Fuzzy Set Values Research Articles

(L, M)‐Fuzzy σ‐Algebras

The notion of (L, M)‐fuzzy σ‐algebras is introduced in the lattice value fuzzy set theory. It is a generalization of Klement′s fuzzy σ‐algebras. In our definition of (L, M)‐fuzzy σ‐algebras, each L‐fuzzy subset can be regarded as an L‐measurable set to some degree.

The variety generated by the truth value algebra of type-2 fuzzy sets

This paper addresses some questions about the variety generated by the algebra of truth values of type-2 fuzzy sets. Its principal result is that this variety is generated by a finite algebra, and in particular is locally finite. This provides an algorithm for determining when an equation holds in this variety. It also sheds light on the question of determining an equational axiomatization of this variety, although this problem remains open.

Fuzzy multiple attributes group decision-making based on the ranking values and the arithmetic operations of interval type-2 fuzzy sets

In this paper, we present a new method to handle fuzzy multiple attributes group decision-making problems based on the ranking values and the arithmetic operations of interval type-2 fuzzy sets. First, we present the arithmetic operations between interval type-2 fuzzy sets. Then, we present a fuzzy ranking method to calculate the ranking values of interval type-2 fuzzy sets. We also make a comparison of the ranking values of the proposed method with the existing methods. Based on the proposed fuzzy ranking method and the proposed arithmetic operations between interval type-2 fuzzy sets, we present a new method to handle fuzzy multiple attributes group decision-making problems. The proposed method provides us with a useful way to handle fuzzy multiple attributes group decision-making problems in a more flexible and more intelligent manner due to the fact that it uses interval type-2 fuzzy sets rather than traditional type-1 fuzzy sets to represent the evaluating values and the weights of attributes.

The Concepts of Tightness for Fuzzy Set Valued Random Variables

In this paper, we introduce several concepts of tightness for a sequence of random variables taking values in the space of normal and upper-semicontinuous fuzzy sets with compact support in <TEX>$R^p$</TEX> and give some characterizations of their concepts. Also, counter-examples for the relationships between the concepts of tightness are given.

Some general comments on fuzzy sets of type-2

This paper contains some general comments on the algebra of truth values of fuzzy sets of type 2. It details the precise mathematical relationship with the algebras of truth values of ordinary fuzzy sets and of interval-valued fuzzy sets. Subalgebras of the algebra of truth values and t-norms on them are discussed. There is some discussion of finite type-2 fuzzy sets. © 2008 Wiley Periodicals, Inc.

AUTOMORPHISMS OF THE ALGEBRA OF FUZZY TRUTH VALUES II

This paper continues an earlier investigation of automorphisms of the algebra of truth values of type-2 fuzzy sets. Several significant problems left unresolved are settled, the most basic being that the automorphism group of the algebra of fuzzy truth values is isomorphic in a natural way to the product of two copies of the automorphism group of the unit interval. This theorem has several corollaries concerning characteristic subalgebras.

Sets with type-2 operations

The algebra of truth values of type-2 fuzzy sets consists of all mappings of the unit interval to itself, with type-2 operations that are convolutions of ordinary max and min operations. This paper is concerned with a special subalgebra of this truth value algebra, namely the set of nonzero functions with values in the two-element set {0, 1}. This algebra can be identified with the set of all non-empty subsets of the unit interval, but the operations are not the usual union and intersection. We give simplified descriptions of the operations and derive the basic algebraic properties of this algebra, including the identification of its automorphism group. We also discuss some subalgebras and homomorphisms between them and look briefly at t-norms on this algebra of sets.

Lattices of convex normal functions

The algebra of truth values of type-2 fuzzy sets is the set of all functions from the unit interval into itself, with operations defined in terms of certain convolutions of these functions with respect to pointwise max and min. This algebra has been studied rather extensively, both from a theoretical and from a practical point of view. It has a number of interesting subalgebras, and this paper is about the subalgebra of all convex normal functions, and closely related ones. These particular algebras are De Morgan algebras, and our concern is principally with their completeness as lattices. A special feature of our treatment is a representation of these algebras as monotone functions with pointwise order, making the operations more intuitive.

Fuzzy interpolative reasoning for sparse fuzzy rule-based systems based on the ranking values of fuzzy sets

Fuzzy interpolative reasoning is an important research topic of sparse fuzzy rule-based systems. In recent years, some methods have been presented for dealing with fuzzy interpolative reasoning. However, the involving fuzzy sets appearing in the antecedents of fuzzy rules of the existing fuzzy interpolative reasoning methods must be normal and non-overlapping. Moreover, the reasoning conclusions of the existing fuzzy interpolative reasoning methods sometimes become abnormal fuzzy sets. In this paper, in order to overcome the drawbacks of the existing fuzzy interpolative reasoning methods, we present a new fuzzy interpolative reasoning method for sparse fuzzy rule-based systems based on the ranking values of fuzzy sets. The proposed fuzzy interpolative reasoning method can handle the situation of non-normal and overlapping fuzzy sets appearing in the antecedents of fuzzy rules. It can overcome the drawbacks of the existing fuzzy interpolative reasoning methods in sparse fuzzy rule-based systems.

A hybrid object based model combining probability and fuzzy set theories

Although there have been many fuzzy object-oriented data model proposed, and a bit less for probabilistic ones, models combining the relevance and strength of both fuzzy set theory and probability theory appear to be sporadic. This paper introduces our extension of Eiter et al.'s probabilistic object base model with two key features: uncertain and imprecise attribute values are represented as probability distributions on a set of fuzzy set values; class methods with uncertain and imprecise input and output arguments are formally integrated into the new model. A probabilistic interpretation of relations on fuzzy set values is proposed for their combination with probability degrees. Then the syntax and semantics of fuzzy-probabilistic object base schemas, instances, and selection operation are defined.

Fuzzy set-valued Gaussian processes and Brownian motions

Gaussian processes and Brownian motion are concepts and tools in modelling important uncertain systems in many areas. In view of uncertainty complexity in many real-world problems, we extend these tools to the case where stochastic processes can take on fuzzy sets as values. In this paper, we discuss fuzzy set-valued Gaussian processes based on the results of [S. Li, Y. Ogura, V. Kreinovich, Limit Theorems and Applications of Set-Valued and Fuzzy Set-Valued Random Variables, Kluwer Academic Publishers, Dordrecht, 2002; S. Li, Y. Ogura, H.T. Nguyen, Gaussian processes and martingales for fuzzy valued variables with continuous parameter, Inform. Sci. 133 (2001) 7–21; S. Li, Y. Ogura, F.N. Proske, M.L. Puri, Central limit theorems for generalized set-valued random variables, J. Math. Anal. Appl. 285 (2003) 250–263; N.N. Lyashenko, On limit theorems for sums of independent compact random subsets in the Euclidean space, J. Soviet Math. 20 (1982) 2187–2196] and [M.L. Puri, D.A. Ralescu, The concept of normality for fuzzy random variables, Ann. Probab. 13 (1985) 1373–1379]. We also introduce the concept of fuzzy set-valued Brownian motion, and then prove several properties of such processes.

AUTOMORPHISMS OF THE ALGEBRA OF FUZZY TRUTH VALUES

This paper is an investigation of the automorphisms of the algebra of truth values of type-2 fuzzy sets. This algebra contains isomorphic copies of the truth value algebras of type-1 and of interval-valued fuzzy sets. It is shown that these subalgebras are characteristic; that is, are carried onto themselves by automorphisms of the containing algebra of truth values of fuzzy sets. Some other relevant subalgebras are proved characteristic, including the subalgebra of convex normal functions. The principal tool in this study is the determination of various irreducible elements.

Multivariate Fuzzy Multinomial Control Charts

Two approaches for constructing control charts to monitor multivariate attribute processes when data set is presented in linguistic form are suggested. Two monitoring statistics T2f and W2 are developed based on fuzzy and probability theories. The first is similar to the Hotelling’s T2 statistic and is based on representative values of fuzzy sets. The distribution of W2 statistic, being a linear combination of dependent chi-square variables, is derived using Satterthwaite’s approximation. Resulting multivariate control charts are compared based on the average run length (ARL). A numerical example is given to illustrate the application of the proposed multivariate control charts and the interpretation of out-of-control signals.

On the Design of Simple-structured Adaptive Fuzzy Logic Controllers

One of the methods to simplify the design process for a fuzzy logic controller (FLC) is to reduce the number of variables representing the rule antecedent. This in turn decreases the number of control rules, membership functions, and scaling factors. For this purpose, we designed a single-input FLC that uses a sole fuzzy input variable. However, it is still deficient in the capability of adapting some varying operating conditions although it provides a simple method for the design of FLC's. We here design two simple-structured adaptive fuzzy logic controllers (SAFLC's) using the concept of the single-input FLC. Linguistic fuzzy control rules are directly incorporated into the controller by a fuzzy basis function. Thus some parameters of the membership functions characterizing the linguistic terms of the fuzzy control rules can be adjusted by an adaptive law. In our controllers, center values of fuzzy sets are directly adjusted by an adaptive law. Two SAFLC's are designed. One of them uses a Hurwitz error dynamics and the other a switching function of the sliding mode control (SMC). We also prove that 1） their closed-loop systems are globally stable in the sense that all signals involved are bounded and 2） their tracking errors converge to zero asymptotically. We perform computer simulations using a nonlinear plant.

A deductive probabilistic and fuzzy object-oriented database language

We introduce a deductive probabilistic and fuzzy object-oriented model where a class property (i.e., an attribute or a method) can contain fuzzy set values, and uncertain class membership and property applicability are measured by lower and upper bounds on probability. Each uncertainly applicable property is interpreted as a default probabilistic logic rule, which is defeasible, and probabilistic default reasoning on fuzzy events is proposed for uncertain property inheritance and class recognition. This provides a formal basis for the design and implementation of FRIL++, the object-oriented extension of FRIL, a logic programming language dealing with both probability and fuzziness. The basic features of FRIL++ and its application as a programming language for deductive probabilistic and fuzzy object-oriented databases are presented.

Fuzzy convex sets in median algebras

The fuzzification of convex sets in median algebras is considered, and some of their properties are investigated. A characterization of finite valued fuzzy convex set is given.

GIS Integration of Remote Sensing and Topographic Data Using Fuzzy Logic for Ground Water Assessment in Midnapur District, India

A GIS based approach is proposed for the integration of three thematic maps viz. geomorphology, drainage density and slope using fuzzy logic for the assessment of ground water resource potential of a soft rock terrain of Midnapur District, West Bengal, India. The geomorphology and drainage density maps of the area are prepared from IRS‐1B LISS‐II data, and the slope map is obtained from the contours depicted on the topographic map of Survey of India. Each feature of all the thematic maps is assigned with individual fuzzy set values within a range between 0 to 1 according to their relative importance in the prediction of ground water occurrence. The maps are then integrated through fuzzy operation to model the ground water potential zone of the study area. The evolved model while verified with surface geophysical results is found to be in good agreement.

HAUSDORFF INTERVAL VALUED FUZZY FILTERS

The notion of Interval Valued Fuzzy Sets (IVF sets) was introduced by T. K. Mondal. In this paper a notion of IVF filter is introduced and studied. A new notion of Hausdorffness, which can not be defined in crisp theory of filters, is defined on IVF filters and their properties are studied.

Approximate reasoning based on linguistic truth value with α-operator

In this paper we discuss the approximate reasoning based on linguistic truth value (RLV) with the α-operator. A new method of RLV is proposed. It is equivalent to the original RLV method, but without using the linguistic truth value fuzzy set. Under given conditions, we convert the problem of calculating sup( a′ ∧ ( aαb)) into the problem of solving the equation of a′ Ta = b. This significantly simplifies the computations of RLV. Some properties of the RLV are studied with examples, such as the “fuzziness” of the RLV based on deferent α-operators. A graphical method of RLV is also given to provide a convenient tool to implement the proposed method.

A Fuzzy Multiple Model Approach to Manoeuvring Target Tracking

A multiple model algorithm based on fuzzy sets is introduced as an alternative to the Interacting Multiple Model Algorithm(IMMA) and the Adaptive Interacting Multiple Model Algorithm(AIMMA) for tracking manoeuvring targets. An augmented state filter provides estimates of the target acceleration, and using these estimates model weightings are obtained from membership values of fuzzy sets. By arranging the overlap of these fuzzy sets, the number of filters operating, and hence the computational load, can be significantly reduced. Simulation results presented suggest comparable performance to IMMA and AIMMA during manoeuvres, with improved performance during nonmanoeuvre, constant velocity periods.