Showing papers on "Fuzzy mathematics published in 2001"

Effect of rule weights in fuzzy rule-based classification systems

Abstract: This paper examines the effect of rule weights in fuzzy rule-based classification systems. Each fuzzy IF-THEN rule in our classification system has antecedent linguistic values and a single consequent class. We use a fuzzy reasoning method based on a single winner rule in the classification phase. The winner rule for a new pattern is the fuzzy IF-THEN rule that has the maximum compatibility grade with the new pattern. When we use fuzzy IF-THEN rules with certainty grades, the winner is determined as the rule with the maximum product of the compatibility grade and the certainty grade. In this paper, the effect of rule weights is illustrated by drawing classification boundaries using fuzzy IF-THEN rules with/without certainty grades. It is also shown that certainty grades play an important role when a fuzzy rule-based classification system is a mixture of general rules and specific rules. Through computer simulations, we show that comprehensible fuzzy rule-based systems with high classification performance can be designed without modifying the membership functions of antecedent linguistic values when we use fuzzy IF-THEN rules with certainty grades.

Statistical Modeling, Analysis and Management of Fuzzy Data

Abstract: 1. Fuzziness and Randomness.- Fuzziness and randomness.- 2. Fuzzy-Valued Random Elements.- On the variance of random fuzzy variables.- f-inequality indices for fuzzy random variables.- Traditional techniques to prove some limit theorems for fuzzy random variables.- Convergence in graph for fuzzy valued martingales and smartingales.- Remarks on Korovkin-type approximation of fuzzy random variables.- Several notions of differentiability for fuzzy set-valued mappings.- 3. Possibility, Probability and Fuzzy Measures.- Average level of a fuzzy set.- Second order possibility measure induced by a fuzzy random variable.- Measure extension from meet-systems and falling measures representation.- The structure of fuzzy measure families induced by upper and lower probabilities.- Statistical classes and fuzzy set theoretical classification of probability distributions.- 4. Statistics and Fuzzy Data Analysis.- Statistics with one-dimensional fuzzy data.- Testing fuzzy hypotheses with vague data.- Possibilistic interpretation of fuzzy statistical tests.- Possibilistic regression analysis.- Linear regression in a fuzzy context. The least square method.- Linear regression with random fuzzy observations.

The shape of fuzzy sets in adaptive function approximation

Abstract: The shape of if-part fuzzy sets affects how well feedforward fuzzy systems approximate continuous functions. We explore a wide range of candidate if-part sets and derive supervised learning laws that tune them. Then we test how well the resulting adaptive fuzzy systems approximate a battery of test functions. No one shape emerges as the best. The sine function often does well and has tractable learning, but its undulating side-lobes may have no linguistic meaning. This suggests that function-approximation accuracy may sometimes have to outweigh linguistic or philosophical interpretations. We divide the if-part sets into two large classes. The first consists of n-dimensional joint sets that factor into n scalar sets. These sets ignore the correlations among input vector components. Fuzzy systems suffer in general from exponential rule explosion in high dimensions when they blindly approximate functions. The factorable fuzzy sets themselves also suffer from a curse of dimensionality: they tend to become binary spikes in high dimension. The second class consists of the more general but less common n-dimensional joint sets that do not factor into n scalar fuzzy sets. We present a method for constructing such unfactorable joint sets from scalar distance measures. Fuzzy systems that use unfactorable sets need not suffer from exponential rule explosion but their increased complexity may lead to intractable learning and inscrutable if-then rules. We prove that some of these sets still suffer from spikiness.

A survey of defuzzification strategies

Abstract: Defuzzification is an important operation in the theory of fuzzy sets. It transforms a fuzzy set information into a numeric data information. This operation along with the operation of fuzzification is critical to the design of fuzzy systems as both of these operations provide nexus between the fuzzy set domain and the real-valued scalar domain. We need the synergy of both of these domains to solve many of our ill-posed problems effectively. In this paper, we address the problem of defuzzification, we present merits and demerits of various defuzzification strategies that are used in the theory and practice, and in design and implementation of applications involving fuzzy theory, fuzzy control, and fuzzy rule base, and fuzzy inference-based systems. We also present in this paper a simple and yet a novel defuzzification mechanism. © 2001 John Wiley & Sons, Inc.

Discovering knowledge from fuzzy concept lattice

Abstract: Since its inception, association rule mining has become one of the core data mining tasks, and has attracted tremendous interest among researchers and practitioners. Many efficient algorithms have been proposed in the literature, e.g., Apriori, Partition, DIC, for mining association rules in the context of marketbasket analysis. They are all based on apriori methods, i.e., pruning the itemset lattice, and requires multiple database accesses. However, research so far has mainly focused on mining over binary data, i.e., either an item is present in a transaction or not. Little attention was paid to mining over data where the quantity of items is considered. In this paper, we propose to address the problem of mining fuzzy association rules, by considering the quantity of items in the transactions. After the fuzzification of the transaction database, we apply a new efficient algorithm, called FARD (Fuzzy Association Rule Discovery), for mining fuzzy association rules. FARD is based on the pruning of the fuzzy concept lattice, and can be applied equally to classical or fuzzy databases, by scanning the database only once.

Fuzzy entropy: a brief survey

Abstract: This paper presents a survey about different types of fuzzy information measures. A number of schemes have been proposed to combine the fuzzy set theory and its application to the entropy concept as a fuzzy information measurements. The entropy concept, as a relative degree of randomness, has been utilized to measure the fuzziness in a fuzzy set or system. However, a major difference exists between the classical Shannon entropy and the fuzzy entropy. In fact while the later deals with vagueness and ambiguous uncertainties, the former tackles probabilistic uncertainties (randomness).

Using statistical viewpoint in developing correlation of intuitionistic fuzzy sets

Abstract: In this paper, we propose a method to calculate the correlation coefficient of intuitionistic fuzzy sets by means of mathematical statistics. This value obtained from our formula tell us not only the strength of relationship between the intuitionistic fuzzy sets, but also whether the intuitionistic fuzzy sets are positively or negatively related. This approach looks better than previous methods which only evaluate the strength of the relation. Furthermore, we extend the proposed method to interval-valued intuitionistic fuzzy sets. The value of the correlation coefficient between interval-valued intuitionistic fuzzy sets lies in the interval [-1, 1], as computed from our formula.

On complex fuzzy sets

Abstract: The innovative concept of complex fuzzy sets is introduced. The novelty of the complex fuzzy set lies in the range of values its membership function may attain. In contrast to a traditional fuzzy membership function, this range is not limited to [0,1] but extended to the unit circle in the complex plane. Thus, the complex fuzzy set provides a mathematical framework for describing membership in a set in terms of a complex number. A study of this original concept is presented, including examples of possible applications, which demonstrate the new theory.

Data Engineering: Fuzzy Mathematics in Systems Theory and Data Analysis

Abstract: From the Publisher: There are many situations in science and engineering where complex output data from a given system is used to formulate a model of how that system operates, or to simulate its response to different inputs. Applications include control, decision theory, and the emerging fields of bioinformatics. A key advance in this general area is the use of fuzzy mathematics in the development of models.

Fuzzy Cooperative Games: Cooperation with Vague Expectations

Abstract: 1 Introduction.- 2 Fuzzy Quantities.- 3 Deterministic Coalition Games.- 4 Vagueness and Its Processing.- 5 Fuzzy Additivity and Related Topics.- 6 Fuzzy Core and Effective Coalitions.- 7 Fuzzy Convexity.- 8 Fuzzy Balancedness.- 9 Fuzzy Shapley Value.- 10 Fuzzy Superoptimum.- 11 Fuzzy and Imputational Additivity-Like Relations.- 12 Fuzzy Core and Effectivity in Games Without Side-Payments.- 13 Linear Coalition Games.- 14 A Modified Model of Domination and Superoptimum.- 15 Strategic Background of Fuzzy Cooperation.- 16 Generation of Fuzzy Quantities.- V Concluding Remarks.- References.

Generalized Concavity in Fuzzy Optimization and Decision Analysis

Abstract: Preface. Acknowledgments. Part I: Theory. 1. Preliminaries. 2. Generalized Convex Sets. 3. Generalized Concave Functions. 4. Triangular Norms and T-Quasiconcave Functions. 5. Aggregation Operators. 6. Fuzzy Sets. Part II: Applications. 7. Fuzzy Multi-Criteria Decision Making. 8. Fuzzy Mathematical Programming. 9. Fuzzy Linear Programming. 10. Fuzzy Sequencing and Scheduling.

Learning Fuzzy Classification Rules from Data

Abstract: Automatic design of fuzzy rule-based classification systems based on labeled data is considered. It is recognized that both classification performance and interpretability are of major importance and effort is made to keep the resulting rule bases small and comprehensible. An iterative approach for developing fuzzy classifiers is proposed. The initial model is derived from the data and subsequently, feature selection and rule base simplification are applied to reduce the model, and a GA is used for model tuning. An application to the Wine data classification problem is shown.

Connectionist Structures of Type 2 Fuzzy Inference Systems

Abstract: In Fuzzy Inference Systems (FIS) the rule base consists of fuzzy relations between antecedents and consequents represented by classical fuzzy sets. Because their membership grades are exact real numbers in the unit interval [0, 1], there is no uncertainty in this sort of specification. In many applications there is some uncertainty as to the memberships, hence they can be stated as ordinary fuzzy sets of type 1 and can constitute type 2 fuzzy sets.In the world literature exists a global model of type 2 FIS. However it consists of an enormous number of embedded subsystems of type 1 and with regard to this model it has not found any use in connectionist realizations. In this paper we derive connectionist structures of type 2 FIS.

Fuzzy association rules for handling continuous attributes

Abstract: Association rules are frequently used for describing association (i.e., co-occurrence) among attribute values in the field of data mining. When an attribute is continuous (i.e., real-valued) such as height, length and weight, its domain is usually discretized into several intervals. Fuzzy rules are recognized as a convenient tool for handling continuous attributes in a human understandable manner. When we use fuzzy rules, the domain of each continuous attribute is discretized into several fuzzy sets. A linguistic label is usually associated with each fuzzy set especially when linguistic interpretations of fuzzy rules are required. In this paper, we first fuzzify the concept of association rules. That is, we show fuzzy versions of two measures (i.e., confidence and support) that are used for evaluating each association rule in the field of data mining. Then we illustrate these two measures of fuzzy rules for function approximation and pattern classification problems. Finally we examine the relation between the classification performance of fuzzy rules and the definition of their certainty grades through computer simulations. Simulation results show that the direct use of confidence as a certainty grade is not always appropriate from the viewpoint of classification performance.