A GA-based method for constructing fuzzy systems directly from numerical data

A method based on the concepts of genetic algorithm (GA) and recursive least-squares method is proposed to construct a fuzzy system directly from some gathered input-output data of the discussed problem. The proposed method can find an appropriate fuzzy system with a low number of rules to approach an identified system under the condition that the constructed fuzzy system must satisfy a predetermined acceptable performance. In this method, each individual in the population is constructed to determine the number of fuzzy rules and the premise part of the fuzzy system, and the recursive least-squares method is used to determine the consequent part of the constructed fuzzy system described by this individual. Finally, three identification problems of nonlinear systems are utilized to illustrate the effectiveness of the proposed method.     

SIGNIFICANT FUZZY RULES EXTRACTION BY AN SVD–QR-BASED APPROACH

ABSTRACT

In this article, an SVD–QR-based approach is proposed to extract the important fuzzy rules from a rule base with several fuzzy rule tables to design an appropriate fuzzy system directly from some input-output data of the identified system. A fuzzy system with fuzzy rule tables is defined to approach the input-output pairs of an identified system. In the rule base of the defined fuzzy system, each fuzzy rule table corresponds to a partition of an input space. In order to extract the important fuzzy rules from the rule base of the defined fuzzy system, a firing strength matrix determined by the membership functions of the premise fuzzy sets is constructed. According to the firing strength matrix, the number of important fuzzy rules is determined by the Singular Value Decomposition SVD, and the important fuzzy rules are selected by the SVD–QR-based method. Consequently, a reconstructed fuzzy rule base composed of significant fuzzy rules is determined by the firing strength matrix. Furthermore, the recursive least-squares method is applied to determine the consequent part of the reconstructed fuzzy system according to the gathered input-output data so that a fine fuzzy system is determined by the proposed method. Finally, three nonlinear systems illustrate the efficiency of the proposed method.     

MULTITUNING FUZZY CONTROL SYSTEMS DESIGN

A multituning fuzzy control system structure that involves two simple, but effective tuning mechanisms, is proposed: one is called fuzzy control rule tuning mechanism (FCRTM); the other is called dynamic scalar tuning mechanism (DSTM). In FCRTM, it is used to generate the necessary control rules with a center extension method. In DSTM, it contains three fuzzy IF-THEN rules for determining the appropriate scaling factors for the fuzzy control system. In this paper, a method based on the genetic algorithm (GA) is proposed to simultaneously choose the appropriate parameters in FCRTM and DSTM. That is, the proposed GA-based method can automatically generate the required rule base of fuzzy controller and efficiently determine the appropriate map for building the dynamic scalars of fuzzy controller. A multiobjective fitness function is proposed to determine an appropriate parameter set such that not only the selected fuzzy control structure has fewer fuzzy rules, but also the controlled system has a good control performance. Finally, an inverted pendulum control problem is given to illustrate the effectiveness of the proposed control scheme.     

A hybrid clustering and gradient descent approach for fuzzymodeling

In this paper, a hybrid clustering and gradient descent approach is proposed for automatically constructing a multi-input fuzzy model where only the input-output data of the identified system are available. The proposed approach is composed of two steps: structure identification and parameter identification. In the process of structure identification, a clustering method is proposed to provide a systematic procedure to determine the number of fuzzy rules and construct an initial fuzzy model from the given input-output data. In the process of parameter identification, the gradient descent method is used to tune the parameters of the constructed fuzzy model to obtain a more precise fuzzy model from the given input-output data. Finally, two examples of nonlinear system are given to illustrate the effectiveness of the proposed approach.     

Orthogonal Fuzzy Rule-Based Systems: Selection of Optimum Rules

In this paper, the concept of orthogonal fuzzy rule-based systems is introduced. Orthogonal rules are an extension to the definition of orthogonal vectors when the vectors are vectors of membership functions in the antecedent part of rules. The number and combination of rules in a fuzzy rule-based system will be optimised by applying orthogonal rules. The number of rules, and subsequently the complexity of the fuzzy rule-based systems, are directly associated with the number of input variables and distinguishable membership functions for each individual input variable. A subset of rules can be used if it is known which subset provides closer behaviour to the case when all rules are used. Orthogonal fuzzy rule-based systems are proposed as a judgment as to whether the optimal rules are selected. The application of orthogonal fuzzy rules becomes essential when fuzzy rule-based systems containing many inputs are used. An illustrative example is presented to create a model for the solder paste printing stage of surface mount tech-nology.     

Selecting fuzzy if-then rules for classification problems usinggenetic algorithms

This paper proposes a genetic-algorithm-based method for selecting a small number of significant fuzzy if-then rules to construct a compact fuzzy classification system with high classification power. The rule selection problem is formulated as a combinatorial optimization problem with two objectives: to maximize the number of correctly classified patterns and to minimize the number of fuzzy if-then rules. Genetic algorithms are applied to this problem. A set of fuzzy if-then rules is coded into a string and treated as an individual in genetic algorithms. The fitness of each individual is specified by the two objectives in the combinatorial optimization problem. The performance of the proposed method for training data and test data is examined by computer simulations on the iris data of Fisher     

A multiCell visual tracking algorithm using multi-task particle swarm optimization for low-contrast image sequences

In terms of the varying number of cell population, shape deformation, collision and uneven movement, a novel method based on multi-task particle swarm optimization (PSO) algorithm without explicit detection module, named MTPSO tracking method, is developed for automatic tracking of biological cells in time-lapse low-contrast microscopy image sequences. For tracking existing cells from the previous frames, a PSO-based tracking module is firstly implemented to give the initial positions of existing cells according to the previous estimated state of each cell, then a PSO-based contour module is proposed to determine the corresponding contour of each cell and finally achieve a precise position tracking by an iterative centroid updating process. For tracking new appearing cells at the current frame, a PSO-based discovery module, followed by the aforementioned PSO-based contour module, is proposed to search for new potential cells through appropriate initialization of particle swarm and searching mechanism. MTPSO tracking method is tested over a number of different real cell image sequences and is shown to provide high accuracy both in position and contour estimate of each cell in various challenging cases. Furthermore, it is more competitive against the state-of-the-art multi-object tracking methods in terms of performance measures such as FAR, FNR, LTR, and LSR.     

PSO-based intelligent integration of design and control for one kind of curing process

In this paper, a PSO-based intelligent integration of design and control is proposed for one kind of nonlinear curing process. This method combines the merits of both fuzzy modeling/control and PSO method, where fuzzy modeling/control is proposed to approximate/control the nonlinear process in a large operating region and the PSO-based intelligent optimization method is developed to solve non-convex and non-differential integration problem with design and control optimized simultaneously. Finally, the proposed method is compared with the traditional sequential method on controlling the temperature profile of a nonlinear curing process.     

A new approach to fuzzy modeling of nonlinear dynamic systems with noise: relevance vector learning mechanism

This paper presents a new fuzzy inference system for modeling of nonlinear dynamic systems based on input and output data with measurement noise. The proposed fuzzy system has a number of fuzzy rules and parameter values of membership functions which are automatically generated using the extended relevance vector machine (RVM). The RVM has a probabilistic Bayesian learning framework and has good generalization capability. The RVM consists of the sum of product of weight and kernel function which projects input space into high dimensional feature space. The structure of proposed fuzzy system is same as that of the Takagi-Sugeno fuzzy model. However, in the proposed method, the number of fuzzy rules can be reduced under the process of optimizing a marginal likelihood by adjusting parameter values of kernel functions using the gradient ascent method. After a fuzzy system is determined, coefficients in consequent part are found by the least square method. Examples illustrate effectiveness of the proposed new fuzzy inference system.     

Modeling of a Fuzzy Expert System for Choosing an Appropriate Supply Chain Collaboration Strategy

Nowadays, there has been a great interest for business enterprises to work together or collaborate in the supply chain. It is thus possible for them to gain a competitive advantage in the marketplace. However, determining the right collaboration strategy is not an easy task. Namely, there are several factors that need to be considered at the same time. In this regard, an expert system based on fuzzy rules is proposed to choose an appropriate collaboration strategy for a given supply chain. To this end, firstly, the factors that are significant for supply chain collaboration are extracted via an extensive review of literature. Then, a simulation model of a supply chain is constructed to reveal the performance of collaborative practices under various scenarios. Thereby, it is made possible to establish fuzzy rules for the expert system. Finally, feasibility and practicability of our proposed model is verified with an illustrative case.     

Performance evaluation of fuzzy rule-based classification systems obtained by multi-objective genetic algorithms

In this paper, we examine the classification performance of fuzzy if-then rules selected by a GA-based multi-objective rule selection method. This rule selection method can be applied to high-dimensional pattern classification problems with many continuous attributes by restricting the number of antecedent conditions of each candidate fuzzy if-then rule. As candidate rules, we only use fuzzy if-then rules with a small number of antecedent conditions. Thus it is easy for human users to understand each rule selected by our method. Our rule selection method has two objectives: to minimize the number of selected fuzzy if-then rules and to maximize the number of correctly classified patterns. In our multi-objective fuzzy rule selection problem, there exist several solutions (i.e., several rule sets) called “non-dominated solutions” because two conflicting objectives are considered. In this paper, we examine the performance of our GA-based rule selection method by computer simulations on a real-world pattern classification problem with many continuous attributes. First we examine the classification performance of our method for training patterns by computer simulations. Next we examine the generalization ability for test patterns. We show that a fuzzy rule-based classification system with an appropriate number of rules has high generalization ability.     

DESIGN OF FUZZY SYSTEMS WITH FEWER RULES

This article presents an innovative method for designing fuzzy systems composed of fewer fuzzy rules. The conventional approach to fuzzy-system design usually assumes that there exists no correlation among input variables, therefore, grid-type fuzzy partitions are widely adopted. However, in many cases, it is likely that input variables are highly correlated with one another. To avoid the problem of growth of partitioned grids in some complex system, we used an aggregation of hyperrectangulars with different size and different positions to approximate fuzzy partitions that are arbitrarily shaped. The corresponding parameters defining these hyperrectangulars are selected by using genetic algorithms. Furthermore, the number of fuzzy rules of the constructed system can be automatically minimized by choosing a special fitness function that takes this factor into account. Finally, an inverted pendulum control and nonlinear modeling problems are utilized to illustrate the effectiveness of the proposed method.     

A systematic approach to a self-generating fuzzy rule-table forfunction approximation

In this paper, a systematic design is proposed to determine fuzzy system structure and learning its parameters, from a set of given training examples. In particular, two fundamental problems concerning fuzzy system modeling are addressed: 1) fuzzy rule parameter optimization and 2) the identification of system structure (i.e., the number of membership functions and fuzzy rules). A four-step approach to build a fuzzy system automatically is presented: Step 1 directly obtains the optimum fuzzy rules for a given membership function configuration. Step 2 optimizes the allocation of the membership functions and the conclusion of the rules, in order to achieve a better approximation. Step 3 determines a new and more suitable topology with the information derived from the approximation error distribution; it decides which variables should increase the number of membership functions. Finally, Step 4 determines which structure should be selected to approximate the function, from the possible configurations provided by the algorithm in the three previous steps. The results of applying this method to the problem of function approximation are presented and then compared with other methodologies proposed in the bibliography.     

Multi-objective genetic algorithm based approaches for mining optimized fuzzy association rules

Association rules form one of the most widely used techniques to discover correlations among attribute in a database. So far, some efficient methods have been proposed to obtain these rules with respect to an optimal goal, such as: to maximize the number of large itemsets and interesting rules or the values of support and confidence for the discovered rules. This paper first introduces optimized fuzzy association rule mining in terms of three important criteria; strongness, interestingness and comprehensibility. Then, it proposes multi-objective Genetic Algorithm (GA) based approaches for discovering these optimized rules. Optimization technique according to given criterion may be one of two different forms; The first tries to determine the appropriate fuzzy sets of quantitative attributes in a prespecified rule, which is also called as certain rule. The second deals with finding both uncertain rules and their appropriate fuzzy sets. Experimental results conducted on a real data set show the effectiveness and applicability of the proposed approach.     

Approximation theorem of the fuzzy transform in fuzzy reasoning and its application to the scheduling problem

In this paper, the theoretical foundation of fuzzy reasoning is analyzed, and the idea that the fuzzy transform given in the fuzzy reasoning method should be continuous with respect to a certain fuzzy distance is proposed. Also, the fuzzy transforms given in the two fuzzy reasoning methods, the Mamdani method and the III method, are proved to be continuous. Based on the continuity of the fuzzy transform, the approximation theorem of the continuous fuzzy number transform is proven. Then, on the basis of the approximation theorem, a simple fuzzy number transform is constructed in order to implement the fuzzy reasoning based on multiple rules. At last, the fuzzy reasoning based on multiple rules implemented by a simple fuzzy number transform is applied to machine scheduling problems, and numerical computational results of different scale scheduling problems with the objective of minimizing the total number of tardy jobs show that it is more effective than usual heuristics based on rules, and in the computational time it has the obvious advantage over the reasoning by fuzzy rules directly.     

An efficient algorithm for automatically generating multivariable fuzzy systems by Fourier series method

By exploiting the Fourier series expansion, we have developed a new constructive method of automatically generating a multivariable fuzzy inference system from any given sample set with the resulting multivariable function being constructed within any specified precision to the original sample set. The given sample sets are first decomposed into a cluster of simpler sample sets such that a single input fuzzy system is constructed readily for a sample set extracted directly from the cluster independent of the other variables. Once the relevant fuzzy rules and membership functions are constructed for each of the variables completely independent of the other variables, the resulting decomposed fuzzy rules and membership functions are integrated back into the fuzzy system appropriate for the original sample set requiring only a moderate cost of computation in the required decomposition and composition processes. After proving two basic theorems which we need to ensure the validity of the decomposition and composition processes of the system construction, we have demonstrated a constructive algorithm of a multivariable fuzzy system. Exploiting an implicit error bound analysis available at each of the construction steps, the present Fourier method is capable of implementing a more stable fuzzy system than the power series expansion method of ParNeuFuz and PolyNeuFuz, covering and implementing a wider range of more robust applications.     

基于模糊控制的双闭环直流可逆调速系统的设计

对于双闭环直流可逆调速系统,提出了一种将模糊控制与常规PI控制相结合应用在转速环调节器设计的方法。根据工程经验与专家知识所确定的模糊控制规则,进行模糊推理,实现转速环调节器参数的动态整定。应用Matlab软件构建了双闭环直流可逆调速系统的仿真模型,并对转速环分别采用模糊PI控制器和常规PI控制器的直流可逆调速系统分别进行仿真实验并对比结果。从仿真结果可以得出采用模糊控制可以对直流可逆调速系统的动态与静态特性、抗扰性能、恢复性能以及跟踪性能有比较明显的改善与提高。     

一种基于聚类的超闭球模糊神经网络

针对一类不确定非线性多输入多输出复杂系统,根据系统的输入输出数据对,提出一种基于聚类的超闭球模糊神经网络系统．该系统通过改进的模糊聚类方法（FCM）确定模糊规则数,采用高维隶属度函数取代常规的单维隶属度函数,并对隶属度函数中心值和隶属度函数参数采用一步通过算法,所提方法可降低系统的模糊规则数,简化网络计算．此外,当系统的输入输出发生变化时,可实现模糊规则库的在线修改．仿真实例验证了所提方法的有效性．     

A genetic approach to the design of autonomous agents for futures trading

We propose a genetic algorithm-based method for designing an autonomous trader agent. The task of the proposed method is to find an optimal set of fuzzy if–then rules that best represents the behavior of a target trader agent. A highly profitable trader agent is used as the target in the proposed genetic algorithm. A trading history for the target agent is obtained from a series of futures trading. The antecedent part of fuzzy if–then rules considers time-series data of spot prices, while the consequent part indicates the order of trade (Buy, Sell, or No action) with its degree of certainty. The proposed method determines the antecedent part of fuzzy if–then rules. The consequent part of fuzzy if–then rules is automatically determined from the trading history of the target trader agent. The autonomous trader agent designed by the proposed genetic algorithm consists of a fixed number of fuzzy if–then rules. The decision of the autonomous trader agent is made by fuzzy inference from the time-series data of spot prices. This work was presented in part at the 11th International Symposium on Artificial Life and Robotics, Oita, Japan, January 23–25, 2006