Fuzzy measure of fuzzy events defined by fuzzy integrals

We define the concept of fuzzy measure of a fuzzy event by using a general form of fuzzy integral proposed by Murofushi, called fuzzy t-conorm integral, encompassing previous definitions. Zadeh defined the probability measure of a fuzzy event, and later the possibility measure of fuzzy event. Using a duality property of fuzzy t-conorm integral, we propose a general definition of fuzzy measure of fuzzy events, which is compatible with previous definitions of Zadeh, and possesses all properties of a fuzzy measure, in particular the duality property. Using our definition, we examine the case of decomposable measures and belief functions. A comparison with previous works is provided.     

A generalization of local divergence measures

In this paper we propose a generalization of the concept of the local property for divergence measures. These new measures will be called g-local divergence measures, and we study some of their properties. Once this family is defined, a characterization based on Ling’s theorem is given. From this result, we obtain the general form of g-local divergence measures as a function of the divergence in each element of the reference set; this study is divided in three parts according to the cardinality of the reference set: finite, infinite countable or non-countable. Finally, we study the problem of componible divergence measures as a dual concept of g-local divergence measures.     

Fuzzy risk adjusted performance measures: Application to hedge funds

In this paper, following the notion of probabilistic risk adjusted performance measures, we introduce that of fuzzy risk adjusted performance measures (FRAPM). In order to deal efficiently with the closing-based returns bias induced by market microstructure noise, as well as to handle their uncertain variability, we combine fuzzy set theory and probability theory. The returns are first represented as fuzzy random variables and then used in defining fuzzy versions of some adjusted performance measures. Using a recent ordering method for fuzzy numbers, we propose a ranking of funds based on these fuzzy performance measures. Finally, empirical studies carried out on fifty French hedge funds confirm the effectiveness and give the benefits of our approach over the classical performance ratios.     

A generalization of local divergence measures

In this paper we propose a generalization of the concept of the local property for divergence measures. These new measures will be called g-local divergence measures, and we study some of their properties. Once this family is defined, a characterization based on Ling’s theorem is given. From this result, we obtain the general form of g-local divergence measures as a function of the divergence in each element of the reference set; this study is divided in three parts according to the cardinality of the reference set: finite, infinite countable or non-countable. Finally, we study the problem of componible divergence measures as a dual concept of g-local divergence measures.     

Adjacency on the order polytope with applications to the theory of fuzzy measures

In this paper we study the adjacency structure of the order polytope of a poset. For a given poset, we determine whether two vertices in the corresponding order polytope are adjacent. This is done through filters in the original poset. We also prove that checking adjacency between two vertices can be done in quadratic time on the number of elements of the poset. As particular cases of order polytopes, we recover the adjacency structure of the set of fuzzy measures and obtain it for the set of p-symmetric measures for a given indifference partition; moreover, we show that the set of p-symmetric measures can be seen as the order polytope of a quotient set of the poset leading to fuzzy measures. From this property, we obtain the diameter of the set of p-symmetric measures. Finally, considering the set of p-symmetric measures as the order polytope of a direct product of chains, we obtain some other properties of these measures, as bounds on the volume and the number of vertices on certain cases.     

Distances in evidence theory: Comprehensive survey and generalizations

The purpose of the present work is to survey the dissimilarity measures defined so far in the mathematical framework of evidence theory, and to propose a classification of these measures based on their formal properties. This research is motivated by the fact that while dissimilarity measures have been widely studied and surveyed in the fields of probability theory and fuzzy set theory, no comprehensive survey is yet available for evidence theory. The main results presented herein include a synthesis of the properties of the measures defined so far in the scientific literature; the generalizations proposed naturally lead to additions to the body of the previously known measures, leading to the definition of numerous new measures. Building on this analysis, we have highlighted the fact that Dempster’s conflict cannot be considered as a genuine dissimilarity measure between two belief functions and have proposed an alternative based on a cosine function. Other original results include the justification of the use of two-dimensional indexes as (cosine; distance) couples and a general formulation for this class of new indexes. We base our exposition on a geometrical interpretation of evidence theory and show that most of the dissimilarity measures so far published are based on inner products, in some cases degenerated. Experimental results based on Monte Carlo simulations illustrate interesting relationships between existing measures.     

Fuzzy variables

The purpose of this study is to explore a possible axiomatic framework from which a rigorous theory of fuzziness may be constructed. The approach we propose is analogous to the sample space concept of probability theory. A fuzzy variable is a mapping from an abstract space (called the pattern space) onto the real line. The membership function is obtained as the extension of a special type of capacity (called a scale) from the pattern space to the real line via the fuzzy variable. In essence we are proposing an entirely new definition of a fuzzy set on the line as a mapping to the line rather than on the line. The current definition of a transformation of a fuzzy set is obtained as a derived result of our model. In addition, we derive the membership function of sums and products of fuzzy sets and present an example which reinforces the credibility of our approach.     

Interpretability assessment of fuzzy knowledge bases: A cointension based approach

Computing with words (CWW) relies on linguistic representation of knowledge that is processed by operating at the semantical level defined through fuzzy sets. Linguistic representation of knowledge is a major issue when fuzzy rule based models are acquired from data by some form of empirical learning. Indeed, these models are often requested to exhibit interpretability, which is normally evaluated in terms of structural features, such as rule complexity, properties on fuzzy sets and partitions. In this paper we propose a different approach for evaluating interpretability that is based on the notion of cointension. The interpretability of a fuzzy rule-based model is measured in terms of cointension degree between the explicit semantics, defined by the formal parameter settings of the model, and the implicit semantics conveyed to the reader by the linguistic representation of knowledge. Implicit semantics calls for a representation of user’s knowledge which is difficult to externalise. Nevertheless, we identify a set of properties - which we call “logical view” - that is expected to hold in the implicit semantics and is used in our approach to evaluate the cointension between explicit and implicit semantics. In practice, a new fuzzy rule base is obtained by minimising the fuzzy rule base through logical properties. Semantic comparison is made by evaluating the performances of the two rule bases, which are supposed to be similar when the two semantics are almost equivalent. If this is the case, we deduce that the logical view is applicable to the model, which can be tagged as interpretable from the cointension viewpoint. These ideas are then used to define a strategy for assessing interpretability of fuzzy rule-based classifiers (FRBCs). The strategy has been evaluated on a set of pre-existent FRBCs, acquired by different learning processes from a well-known benchmark dataset. Our analysis highlighted that some of them are not cointensive with user’s knowledge, hence their linguistic representation is not appropriate, even though they can be tagged as interpretable from a structural point of view.     

A novel approach to multi attribute group decision making based on trapezoidal interval type-2 fuzzy soft sets

Soft set theory, originally proposed by Molodtsov, has become an effective mathematical tool to deal with uncertainty. A type-2 fuzzy set, which is characterized by a fuzzy membership function, can provide us with more degrees of freedom to represent the uncertainty and the vagueness of the real world. Interval type-2 fuzzy sets are the most widely used type-2 fuzzy sets. In this paper, we first introduce the concept of trapezoidal interval type-2 fuzzy numbers and present some arithmetic operations between them. As a special case of interval type-2 fuzzy sets, trapezoidal interval type-2 fuzzy numbers can express linguistic assessments by transforming them into numerical variables objectively. Then, by combining trapezoidal interval type-2 fuzzy sets with soft sets, we propose the notion of trapezoidal interval type-2 fuzzy soft sets. Furthermore, some operations on trapezoidal interval type-2 fuzzy soft sets are defined and their properties are investigated. Finally, by using trapezoidal interval type-2 fuzzy soft sets, we propose a novel approach to multi attribute group decision making under interval type-2 fuzzy environment. A numerical example is given to illustrate the feasibility and effectiveness of the proposed method.     

Decision tree search methods in fuzzy modeling and classification

This paper proposes input selection methods for fuzzy modeling, which are based on decision tree search approaches. The branching decision at each node of the tree is made based on the accuracy of the model available at the node. We propose two different approaches of decision tree search algorithms: bottom-up and top-down and four different measures for selecting the most appropriate set of inputs at every branching node (or decision node). Both decision tree approaches are tested using real-world application examples. These methods are applied to fuzzy modeling of two different classification problems and to fuzzy modeling of two dynamic processes. The models accuracy of the four different examples are compared in terms of several performance measures. Moreover, the advantages and drawbacks of using bottom-up or top-down approaches are discussed.     

Dominance-based fuzzy rough set analysis of uncertain and possibilistic data tables

In this paper, we propose a dominance-based fuzzy rough set approach for the decision analysis of a preference-ordered uncertain or possibilistic data table, which is comprised of a finite set of objects described by a finite set of criteria. The domains of the criteria may have ordinal properties that express preference scales. In the proposed approach, we first compute the degree of dominance between any two objects based on their imprecise evaluations with respect to each criterion. This results in a valued dominance relation on the universe. Then, we define the degree of adherence to the dominance principle by every pair of objects and the degree of consistency of each object. The consistency degrees of all objects are aggregated to derive the quality of the classification, which we use to define the reducts of a data table. In addition, the upward and downward unions of decision classes are fuzzy subsets of the universe. Thus, the lower and upper approximations of the decision classes based on the valued dominance relation are fuzzy rough sets. By using the lower approximations of the decision classes, we can derive two types of decision rules that can be applied to new decision cases.     

Estimating a fuzzy term structure of interest rates using fuzzy regression techniques

Several papers in the financial literature propose using fuzzy numbers (FNs) to model interest rate uncertainty. However, in our opinion, the first problem to be solved is how to estimate these rates with FNs. In this paper, we attempt to provide a solution to this question with a method for adjusting the temporal structure of interest rates (TSIR) that is based on fuzzy regression techniques. This method will enable to quantify the anticipated rates in the fixed income markets for the future with FNs. In particular, we discuss how to estimate the TSIR with triangular fuzzy numbers because of their desirable properties.     

On the use of fuzzy inference techniques in assessment models: part II: industrial applications

In this paper, we study the applicability of the monotone output property and the output resolution property in fuzzy assessment models to two industrial Failure Mode and Effect Analysis (FMEA) problems. First, the effectiveness of the monotone output property in a single-input fuzzy assessment model is demonstrated with a proposed fuzzy occurrence model. Then, the usefulness of the two properties to a multi-input fuzzy assessment model, i.e., the Bowles fuzzy Risk Priority Number (RPN) model, is assessed. The experimental results indicate that both the fuzzy occurrence model and Bowles fuzzy RPN model are able to fulfill the monotone output property, with the derived conditions (in Part I) satisfied. In addition, the proposed rule refinement technique is able to improve the output resolution property of the Bowles fuzzy RPN model.     

SH型区间值模糊理想

给出SH型区间值模糊左(右)理想和SH型区间值模糊理想的概念,研究了它们的一些性质,并应用区间值模糊集合的扩展原理,讨论了它的同态像和逆像问题,同时对于SH型区间值模糊左(右)特征理想及其性质进行了初步的讨论。     

Fuzzy映射和乘积Fuzzy测度

从研究Ｆｕｚｚｙ集合的直积和Ｆｕｚｚｙ关系出发,讨论了Ｆｕｚｚｙ关系和Ｆｕｚｚｙ映射的一些性质,最后利用Ｆｕｚｚｙ关系的截影性质给出了乘积Ｈ型Ｆｕｚｚｙ测度的存在性定理。     

主客观组合赋权的灰色决策模型及其在消费者决策中的运用

现实中许多抽象系统的数值并不能精确地得到,在多属性决策过程中,其属性值多以模糊数或者区间灰数的形式表示,属性权重也受到主观因素和客观因素的影响,针对此类问题,以属性值为区间灰数的多属性决策问题为出发点,采用主客观结合的赋权方式,提出了基于区间距离的灰色相对关联度分析方法,构建了灰色决策模型,并通过消费者决策实例进行了分析,模型是对传统灰色决策模型的拓展,具有更广泛的应用性和实用性.     

Fuzzy criteria for feature selection

The presence of less relevant or highly correlated features often decrease classification accuracy. Feature selection in which most informative variables are selected for model generation is an important step in data-driven modeling. In feature selection, one often tries to satisfy multiple criteria such as feature discriminating power, model performance or subset cardinality. Therefore, a multi-objective formulation of the feature selection problem is more appropriate. In this paper, we propose to use fuzzy criteria in feature selection by using a fuzzy decision making framework. This formulation allows for a more flexible definition of the goals in feature selection, and avoids the problem of weighting different goals is classical multi-objective optimization. The optimization problem is solved using an ant colony optimization algorithm proposed in our previous work. We illustrate the added value of the approach by applying our proposed fuzzy feature selection algorithm to eight benchmark problems.     

A generalized real-valued measure of the inequality associated with a fuzzy random variable

Fuzzy random variables have been introduced by Puri and Ralescu as an extension of random sets. In this paper, we first introduce a real-valued generalized measure of the “relative variation” (or inequality) associated with a fuzzy random variable. This measure is inspired in Csiszár's f-divergence, and extends to fuzzy random variables many well-known inequality indices. To guarantee certain relevant properties of this measure, we have to distinguish two main families of measures which will be characterized. Then, the fundamental properties are derived, and an outstanding measure in each family is separately examined on the basis of an additive decomposition property and an additive decomposability one. Finally, two examples illustrate the application of the study in this paper.