Department of Computer Science. and Articial Intelligence

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1 DECSAI Department of Computer Science and Articial Intelligence A Two-Stage Evolutionary Process for Designing TSK Fuzzy Rule-Based Systems O. Cordon, F. Herrera Technical Report #DECSAI-9711 May, 1996 ETS de Ingeniera Informatica. Universidad de Granada Granada - Spain - Phone , Fax

2 A Two-Stage Evolutionary Process for Designing TSK Fuzzy Rule-Based Systems Oscar Cordon and Francisco Herrera Abstract Nowadays, Fuzzy Rule-Based Systems are successfully applied to many dierent real-world problems. Unfortunatelly, relatively few well-structured methodologies exist for designing them and, in many cases, human experts are not able to express the knowledge needed to solve the problem in the form of fuzzy rules. TSK Fuzzy Rule-Based Systems were enunciated in order to solve this design problem because they are usually identied using numerical data. In this paper we present a two-stage evolutionary process for designing TSK Fuzzy Rule-Based Systems from examples combining a generation stage based on a (; )-Evolution Strategy, in which the fuzzy rules with dierent consequents compete among themselves to form part of a preliminary Knowledge Base, and a renement stage, in which both the antecedent and consequent parts of the fuzzy rules in this previous Knowledge Base are adapted by a hybrid evolutionary process composed of a Genetic Algorithm and an Evolution Strategy to obtain the nal Knowledge Base whose rules cooperate in the best possible way. Some aspects make this process dierent from others proposed till now: the design problem is addressed in two dierent stages, the use of an angular coding of the consequent parameters that allows us to search across the whole space of possible solutions, and the use of the available knowledge about the system under identication to generate the initial populations of the Evolutionary Algorithms that allow us to speed up the search process, obtaining good solutions more quickly. The performance of the method proposed is shown by obtaining several TSK Fuzzy Rule-Based Systems for fuzzy modeling dierent three-dimensional surfaces and comparing them with other TSK and Mamdani-type ones generated by using other evolutionary design processes. Keywords TSK Fuzzy Rule-Based Systems, TSK Knowledge Base, Evolutionary Algorithms, Evolution Strategies, Genetic Algorithms, Learning. I. Introduction FUZZY Rule-Based Systems (FRBSs) are now considered as one of the most important applications of fuzzy set theory suggested by Zadeh in 196 [1]. These kind of systems constitute an extension of the classical Rule-Based Systems because they deal with fuzzy rules instead of classical logic rules. Thanks to this, they have been succesfully applied to a wide range of problems presenting uncertainty and vagueness in dierent ways []. In particular, the most promising results have been obtained by the Fuzzy Logic Controllers [3], the FRBSs for control problems. Several tasks have to be performed in order to design an intelligent system of this kind for a concrete application. They can be grouped into two main tasks: to design the FRBS Inference System, i. e., to select the fuzzy operators considered to make inference, and to obtain an accurate Knowledge Base (KB) comprising the known knowledge about the problem being solved. The latter used to be the most important and dicult, due to the fact that human experts are not usually able to express their knowledge in the form of fuzzy if-then rules. This has forced researchers to develop automatic techniques for performing this task. Over the last few years, many dierent approaches have been presented taking Genetic Algorithms (GAs) [4] as their base, obtaining the so called Genetic Fuzzy Systems (GFSs) [] or, more generically, Evolutionary Fuzzy Systems (EFSs) when an Evolutionary Algorithm (EA) [6] is used instead of a GA. EAs are applied to modify/learn the Data Base (DB), i. e., the membership function denitions, and/or the Rule Base (RB), i. e., the fuzzy rules themselves. Therefore, they may act on one or both The authors are from the Department of Computer Science and Articial Intelligence. E.T.S. Ingeniera Informatica. University of Granada, Granada, Spain. ocordon, herrera@decsai.ugr.es. This research has been supported by CICYT TIC

3 KB components (DB and RB) and it is possible to distinguish three dierent groups according to the components included in the learning process: 1. Genetic denition of the FRBS Data Base. Genetic derivation of the FRBS Rule Base 3. Genetic learning of the FRBS Knowledge Base For a wider description of some approaches belonging to each one of these see [], [7], and for an extensive bibliography see [8]. TSK-type FRBS are easier to design for two main reasons. On the one hand, they always make inference in the same way, and we only need to choose the t-norm used to combine the antecedent values to set up the FRBS Inference System. On the other hand, these kinds of systems were originally designed to be identied from numerical data, which makes automatic design easier. In this paper, we present a two-stage evolutionary process belonging to the third aforementioned family. The learning process is thus divided into two stages: the generation and the renement stages. The rst one, based on the combination of an inductive algorithm and a (; )-Evolution Strategy ((; )-ES) [6], [9], will allow us to automatically generate a preliminary TSK-type KB for a concrete problem when a training data set representing its behavior is available. It is able to decide the number of rules composing the KB and to determine their consequent parameters by inducing competition on the single fuzzy rules, and generates a locally optimal KB. The second stage is addressed by means of a hybrid GA-ES process that works with a population of KBs, taking the preliminary denition obtained in the previous stage as a base, to obtain one presenting the best possible cooperation among the fuzzy rules composing it and an optimal global behavior. The performance of the EFS proposed is compared with other Mamdani and TSK-type FRBSs, obtained from dierent evolutionary design processes: a Mamdani-type two-stage EFS based on the Wang and Mendel fuzzy rule generation method [1], a three-stage Mamdani-type EFS [11], [1], and a TSK-type EFS [13], [14], in the problem of fuzzy modeling three three-dimensional surfaces. All the evolutionary design processes selected belong to the third family as well. In order to put this into eect, this paper is set up as follows. The next Section presents some guidelines about the TSK FRBS and its design. In Section 3, a new coding to represent TSK fuzzy model rule consequents is introduced, allowing us to explore the whole possible solution space when using EAs. Section 4 introduces the problems existing when designing an EFS, analyzes the possible solutions, and shows and justies the structure of the EFS proposed in this paper. Both stages composing it are described in Sections and 6 respectively. Section 7 shows the experiments developed and the dierent results obtained. In Section 8, some concluding remarks are pointed out. Finally, an Appendix describing briey the EAs used in this work, GAs and ESs, is presented. II. TSK Fuzzy Rule-Based Systems A. Structure and inference process of a TSK Fuzzy Rule-Based System The TSK fuzzy model was rst presented in [1]. It is based on rules in which the consequent is not a linguistic variable, as in the Mamdani-type fuzzy model, but a function of the input variables. This kind of rule usually presents the following form: If X 1 is A 1 and : : : and X n is A n then Y = p 1 X 1 + : : : + p n X n + p where X i are the system input variables, A i are fuzzy sets specifying their meaning, and Y is the system output variable. The output of a FRBS using a KB composed of m TSK rules is computed as the weighted average of the individual rule outputs, Y i, i = 1 : : : m, in the following way: P m i=1 h i Y i P m i=1 Y i

4 where h i = T (A 1 (x 1 ); : : : ; A n (x n )) is the matching between the antecedent part of the rule i and the current system inputs, x = (x 1 ; : : : ; x n ), with T being a t-norm. Therefore, this fuzzy model is based on rst dividing the (uni or multidimensional) input space into some (uni or multidimensional) fuzzy subspaces and then building a linear input-output relation in each subspace. In the inference process, these partial relations are combined in this way, taking into account their dominance in their respective area of application and the conict existing in the overlapping areas, to obtain the global input-output relationship [1]. As may be observed, the form of the partial linear relations depends directly on the space dimension, i. e., on the number of system input variables. When the system only presents one input variable, each TSK rule output, Y = p 1 X + p, represents a straight line in the part of the plane determined by its fuzzy input subspace. In the case of a system with two inputs, each output, Y = p 1 X 1 + p X + p, determines a plane in one part of the three-dimensional space R 3. When a number iv greater than two inputs is considered, each output relation corresponds to a hyperplane of dimension iv in one part of the (iv + 1)-dimensional space R i+1. These geometrical properties will be taken into account in order to design the angular coding presented in the next Section. B. Design of TSK Fuzzy Rule-Based Systems As commented on in the introduction, one of the main tasks to be developed to design a FRBS is to build an accurate KB comprising the known knowledge about the problem being solved. The diculty presented by human experts in linguistically describing what kind of action they take in a particular situation, and the absence of any assurances showing that they develop an optimal decision-making process were the assumptions that Takagi and Sugeno took as a basis for proposing the TSK fuzzy model: "it is quite useful to give a way to model control actions using numerical data about the system behavior" [1]. Many dierent techniques have been employed till now for performing this task since Takagi and Sugeno rst presented a process based on the least squares method in [1]. For example, Neural networks [16], [17]; gradient [18], and hybrid methods (learning rule composed of the least squares and the gradient method) [19] have been employed. The use of EAs, either specic, GAs [13], [] and EEs [1]; or hybrid [], [1], has increased over the last few years. III. A new coding scheme to represent TSK rule consequents There is a problem when designing TSK FRBSs using EAs. Usually, an EA needs to know the intervals in which each problem variable is dened to solve a specic problem. This information is necessary to dene the genetic coding of the possible solutions and to perform evolution on them using the genetic operators. Unfortunately, this information is not available in the problem of learning the TSK rule consequent parameters. This problem has usually been solved by the authors [13], [14], [1], [], [] by xing suciently large values for the low and high interval extremes. This is not a bad idea because the powerful search of the EA allows us to obtain good solutions working in this way but presents the drawback that not all the solution space is considered, so it may not be possible to nd the global problem solution since the value of some of the parameters may lie outside the intervals considered. In this Section, we propose a new coding scheme, called angular coding, which was rst presented in [3]. It is based on encoding the values of the angles instead of the tangent values for each TSK rule consequent parameter, that allows us to have all the variables lying in the same xed interval and to represent the whole space of possible solutions. As commented on in Section, the partial linear relation dened by the consequent of a TSK rule determines a geometrical gure in the corresponding hyperspace. For example, when working with a system with a single input variable, each TSK rule output, Y = p 1 X + p, represents a straight line in one part of the plane. Bearing this in mind, we know that the real value p 1 is simply the tangent

5 of the angle existing between this line and the X axis. Thus, if we code the angle value instead of the tangent one by means of the function C : R! (? ; ) C(x) = arctan(x) ; all the possible values of the parameter p 1 lie inside the interval (? ; ). Figure 1 shows some examples of this idea. Y p1=3 C(p1) = -71.6º p1=1 C(p1) = 4º X C(p1) = º p1=- Fig. 1. Examples of angular coding As may be observed graphically in the Figure, using very short intervals, a very large part of the possible solution space may be represented. For example, when considering the interval [?; ], we work with the angular interval [?87:13 o ; 87:13 o ]. Thus, approximately only 3:3 percent of the search space (more or less :74 o ) is not taken into account. This justies the fact that the EA-based design processes which consider xed intervals allow us to obtain good results. Anyway, it seems more appropriate to represent the whole possible solution space when performing a search towards a global solution. Y 8 X + 3 X 3 X 1 Fig.. Geometrical interpretation of the parameter p As regards the parameter p, when working in the plane it determines the movement of the straight line from the origin along the Y axis, as shown in Figure. Since the values of the parameter p may be very dierent from one TSK rule to another, the consideration of a xed interval is not a good solution for their evolutionary learning, and angular coding becomes a powerful tool to solve the problem. In this case, there is no geometric interpretation in the coding (remember that p does not correspond to the tangent of any angle in the concrete hyperspace), we only use angular coding to translate an interval with undened extents, R, to another with dened ones, (? ; ). Therefore, with this transformation we can use the EA to adequately search in the solution space to learn the values of these parameters.

6 IV. Structure and Genetic Learning Approach of the proposed EFS In this Section, we introduce the structure and components of the EFS presented for designing TSK FRBSs from examples. In order to do so, we rst focus on the existing problems when building an EFS along with the possible solutions we can use, and nally we present the structure selected and justify this choice according to the aforesaid aspects. There is one major problem that appears when designing FRBSs by means of EAs which has to be solved adequately in order to obtain accurate FRBS from the design process. As with other system design processes, EFSs try to combine the main aspects of the design tool and of the system being designed, an EA and an FRBS, respectively, in this case. In this way, they take one of the most interesting features of the FRBSs into account, the interpolative reasoning they develop, which is a consequence of the cooperation between the fuzzy rules composing the KB; and of the EAs, the competition induced between members of each population of possible solutions to the problem being solved, which allows us to obtain better ones. Therefore, EFSs work by combining both features, that is, they induce competition between the possible solutions to get the best possible cooperation in the nal one. The problem is to nd the best way to put this into eect. This problem is referred to as cooperation vs. competition problem (CCP) [4]. There are dierent ways to approach the problem and the diculty in solving it usually depends directly on the genetic learning approach followed by the EFS. In the last few years, some authors have proposed a new genetic learning approach, the iterative rule learning approach (IRL) [] in order to combine the advantages and solve the drawbacks presented by classical genetic learning approaches, the Michigan and Pittsburgh ones [6], when applied to the problem of designing Rule-Based Systems. There are dierent EFSs based on the IRL in the specialized literature (see [7], [1], [8], [9]). These kind of FRBS evolutionary design processes are based on three main aspects [], [7]: 1. As in the Michigan approach, each chromosome in the population represents a single fuzzy rule, but only the best individual is considered to form part of the nal KB. Therefore, in this approach the EA provides a partial solution to the problem of learning, and, contrary to both classical ones, it is run several times to obtain the complete KB. This is put into eect by including it in an iterative scheme based on obtaining the best current fuzzy rule for the system, incorporating this rule into the nal KB, and penalizing it before repeating the process. It ends when the KB is able to represent the system adequately.. This scheme is usually employed in EFSs based on inductive learning, in which the penalization of the fuzzy rules already generated uses to be developed by removing from the training data set all those examples that are already covered by the KB obtained till then. 3. On the other hand, as the EFSs using it do not envisage any relationship between the fuzzy rules generated, it is usual to employ a postprocessing method to simplify and/or adjust the KB obtained, thus forming a multi-stage EFS. Therefore, multi-stage EFSs based on the IRL try to solve adequately the CCP at the same time as reducing the search space by encoding a single fuzzy rule in each chromosome. To put this into eect, these processes follow the usual problem partitioning and divide the genetic learning process into, at least, two stages. Therefore, the CCP is solved in two steps acting at two dierent levels, with competition between rules predominating in the rst one, the generation stage, and cooperation between these generated fuzzy rules in the second one, the postprocessing stage. Hence, the generation stage forces competition between fuzzy rules, as the genetic learning processes based on the Michigan approach, to obtain a KB composed of the best possible fuzzy rules, and, on the other hand, the postprocessing stage forces cooperation between the fuzzy rules generated in the previous stage by rening or eliminating the redundant or unnecessary fuzzy rules from it to obtain the best possible KB. The EFS presented in this paper has been designed according to this genetic learning approach. Thus, it is composed of the following two stages:

7 1. An evolutionary generation process for learning a preliminary TSK KB from examples. This rst process is based on an iterative algorithm that equally divides the input space into a number of fuzzy subspaces determined by the EFS designer, and studies the existence of data in them. Each time data are located in a specic fuzzy input subspace, the process applies a TSK rule consequent learning method to determine the existing partial linear input-output relation, taking the data located in this input subspace, a subset of the global data set, as a base. The latter method is based on a (; )-ES using the angular coding proposed in the previous Section and a local measure of error, and takes into account the knowledge contained in this training data subset to improve the search process. Thus, this rst stage induces competition between the fuzzy rules that will nally compose the preliminary TSK KB obtained as output from it by considering only the subset of input-output data pairs located in each fuzzy input subspace and guiding the search process by using a local error measure.. An evolutionary renement process for adjusting both the consequent and the antecedent parts of the fuzzy rules in the preliminary KB obtained from the rst stage. The second process is composed of a special real-coded GA which includes an (1 + 1)-ES as a genetic operator to improve the search process. The algorithm works on chromosomes encoding the whole preliminary denition of the KB obtained and adjusts this denition in order to obtain the best possible cooperation between the fuzzy rules. The tness function considered in this case is based on a global error measure, more adequate for the purpose followed. The available knowledge is again considered for generating the initial population of the GA. In this case, the preliminary denition of the TSK KB is taken into account for this generation. Thus, this second stage induces cooperation in the preliminary denition of the KB in order to obtain the best possible FRBS as output from the multi-stage EFS. As will be observed in the following two Sections, which will present each one of the EFS stages, respectively, the proposed EFS presents some important aspects that should be pointed out: There is no need to penalyze the fuzzy rules already obtained in the generation stage of our EFS. As has been commented above, the working way followed by this process takes into account the fuzzy input subspaces dened by the EFS designer and determines the need for a fuzzy rule in each subspace by studying the existence of input-output data of the training set in it. Thus, we do not need to penalize each generated fuzzy rule because the fuzzy input subspace constituting the rule antecedent will not be considered for subsequent runs. Another consequence of the previous aspect is that there is no need for a postprocessing stage that simplies the preliminary KB obtained by removing the redundant rules generated from it. Therefore, the aim of the second stage is reduced to nding the best antecedent membership function and consequent parameter denition in order to obtain a nal KB presenting the best possible cooperation between the rules composing it, and thus the best possible behavior. The generation of the initial populations of EAs constituting each one of the EFS stages is made taking into account the knowledge available about the problem being solved. The idea of using the a priori knowledge available to improve the design process has been previously applied in the eld of TSK EFS. In [14], the authors put it into eect in two dierent ways, via the structural representation of the fuzzy system, i. e., considering that the problem with which they deal is of a symmetrical nature and reducing the search space accordingly, and via the initial settings of the fuzzy system parameters. This second way is considered in [3] for designing Mamdani-type EFSs as well. The knowledge available in the input-output data set used to dene the TSK rule consequent parameters will be employed to generate the initial population of the ES in the rst stage. As regards the second one, the preliminary denition of the TSK KB obtained from the previous process will be considered for this task. As commented in [14], biasing the initial populations in

8 this way does not restrict the EA to search for solutions in the neighbourhood of the initial ones, because the evolution performed is suciently strong to search points far from the initial ones. If the proposed initial solutions are approximately correct, this procedure will allow us to speed up the search process, obtaining good solutions more quickly. The tness function considered in each process is designed by taking into account the aim of each stage: competition or cooperation. In [] the authors proposed two dierent error criteria for designing EFS for TSK FRBSs, one of them considering the local behavior of the FRBS and the other one taking into account the global accuracy of the system, and concluded that an accurate TSK FRBS should present good behavior with respect to both measures. Analyzing these criteria we may conclude that they are really adequate for inducing cooperation at the level of single fuzzy rules, and to induce cooperation at the level of whole KB, respectively. Thus, using each one of them in the denition of the tness functions considered in each one of our EFS stages, we will obtain a high-performance TSK FRBS as ouput. Finally, focusing on the EA search, we need to make use of suitable techniques to develop an accurate trek on the search spaces tackled in each stage for obtaining the best possible solutions. Several factors have to be considered in order to reduce the search space complexity and to perform an adequate exploration and exploitation over it to allow the search process to be eective. A good analysis of these factors in EFS design is presented in [31]. Amongst the techniques usually employed in genetic learning processes (as well as in other genetic processes) we shall consider the following ones: to choose an adequate representation of the individuals, encoding as much information as possible, to design specic operators to perform a robust trek on the search space, with a suitable exploration-exploitation rate, and to use the knowledge available to set the initial population in order to speed up the search process (in our case, to initialize the individuals in the rst population by using the knowledge existing in the input-output training data set). V. The Evolutionary Generation Process In this Section, we introduce the evolutionary generation process that was rst presented as a single design process in [3]. First of all, the TSK rule consequent learning method is introduced. Then we propose the use of the knowledge contained in the training data set to improve the search process. Finally we present the algorithm of the generation process, which makes use of the two previous aspects. A. The TSK rule consequent learning method In this method, the (; )-ES (see the Appendix) is considered to dene TSK rule consequent parameters. The dimension n of the object variable vector ~x is determined by the number of input variables in the problem under control. When there are iv input variables, there are n = iv + 1 parameters to learn in the TSK rule consequent. The ~x part of the individuals forming the (; )-ES population is built by encoding the possible values using angular coding. EA evolution is guided by a tness function composed of a local measure of error. This will allow us to obtain an optimal TSK rule consequent in the fuzzy subspace dened by rule antecedents. The expression of the measure used is the following []: X e l E h (ey l? S(ex l )) where E is the set of input-output data pairs e l = (ex l 1; : : : ; ex l iv; ey l ) located in the fuzzy input subspace dened by the rule antecedent, h = T (A 1 (x 1 ); : : : ; A iv (x iv )) is the matching between the antecedent part of the rule and the input part of the current data pair ex l, and S(ex l ) is the ouput provided by the TSK FRBS when it receives ex l as input. The object variables of the individuals in the rst population are generated in the way shown in the next subsection, taking into account the knowledge contained in the input-output data set. As regards

9 the composition of the remaining vectors, the components of ~ are initiated to :1, and the ones in ~, when considered, are set to arctan (1). B. Using available knowledge in the design process To develop the knowledge-based generation of the initial population, we compute the following indices and obtain the following set from the input-output data set E: Pe l E eyl y med = ; y jej min = Min el Efey l g y max = Max el Efey l g h max = Max el Efh l g; h l = T (A 1 (ex l 1; ); : : : ; A iv (ex l iv)) E = fe l E=h l h max g Therefore, we generate the initial population of the proposed ES in three steps as follows: 1. Generate 1 individual initiating parameters p i, i = 1; : : : ; iv, to zero, and parameter p to the angular coding of y med.. Generate individuals, with f; : : : ;? 1g dened by the EFS designer, initiating parameters p i, i = 1; : : : ; iv, to zero, and p to the angular coding of a value computed at random in the interval [y min ; y max ]. 3. Generate the remaining?( +1) individual initiating parameters p i, i = 1; : : : ; iv, to the angular coding of values computed at random in the interval (? ; ), and p to the angular coding of a value computed from a randomly selected element e in E ( [:; 1] is provided by the EFS designer as well) in such a way that e belongs to the hyperplane dened by the TSK rule consequent generated. Thus, we shall ensure that this hyperplane intersects with the swarm of points contained in E, the most signicative ones from E. Since with small angular values, large search space zones are covered, it seems interesting to generate small values for the parameters p i in this third step. To do this, we make use of a modier function that assigns greater probability of appearance to the smaller angles according to a parameter q, also provided by the EFS designer. We use the following function: f : [; 1] f?1; 1g! (? ; ) f(x; z) = z xq Hence, the generation of the individuals is performed in this third step as follows: For j = 1; : : : ;? ( + 1) do a) For i = 1; : : : ; iv do a.1) Generate x at random in [; 1]. a.) Generate z at random in f?1; 1g. a.3) Set p i to f(x; z). b) Generate the value of p : b.1) Select e at random P from E. iv b.) Set p to ey? k=1 C?1 (p k ) ex k, where C?1 () = tan() is the inverse of C. C. Algorithm of the Evolutionary Generation Process The generation process proposed is developed by means of the following steps: 1. Perform a fuzzy partition of the input variable spaces dividing each universe of discourse into a number of equal or unequal partitions. Select a kind of membership function and assign one fuzzy set to each subspace. In this paper, we will work with symmetrical fuzzy partitions of triangular membership functions.. For each multidimensional fuzzy subspace obtained by combining the individual input variable subspaces using the and conjunction do:

10 (a) Build the set E composed of the input-ouput data pairs e E that are located in this subspace. (b) If jej 6=, i. e., if there is any data in this space zone, apply the TSK rule consequent learning method over the data set E to determine the partial linear input-output relation existing in this subspace. Therefore, no rules are considered in the fuzzy subspaces in which no data are located. (c) Add the generated rule to the preliminary KB. VI. The Evolutionary Refinement Process The evolutionary renement process is a tuning process that takes a TSK KB as input and adjusts the preliminary denitions of the antecedent membership functions and consequent parameters according to the global behavior of the KB evolved in the problem being solved, represented as a training data set. It is composed of a special real-coded GA including an (1 + 1)-ES as another genetic operator to improve the search process, guided by a global error measure over the training data set. We describe the hybrid EA components below. A. Representation A chromosome C encoding a TSK KB denition is composed of two dierent parts, C 1 and C, each one corresponding to the denition of the fuzzy membership functions considered in the antecedent part of the dierent fuzzy rules in the KB, and the other to the consequent parameters. NB NM NS ZR PS PM PB. m Fig. 3. Fuzzy partition used M We consider every fuzzy set contained in the input variable initial fuzzy partitions associated with a normalized triangular membership function (see Figure 3). A computational way to characterize it is by using a parametric representation achieved by means of the 3-tuple (a; b; c). Therefore, a primary fuzzy partition can be represented by an array composed by N 3-tuples (3 N real values) (a l ; b l ; c l ), l = 1; : : : ; N, with N being the number of terms forming the linguistic variable term set. The complete denition of all the input variable fuzzy partitions for a problem in which iv input variables are involved is encoded into the rst part C 1 of each chromosome C j in the population. C 1 is built by joining the partial representations of each one of the iv input variable fuzzy partitions as is shown below: C 1 i = (a i1 ; b i1 ; c i1 ; : : : ; a ini ; b ini ; c ini ) ; C 1 = C 1 1 C 1 ::: C iv 1 : Each one of the triangular fuzzy sets C ij = (a ij ; b ij ; c ij ), i = 1; : : : ; iv, j = 1; : : : ; N i, dening these preliminary fuzzy partitions are allowed to vary freely in any meaningful way in an interval of performance [C min ij ; C max ij ]. The extremes of these intervals are computed before running the renement process according to the preliminary fuzzy partition denitions provided by the FRBS designer, in the following way: [C min ij ; C max ij ] = [a ij? b ij? a ij ; c ij + c ij? b ij ] Therefore, the interval of performance of each gen in C 1 will depend on the fuzzy membership function to which it is associated. Each one of these intervals of performance will be the interval of adjustment

11 for the corresponding gen, c t [c l t; c r t]. If (t mod 3) = 1 then c t is the left value of the support of a fuzzy set, which is dened by the three parameters (c t, c t+1, c t+ ) and the intervals of performance are the following: c t [c l t; c r t] = [C min ; c t+1 ] c t+1 [c l t+1 ; cr t+1 ] = [c t; c t+ ] c t+ [c l t+; c r t+] = [c t+1 ; C max ] with C min and C max being the extremes of the interval of performance in the fuzzy set dened by the 3-tuple (c t, c t+1, c t+ ). These values are the only ones dening the intervals of adjustment of the c t 's that remain constant during the GA run. Figure 4 shows an example of these intervals. C c l t min c t c l t+1 c t+1 c r t+1 c c r t l c t+ t+ C max c r t+ Fig. 4. Example of triangular membership function and intervals of performance for the renement process As regards the second part of the chromosome, C, it encodes the consequent parameters of each fuzzy rule in the preliminary denition of the TSK KB. Thus, it is composed of m (iv + 1) genes, where m stands for the number of rules in the KB and iv + 1 for the number of consequent parameters for TSK fuzzy rule: C i = (p i ; p i1 ; : : : ; p i iv ) ; i = 1; : : : ; m ; C = C 1 C ::: C m : Since all of these parameters are encoded by using the proposed angular coding, the interval of performance of all the genes in C is the same, [? ; ]. Now, the fundamental underlying mechanisms of a GA, formation of an initial gene pool, tness function, and genetic operators are developed. B. Initial Gene Pool The second stage uses the available knowledge to initialize the rst population as well. In this case, we make use of the preliminary denition of the KB being optimized in order to perform this task. With M being the GA population size, the initial population generation process is performed in three steps as follows: 1. The preliminary denition of the KB taken as process input is encoded directly into a chromosome, denoted as C 1.. The following M? 1 chromosomes are initiated by generating, at random, the rst part, C 1, with each gene being in its respective interval of performance, and by encoding the preliminary detion of the rule consequent parameters in C. 3. The latter M are set up by generating C 1 in the same way followed in the previous step, and by generating the values for C by adding a random value distributed following a normal distribution N(; d) to the values in the C part of the previous chromosomes.

12 C. Evaluation of Individual Fitness The tness function is based on using a training input-output data set, E, and a global error measure, the mean square error (MSE). In this way, the adaption value associated to an individual C j is obtained by computing the error between the outputs given by the TSK FRBS using the KB encoded in the chromosome and those contained in the training data set. The tness function is thus represented by the following expression: F (C j ) = 1 jej with the same equivalences presented in Section. D. Genetic Operators X e l E (ey l? S(ex l )) The genetic operators described in the following subsections are going to be used in the evolutionary renement process. D.1 Mutation We shall use Michalewicz's non-uniform mutation operator [6], which has demonstrated an accurate behavior when working with real coding schemes. This operator works as follows: If C t v = (c 1 ; :::; c k ; :::; c H ) is a chromosome and the element c k was selected for this mutation (the domain of c k is [c kl ; c kr ]), the result is a vector C t+1 v = (c 1 ; :::; c k; :::; c H ), with k 1; :::; H, and c k = ( ck + 4(t; c kr? c k ) if a =, c k? 4(t; c k? c kl ) if a = 1, where a is a random number that may have a value of zero or one, and the function 4(t; y) returns a value in the range [; y] such that the probability of 4(t; y) being close to increases as t increases: 4(t; y) = y(1? r (1? t T )b ) where r is a random number in the interval [; 1], T is the maximum number of generations and b is a parameter chosen by the user, which determines the degree of dependency with the number of iterations. This property causes this operator to make a uniform search in the initial space when t is small, and a very local one in later stages. D. Crossover In this case we shall work with another genetic operator which has shown good behavior for RCGAs, the max-min-arithmetical crossover. This crossover operator was proposed in [3] and has been widely used in the eld of EFSs [33], [9], [7], [1]. It works in the way shown below. If C t v = (c 1 ; :::; c k ; :::; c H ) and C t w = (c 1; :::; c k; :::; c H) are to be crossed, the following four ospring are generated C t+1 1 = ac t w + (1? a)c t v C t+1 = ac t v + (1? a)c t w C t+1 3 with c t+1 3k = minfc k ; c kg C t+1 4 with c t+1 4k = maxfc k ; c k g This operator can use a parameter a which is either a constant, or a variable whose value depends on the age of the population. The resulting descendents are the two best of the four aforesaid ospring.

13 D.3 Evolution Strategy The last genetic operator to be applied consists of an (1+1)-ES. This optimization technique has been selected and integrated into the genetic recombination process in order to perform a local tuning of the best population individuals in each run. Each time a GA generation is performed, the ES will be applied over a percentage of the best dierent population individuals existing in the current genetic population. This idea has already been applied in the eld of EFSs in an evolutionary generation process of a multi-stage EFS presented in [7]. Anther variant of application of this operator to the design of EFSs not including it into a GA scheme is to be found in [1]. The basis of the ES employed are to be briey presented in the Appendix. The coding scheme and the tness function considered in the (1 + 1)-ES are the same as those used in the GA. Thus, the only changes to be performed have to be done in the generic ES mutation scheme, and the great majority of them only when working with the rst part of the chromosome, C 1. In this case, the following two changes have to be put into eect: Denition of multiple step sizes: Since the mutation strength depends directly on the value of the parameter, which determines the standard deviation of the normally distributed random variable z i. In our case, the step size cannot be a single value because the membership functions encoded in the rst part of the chromosome are dened over dierent universes and so require dierent order mutations. Therefore, a step size i = s i for each component in C 1 is going to be used in the (1+1)-ES. Anyway the relations of all i were xed by the values s i and only the common factor is adapted following the assumptions presented in [6]. Incremental optimization of the individual parameters: Usually, the dierent parent components are not related and the ES is used in its usual working mode in which all of them are adapted at the same time. Unfortunately, in our problem each three correlative parameters, (x ; x 1 ; x ), in C 1 dene a triangular-shaped membership function, and the property x x 1 x must be veried in order to obtain meaningful fuzzy sets. Therefore, there is a need to develop an incremental optimization of the individual parameters because the intervals of performance for each one of them will depend on the value of any of the others. As we have commented in the description of the coding scheme, a global interval of performance (in which the three parameters dening the membership function may vary freely) is dened for each fuzzy set involved in the optimization process. With C ij = (x ; x 1 ; x ) being the membership function currently adapted, the associated interval of performance is [C min ij ; C max ij ] = [x? x 1?x ; x + x?x1 ]. The incremental adaptation is based on generating the mutated fuzzy set C = ij (x ; x ; 1 x ) by rst adapting the modal point x 1 obtaining the mutated value x 1 dened in the interval [x ; x ], and then adapting the left and right points x and x obtaining the values x and x dened respectively in the intervals [C min ij ; x 1] and [x 1; C max ij ]. It may be clearly observed that the progressive application of this process allows us to obtain fuzzy sets freely dened in the said interval of performance. The value of the parameter s(x i ) determining the particular step sizes, i = s(x i ), is computed each time the component x i is going to be mutated. When i = 1, i. e., the modal point is being adapted, and then s(x 1 ) is equal to M in(x 1?x;x?x1). In the other two cases, i = and i =, s(x ) = M in(x?cmin ij ;x 1?x ) and s(x ) = M in(x?x 1 ;Cmax ij?x). Hence when takes value 1 at the rst ES generation, the obtaining of a large quantity of z i normal values performing a successful x i mutation (i. e., the corresponding x i = x i + z i with z i N i (; i ) lying in the expected interval for x i ) is ensured. If the mutated value lies outside, it is assigned the value of the interval extent closest to x i + z i. The next algorithm summarizes the application of the adaptation process on a membership function encoded in the parent. With C ij = (x ; x 1 ; x ) being the fuzzy set currently adapted, the steps to follow are: 1. Compute the step size of the central point, s(x 1 ) M infx1?x;x?x1g.

14 . Generate z 1 N(; 1) and compute x 1 in the following way: x 1 8>< >: x 1 + z 1, if x 1 + z 1 [x ; x ] x, if x 1 + z 1 < x x, if x 1 + z 1 > x 3. Adapt the remaining two points: M infx?c min ;x 1?x g ij (a) s(x ) Generate z N(; ) x 8>< >: x l + z, if x + z [Ci; x 1] Ci, l if x + z < C l i x 1, if x + z > x 1 M infx?x 1 ;Cmax?xg ij (b) s(x ) Generate z N(; ) x 8>< >: x + z, if x + z [x1; C r i ] x 1, if x + z < x 1 C r i, if x + z > C r i When working with the second part of the chromosome, C, the latter problem does not appear. In this case, the dierent components are not related and the mutation can be performed in its usual way. The only change that has to be made is to adapt the step size to the components in C. As all of them are dened over the same interval of performance, [? ; ], they all will use the same step size i = s i with s i = :1. With regards to the selection procedure, it is Baker's stochastic universal sampling, in which the number of any structure ospring is limited by the oor and ceiling of the expected number of ospring, together with the elitist selection. VII. Experiments developed and results obtained A. Description of the experiments developed In order to analyze the accuracy of the method proposed, we are going to perform a fuzzy modeling of three three-dimensional surfaces presenting dierent characteristics. The associated functions and the variable universes of discourse considered are shown below. The spherical model, F 1, is a smooth unimodal function, F is another smooth one presenting discontinuities in (; ) and (1; 1); while the third function, F 3, is an intermediate multimodal function being very smooth in the great majority of its denition space and presenting strong nonlinearities in some space zones; as may be observed in their graphical representations (Figure ). F 1 (x 1 ; x ) = x 1 + x ; x 1 ; x [?; ]; F 1 (x 1 ; x ) [; ] x1?x1x x1?x1x+x ; F (x 1 ; x ) = 1 x 1 ; x [; 1]; F (x 1 ; x ) [; 1] F 3 (x 1 ; x ) = e x1 sin x + e x sin x 1 ; x 1 ; x [?8; 8]; F 3 (x 1 ; x ) [; 836] These surfaces will be approximated by dierent fuzzy models derived from several evolutionary design methods. We will consider the following four processes, the rst two for generating Mamdanitype FRBSs, and the second two for designing TSK-type ones:

15 Fig.. Graphical representations of F1 (at the top), F (bottom left), and F3 (bottom right) M1. a two-stage EFS based on obtaining a complete KB by deriving the RB by means of the widely employed Wang and Mendel (WM) method [1]in the rst stage, and dening the DB by means of the descriptive genetic tuning process presented in [11], [1] in the second, M. a three-stage descriptive Mamdani-type EFS presented in [11], [1], T1. a single-stage EFS presented in [13], [14], and T. the two-stage EFS proposed in this paper. For each function, a training data set uniformly distributed in the three-dimensional denition space has been obtained experimentally. In this way, three data sets with 1681, 674 and 189 values has been generated for function F 1 to F 3 by taking, respectively, 41, 6 and 33 values for each one of the two state variables considered to be uniformly distributed in their respective intervals. We have to note that the training data set for F is composed of 674 values (instead of 676) because this function is not dened in two space points. Three other data sets have been generated for their use as test sets for evaluating the performance of the learning method, avoiding any possible bias related to the data in the training set. The size of these data sets is a percentage of the corresponding training set, ten percent to be precise. The data are obtained by generating the state variable values at random in the concrete universes of discourse for each one of them, and computing the associated output variable value. Hence, three test sets formed respectively by 168, 67 and 18 data are used to measure the accuracy of the FRBSs designed by computing the SE for them (see Section 6). The initial DB used in processes M1, M and T is constituted by three primary, equally partitioned fuzzy partitions (two corresponding to the input variables and one associated to the output one, the latter in the case of both Mamdani-type EFSs) formed by seven linguistic terms with triangular-shaped fuzzy sets giving meaning to them (as shown in Figure 3), and the adequate scaling factors to translate the generic universe of discourse into the one associated with each problem variable. As regards process T1, we should point out that we have made some changes on the initial proposal presented in [13], [14]. With the aim of improving the behavior of the EFS, but maintaining the same process working way, we have modied the coding scheme from the binary coding with eight bits

16 TABLE I Fuzzy modeling of F1 using EFS T PARAMETERS GENERATION REFINEMENT AVERAGE n n Run SE g tra SE g tst SE r tra SE r tst SE avg tra SE avg tst used by the authors to the real coding issue, which seems more adequate to deal with a problem in which the variables are dened over continuous universes of discourse. It is clear the need for genetic operators managing adequatelly the real coding issue in the implementation considered. Therefore, both crossover and mutation operators used in the evolutionary renement process presented in Section 6, the max-min-arithmetical crossover and the non-uniform mutation, are employed. On the other hand, the tness function has been adapted to the problem being solved, and the same function considered for the renement process has been adopted. Apart from this, the remaining components of the GA have remained unchanged, using the same genetic representation of the possible TSK FRBSs, the same initialization of the rst genetic population based on the a priori knowledge contained in the training data set, and the same GA parameters. To design the Inference System in the Mamdani-type FRBSs generated by means of the rst two processes, we have selected the Minimum t-norm playing the role of the implication and conjunctive operators, and the Center of Gravity weighted by the matching strategy acting as the defuzzication operator [34]. In the TSK-type ones obtained from processes T1 and T, the role of conjunctive operator is played by the Minimum t-norm as well. We have performed dierent runs of the proposed process, T, using two of the four usual combinations of dimensions of vectors ~ and ~ (see the Appendix), (~,~) = f(n; ); (n; n(n?1) )g,and three dierent values for the parameter, = f; :1; :g, dening the percentage of population individuals to which the ES is applied in the renement stage. The remaining parameter values for EFS T are the following: 1. Evolutionary Generation Process: iterations, = 1, = 1, = : = 3, = :7, q =, ~r = (r ~x ; r ~ ; r ~ ) = (3; ; ), and ~ = ( ~x ; ~ ; ~ ) = (; ; 1).. Evolutionary Renement Process: 1 GA iterations, N = 61, P c = :6, P m = :1 (per individual), a = :3, b =, (1 + 1)? ES, d = :1.

17 B. Analysis of the results obtained using the proposed EFS The results obtained in the dierent experiments developed with design process T are collected in Tables 1, and 3, where SE x and tra SEx tst stand for values obtained by the specic TSK FLC designed in the SE measure computed over the training and test data sets, respectively (x can be equal to g and r standing for the generation and renement stages). All the KBs learnt are composed of 49 fuzzy rules. The nal values included in the last two columns of each table, noted as SE avg tra and SE avg tst respectively, are computed as an average of three EFS runs with dierent values for the random seed in order to give us more information about the process accuracy. TABLE II Fuzzy modeling of F using EFS T PARAMETERS GENERATION REFINEMENT AVERAGE n n Run SE g tra SE g tst SE r tra SE r tst SE avg tra SE avg tst In view of the dierent results obtained, some conclusions can be drawn with respect to the dierent parameters existing in the proposed EFS. First, it can be observed that, in the great majority of the cases, the best results are obtained when not considering the angle vector in the evolutionary generation process (; )? ES, i. e., when using the combination (~, ~) = (n; ). On the other hand, it is not so easy to decide the best value for the the parameter in the second stage, because the behavior of the learning process depends on the concrete application when varying the value of this parameter. There are no signicative dierences among the results obtained in the fuzzy modeling of F 1 with the three dierent values proposed, = f; :1; :g. The best individual result is obtained with = and the best average one with = :1, but we have to remark again that the dierences are not signicative. The conclusions are dierent when analyzing the results obtained in the second of the applications, the fuzzy modeling of F. In this case, the bigger the value of is, i. e., the more the (1+1)?ES is applied to optimize the best individuals in the GA populations, the better the TSK FRBS designed models the data contained in the training data set but the worse it generalizes the data in the test set. In this case, the application of the (1 + 1)? ES provokes an undesirable overlearning in the learning process. The best individual and average behavior are presented by FRBSs designed with =. Finally, when analyzing the results obtained in the third application, we nd exactly the opposite. Now, the