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In a recent paper by the authors, Jensen's inequality for Choquet integral was given, and a wrong assertion—"Jensen's inequality does not hold for asymmetric Choquet integral" was made. This paper can be viewed as a continuation of the previous one, Jensen's inequality for asymmetric Choquet integral is proved, the error is corrected. As its generalization, Jensen's inequality for generalized asymmetric Choquet integral is obtained. [ABSTRACT FROM AUTHOR]

As powerful tools for modeling nondeterministic problems, Choquet integrals have attracted much attention in recent years. This study generalizes the existing set-valued Choquet integrals and formulates the theory of Choquet integrals with respect to set-valued fuzzy measures. First, a set-valued Choquet integral of a real-valued function is defined using the set of Choquet integrals of real-valued functions with respect to fuzzy measure selections of a set-valued fuzzy measure. The integrand is then extended to set-valued functions, after which various properties, convergence theorems, and set-valued Jensen's inequalities are obtained. Second, following how single-valued Choquet integrals are defined, an alternative set-valued Choquet integral of real-valued function is formulated using Aumann integrals. Through these steps, the existing set-valued Choquet integrals are covered. [ABSTRACT FROM AUTHOR]

Due to many applications, the Choquet integral as a powerful tool for modeling non-deterministic problems needs to be further extended. Therefore the paper is devoted to a generalization of the Choquet integral. As a basis, the pseudo-integral for bounded integrand is extended to the case for nonnegative integrands at first, and then the generalized Choquet integral is defined. As special cases, pseudo-Choquet Stieltjes integrals, pseudo-fuzzy Stieltjes integrals, g -Choquet integrals, pseudo-(N)fuzzy integrals and pseudo-(S)fuzzy integrals are obtained, and various kinds of properties and convergence theorems are shown, meanwhile Markov, Jensen, Minkowski and Hölder inequalities are proved. In the end, the generalized discrete Choquet integral is discussed. The obtained results for the generalized Choquet integral cover some previous results on different types of nonadditive integrals. [ABSTRACT FROM AUTHOR]

In this paper, it is shown that Jensen's inequality for Choquet integral given by R. Wang ten years ago is incorrect, then it is revisited, the modified Jensen's and reverse Jensen's inequalities for Choquet integral are proved. Then Jensen's inequality for generalized Choquet integral, obtained by the authors in a recent paper, is modified accordingly. [ABSTRACT FROM AUTHOR]

It is remarked that the generalization of Jensen's inequality for pseudo-integrals (Pap and Štrboja [14]) is not a complete generalization of the classical Jensen's inequality, and a generalized Jensen's inequality for pseudo-integral with respect to (⊕ , ⊗) -pseudo-convex function is given in [28]. The present article is a continuation of the previous work. A new notion of (⊕ 1 , ⊗ 1) , (⊕ 2 , ⊗ 2) -pseudo-convex function is introduced, which generalize the notion of (⊕ , ⊗) -pseudo-convex function and many other previously generalizations. Motivated by the work of Kaluszka et al. [6] , related to Jensen's inequality with respect to different generalized fuzzy integrals, a new generalized Jensen's inequality between a pseudo-integral and general fuzzy integral, as well as between two different pseudo-integrals, with respect to ((⊕ 1 , ⊗ 1) , (⊕ 2 , ⊗ 2)) -pseudo-convex functions are proved. These results cover all previously obtained Jensen's inequalities for pseudo-integrals (Zhang and Pap [28]) as well as the classical Jensen's inequality. [ABSTRACT FROM AUTHOR]

Being an important part of classical analysis, Jensen's inequality has drawn much attention recently. Due to its generality, the inequality based on non-additive integrals appears in many forms, such as Sugeno integrals, Choquet integrals and pseudo-integrals. As a well-known generalization of classical one, the set-valued analysis is frequently applied to the research of mathematical economy, control theory and so on. Thus, it is of great necessity to generalize the set-valued case. Motivated by the pioneering work of Costa's Jensen's fuzzy-interval-valued inequality and Štrboja et al.'s Jensen's set-valued inequality based on Aumann integrals and pseudo-integrals respectively, this paper focuses particularly on proving certain kinds of Jensen's set-valued inequalities and fuzzy set-valued inequalities. These inequalities consist of two families: the related convex (or concave) function is a set-valued or fuzzy set-valued function and the integrand is a real-valued function; the related convex (or concave) function is a real-valued function and the integrand is a set-valued or fuzzy set-valued function. Particularly, Jensen's interval-valued and fuzzy-interval-valued inequalities, including Costa's, are obtained as corollaries. [ABSTRACT FROM AUTHOR]

Abstract: Fuzzy similarity relations have been considered as kernels in the domain of machine learning in previous works. As a supplementary of this consideration, this paper focuses on construction of feature space related to fuzzy similarity relations as indefinite kernels. First we generalize the Mercer Theorem from positive definite kernels to fuzzy similarity relations. With this generalization we construct the feature space related to a fuzzy similarity relation as a particular Krein space. The proposed feature space is a naturally generalization of the existing Pseudo-Euclidean space from finite to infinite universe of discourses. At the end of this paper we develop an open problem to set up the connection between transitivity and the positive definiteness of a fuzzy similarity relation. Results in this paper complete the mathematical foundation for further applications of fuzzy similarity relations under the framework of kernel tricks. [Copyright &y& Elsevier]

This paper performs a further investigation on the set-valued Choquet integral given by Jang and Kwon (Fuzzy Sets and Systems 112 (2000) 233), some incorrect properties are revised. It is pointed out that the definition for convergence of sequence of sets or set-valued functions is unreasonable. By using the usual Kuratowski''s convergence, some new convergence theorems for the set-valued Choquet integral are shown. Meanwhile some other properties are obtained. [Copyright &y& Elsevier]

Attribute reduction with fuzzy rough set is an effective technique for selecting most informative attributes from a given real-valued dataset. However, existing algorithms for attribute reduction with fuzzy rough set have to re-compute a reduct from dynamic data with sample arriving where one sample or multiple samples arrive successively. This is clearly uneconomical from a computational point of view. In order to efficiently find a reduct from such datasets, this paper studies incremental attribute reduction with fuzzy rough sets. At the arrival of one sample or multiple samples, the relative discernibility relation is updated for each attribute. On the basis of the updated relation, an insight into the incremental process of attribute reduction with fuzzy rough sets is gained to reveal how to add new attributes into the current reduct and delete existing attributes from the current reduct. Applying the incremental process, two incremental algorithms for attribute reduction with fuzzy rough sets are presented for one incoming sample and multiple incoming samples, respectively. Experimental comparisons with several non-incremental algorithms and the proposed incremental algorithm for one incoming sample show that our proposed incremental algorithm for multiple incoming samples can efficiently find one reduct with a comparable classification accuracy. [ABSTRACT FROM AUTHOR]