Showing papers on "Membership function published in 1999"

Intuitionistic Fuzzy Sets: Theory and Applications

Abstract: The basic definitions and properties of the Intuitionistic Fuzzy Sets (IFSs) are introduced in the book. The IFSs are substantial extensions of the ordinary fuzzy sets. IFSs are objects having degrees of membership and of non-membership, such that their sum is exactly 1. The most important property of IFS not shared by the fuzzy sets is that modal-like operators can be defined over IFSs. The IFSs have essentially higher describing possibilities than fuzzy sets. In this book, readers will find discussions on some of the IFS extensions (for example, interval-values IFSs, temporal IFSs and others) and applications (e.g. intuitionistic fuzzy expert systems, intuitionistic fuzzy neural networks, intuitionistic fuzzy systems, intuitionistic fuzzy generalized nets, and other).

Performance evaluation of fuzzy classifier systems for multidimensional pattern classification problems

Abstract: We examine the performance of a fuzzy genetics-based machine learning method for multidimensional pattern classification problems with continuous attributes In our method, each fuzzy if-then rule is handled as an individual, and a fitness value is assigned to each rule Thus, our method can be viewed as a classifier system In this paper, we first describe fuzzy if-then rules and fuzzy reasoning for pattern classification problems Then we explain a genetics-based machine learning method that automatically generates fuzzy if-then rules for pattern classification problems from numerical data Because our method uses linguistic values with fixed membership functions as antecedent fuzzy sets, a linguistic interpretation of each fuzzy if-then rule is easily obtained The fixed membership functions also lead to a simple implementation of our method as a computer program The simplicity of implementation and the linguistic interpretation of the generated fuzzy if-then rules are the main characteristic features of our method The performance of our method is evaluated by computer simulations on some well-known test problems While our method involves no tuning mechanism of membership functions, it works very well in comparison with other classification methods such as nonfuzzy machine learning techniques and neural networks

Implementation of evolutionary fuzzy systems

Abstract: Evolutionary fuzzy systems are discussed in which the membership function shapes and types and the fuzzy rule set including the number of rules inside it are evolved using a genetic (evolutionary) algorithm. In addition, the genetic parameters (operators) of the evolutionary algorithm are adapted via a fuzzy system. Benefits of the methodology are illustrated in the process of classifying the iris data set. Possible extensions of the methods are summarized.

Semantic constraints for membership function optimization

Abstract: The optimization of fuzzy systems using bio-inspired strategies, such as neural network learning rules or evolutionary optimization techniques, is becoming more and more popular. In general, fuzzy systems optimized in such a way cannot provide a linguistic interpretation, preventing us from using one of their most interesting and useful features. This paper addresses this difficulty and points out a set of constraints that when used within an optimization scheme obviate the subjective task of interpreting membership functions. To achieve this a comprehensive set of semantic properties that membership functions should have is postulated and discussed. These properties are translated in terms of nonlinear constraints that are coded within a given optimization scheme, such as backpropagation. Implementation issues and one example illustrating the importance of the proposed constraints are included.

Fuzzy sets in decision analysis, operations research and statistics

Abstract: Series Foreword D Dubois, H Prade Preface R Slowinski Part I: Decision Making 1 Fuzzy Preference Modeling P Perny, M Roubens 2 Fuzzy Aggregation of Numerical Preferences M Grabisch, et al 3 The Use of Fuzzy Preference Models in Multiple Criteria Choice, Ranking and Sorting J Fodor, et al 4 Group Decision Making Under Fuzziness J Kacprzyk, H Nurmi 5 Elements of Fuzzy Game Theory A Billot Part II: Mathematical Programming 6 Fuzzy Linear Programming with Single or Multiple Objective Functions H Rommelfanger, R Slowinski 7 Fuzzy Nonlinear Programming with Single or Multiple Objective Functions M Sakawa 8 Discrete Fuzzy Optimization S Chanas, D Kuchta 9 Fuzzy Dynamic Programming AO Esogbue, J Kacprzyk Part III: Statistics and Data Analysis 10 Fuzzy Set-Theoretic Methods in Statistics J Gebhardt, et al 11 Fuzzy Regression Analysis P Diamond, H Tanaka Part IV: Reliability, Maintenance and Replacement 12 Reliability E Kerre, et al 13 Maintenance and Replacement Models under a Fuzzy Framework AO Esogbue, WE Hearnes II Index

Reduction of fuzzy rule base via singular value decomposition

Abstract: Introduces a singular value-based method for reducing a given fuzzy rule set. The method conducts singular value decomposition of the rule consequents and generates certain linear combinations of the original membership functions to form new ones for the reduced set. The present work characterizes membership functions by the conditions of sum normalization (SN), nonnegativeness (NN), and normality (NO). Algorithms to preserve the SN and NN conditions in the new membership functions are presented. Preservation of the NO condition relates to a high-dimensional convex hull problem and is not always feasible in which case a closed-to-NO solution may be sought. The proposed method is applicable regardless of the adopted inference paradigms. With product-sum-gravity inference and singleton support fuzzy rule base, output errors between the full and reduced fuzzy set are bounded by the sum of the discarded singular values. The work discusses three specific applications of fuzzy reduction: fuzzy rule base with singleton support, fuzzy rule base with nonsingleton support (which includes the case of missing rules), and the Takagi-Sugeno-Kang (TSK) model. Numerical examples are presented to illustrate the reduction process.

Fuzzy method in group decision making

Abstract: This paper investigates the problems of decision making with multiple judge, multiple criteria in a fuzzy environment, where the performance of alternatives and the importance of criteria are imprecisely defined and represented by fuzzy sets. A fuzzy model associated with the solution algorithm is proposed on the basis of an α-level weighted, fuzzy preference relation. A numerical example is solved for illustration.

A new method for constructing membership functions and fuzzy rules from training examples

Abstract: To extract knowledge from a set of numerical data and build up a rule-based system is an important research topic in knowledge acquisition and expert systems. In recent years, many fuzzy systems that automatically generate fuzzy rules from numerical data have been proposed. In this paper, we propose a new fuzzy learning algorithm based on the /spl alpha/-cuts of equivalence relations and the /spl alpha/-cuts of fuzzy sets to construct the membership functions of the input variables and the output variables of fuzzy rules and to induce the fuzzy rules from the numerical training data set. Based on the proposed fuzzy learning algorithm, we also implemented a program on a Pentium PC using the MATLAB development tool to deal with the Iris data classification problem. The experimental results show that the proposed fuzzy learning algorithm has a higher average classification ratio and can generate fewer rules than the existing algorithm.

Fuzzy Weights of Evidence Method and Its Application in Mineral Potential Mapping

Abstract: This paper proposes a new approach of weights of evidence method based on fuzzy sets and fuzzy probabilities for mineral potential mapping. It can be considered as a generalization of the ordinary weights of evidence method, which is based on binary or ternary patterns of evidence and has been used in conjunction with geographic information systems for mineral potential mapping during the past few years. In the newly proposed method, instead of separating evidence into binary or ternary form, fuzzy sets containing more subjective genetic elements are created; fuzzy probabilities are defined to construct a model for calculating the posterior probability of a unit area containing mineral deposits on the basis of the fuzzy evidence for the unit area. The method can be treated as a hybrid method, which allows objective or subjective definition of a fuzzy membership function of evidence augmented by objective definition of fuzzy or conditional probabilities. Posterior probabilities calculated by this method would depend on existing data in a totally data-driven approach method, but depend partly on expert's knowledge when the hybrid method is used. A case study for demonstration purposes consists of application of the method to gold deposits in Meguma Terrane, Nova Scotia, Canada.

Fuzzy adaptive output tracking control of nonlinear systems

Abstract: A stable adaptive fuzzy output tracking control scheme is developed for a single-input-single-output unknown nonlinear system. The main characteristics of the proposed adaptive fuzzy control are: i) it does not need the assumption that all the states of the system are available for full feedback, but introduces a high gain observer to estimate them; ii) it is composed of a robust control term and an equivalence fuzzy control so that it not only ensures the stability of the closed-loop system, but also attenuates the effect of fuzzy approximation error on the tracking error of the system to an arbitrary small level; and iii) it is proved that the designed output feedback adaptive fuzzy control can recover the performance achieved under the state feedback controller.

Uncertainty Management for Spatial Data in Databases: Fuzzy Spatial Data Types

Abstract: In many geographical applications there is a need to model spatial phenomena not simply by sharply bounded objects but rather through vague concepts due to indeterminate boundaries. Spatial database systems and geographical information systems are currently not able to deal with this kind of data. In order to support these applications, for an important kind of vagueness called fuzziness, we propose an abstract, conceptual model of so-called fuzzy spatial data types (i.e., a fuzzy spatial algebra) introducing fuzzy points, fuzzy lines, and fuzzy regions. This paper focuses on defining their structure and semantics. The formal framework is based on fuzzy set theory and fuzzy topology.

Robust adaptive fuzzy-neural controllers for uncertain nonlinear systems

Abstract: A robust adaptive fuzzy-neural controller for a class of unknown nonlinear dynamic systems with external disturbances is proposed. The fuzzy-neural approximator is established to approximate an unknown nonlinear dynamic system in a linearized way. The fuzzy B-spline membership function (BMF) which possesses a fixed number of control points is developed for online tuning. The concept of tuning the adjustable vectors, which include membership functions and weighting factors, is described to derive the update laws of the robust adaptive fuzzy-neural controller. Furthermore, the effect of all the unmodeled dynamics, BMF modeling errors and external disturbances on the tracking error is attenuated by the error compensator which is also constructed by fuzzy-neural inference. We prove that the closed-loop system which is controlled by the robust adaptive fuzzy-neural controller is stable and the tracking error will converge to zero under mild assumptions. Several examples are simulated in order to confirm the effectiveness and applicability of the proposed methods.

Differential Equations With Fuzzy Parameters

Abstract: This paper is concerned with systems of ordinary differential equations with fuzzy parameters. Applying the Zadeh extension principle to the equations, we introduce the notions of fuzzy solutions and of componentwise fuzzy solutions. The fuzzy extension of the solution operator is shown to provide the unique fuzzy solution as well as the maximal componentwise fuzzy solution. A numerical algorithm based on monotonicity properties of membership functions is presented, together with a proof of convergence. In an interplay of interval analysis and possibility theory, these methods allow to process subjective information on the possible fluctuations of parameters in models involving ordinary differential equations. This is demonstrated in two engineering applications: a queueing model for earthwork and a model of oscillations of bell-towers.

Choquet fuzzy integral-based hierarchical networks for decision analysis

Abstract: A Choquet fuzzy integral-based approach to hierarchical network implementation is investigated. In this approach, we generalized the fuzzy integral as an excellent component for decision analysis. The generalization involves replacing the max (or min) operator in information aggregation with a fuzzy integral-based neuron, resulting in increased flexibility. The characteristics of the Choquet fuzzy integral are studied and a network-based decision-analysis framework is proposed. The trainable hierarchical network can be implemented utilizing the fuzzy integral-based neurons and connectives. The training algorithms are derived and several examples given to illustrate the behaviors of the networks. Also, we present a decision making experiment using the proposed network to learn appropriate functional relationships in the defective numeric fields detection domain.

Fuzzy relational classifier trained by fuzzy clustering

Abstract: A novel approach to nonlinear classification is presented, in the training phase of the classifier, the training data is first clustered in an unsupervised way by fuzzy c-means or a similar algorithm. The class labels are not used in this step. Then, a fuzzy relation between the clusters and the class identifiers is computed. This approach allows the number of prototypes to be independent of the number of actual classes. For the classification of unseen patterns, the membership degrees of the feature vector in the clusters are first computed by using the distance measure of the clustering algorithm. Then, the output fuzzy set is obtained by relational composition. This fuzzy set contains the membership degrees of the pattern in the given classes. A crisp decision is obtained by defuzzification, which gives either a single class or a "reject" decision, when a unique class cannot be selected based on the available information. The principle of the proposed method is demonstrated on an artificial data set and the applicability of the method is shown on the identification of live-stock from recorded sound sequences. The obtained results are compared with two other classifiers.

Design of fuzzy systems using neurofuzzy networks

Abstract: Introduces a systematic approach for fuzzy system design based on a class of neural fuzzy networks built upon a general neuron model. The network structure is such that it encodes the knowledge learned in the form of if-then fuzzy rules and processes data following fuzzy reasoning principles. The technique provides a mechanism to obtain rules covering the whole input/output space as well as the membership functions (including their shapes) for each input variable. Such characteristics are of utmost importance in fuzzy systems design and application. In addition, after learning, it is very simple to extract fuzzy rules in the linguistic form. The network has universal approximation capability, a property very useful in, e.g., modeling and control applications. Here we focus on function approximation problems as a vehicle to illustrate its usefulness and to evaluate its performance. Comparisons with alternative approaches are also included. Both, non-noisy and noisy data have been studied and considered in the computational experiments. The neural fuzzy network developed here and, consequently, the underlying approach, has shown to provide good results from the accuracy, complexity, and system design points of view.

Design of fuzzy logic controllers using genetic algorithms

Abstract: Since membership functions and fuzzy control rules are interdependent in designing a fuzzy logic controller (FLC), a GA-based approach is proposed for simultaneous design of these two components. With triangular membership functions, the left and right widths of these functions, the locations of their peaks, and the output fuzzy set corresponding to every possible combination of input fuzzy sets are then chosen as parameters to be optimized. In a proportional scaling method, these parameters are then transformed into real-coded chromosomes, over which arithmetical crossover and nonuniform mutation are implemented. Meanwhile, enlarged sampling space and a ranking mechanism are also be used in the evolution process. To show the application of the proposed method, a cart-centering example is given. From the simulation results, we find that the designed FLC is robust and can drive the cart system from any given initial state to the desired final state, which verifies the feasibility and validity of the proposed method.

Change detection assessment using fuzzy sets and remotely sensed data: an application of topographic map revision

Abstract: This paper explores a methodology for computing the amount of changes that have occurred within an area by using remotely sensed technologies and fuzzy modelling. The discussion concentrates on the formulation of a standard procedure that, using the concept of fuzzy sets and fuzzy logic, can define the likelihood of changes detected from remotely sensed data. Furthermore, an example of how fuzzy visualisation of areas undergoing changes can be incorporated into a decision support system for prioritisation of areas requiring topographic map revision and updating is presented. By adapting the membership function of the fuzzy model to fit the shape of the histogram characterising the change image (derived from any of the common pre-classification methods of change detection), areas can be identified according to their likelihood of having undergone change during the period of observation.