Showing papers on "Fuzzy associative matrix published in 1999"

Mathematical Principles of Fuzzy Logic

Abstract: Preface. 1. Fuzzy Logic: What, Why, for Which? 2. Algebraic Structures for Logical Calculi. 3. Logical Calculi and Model Theory. 4. Fuzzy Logic in Narrow Sense. 5. Functional Systems in Fuzzy Logic Theories. 6. Fuzzy Logic in Broader Sense. 7. Topoi and Categories of Fuzzy Sets. 8. Few Historical and Concluding Remarks. References. Index.

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

Mathematics Behind Fuzzy Logic

Abstract: Residuated Lattices: Lattices and Equivalence Relations Lattice Filters Residuated Lattices BL-Algebras Exercises.- MV-Algebras: MV-Algebras and Wajsberg Algebras Complete MV-Algebras Pseudo-Boolean Algebras Exercises.- Fuzzy Propositional Logic: Semantics of Fuzzy Propositional Logic Exercises.- Fuzzy Relations: Solvability of Fuzzy Relation Equations On Fuzzy Similiarity Relations Fuzzy Similiarity and Approximate Reasoning Maximal Similiarity and Fuzzy Reasoning Exercises.- Solutions to Exercises.

On stability of fuzzy systems expressed by fuzzy rules with singleton consequents

Abstract: Addresses the issue of stability of a fuzzy system described by fuzzy rules with singleton consequents. It first presents two canonical forms of a fuzzy system: a parametric expression and a state-space expression. A fuzzy system with singleton consequents is found to be a piecewise-polytopic-affine system. Then the paper gives stability theorems for a discrete-time system as well as for a continuous time system. It also gives a brief survey on the stability issues of fuzzy control systems.

Analysis of direct action fuzzy PID controller structures

Abstract: The majority of the research work on fuzzy PID controllers focuses on the conventional two-input PI or PD type controller proposed by Mamdani (1974). However, fuzzy PID controller design is still a complex task due to the involvement of a large number of parameters in defining the fuzzy rule base. This paper investigates different fuzzy PID controller structures, including the Mamdani-type controller. By expressing the fuzzy rules in different forms, each PLD structure is distinctly identified. For purpose of analysis, a linear-like fuzzy controller is defined. A simple analytical procedure is developed to deduce the closed form solution for a three-input fuzzy inference. This solution is used to identify the fuzzy PID action of each structure type in the dissociated form. The solution for single-input-single-output nonlinear fuzzy inferences illustrates the effect of nonlinearity tuning. The design of a fuzzy PID controller is then treated as a two-level tuning problem. The first level tunes the nonlinear PID gains and the second level tunes the linear gains, including scale factors of fuzzy variables. By assigning a minimum number of rules to each type, the linear and nonlinear gains are deduced and explicitly presented. The tuning characteristics of different fuzzy PID structures are evaluated with respect to their functional behaviors. The rule decoupled and one-input rule structures proposed in this paper provide greater flexibility and better functional properties than the conventional fuzzy PHD structures.

Analysis and design of hierarchical fuzzy systems

Abstract: In this letter, the hierarchical fuzzy systems are analyzed and designed. In the analysis part, we prove that the hierarchical fuzzy systems are universal approximators and analyze the sensitivity of the fuzzy system output with respect to small perturbations in its inputs. In the design part, we derive a gradient descent algorithm for tuning the parameters of the hierarchical fuzzy system to match the input-output pairs. The algorithm is simulated for two examples and the results show that the algorithm is effective and the hierarchical structure gives good approximation accuracy.

Multicriteria analysis with fuzzy pairwise comparison

Abstract: Presents an approach for solving qualitative multicriteria analysis (MA) problems using fuzzy pairwise comparison. Fuzzy numbers are used to approximate the decision-maker's (DM's) subjective assessments in assessing alternative performance and criteria importance. The concept of fuzzy extent analysis is applied for solving the reciprocal judgement matrices. To avoid the complex and unreliable process of comparing fuzzy utilities, the /spl alpha/-cut technique is applied to transform the fuzzy performance matrix into an interval matrix. Incorporated with the DM's attitude towards risk, an overall performance index is obtained for each alternative across all criteria in line with the ideal solution concept. An empirical study of a tender selection problem in Australia is conducted. The result shows that the approach developed is simple and comprehensible in concept, efficient in computation, and robust and flexible in modeling the human evaluation process, thus making it of general use for solving practical MA problems.

On generating FC/sup 3/ fuzzy rule systems from data using evolution strategies

Abstract: Sophisticated fuzzy rule systems are supposed to be flexible, complete, consistent and compact (FC/sup 3/). Flexibility, and consistency are essential for fuzzy systems to exhibit an excellent performance and to have a clear physical meaning, while compactness is crucial when the number of the input variables increases. However, the completeness and consistency conditions are often violated if a fuzzy system is generated from data collected from real world applications. A systematic design paradigm is proposed using evolution strategies. The structure of the fuzzy rules, which determines the compactness of the fuzzy systems, is evolved along with the parameters of the fuzzy systems. Special attention has been paid to the completeness and consistency of the rule base. The completeness is guaranteed by checking the completeness of the fuzzy partitioning of input variables and the completeness of the rule structure. An index of inconsistency is suggested with the help of a fuzzy similarity which can prevent the algorithm from generating rules that seriously contradict with each other or with the heuristic knowledge. In addition, soft T-norm and BADD defuzzification are introduced and optimized to increase the flexibility of the fuzzy system. The proposed approach is applied to the design of a distance controller for cars. It is verified that a FC/sup 3/ fuzzy system works very well both, for training and test driving situations, especially when the training data are insufficient.

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.

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.

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.

Analysis and design of fuzzy control systems using dynamic fuzzy-state space models

Abstract: A discrete-time fuzzy control system which is composed of a dynamic fuzzy model and a fuzzy-state feedback controller is proposed. Stability of the fuzzy control system is discussed and two sufficient conditions to guarantee the stability of the system are given in terms of uncertain linear system theory. An algorithm is developed to check the stability condition. The controller design method is divided into two procedures, one is to get the state feedback matrix by linear system theory in every local rule map; the other is to determine conditions of global stability by using a nonlinear analysis method. Two examples are used to show the design method.

Improving the performance of fuzzy classifier systems for pattern classification problems with continuous attributes

Abstract: In this paper, various methods are introduced for improving the ability of fuzzy classifier systems to automatically generate fuzzy if-then rules for pattern classification problems with continuous attributes. First, we describe a simple fuzzy classifier system where a randomly generated initial population of fuzzy if-then rules is evolved by typical genetic operations, such as selection, crossover, and mutation. By computer simulations on a real-world pattern classification problem with many continuous attributes, we show that the search ability of such a simple fuzzy classifier system is not high. Next, we examine the search ability of a hybrid algorithm where a learning procedure of fuzzy if-then rules is combined with the fuzzy classifier system. Then, we introduce two heuristic procedures for improving the performance of the fuzzy classifier system. One is a heuristic rule generation procedure for an initial population where initial fuzzy if-then rules are directly generated from training patterns. The other is a heuristic population update procedure where new fuzzy if-then rules are generated from misclassified and rejected training patterns, as well as from existing fuzzy if-then rules by genetic operations. By computer simulations, we demonstrate that these two heuristic procedures drastically improve the search ability of the fuzzy classifier system. We also examine a variant of the fuzzy classifier system where the population size (i.e., the number of fuzzy if-then rules) varies depending on the classification performance of fuzzy if-then rules in the current population.

Comparison of necessary conditions for typical Takagi-Sugeno and Mamdani fuzzy systems as universal approximators

Abstract: Both Takagi-Sugeno (TS) and Mamdani fuzzy systems are known to be universal approximators. We investigate whether one type of fuzzy approximators is more economical than the other. The TS fuzzy systems are the typical two-input single-output TS fuzzy systems. We first establish necessary conditions on minimal system configuration of the TS fuzzy systems as function approximators. We show that the number of the input fuzzy sets and fuzzy rules needed by the TS fuzzy systems depend on the number and locations of the extrema of the function to be approximated. The resulting conditions reveal the strength of the TS fuzzy approximators. The drawback, though, is that a large number of fuzzy rules must be employed to approximate periodic or highly oscillatory functions. We then compare these necessary conditions with the ones that we established for the general Mamdani fuzzy systems in our previous papers. Results of the comparison unveil that the minimal system configurations of the TS and Mamdani fuzzy systems are comparable. Finally, we prove that the minimal configuration of the TS fuzzy systems can be reduced and becomes smaller than that of the Mamdani fuzzy systems if nontrapezoidal or nontriangular input fuzzy sets are used. We believe that all the results in present paper hold for the TS fuzzy systems with more than two input variables but the proof seems to be mathematically difficult. Our new findings are valuable in designing more compact fuzzy systems, especially fuzzy controllers and models which are two most popular and successful applications of the fuzzy approximators.

A two-stage evolutionary process for designing TSK fuzzy rule-based systems

Abstract: Nowadays, fuzzy rule-based systems are successfully applied to many different real-world problems. Unfortunately, relatively few well-structured methodologies exist for designing and, in many cases, human experts are not able to express the knowledge needed to solve the problem in the form of fuzzy rules. Takagi-Sugeno-Kang (TSK) fuzzy rule-based systems were enunciated in order to solve this design problem because they are usually identified 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 (/spl mu/, /spl lambda/)-evolution strategy, in which the fuzzy rules with different consequents compete among themselves to form part of a preliminary knowledge base, and a refinement 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 final Knowledge base whose rules cooperate in the best possible way. Some aspects make this process different from others proposed until now: the design problem is addressed in two different 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 identification to generate the initial populations of the Evolutionary Algorithms that causes the search process to obtain good solutions more quickly. The performance of the method proposed is shown by solving two different problems: the fuzzy modeling of some three-dimensional surfaces and the computing of the maintenance costs of electrical medium line in Spanish towns. Results obtained are compared with other kind of techniques, evolutionary learning processes to design TSK and Mamdani-type fuzzy rule-based systems in the first case, and classical regression and neural modeling in the second.

Adaptive fuzzy controllers based on variable universe

Abstract: Adaptive fuzzy controllers by means of variable universe are proposed based on interpolation forms of fuzzy control. First, monotonicity of control rules is defined, and it is proved that the monotonicity of interpolation functions of fuzzy control is equivalent to the monotonicity of control rules. This means that there is not any contradiction among the control rules under the condition for the control rules being monotonic. Then structure of the contraction-expansion factor is discussed. At last, three models of adaptive fuzzy control based on variable universe are given which are adaptive fuzzy control model with potential heredity, adaptive fuzzy control model with obvious heredity and adaptive fuzzy control model with successively obvious heredity.

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.

A hybrid clustering and gradient descent approach for fuzzy modeling

Abstract: 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.

FARM: a data mining system for discovering fuzzy association rules

Abstract: In this paper, we introduce a novel technique, called FARM, for mining fuzzy association rules. FARM employs linguistic terms to represent the revealed regularities and exceptions. The linguistic representation is especially useful when those rules discovered are presented to human experts for examination because of the affinity with the human knowledge representations. The definition of linguistic terms is based on fuzzy set theory and hence we call the rules having these terms fuzzy association rules. The use of fuzzy technique makes FARM resilient to noises such as inaccuracies in physical measurements of real-life entities and missing values in the databases. Furthermore, FARM utilizes adjusted difference analysis which has the advantage that it does not require any user-supplied thresholds which are often hard to determine. In addition to this interestingness measure, FARM has another unique feature that the conclusions of a fuzzy association rule can contain linguistic terms. Our technique also provides a mechanism to allow quantitative values be inferred from fuzzy association rules. Unlike other data mining techniques that can only discover association rules between different discretized values, FARM is able to reveal interesting relationships between different quantitative values. Our experimental results showed that FARM is capable of discovering meaningful and useful fuzzy association rules in an effective manner from a real-life database.

Mining fuzzy quantitative association rules

Abstract: Given a relational database and a set of fuzzy terms defined for some attributes we consider the problem of mining fuzzy quantitative association rules that may contain crisp values, intervals, and fuzzy terms in both antecedent and consequent. We present an algorithm extended from the equi-depth partition (EDP) algorithm for solving this problem. Our approach combines interval partition with pre-defined fuzzy terms and is more general.

A Soft Computing Approach for Modelling the Supervisor of Manufacturing Systems

Abstract: The development of a novel soft computing approach to model the supervisor of manufacturing systems is described, it is named Fuzzy Cognitive Maps (FCMs) and it is used to model the behaviour of complex systems. Fuzzy cognitive maps combine characteristics of both fuzzy logic and neural networks. The description and the construction of fuzzy cognitive maps are examined, a new methodology for developing fuzzy cognitive maps is proposed here and as an example the fuzzy cognitive map for a simple plant is developed. A hierarchical two-level structure for supervision of manufacturing systems is presented, where the supervisor is modelled as a fuzzy cognitive map. The fuzzy cognitive map model for the failure diagnosis part of the supervisor for a simple chemical process is constructed.