Showing papers on "Fuzzy number published in 1992"

Generating fuzzy rules by learning from examples

Abstract: A general method is developed to generate fuzzy rules from numerical data. The method consists of five steps: divide the input and output spaces of the given numerical data into fuzzy regions; generate fuzzy rules from the given data; assign a degree of each of the generated rules for the purpose of resolving conflicts among the generated rules; create a combined fuzzy rule base based on both the generated rules and linguistic rules of human experts; and determine a mapping from input space to output space based on the combined fuzzy rule base using a defuzzifying procedure. The mapping is proved to be capable of approximating any real continuous function on a compact set to arbitrary accuracy. Applications to truck backer-upper control and time series prediction problems are presented. >

Fuzzy basis functions, universal approximation, and orthogonal least-squares learning

Abstract: Fuzzy systems are represented as series expansions of fuzzy basis functions which are algebraic superpositions of fuzzy membership functions. Using the Stone-Weierstrass theorem, it is proved that linear combinations of the fuzzy basis functions are capable of uniformly approximating any real continuous function on a compact set to arbitrary accuracy. Based on the fuzzy basis function representations, an orthogonal least-squares (OLS) learning algorithm is developed for designing fuzzy systems based on given input-output pairs; then, the OLS algorithm is used to select significant fuzzy basis functions which are used to construct the final fuzzy system. The fuzzy basis function expansion is used to approximate a controller for the nonlinear ball and beam system, and the simulation results show that the control performance is improved by incorporating some common-sense fuzzy control rules. >

Fuzzy ARTMAP: A neural network architecture for incremental supervised learning of analog multidimensional maps

Abstract: A neural network architecture is introduced for incremental supervised learning of recognition categories and multidimensional maps in response to arbitrary sequences of analog or binary input vectors, which may represent fuzzy or crisp sets of features. The architecture, called fuzzy ARTMAP, achieves a synthesis of fuzzy logic and adaptive resonance theory (ART) neural networks by exploiting a close formal similarity between the computations of fuzzy subsethood and ART category choice, resonance, and learning. Four classes of simulation illustrated fuzzy ARTMAP performance in relation to benchmark backpropagation and generic algorithm systems. These simulations include finding points inside versus outside a circle, learning to tell two spirals apart, incremental approximation of a piecewise-continuous function, and a letter recognition database. The fuzzy ARTMAP system is also compared with Salzberg's NGE systems and with Simpson's FMMC system. >

Fuzzy systems are universal approximators

Abstract: The author proves that fuzzy systems are universal approximators. The Stone-Weierstrass theorem is used to prove that fuzzy systems with product inference, centroid defuzzification, and a Gaussian membership function are capable of approximating any real continuous function on a compact set to arbitrary accuracy. This result can be viewed as an existence theorem of an optimal fuzzy system for a wide variety of problems. >

Multilayer perceptron, fuzzy sets, and classification

Abstract: A fuzzy neural network model based on the multilayer perceptron, using the backpropagation algorithm, and capable of fuzzy classification of patterns is described. The input vector consists of membership values to linguistic properties while the output vector is defined in terms of fuzzy class membership values. This allows efficient modeling of fuzzy uncertain patterns with appropriate weights being assigned to the backpropagated errors depending upon the membership values at the corresponding outputs. During training, the learning rate is gradually decreased in discrete steps until the network converges to a minimum error solution. The effectiveness of the algorithm is demonstrated on a speech recognition problem. The results are compared with those of the conventional MLP, the Bayes classifier, and other related models. >

Learning and tuning fuzzy logic controllers through reinforcements

Abstract: A method for learning and tuning a fuzzy logic controller based on reinforcements from a dynamic system is presented. It is shown that: the generalized approximate-reasoning-based intelligent control (GARIC) architecture learns and tunes a fuzzy logic controller even when only weak reinforcement, such as a binary failure signal, is available; introduces a new conjunction operator in computing the rule strengths of fuzzy control rules; introduces a new localized mean of maximum (LMOM) method in combining the conclusions of several firing control rules; and learns to produce real-valued control actions. Learning is achieved by integrating fuzzy inference into a feedforward network, which can then adaptively improve performance by using gradient descent methods. The GARIC architecture is applied to a cart-pole balancing system and demonstrates significant improvements in terms of the speed of learning and robustness to changes in the dynamic system's parameters over previous schemes for cart-pole balancing. >

Self-learning fuzzy controllers based on temporal backpropagation

Abstract: A generalized control strategy that enhances fuzzy controllers with self-learning capability for achieving prescribed control objectives in a near-optimal manner is presented. This methodology, termed temporal backpropagation, is model-sensitive in the sense that it can deal with plants that can be represented in a piecewise-differentiable format, such as difference equations, neural networks, GMDH structures, and fuzzy models. Regardless of the numbers of inputs and outputs of the plants under consideration, the proposed approach can either refine the fuzzy if-then rules of human experts or automatically derive the fuzzy if-then rules if human experts are not available. The inverted pendulum system is employed as a testbed to demonstrate the effectiveness of the proposed control scheme and the robustness of the acquired fuzzy controller. >

Fuzzy min-max neural networks. I. Classification

Abstract: A supervised learning neural network classifier that utilizes fuzzy sets as pattern classes is described. Each fuzzy set is an aggregate (union) of fuzzy set hyperboxes. A fuzzy set hyperbox is an n-dimensional box defined by a min point and a max point with a corresponding membership function. The min-max points are determined using the fuzzy min-max learning algorithm, an expansion-contraction process that can learn nonlinear class boundaries in a single pass through the data and provides the ability to incorporate new and refine existing classes without retraining. The use of a fuzzy set approach to pattern classification inherently provides a degree of membership information that is extremely useful in higher-level decision making. The relationship between fuzzy sets and pattern classification is described. The fuzzy min-max classifier neural network implementation is explained, the learning and recall algorithms are outlined, and several examples of operation demonstrate the strong qualities of this new neural network classifier. >

Fuzzy Logic for the Management of Uncertainty

Abstract: Partial table of contents: ISSUES IN THE MANAGEMENT OF UNCERTAINTY A Survey of Uncertain and Approximate Inference (R. Neapolitan) Rough Sets: A New Approach to Vagueness (Z. Pawlak) ASPECTS OF FUZZY LOGIC: THEORY AND IMPLEMENTATIONS LT-Fuzzy Logics (H. Rasiowa & N. Cat Ho) On Fuzzy Intuitionistic Logic (E. Turunen) On Modifier Logic (J. Mattila) FUZZY LOGIC FOR APPROXIMATE REASONING Presumption, Prejudice, and Regularity in Fuzzy Material Implication (T. Whalen & B. Schott) Inference for Information Systems Containing Probabilistic and Fuzzy Uncertainties (J. Baldwin) FUZZY LOGIC FOR KNOWLEDGE REPRESENTATION AND ELICITATION Approximate Reasoning in Diagnosis, Therapy, and Prognosis (A. Rocha, et al.) Elementary Learning in a Fuzzy Expert System (J. Buckley) KNOWLEDGE-BASED SYSTEMS USING FUZZY LOGIC Structured Local Fuzzy Logics in MILORD (J. Agustm, et al.) The Validation of Fuzzy Knowledge-Based Systems (A. Chang & L. Hall) FUZZY LOGIC FOR INTELLIGENT DATABASE MANAGEMENT SYSTEMS Fuzzy Querying in Conventional Databases (P. Bosc & O. Pivert) Index.

An Introduction to Fuzzy Logic Applications in Intelligent Systems

Abstract: Knowledge Representation in Fuzzy Logic LA Zadeh Expert Systems Using Fuzzy Logic RR Yager Fuzzy Rules in Knowledge-Based Systems D Dubois, H Prade Fuzzy Logic Controllers H Berenji Methods and Applications of Fuzzy Mathematical Programming HJ Zimmermann Fuzzy Set Methods in Computer Vision JM Keller, R Krishnapuram Fuzziness, Image Information and Scene Analysis SK Pal Fuzzy Sets in Natural Language Processing V Novak Fuzzy-Set-Theoretic Applications in Modeling of Man-Machine Interactions W Karwowski, G Salvendy Questionnaires and Fuzziness B Bouchon-Meunier Fuzzy Logic Knowledge Systems and Artificial Neural Networks in Medicine and Biology E Sanchez The Representation and Use of Uncertainty and Metaknowledge in Milord R Lopez de Montaras, C Sierra, J Augusti Fuzzy Logic with Linguistic Quantifiers in Group Decision Making J Kacprzyk, M Fedrizzi, H Nurmi Learning in Uncertain Environments M Botta, A Giordana, L Saitta Evidential Reasoning Under Probabilistic and Fuzzy Uncertainties JF Baldwin Probabilistic Sets-Probabilistic Extension of Fuzzy Sets K Hirota Index

Putting Rough Sets and Fuzzy Sets Together

Abstract: In this paper we argue that fuzzy sets and rough sets aim to different purposes and that it is more natural to try to combine the two models of uncertainty (vagueness for fuzzy sets and coarseness for rough sets) in order to get a more accurate account of imperfect information. First, the upper and lower approximations of a fuzzy set are defined, when the universe of discourse of a fuzzy sets is coarsened by means of an equivalence relation. We then come close to Caianiello’s C-calculus. Shafer’s concept of coarsened belief functions also belongs to the same line of thought and is reviewed here. Another idea is to turn the equivalence relation relation into a fuzzy similarity relation, for a more expressive modeling of coarseness. New results on the representation of similarity relations by means of a fuzzy partition of fuzzy clusters of more or less indiscernible points are surveyed. The properties of upper and lower approximations of fuzzy sets by similarity relations are thoroughly studied. Lastly the potential usefulness of the fuzzy rough set notions for logical inference in the presence of both fuzzy predicates and graded indiscernibility is indicated. Especially fuzzy rough sets may provide a nice semantic background for modal logic involving fuzzy modalities and/or fuzzy sentences.

Fuzzy Min-Max Neural Networks-Part 1 : Classification

Abstract: A supervised learning neural network classifier that utilizes fuzzy sets as pattern classes is described. Each fuzzy set is an aggregate (union) of fuzzy set hyperboxes. A fuzzy set hyperbox is an n-dimensional box defined by a min point and a max point with a corresponding membership function. The min-max points are determined using the fuzzy min-max learning algorithm, an expansionxontraction process that can learn nonlinear class boundaries in a single pass through the data and provides the ability to incorporate new and refine existing classes without retraining. The use of a fuzzy set approach to pattern classification inherently provides degree of membership information that is extremely useful in higher level decision mak- ing. This paper will describe the relationship between fuzzy sets and pattern classification. It explains the fuzzy min-max classifier neural network implementation, it outlines the learning and recall algorithms, and it provides several examples of operation that demonstrate the strong qualities of this new neural network classifier. AmRN classification is a key element to many engi- P neering solutions. Sonar, radar, seismic, and diagnostic applications all require the ability to accurately classify a situation. Control, tracking, and prediction systems will often use classifiers to determine input-output relationships. Because of this wide range of applicability, pattern classification has been studied a great deal (13), (15), (19). This paper describes a neural network classifier that creates classes by aggregating several smaller fuzzy sets into a single fuzzy set class. This technique, introduced in (42) as an extension of earlier work (41), can learn pattern classes in a single pass through the data, it can add new pattern classes on the fly, it can refine existing pattern classes as new information is received, and it uses simple operations that allow for quick execution. Fuzzy min-max classification neural networks are built using hyperbox fuzzy sets. A hyperbox defines a region of the n-dimensional pattern space that has patterns with full class membership. A hyperbox is completely defined by its min point and its max point, and a membership function is defined with respect to these hyperbox min-max points. The min-max (hyperbox) membership function combination defines a fuzzy set, hyperbox fuzzy sets are aggregated to form a single fuzzy set class, and the resulting structure fits naturally into a neural network framework; hence this classification system is called a fuzzy min-max classification neural network. Learning in the fuzzy min-max classification neural network is performed by properly placing and adjusting hyperboxes in the pattern space.

A decision theoretic framework for approximating concepts

Abstract: This paper explores the implications of approximating a concept based on the Bayesian decision procedure, which provides a plausible unification of the fuzzy set and rough set approaches for approximating a concept. We show that if a given concept is approximated by one set, the same result given by the α-cut in the fuzzy set theory is obtained. On the other hand, if a given concept is approximated by two sets, we can derive both the algebraic and probabilistic rough set approximations. Moreover, based on the well known principle of maximum (minimum) entropy, we give a useful interpretation of fuzzy intersection and union. Our results enhance the understanding and broaden the applications of both fuzzy and rough sets.

Stable adaptive fuzzy control of nonlinear systems

Abstract: The author develops a direct adaptive fuzzy controller which does not require an accurate mathematical model of the system under control, is capable of incorporating fuzzy control rules directly into the controllers, and guarantees the global stability of the resulting closed-loop system in the sense that all signals involved are uniformly bounded. The specific formula of the bounds is provided so that controller designers can determine the bounds based on their requirements. The direct adaptive fuzzy controller is used to regulate an unstable system to the origin. Simulation results show that the direct adaptive fuzzy controller can perform successful control without using any fuzzy control rules. After incorporating some fuzzy control rules into the controllers the adaptation speed became much faster. The author also showed explicitly how the supervisory control forced the state to be within the constraint set and how the adaptive fuzzy controller learned to regain control. >

Back-propagation fuzzy system as nonlinear dynamic system identifiers

Abstract: The authors develop a training algorithm, similar to the backpropagation algorithm for neural networks, to train fuzzy systems to match desired input-output pairs. The key ideas in developing this training algorithm are to view a fuzzy system as a three-layer feedforward network, and to use the chain rule to determine gradients of the output errors of the fuzzy system with respect to its design parameters. It is shown that this training algorithm performs an error backpropagation procedure: hence, the fuzzy system equipped with the backpropagation training algorithm is called the backpropagation fuzzy system (BP FS). An online initial parameter choosing method is proposed for the BP FS, and it is shown that it is straightforward to incorporate linguistic if-then rules into the BP FS. Two examples are presented which demonstrate (1) how the fuzzy system learns to match an unknown nonlinear mapping as training progresses and (2) that performance is improved by incorporating linguistic rules. >

Fuzzy systems as universal approximators

Abstract: The author shows that an additive fuzzy system can approximate any continuous function on a compact domain to any degree of accuracy. Fuzzy systems are dense in the space of continuous functions. The fuzzy system approximates the function by covering its graph with fuzzy patches in the input-output state space. Each fuzzy rule defines a fuzzy patch and connects commonsense knowledge with state-space geometry. Neural or statistical clustering algorithms can approximate the unknown fuzzy patches and generate fuzzy systems from training data. >

Multiobjective fuzzy linear regression analysis for fuzzy input-output data

Abstract: Fuzzy linear regression models, where both input data and output data are fuzzy numbers, are introduced by using three indices for equalities between fuzzy numbers. By considering the conflict between the fuzzy threshold for the three indices and the fuzziness of the fuzzy linear regression model, three types of multiobjective programming problems for obtaining fuzzy linear regression models are formulated corresponding to the three indices. Then a linear programming based interactive decision making method to derive the satisficing solution of the decision maker for the formulated multiobjective programming problems is developed. A numerical example demonstrates the appropriateness and efficiency of the proposed method.

A fuzzy linear programming approach to the reactive power/voltage control problem

Abstract: The integration of traditional and heuristic techniques is considered for the reactive power/voltage control program. The steady-state reactive power problem is addressed. Fuzzy sets are used to modify the linear programming (LP) approach to voltage control and to incorporate some heuristic concepts of the expert system approach. Multiple objectives and soft constraints are modeled using fuzzy sets. Piecewise linear convex membership functions for the fuzzy sets are defined. Under this definition, the fuzzy optimization problem is reformulated as a standard linear programming problem. The objective function represents the compromise among the original competing objectives and the soft constraints. In addition, discrete constraints are considered. Numerical examples demonstrate the approach. >