Implementation Of Evolutionary Fuzzy Systems - Fuzzy .
IEEE TRANSACTIONS ON FUZZY SYSTEMS, VOL. 7, NO. 2, APRIL 1999109Implementation of Evolutionary Fuzzy SystemsYuhui Shi, Senior Member, IEEE, Russell Eberhart, Senior Member, IEEE, and Yaobin Chen, Member, IEEEAbstract— In this paper, evolutionary fuzzy systems are discussed in which the membership function shapes and types andthe fuzzy rule set including the number of rules inside it areevolved using a genetic (evolutionary) algorithm. In addition,the genetic parameters (operators) of the evolutionary algorithmare adapted via a fuzzy system. Benefits of the methodology areillustrated in the process of classifying the iris data set. Possibleextensions of the methods are summarized.Index Terms—Fuzzy expert systems, genetic algorithm, membership.I. INTRODUCTIONA. BackgroundFUZZY systems are being used successfully in an increasing number of application areas; they use linguisticrules to describe systems. These rule-based systems are moresuitable for complex system problems where it is very difficult,if not impossible, to describe the system mathematically.One of the most important considerations in designing anyfuzzy system is the generation of the fuzzy rules as wellas the membership functions for each fuzzy set. In mostexisting applications, the fuzzy rules are generated by expertsin the area, especially for control problems with only a fewinputs. With an increasing number of variables, the possiblenumber of rules for the system increases exponentially, whichmakes it difficult for experts to define a complete rule set forgood system performance. An automated way to design fuzzysystems might be preferable.There are many ways to attack this problem. A straightforward approach is to use clustering algorithms (like the c-meansclustering algorithm, fuzzy c-means clustering algorithm, etc.[2]) or similar methods to partition the pattern space intomany subspaces with or without overlaps among them, thenmap the center of each cluster into a rule according to thedefinitions of fuzzy variables [1], [24]. One disadvantageof this approach is that the extracted rules are independentof the membership functions so there is no guarantee thatthe fuzzy system obtained will have sufficiently good performance, especially for a complex system problem with alarge number of input variables. In many cases, however, thesystem’s performance can be improved by further tuning theManuscript received September 23, 1997; revised September 22, 1998. Thiswork was supported by Delphi Engine and Energy Management.The authors are with the Purdue School of Engineering and Technology,Indiana University/Purdue University Indianapolis, Indianapolis, IN 46202USA.Publisher Item Identifier S 1063-6706(99)02801-5.membership functions and selecting suitable fuzzification anddefuzzification methods.The design of a fuzzy system can be formulated as asearch problem in high-dimensional space where each pointrepresents a rule set, membership functions, and the corresponding system’s behavior. Given some performance criteria,the performance of the system forms a hypersurface in thespace. Developing the optimal fuzzy system design is equivalent to finding the optimal location of this hypersurface. Thehypersurface has the following characteristics. The hypersurface is infinitely large since the number ofpossible fuzzy sets for each variable is unbounded. The hypersurface is nondifferentiable since changes inthe number of fuzzy sets are discrete and can have adiscontinuous effect on the fuzzy system’s performance. The hypersurface is complex and noisy since the mappingfrom a fuzzy rule set to its performance is indirect anddependent on the evaluation method used. The hypersurface is multimodal since different fuzzyrule sets and/or membership functions may have similarperformance. The hypersurface is deceptive since similar fuzzy rulesets and membership functions may have quite differentperformances.These characteristics seem to make evolutionary algorithmssuch as genetic algorithms (GA’s) better candidates for searching the hypersurface than conventional methods such as hillclimbing search methods.B. Previous ApproachesGA’s are commonly used evolutionary algorithms that provide a way to search poorly understood, irregular spaces. Oneof the key issues in the evolutionary design of fuzzy systemsusing GA’s is their genotype representation; that is, what isencoded into the chromosomes.Thrift [23] and Hwang and Thompson [11] encode allthe rules into the chromosome while fixing the membershipfunctions. Using several critical points to represent each membership function while using all the possible rules, Karr [15]and Karr and Gentry [16] use GA’s to evolve these criticalpoints; that is, to adjust the membership functions. Since ina fuzzy system the membership functions and rule set arecodependent, they should be designed or evolved at the sametime. Homaifar and McCormick [10] use GA’s to tune themembership functions and evolve the rule set at the sametime. The base length of each triangular membership functionand all possible rules are encoded into the chromosomes.Similar to [10], Lee and Takagi [17] also encode membership1063–6706/99 10.00 1999 IEEE
110functions and all the rules into the chromosome, but have adifferent way to encode the triangular membership functions.They restrict adjacent membership functions to fully overlapand also constrain one membership function to have its centerresting at the lower boundaries of the input range. By usingmembership function centers need tothis coding, onlybe encoded, where is the maximum number of partitions fora given dimension. The above-mentioned methods encode allpossible rules into the chromosome. There are some drawbacksby doing so [3]: first, the computational efficiency associatedwith fuzzy logic is lost using a high number of rules andsecond, the robustness decreases with increasing the numberof rules. This is especially true when the dimension of theinputs and the number of fuzzy sets for each input variablebecome great since the number of possible rules exponentiallyincreases with the these numbers.In most applications, not all possible rules need to be used;only a portion of the rules are needed. So only this portionof rules should be encoded into the chromosome and evolved.By doing so, the length of the chromosome will be reducedgreatly and, therefore, will be suitable for larger problems.Karr [14] considers a very special case where the number ofrules is provided by an expert, together with many completerules forming the rule set and the antecedents for the remainingones so only the consequent parts of the latter type need to beevolved and, therefore, need to be encoded in the chromosome.This is not the case for many applications. Most of the time,it will be difficult, if not impossible, to know a priori exactlyhow many rules are required to be included in the rule set;only a maximal number can be guessed or estimated.It is better to encode the number of rules to be included inthe rule set together with rules and/or membership functionsinto the chromosome to be evolved. There are several waysto do this. Lee and Takagi [18], [19] proposed encodingmembership functions and fitness functions in chromosomes.Shimojima et al. [22] and Inoue et al. [12] defined membershipfunctions for each rule and encoded effectiveness informationfor each rule and membership function. Shimojima et al. usedfitness functions that encouraged minimizing the number ofrules; Inoue et al. used a method they called ‘‘forgetting.’’Due to the highly complex and nonlinear characteristic ofthe problem space, uniform distribution of the fuzzy sets isusually not optimal. The performance of a fuzzy classificationsystem based on fuzzy if–then rules depends on the choiceof a fuzzy partition. If a fuzzy partition is too coarse, theperformance may be low. If a fuzzy partition is too fine,many fuzzy if–then rules cannot be generated because of thelack of training patterns in the corresponding fuzzy subspaces.For a problem, some parts of pattern space might requirefine partition, while other parts require only coarse partition.Therefore, the choice of an appropriate fuzzy partition isimportant and difficult. To cope with this difficulty, Ishibuchi,Nozaki et al. [13] introduce the concept of distributed fuzzyif–then rules. They encode all fuzzy if–then rules corresponding to several different fuzzy partitions into a tri-valueand apply GA’s to remove the unnecessarystringrules from fuzzy if–then rules corresponding to the differentfuzzy partitions. Since each possible rule for each subspaceIEEE TRANSACTIONS ON FUZZY SYSTEMS, VOL. 7, NO. 2, APRIL 1999is coded into the chromosome, the length of the chromosomeis very large when the number of input dimensions and/orof different partitions is large. Other ways to tackle thenonlinear distribution problem should be sought. A naturaland better way is to employ nonlinear functions in addition tolinear functions as membership functions. Natural choices areGaussian functions, sigmoid functions, etc. Through inclusionof linear and nonlinear functions, the type of membershipfunction for each fuzzy set will not be predetermined, butinstead be evolved during the design process.GA behavior is determined by the exploitation and exploration relationships kept throughout the run [8]. Given fixedsettings for parameters such as crossover and mutation ratesthrough the run, the GA may have its exploitation/explorationrelationship (EER) disproportioned and produce a lack ofdiversity in the population [8]. Accordingly, GA parametersettings should be adapted through the run. Since the interaction between GA parameter settings and GA performance iscomplex and unknown, finding algorithms to achieve optimaladaptive parameter settings is very difficult, if not impossible.This suggests the use of fuzzy systems for adapting GAparameters. The main idea is to use a fuzzy system whoseinputs are any combination of GA performance measures orcurrent control parameters and whose outputs are GA controlparameters [8]. Lee and Takagi [17] propose an automaticlearning technique to design fuzzy rules for tuning GA’s. Dueto the heavy computation requirement, they first apply thistechnique to design a fuzzy system to tune a GA for solvingthe simple DeJong F1 function, then apply the obtained fuzzysystem to tune the GA to solve other different and morecomplex problems. This technique is very similar to the metaGA of Grefenstette [7]. This implies that the robustness of theobtained fuzzy rules strongly depends on the problems to besolved and the performance measures used by the technique.As we know from the literature, there exist a lot of adaptiveGA’s to tackle the lack of diversity problem and a certain bodyof expertise, experience, and knowledge on GA’s has becomeavailable as a result of empirical studies conducted over anumber of years [9]. This human expertise and knowledge isvery useful and should be the first choice for designing a fuzzysystem to tune a GA to reach a suitable EER for avoidingpremature convergence and improving GA behavior.C. Current ApproachIn this paper, a GA-based method to evolve a fuzzy expertsystem is discussed. It not only can evolve the rule set(including the optimal number of rules inside the rule set),tune the membership functions, and evolve the membershipfunction types, but also scales well and is, therefore, useful forlarge complex problems. In addition, a fuzzy expert system isdesigned from our experience and knowledge and is used toadapt the genetic parameters of the GA.The paper is organized as follows. Section II describesthe GA. Section III describes the fuzzy expert system. InSection IV, details are given on how to design the fuzzysystem using a GA. In Sections II–IV, the implementationsdescribed are related to the simulation example of Section V.Example results are given in Section V, which demonstrate
SHI et al.: IMPLEMENTATION OF EVOLUTIONARY FUZZY SYSTEMSthat the method discussed in this paper is effective andefficient.II. GENETIC ALGORITHMSGA’s are search algorithms that reflect in a primitive waysome of the processes of natural evolution including crossover,mutation, and survival of the fittest. They are analogousto neural networks’ status as primitive approximations tobiological neural processing. GA’s provide powerful searchmechanisms that can be used in optimization or classificationapplications. While stochastic in nature, GA’s perform ahighly effective search of the problem hyperspace, efficientlydirecting the search to promising regions. GA paradigmsare effective in a wide variety of applications; they are notdesigned to solve only a narrow class of problems. GA’swork with a population of points rather than a single point.Each ‘‘point’’ is a vector in hyperspace representing onepotential (or candidate) solution to the optimization problem.A population is, thus, just an ensemble or set of hyperspacevectors. Each vector is called a chromosome in the population.The number of elements in each vector (chromosome) dependson the number of parameters in the optimization problemand the way to represent the problem. How to represent theproblem as a string of elements is one of the critical factors insuccessfully applying a GA (or other evolutionary algorithm)to a problem.GA paradigms do not require information that is auxiliaryor related to the problem such as function derivatives, whilemany hill-climbing search paradigms, for example, requirethe calculation of derivatives in order to successfully explorethe local maximum or minimum. So GA’s can be appliedto wider areas, especially those difficult for traditional hillclimbing methods. A typical series of operations carried outwhen implementing a GA paradigm is:1) initialize the population;2) calculate fitness for each chromosome in population;3) reproduce selected chromosomes to form a new population;4) perform crossover and mutation on the population;5) loop to step 2) until some condition is met.Initialization of the population is commonly done by seedingthe population with random values. The fitness value is proportional to the performance measurement of the function beingoptimized. The calculation of fitness values is conceptuallysimple. It can, however, be quite complex to implement in away that optimizes the efficiency of the GA’s search of theproblem space. It is this fitness that guides the search of theproblem space.It is not unusual for most (if not all) of the fitness valuesafter, say, a few dozen to a few hundred generations, to bequite high. In cases where the fitness value can range from 0to 1, for example, most or all of the fitness values may be 0.9or higher. This lowers the differential between fitnesses thatprovides the impetus for effective reproduction, i.e., ensuringthat higher fitness values have a significantly higher probabilityof reproduction. One way around this problem is to shift thefitness values in some manner.111After fitness calculation, the next step is reproduction.Reproduction comprises forming a new population, usuallywith the same total number of chromosomes, by selectingfrom members of the current population using a stochasticprocess that is weighted by each of their fitness values. Thehigher the fitness, the more likely it is that the chromosomewill be selected for the new generation. One commonly usedway is a ‘‘roulette wheel’’ procedure that assigns a portion ofa roulette wheel to each population member where the size ofthe portion is proportional to the fitness value. This procedureis often combined with the elitist strategy, which ensures thatthe chromosome with the highest fitness is always copied intothe next generation.The next operation is called crossover. To many evolutionary computation practitioners, crossover is what distinguishes a GA from other evolutionary computation paradigms.Crossover is the process of exchanging portions of the stringsof two ‘‘parent’’ chromosomes. An overall probability isassigned to the crossover process, which is the probabilitythat given two parents, the crossover process will occur.This probability is often in the range of 0.65–0.80. The finaloperation in the typical GA procedure is mutation. Mutationconsists of changing an element’s value at random, often witha constant probability for each element in the population.The probability of mutation can vary widely according to theapplication and the preference of the person exercising the GA.However, values of between 0.001 and 0.01 are not unusualfor mutation probability.In the example simulation in this paper, the ‘‘roulettewheel’’ procedure with the elitist strategy [6] is used forreproduction, where the portions of the roulette wheel assignedto population members are proportional to the shifted fitnessvalues [4]. The original fitness values are linearly shiftedwith the minimal fitness mapping to 0.1. The crossover operator used is two-point crossover with a default crossoverprobability of 0.75 [6]. The mutation operator used in thispaper depends on our chromosome representation and will beexplained later. Note that in our evolutionary fuzzy systemdescribed in Section IV, fuzzy rules can be used to adaptcrossover probability and mutation rate.III. FUZZY EXPERT SYSTEMSFuzzy logic provides a general concept for descriptionand measurement. Most fuzzy logic systems encode humanreasoning into a program to make decisions or control asystem. Fuzzy logic comprises fuzzy sets, which are a wayof representing nonstatistical uncertainty and approximate reasoning, which includes the operations used to make inferencesin fuzzy logic. Unlike traditional Aristotelian two-valued logic,in fuzzy logic, fuzzy set membership occurs by degree over therange [0,1], which is represented by a membership function. Itis this function that is the fuzzy set. The function can be linearor nonlinear. Commonly used are left triangle, right triangle,triangle, Gaussian, and sigmoid functions, as shown in Fig. 1.Definitions of these membership functions as used in this paperare as follows.
112IEEE TRANSACTIONS ON FUZZY SYSTEMS, VOL. 7, NO. 2, APRIL 1999Fig. 2. Membership functions of variable temperature.Fig. 1. Six commonly used membership functions.Left triangle membership functionifififRight triangle membership functionifififTriangle membership functionififififGaussian membership functionwhereSigmoid membership functionwhereReverse sigmoid membership functionOther definitions are possible, of course, but the authorshave found these to be useful for a variety of problems.From the definitions, it can be seen that each membershipandfunction is determined by two values—the start pointthe end point .Theoretically, each fuzzy variable can have many fuzzy setswith each having its own membership function, but commonlyused are three, five, seven, or nine fuzzy sets for each fuzzyvariable. Fig. 2 shows a fuzzy variable temperature havingfive fuzzy sets with triangular membership functions.The general form of a Mamdani-type fuzzy rule in a fuzzyexpert system isIfthenisandisisandisiswhere each is the consequent (output) variable whose valueis an input variable (an antecedent), andis inferred, eachandis a fuzzy set represented by a membershipeachfunction. For simplicity, only Mamdani-type fuzzy rules areconsidered in this paper. A fuzzy system is defined if andonly if its rule set and its membership functions associatedwith its fuzzy sets are defined. An example definition of afuzzy system as used in this paper is given in List I.The number ‘‘5’’ in the first line specifies the number ofrules listed in the rule set. The next line contains the numberof input fuzzy variables (4) followed by the number of outputfuzzy variables (1). Next, the fuzzy sets for all input and outputvariables are defined. In accordance with the second line, wedefine four input and one output fuzzy variables. The nextline, input one 5 0.4 1.0, defines the first fuzzy input variable’sname as
Implementation of Evolutionary Fuzzy Systems Yuhui Shi, Senior Member, IEEE, Russell Eberhart, Senior Member, IEEE, and Yaobin Chen, Member, IEEE Abstract— In this paper, evolutionary fuzzy systems are dis-cussed in which the membership function shapes and types and the fuzzy rule set including the number of rules inside it are
ing fuzzy sets, fuzzy logic, and fuzzy inference. Fuzzy rules play a key role in representing expert control/modeling knowledge and experience and in linking the input variables of fuzzy controllers/models to output variable (or variables). Two major types of fuzzy rules exist, namely, Mamdani fuzzy rules and Takagi-Sugeno (TS, for short) fuzzy .
fuzzy controller that uses an adaptive neuro-fuzzy inference system. Fuzzy Inference system (FIS) is a popular computing framework and is based on the concept of fuzzy set theories, fuzzy if and then rules, and fuzzy reasoning. 1.2 LITERATURE REVIEW: Implementation of fuzzy logic technology for the development of sophisticated
Different types of fuzzy sets [17] are defined in order to clear the vagueness of the existing problems. D.Dubois and H.Prade has defined fuzzy number as a fuzzy subset of real line [8]. In literature, many type of fuzzy numbers like triangular fuzzy number, trapezoidal fuzzy number, pentagonal fuzzy number,
Fuzzy Logic IJCAI2018 Tutorial 1. Crisp set vs. Fuzzy set A traditional crisp set A fuzzy set 2. . A possible fuzzy set short 10. Example II : Fuzzy set 0 1 5ft 11ins 7 ft height . Fuzzy logic begins by borrowing notions from crisp logic, just as
of fuzzy numbers are triangular and trapezoidal. Fuzzy numbers have a better capability of handling vagueness than the classical fuzzy set. Making use of the concept of fuzzy numbers, Chen and Hwang [9] developed fuzzy Technique for Order of Preference by Similarity to Ideal Solution (TOPSIS) based on trapezoidal fuzzy numbers.
ii. Fuzzy rule base: in the rule base, the if-then rules are fuzzy rules. iii. Fuzzy inference engine: produces a map of the fuzzy set in the space entering the fuzzy set and in the space leaving the fuzzy set, according to the rules if-then. iv. Defuzzification: making something nonfuzzy [Xia et al., 2007] (Figure 5). PROPOSED METHOD
2D Membership functions : Binary fuzzy relations (Binary) fuzzy relations are fuzzy sets A B which map each element in A B to a membership grade between 0 and 1 (both inclusive). Note that a membership function of a binary fuzzy relation can be depicted with a 3D plot. (, )xy P Important: Binary fuzzy relations are fuzzy sets with two dimensional
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