulus. Both of these algorithms were coded and acquired results were compared. This paper is organized as follow. In Sect. 2, basic principles and features of Clonalg are explained. In Sect. 3, how the membership functions that are in a given shape can be suitably placed using clonal selection algorithm has discussed. The experimental results and the discussion of them have been given in Sect. 4. Finally, in Sect. 5, we present our conclusions.
2 Clonal Selection Principle and Clonalg Algorithm Clonal selection is the theory used to explain the basic properties of an adaptive immune response to an antigenic stimulus. It establish the idea that only those cells capable of recognizing an antigenic stimulus will proliferate and differentiate into effector cells, thus being selected against those that do not. The mainly features of the clonal selection principle are affinity proportional reproduction and mutation. The higher affinity, the higher number of offspring generated. The mutation suffered by each immune cell during reproduction is inversely proportional to the affinity of the cell receptor with the antigen. The standard genetic algorithm doesn’t account for these immune properties [9]. De Castro& Von Zuben proposed a Clonal selection algorithm named Clonalg, to fulfill these basic processes involved in clonal selection principle. It will be initially proposed to perform machine-learning and pattern recognition tasks, and then adapted to be applied to optimization problems [8]. The algorithm of it for the optimization task is given below. 1. 2.
Generate j antibodies randomly. Repeat a predetermined number of times: a. Determine the affinity of each antibody (Ab). This affinity corresponds to the evaluation of the objective function. b. Select the n highest affinity antibodies. c. The n selected antibodies will be cloned proportionally to their affinities, generating a repertory C of clones: the higher affinity the higher number of clones and vice versa. d. The clones from C are subject to hypermutation process inversely proportional to their antigenic affinity. The higher affinity, the smaller mutation, and vice versa. e. Determine the affinity of the mutated clones C.
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f.
3.
From this set C of clones and antibodies, select the j highest affinity clones to compose the new antibodies’ population. g. Replace the d lowest affinity antibodies by new individuals generated at random. End repeat [7].
3 Computation of the Membership Functions Using Clonalg The most appropriate placement of membership functions with respect to fuzzy variables can be found for a fuzzy system whose rules table and shape of membership functions were given previously. There are no restrictions for shape of membership functions. They can be generally used but it can be mathematical function whose model is known. Then, the issues to be determined are the parameters that define the model. Because of this, the membership optimization problem can be reduced to parameter optimization problem. How the membership functions compute as a parameter optimization problem using Clonalg is descried bellow for single input-output system . It is assumed that there are two different membership functions for input and output of considered system and their shapes are right triangle. The membership functions are called input(x) and output(y); x uses slow and fast; y uses easy and difficult. In this case, the linguistic rules are as follows: Rule 1: If x is slow then y is easy Rule 2: If x is fast then y is difficult If the range of input variable x as [0-7] and the range of output variable y as [0-49] are assumed, then the membership functions of fuzzy system for input and output variables will be as shown in Fig. 1.
Fig. 1. The membership functions of a fuzzy system for input and output
Input : xi = {1, 3, 5, 7} Output : yi = {1, 9, 25, 49}, i=1, 2, 3, 4 Expected from a Clonalg is to find the base lengths of right triangles. If the base length of each membership function is represented by 6-bit, the genes Base1Base2Base3Base4 containing solution of problem has 6*4=24 bits. In this case the maximum value each base can take is 26−1=63. The domain intervals for input and output variables are [0-7] and [0-49], respectively. The base values are reflected under these values. This is formulated in (1).
Optimization of Fuzzy Membership Function Using Clonal Selection
Xi = X
min i
+
d *(X −X ) maxi min i (2 L − 1)
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(1)
L is the length of related variable (in this case, it is Antibody- Ab ) in bits, d is the decimal value of this variable, Xmin is the minimum value of region to be transformed, and Xmax is the maximum value of this region, then Xi is the transformed figure of that variable. Thereafter, the necessary thing to be done is to create the initial antibody population-Ab with 10 antibodies randomly, and let begin Clonalg process. Table 1 expresses the initial antibody population. Columns 2-5, are the decimal values of lengths of related antibodies. Columns 6-9 are the cases of these lengths reflected to domain intervals. Before the computation of affinity, total error is calculated in (2). Total_Erro r = ∑i =1 ( yi − y )2 clona lg i 4
(2)
th
Where yi is the output of i reference input; yClonalgi which is obtained by Clonalg, is output for ith reference input. The aim is to approximate the total error to zero as close as possible. Affinity function is converted to 4480-Total_Error, in this way; minimization process is also converted to maximization process (Column 10). In order to prevent affinity function from getting negative values, the maximum number 4480 is used and this number is also maximum error at the same time. It is evaluated by ∑4 ( yi − 49) 2 ; where 49 is the biggest output value for : i =1
Base1
