Deriving minimal solutions for fuzzy relation equations with max-product composition

This work considers fuzzy relation equations with max-product composition. The critical problem in solving such equations is to determine the minimal solutions when an equation is solvable. However, this problem is NP-hard and difficult to solve [A.V. Markovskii, On the relation between equations with max-product composition and the covering problem, Fuzzy Sets and Systems 153 (2005) 261-273]. This work first examines the attributes of a solvable equation and characteristics of minimal solutions, then reduces the equation to an irreducible form, and converts the problem into a covering problem, for which minimal solutions are correspondingly determined. Furthermore, for theoretical and practical applications, this work presents a novel method for obtaining minimal solutions. The proposed method easily derives a minimal solution, and obtains other minimal solutions from this predecessor using a back-tracking step. The proposed method is compared with an existing algorithm, and some applications are described in detail.     

H-transformation of t-norms

Cancellative t-norms are generated only if they do not posses anomalous pairs and then they are isomorphic to subsemigroups of ([0, ∞], +). A new construction method for t-norms, so-called H-transformation of t-norms, is introduced and studied, generalizing a recent example of Hájek. H-transformation allows to construct non-generated cancellative (left-continuous) t-norms with prescribed number of non-trivial Archimedean components. The unique t-norm stable under H-transformation is also described. In an analogous way, H-transformation of residual implications is given. Finally, a general H-transformation based on cancellative discrete t-conorms is proposed and examined.     

Applications of interval valued t-norms (t-conorms) to fuzzy n-ary sub-hypergroups

In this paper, we study a generalization of group, hypergroup and n-ary group. Firstly, we define interval-valued fuzzy (anti fuzzy) n-ary sub-hypergroup with respect to a t-norm T (t-conorm S). We give a necessary and sufficient condition for, an interval-valued fuzzy subset to be an interval-valued fuzzy (anti fuzzy) n-ary sub-hypergroup with respect to a t-norm T (t-conorm S). Secondly, using the notion of image (anti image) and inverse image of a homomorphism, some new properties of interval-valued fuzzy (anti fuzzy) n-ary sub-hypergroup are obtained with respect to infinitely ∨-distributive t-norms T (∧-distributive t-conorms S). Also, we obtain some results of T-product (S-product) of the interval-valued fuzzy subsets for infinitely ∨-distributive t-norms T (∧-distributive t-conorms S). Lastly, we investigate some properties of interval-valued fuzzy subsets of the fundamental n-ary group with infinitely ∨-distributive t-norms T (∧-distributive t-conorms S).     

An algorithm for solving fuzzy relation equations with max-T composition operator

This paper studies the problem of solving a max-T composite finite fuzzy relation equation, where T is a special class of pseudo-t-norms. If the equation is solvable, then its set of feasible solutions is determined by the greatest solution and a finite number of minimal solutions. Some necessary conditions are presented for the minimal solutions in terms of the maximum solution and zero value. Under these conditions, some minimal solutions of the system can be obtained easily. Some procedures are also proposed in order to simplify the original system. The simplified system is then decomposed (if possible) into several subsystems with smaller dimensions, which are very easy to solve. Furthermore, a method is presented to solve each subsystem. By combining the method and those procedures, an efficient algorithm is proposed to obtain the set of feasible solutions of the original system. Two examples are also given to illustrate the algorithm.     

Solutions of fuzzy relation equations based on continuous t-norms

This study is concerned with fuzzy relation equations with continuous t-norms in the form ATR = B, where A and B are the fuzzy subsets of X and Y, respectively; R ⊂ X × Y is a fuzzy relation, and T is a continuous t-norm. The problem is how to determine A from R and B. First, an “if and only if” condition of being solvable is presented. Novel algorithms are then presented for determining minimal solutions when X and Y are finite. The proposed algorithms generate all minimal solutions for the equations, making them efficient solving procedures.     

On fuzzy relational equations and the covering problem

Previous studies have shown that fuzzy relational equations (FREs) based on either the max-continuous Archimedean t-norm or the max-arithmetic mean composition can be transformed into the covering problem, which is an NP-hard problem. Exploiting the properties common to the continuous Archimedean t-norm and the arithmetic mean, this study proposes a generalization of them as the “u-norm”, enabling FREs that are based on the max-continuous u-norm composition also to be transformed into the covering problem. This study also proposes a procedure for transforming the covering problem into max-product FREs. Consequently, max-continuous u-norm FREs can be solved by extending any procedure for solving either the covering problem or max-product FREs.     

Measuring incompatibility between Atanassov’s intuitionistic fuzzy sets

The aim of this paper is to establish an axiomatic definition of incompatibility measure in the framework of Atanassov’s intuitionistic fuzzy sets and use geometrical methods to build some families of such incompatibility measures. First, we construct several functions to measure incompatibility for an intuitionistic t-norm that can be represented by an adequate t-norm and t-conorm. Additionally, we establish some relations between some particular cases of these functions. Similarly, we then obtain incompatibility measures for a family of non-representable intuitionistic t-norms.     

T-sum of sigmoid-shaped fuzzy intervals

The usual arithmetic operations on real numbers can be extended to arithmetical operations on fuzzy intervals by means of Zadeh’s extension principle based on a t-norm T. A t-norm is called consistent with respect to a class of fuzzy intervals for some arithmetic operation, if this arithmetic operation is closed for this class. It is important to know which t-norms are consistent with particular types of fuzzy intervals. Recently, Dombi and Gy?rbíró [J. Dombi, N. Gy?rbíró, Additions of sigmoid-shaped fuzzy intervals using the Dombi operator and infinite sum theorems, Fuzzy Sets and Systems 157 (2006) 952-963] proved that addition is closed if the Dombi t-norm is used with sigmoid-shaped fuzzy intervals. In this paper, we define a broader class of sigmoid-shaped fuzzy intervals. Then, we study t-norms that are consistent with these particular types of fuzzy intervals. Dombi and Gy?rbíró’s results are special cases of the results described in this paper.     

Combining fuzzy weight average with fuzzy inference system for material substitution selection in electric industry

Material selection is a very important issue for an electronics company as it includes many qualitative or quantification factors. The material selection problem is associated with design and manufacturing problems which have been widely investigated. This study develops a hybrid fuzzy decision-making model which combines the fuzzy weight average (FWA) with the fuzzy inference system (FIS) for material substitution selection in the electronics industry. FWA is employed to select a substitute material in an uncertain environment, while FIS is used for reasoning purposes. FWA with α-cuts arithmetic (FWA_α-cut) is a popularly technology in decision-making problems. However, FWA_α-cut may result in the following unanticipated situations: (1) unclear decision situations; (2) undecided results expressed by fuzzy membership functions; and (3) high computational complexity. Therefore, a fuzzy weight average with the weakest t-norm (FWA_Tω) is designed as an alternative method for group decision making. In contrast to traditional FWA methods, FWA_Tω obtains more visible fuzzy results for decision makers with lower computational complexity, and can provide exacter estimation by the weakest t-norm operations in uncertain environment. Thus, the proposed hybrid fuzzy decision-making model imitates an expert’s experiences and can estimate substitution purchasing in various statuses. A real material substitution selection case is employed to examine the feasibility of the proposed model; experimental results reveal that the proposed model performs better than the traditional FWA model in coping with material substitution selection problems.     

t-Copula generation for control variates

The standard method for generating multi-t vectors is simple and convenient but it has the disadvantage that the generated multi-normal and multi-t vectors are not similar. For t-copula models this destroys much of the variance reduction when using the result of the multi-normal model as external control variate. Therefore we develop a new generation method for multi-t vectors. It is based on the polar method and numerical inversion, and generates multi-normal and multi-t vectors that are very similar. Numerical experiments with simple functions of the weighted sum of t-copula vectors and with pricing European basket options with a t-copula model confirm that the obtained variance reduction factors of the new method are high; 2-100 times higher than when using the standard generation method.     

Matrix elementary transformations in solving systems of fuzzy relation equations

In this paper, we consider a method using matrix elementary transformations, which is similar to the method in linear algebraic systems, for solving systems of fuzzy relation equations. The solution sets of a system of fuzzy relation equations before and after performing some elementary transformations are compared. We also give some necessary and sufficient conditions for some elementary transformations which do not change the solution sets.     

On fuzzy approximation operators in attribute reduction with fuzzy rough sets

Generally speaking, there are four fuzzy approximation operators defined on a general triangular norm (t-norm) framework in fuzzy rough sets. Different types of t-norms specify various approximation operators. One issue whether and how the different fuzzy approximation operators affect the result of attribute reduction is then arisen. This paper addresses this issue from the theoretical viewpoint by reviewing attribute reduction with fuzzy rough sets and then describing and proving some theorems which demonstrate the effects of the fuzzy approximation operators on the results of attribute reduction. First, we review some notions of attribute reduction with fuzzy rough sets, such as positive region, dependency degree and attribute reduction. We then present and prove some theorems which describe how and to what degree fuzzy approximation operators impact the performance of attribute reduction. Finally, we report some experimental simulation results which demonstrate the effectiveness and correctness of the theoretical contributions. One main contribution in this paper is that we have described and proven that each attribute reduction obtained using one type of fuzzy lower approximation operator always contains one reduction obtained using the other type of fuzzy lower approximation operator.     

Fuzzy measures for a correlation coefficient of fuzzy numbers under TW(the weakest t-norm)-based fuzzy arithmetic operations

Recently, the sup-min convolution based on Zadeh’s extension principle has been used by Liu and Kao [Fuzzy measures for correlation coefficient of fuzzy numbers, Fuzzy Sets and Systems 128 (2002) 267-275], to calculate a fuzzy correlation coefficient. They used a mathematical programming approach to derive fuzzy measures based on the classical definition of the correlation coefficient. It is well known that T_W (the weakest t-norm)-based addition and multiplication preserve the shape of L-R fuzzy numbers. In this paper, we consider the computational aspect of the T_W-based extension principle when the principle is applied to a correlation coefficient of L-R fuzzy numbers. We give the exact solution of a fuzzy correlation coefficient without programming or the aid of computer resources.     

Minimizing a linear objective function under a fuzzy max-t norm relation equation constraint

The work examines the feasibility of minimizing a linear objective function subject to a max-t fuzzy relation equation constraint, where t is a continuous/Archimedean t-norm. Conventional methods for solving this problem are significantly improved by, first separating the problem into two sub-problems according to the availability of positive coefficients. This decomposition is thus more easily handled than in previous literature. Next, based on use of the maximum solution of the constraint equation, the sub-problem with non-positive coefficients is solved and the size of the sub-problem with positive coefficients reduced as well. This step is unique among conventional methods, owing to its ability to determine as many optimal variables as possible. Additionally, several rules are developed for simplifying the remaining problem. Finally, those undecided optimal variables are obtained using the covering problem rather than the branch-and-bound methods. Three illustrative examples demonstrate that the proposed approach outperforms conventional schemes. Its potential applications are discussed as well.     

On approximating Euclidean metrics by weighted t-cost distances in arbitrary dimension

In this work, we propose a new class of distance functions called weighted t-cost distances. This function maximizes the weighted contribution of different t-cost norms in n-dimensional space. With proper weight assignment, this class of function also generalizes m-neighbor and octagonal distances. A non-strict upper bound (denoted as R_u in this work) of its relative error with respect to Euclidean norm is derived and an optimal weight assignment by minimizing R_u is obtained. However, it is observed that the strict upper bound of weighted t-cost norm may be significantly lower than R_u. For example, an inverse square root weight assignment leads to a good approximation of Euclidean norm in arbitrary dimension.     

On asymmetric generalised t stochastic volatility models

In stochastic volatility (SV) models, asset returns conditional on the latent volatility are usually assumed to have a normal, Student-t or exponential power (EP) distribution. An earlier study uses a generalised t (GT) distribution for the conditional returns and the results indicate that the GT distribution provides a better model fit to the Australian Dollar/Japanese Yen daily exchange rate than the Student-t distribution. In fact, the GT family nests a number of well-known distributions including the commonly used normal, Student-t and EP distributions. This paper extends the SV model with a GT distribution by incorporating general volatility asymmetry. We compare the empirical performance of nested distributions of the GT distribution as well as different volatility asymmetry specifications. The new asymmetric GT SV models are estimated using the Bayesian Markov chain Monte Carlo (MCMC) method to obtain parameter and log-volatility estimates. By using daily returns from the Standard and Poors (S&P) 500 index, we investigate the effects of the specification of error distributions as well as volatility asymmetry on parameter and volatility estimates. Results show that the choice of error distributions has a major influence on volatility estimation only when volatility asymmetry is not accounted for.     

Extension of the mixture of factor analyzers model to incorporate the multivariate t-distribution

Mixtures of factor analyzers enable model-based density estimation to be undertaken for high-dimensional data, where the number of observations n is small relative to their dimension p. However, this approach is sensitive to outliers as it is based on a mixture model in which the multivariate normal family of distributions is assumed for the component error and factor distributions. An extension to mixtures of t-factor analyzers is considered, whereby the multivariate t-family is adopted for the component error and factor distributions. An EM-based algorithm is developed for the fitting of mixtures of t-factor analyzers. Its application is demonstrated in the clustering of some microarray gene-expression data.     

Toward the existence and uniqueness of solutions of second-order fuzzy differential equations

In this paper the existence and uniqueness of solutions for second-order fuzzy differential equations with initial conditions under generalized H-differentiability is proved. To this end, the concept of second-order generalized differential equation is defined, which is based on an enlargement of the class of differentiable fuzzy mappings.     

A motion compression/reconstruction method based on max t-norm composite fuzzy relational equations

A motion compression/reconstruction method based on max t-norm composite fuzzy relational equations (MCF) is proposed, where into intra-pictures (I-pictures) and predictive-pictures (P-pictures) of the original motion are compressed by uniform and non-uniform coders, respectively. The non-uniform coders of the proposed method can preserve edge information of P-pictures on the compressed image. To perform an effective compression/reconstruction of the P-pictures, a design method of non-uniform coders is proposed based on an overlap level of fuzzy sets and a fuzzy equalization. An experiment using 10 P-pictures confirms that the root mean square error of the reconstructed images obtained by the proposed non-uniform coders is decreased to 89.4% of that one of the uniform coders under the condition that compression rate (the ratio between the file size of compressed image and original one) is 0.0057. Two test motions (‘Tennis’ and ‘Woman’, 100 frames) are compressed and reconstructed by the proposed MCF.