Showing papers on "Membership function published in 1998"

An introduction to fuzzy sets : analysis and design

Abstract: Part 1 Fundamentals of fuzzy sets: basic notions and concepts of fuzzy sets - set membership and fuzzy sets, basic definitions of a fuzzy set, types of membership functions, characteristics of a fuzzy set, basic relationships between fuzzy sets - equality and inclusion, fuzzy sets and sets - the representation theorem, the extension principles, membership function determination, generalizations of fuzzy sets, chapter summary, problems, references fuzzy set operations - set theory operations and their properties, triangular norms, aggregation operations on fuzzy sets, sensitivity of fuzzy sets operators, negations, comparison operations on fuzzy sets, chapter summary, problems, references information-based characterization of fuzzy sets -entropy measures of fuzziness, energy measures of fuzziness, specificity of a fuzzy set, frames of cognition, information encoding and decoding using linguistic landmarks, decoding mechanisms for pointwise data, decoding using membership functions of the linguistic terms of the codebook, general possibility-necessity decoding, distance between fuzzy sets based on their internal, linguistic representation, chapter summary, problems, references fuzzy relations and their calculus -relations and fuzzy relations, operations on fuzzy relations, compositions of fuzzy relations, projections and cylindric extensions of fuzzy relations, binary fuzzy relations, some classes of fuzzy relations, fuzzy-relational equations, estimation and inverse problem in fuzzy relational equations, solving fuzzy-relational equations with the sup-t composition, solutions to dual fuzzy-relational equations, adjoint fuzzy-relational equations, generaliations of fuzzy relational equations, approximate solutions to fuzzy-relational equations, chapter summary, problems, references fuzzy numbers - defining fuzzy numbers, interval analysis and fuzzy numbers, computing with fuzzy numbers, triangular fuzzy numbers and basic operations, general formulas for LR fuzzy numbers, accumulation of fuzziness in computing with fuzzy numbers, inverse problem in computation with fuzzy numbers, fuzzy numbers and approximate operations, chapter summary, problems, references fuzzy modelling - fuzzy models - beyond numerical computations, main phases of system modelling, fundamental design objectives in system modelling, general topology of fuzzy models, compatibility of encoding and decoding modules, classes of fuzzy models, verification and validation of fuzzy models, chapter summary, problems, references. Part 3 Problem solving with fuzzy sets: methodology -analysis and design, fuzzy controllers and fuzzy control, mathematical programming and fuzzy optimization, chapter summary, problems, references case studies - traffic intersection control, distributed traffic control, elevator group control, induction motor control, communication network planning, neurocomputation in fault diagnosis of dynamic systems, multicommodity transportation planning in railways.

Introduction to type-2 fuzzy logic systems

Abstract: This paper introduces a robust fuzzy logic system, one that can handle rule uncertainties. We make use of type-2 fuzzy sets for this purpose. The development of a type-2 fuzzy logic system has led to a new operation that we call type-reduction. In the course of this development, we also study set operations on type-2 sets, properties of membership grades of type-2 sets, type-2 relations and their compositions, and defuzzification.

Mining fuzzy association rules in databases

Abstract: Data mining is the discovery of previously unknown, potentially useful and hidden knowledge in databases. In this paper, we concentrate on the discovery of association rules. Many algorithms have been proposed to find association rules in databases with binary attributes. We introduce the fuzzy association rules of the form, 'If X is A then Y is B', to deal with quantitative attributes. X, Y are set of attributes and A, B are fuzzy sets which describe X and Y respectively. Using the fuzzy set concept, the discovered rules are more understandable to human. Moreover, fuzzy sets handle numerical values better than existing methods because fuzzy sets soften the effect of sharp boundaries.

Shadowed sets: representing and processing fuzzy sets

Abstract: This study introduces a new concept of shadowed sets that can be regarded as a certain operational framework simplifying processing carried out with the aid of fuzzy sets and enhancing interpretation of results obtained therein. Some conceptual links between this idea and some others known in the literature are established. In particular, it is demonstrated how fuzzy sets can induce shadowed sets. Subsequently, shadowed sets reveal interesting conceptual and algorithmic relationships existing between rough sets and fuzzy sets. Detailed computational aspects of shadowed sets are discussed. Several illustrative examples are provided.

A new correlation-based fuzzy logic clustering algorithm for fMRI

Abstract: Fuzzy logic clustering algorithms are a new class of processing strategies for functional MRI (fMRI). In this study, the ability of such methods to detect brain activation on application of a stimulus task is demonstrated. An optimization of the selected algorithm with regard to different parameters is proposed. These parameters include (a) those defining the pre-processing procedure of the data set; (b) the definition of the distance between two time courses, considered as p-dimensional vectors, where p is the number of sequential images in the fMRI data set; and (c) the number of clusters to be considered. Based on the assumption that such a clustering algorithm should cluster the pixel time courses according to their similarity and not their proximity (in terms of distance), cross-correlation-based distances are defined. A clear mathematical description of the algorithm is proposed, and its convergence is proven when similarity measures are used instead of conventional Euclidean distance. The differences between the membership function given by the algorithm and the probability are clearly exposed. The algorithm was tested on artificial data sets, as well as on data sets from six volunteers undergoing stimulation of the primary visual cortex. The fMRI maps provided by the fuzzy logic algorithm are compared to those achieved by the well established cross-correlation technique.

Type-2 fuzzy logic systems: type-reduction

Abstract: Type-reduction in a type-2 fuzzy logic system (FLS) is an "extended" version of the defuzzification operation in a type-1 FLS. In this paper, we briefly review the structure of a type-2 FLS and describe type-reduction in detail. We focus on a center-of-sets type-reducer, and provide some examples to illustrate it. We also provide some practical approximations to type-reduction computations for certain type-2 membership functions.

Type 2 fuzzy sets: an appraisal of theory and applications

Abstract: This paper provides a guide and tutorial to type 2 fuzzy sets. Type 2 fuzzy sets allow for linguistic grades of membership thus assisting in knowledge representation. They also offer improvement on inferencing with type 1 sets. The various approaches to knowledge representation and inferencing are discussed, with worked examples, and some of the applications of type 2 sets are reported.

Integrating fuzzy knowledge by genetic algorithms

Abstract: We propose a genetic algorithm-based fuzzy knowledge integration framework that can simultaneously integrate multiple fuzzy rule sets and their membership function sets. The proposed approach consists of two phases: fuzzy knowledge encoding and fuzzy knowledge integration. In the encoding phase, each fuzzy rule set with its associated membership functions is first transformed into an intermediary representation and then further encoded as a string. The combined strings form an initial knowledge population, which is then ready for integration. In the knowledge-integration phase, a genetic algorithm is used to generate an optimal or nearly optimal set of fuzzy rules and membership functions from the initial knowledge population. Two application domains, the hepatitis diagnosis and the sugarcane breeding prediction, were used to show the performance of the proposed knowledge-integration approach. Results show that the fuzzy knowledge base derived using our approach performs better than every individual knowledge base.

Some comments on interval valued fuzzy sets

Abstract: This article presents a framework for fuzzy set theory in which fuzzy values are intervals. © 1996 John Wiley & Sons, Inc.

Compensatory neurofuzzy systems with fast learning algorithms

Abstract: In this paper, a new adaptive fuzzy reasoning method using compensatory fuzzy operators is proposed to make a fuzzy logic system more adaptive and more effective. Such a compensatory fuzzy logic system is proved to be a universal approximator. The compensatory neural fuzzy networks built by both control-oriented fuzzy neurons and decision-oriented fuzzy neurons cannot only adaptively adjust fuzzy membership functions but also dynamically optimize the adaptive fuzzy reasoning by using a compensatory learning algorithm. The simulation results of a cart-pole balancing system and nonlinear system modeling have shown that: 1) the compensatory neurofuzzy system can effectively learn commonly used fuzzy IF-THEN rules from either well-defined initial data or ill-defined data; 2) the convergence speed of the compensatory learning algorithm is faster than that of the conventional backpropagation algorithm; and 3) the efficiency of the compensatory learning algorithm can be improved by choosing an appropriate compensatory degree.

A Mathematical Programming Approach for Measuring Technical Efficiency in a Fuzzy Environment

Abstract: A three stage approach is proposed to measure technical efficiency in a fuzzy environment. This approach uses the traditional data envelopment analysis framework and then merges concepts developed in fuzzy parametric programming (Carlsson and Korhonen, 1986). In the first stage, vague input and output variables are expressed in terms of their risk-free and impossible bounds and a membership function. This membership function represents the degree to which a production scenario is plausible. In the second stage, conventional efficiency measurement models (Banker, Charnes and Cooper, 1984; Deprins, Simar and Tulkens, 1984) are re-formulated in terms of the risk-free and impossible bounds and the membership function for each of the fuzzy input and output variables. In the third stage, technical efficiency scores are computed for different values of the membership function so as to identify uniquely sensitive decision making units. The approach is illustrated in the context of a preprint and packaging manufacturing line which inserts commercial pamphlets into newspapers.

Minimal fuzzy memberships and rules using hierarchical genetic algorithms

Abstract: A new scheme to obtain optimal fuzzy subsets and rules is proposed. The method is derived from the use of genetic algorithms, where the genes of the chromosome are classified into two different types. These genes can be arranged in a hierarchical form, where one type of gene controls the other. The effectiveness of this genetic formulation enables the fuzzy subsets and rules to be optimally reduced and, yet, the system performance is well maintained. In this paper, the details of formulation of the genetic structure are given. The required procedures for coding the fuzzy membership function and rules into the chromosome are also described. To justify this approach to fuzzy logic design, the proposed scheme is applied to control a constant water pressure pumping system. The obtained results, as well as the associated final fuzzy subsets, are included in this paper. Because of its simplicity, the method could lead to a potentially low-cost fuzzy logic implementation.

How the learning of rule weights affects the interpretability of fuzzy systems

Abstract: Neuro-fuzzy systems have recently gained a lot of interest in research and applications. These are approaches that learn fuzzy systems from data. Many of them use rule weights for this task. In this paper we discuss the influence of rule weights on the interpretability of fuzzy systems. We show how rule weights can be equivalently replaced by modifications in the membership functions of a fuzzy system. We elucidate the effects rule weights have on a fuzzy rule base. Using our neuro-fuzzy model NEFCLASS we demonstrate the problems of using rule weights in a simple example, and we show that learning in fuzzy systems can be done without them.

Genetic algorithms for determining fuzzy measures from data

Abstract: A synthetic evaluation of a given object in terms of multiple factors that contribute to some feature of the object (quality, performance, etc) may be regarded as a system with multiple inputs and one output Traditionally, the output is expressed as the weighted average of the inputs Unfortunately, this method is severely limited as it cannot capture any inherent relation among the factors involved This limitation can be overcome by using the Choquet integral or the fuzzy integral with respect to a fuzzy measure that captures the relation among the factors The crux of this method is to determine the right fuzzy measure In this paper, we describe an efficient genetic algorithm for constructing a suitable fuzzy measure from relevant input-output data This algorithm has a broad applicability in various problem areas, such as decision making, cluster analysis, pattern recognition, image and speech processing, and expert systems

Fuzzy finite-state automata can be deterministically encoded into recurrent neural networks

Abstract: There has been an increased interest in combining fuzzy systems with neural networks because fuzzy neural systems merge the advantages of both paradigms. On the one hand, parameters in fuzzy systems have clear physical meanings and rule-based and linguistic information can be incorporated into adaptive fuzzy systems in a systematic way. On the other hand, there exist powerful algorithms for training various neural network models. However, most of the proposed combined architectures are only able to process static input-output relationships; they are not able to process temporal input sequences of arbitrary length. Fuzzy finite-state automats (FFAs) can model dynamical processes whose current state depends on the current input and previous states. Unlike in the case of deterministic finite-state automats (DFAs), FFAs are not in one particular state, rather each state is occupied to some degree defined by a membership function. Based on previous work on encoding DFAs in discrete-time second-order recurrent neural networks, we propose an algorithm that constructs an augmented recurrent neural network that encodes a FFA and recognizes a given fuzzy regular language with arbitrary accuracy. We then empirically verify the encoding methodology by correct string recognition of randomly generated FFAs. In particular, we examine how the networks' performance varies as a function of synaptic weight strengths.

The Theory of Fuzz Logic and its Application to Real Estate Valuation

Abstract: Fuzzy logic is based on the central idea that in fuzzy sets each element in the set can assume a value from 0 to 1, not just 0 or 1, as in classic set theory. Thus, qualitative characteristics and ...

Fuzzy Rule Based Modeling as a Universal Approximation Tool

Abstract: In some cases, fuzzy rule based model needs tuning. If we have applied some version of fuzzy rule based modeling, and the resulting model is satisfactory, great. But sometimes, the resulting model is not of very high quality: ■ we may have misinterpreted some of the expert’s rules; ■ we may have missed some of the important rules.

Fuzzy Control of Chaos

Abstract: In this chapter a Mamdani fuzzy model based fuzzy control technique is proposed to control chaotic systems, whose dynamics is complex and unknown, to the unstable periodic orbits (UPO). Some empirical tricks are introduced for building up a proper fuzzy rule base and designing a fuzzy controller. Finally, an example of fuzzy control of the Chua’s circuit is presented to illustrate the effectiveness of the proposed approach.

Image thresholding using fuzzy entropies

Abstract: An image can be regarded as a fuzzy subset of a plane. A fuzzy entropy measuring the blur in an image is a functional which increases when the sharpness of its argument image decreases. We generalize and extend the relation "sharper than" between fuzzy sets in view of implementing the properties of a relation "sharper than" between images. We show that there are infinitely many implementations of this relation into an ordering between fuzzy sets (equivalently, images). Relying upon these orderings, we construct classes of fuzzy entropies which are useful for image thresholding by cost minimization. Assuming the image to be a degraded version of an ideal two level image (object/background), a fuzzy entropy can be introduced in a cost functional to force the fitting function to be as close as possible to a two-valued function. The minimization problem is numerically solved, and the results obtained on a synthetic image are reported.