Shilkret Integral Research Articles

Decomposition Integrals of Set-Valued Functions Based on Fuzzy Measures

The decomposition integrals of set-valued functions with regards to fuzzy measures are introduced in a natural way. These integrals are an extension of the decomposition integral for real-valued functions and include several types of set-valued integrals, such as the Aumann integral based on the classical Lebesgue integral, the set-valued Choquet, pan-, concave and Shilkret integrals of set-valued functions with regard to capacity, etc. Some basic properties are presented and the monotonicity of the integrals in the sense of different types of the preorder relations are shown. By means of the monotonicity, the Chebyshev inequalities of decomposition integrals for set-valued functions are established. As a special case, we show the linearity of concave integrals of set-valued functions in terms of the equivalence relation based on a kind of preorder. The coincidences among the set-valued Choquet, the set-valued pan-integral and the set-valued concave integral are presented.

Weakly maxitive set functions and their possibility distributions

The Shilkret integral with respect to a completely maxitive capacity is fully determined by a possibility distribution. In this paper, we introduce a weaker topological form of maxitivity and show that under this assumption the Shilkret integral is still determined by its possibility distribution for functions that are sufficiently regular. Motivated by large deviations theory, we provide a Laplace principle for maxitive integrals and characterize the possibility distribution under certain separation and convexity assumptions. Moreover, we show a maxitive integral representation result for weakly maxitive non-linear expectations. The theoretical results are illustrated by providing large deviations bounds for sequences of capacities, and by deriving a monotone analogue of Cramér's theorem.

The completeness and separability of function spaces in nonadditive measure theory

For a nonadditive measure μ, the space L0(μ) of all measurable functions, the Choquet-Lorentz space Lp,q(μ), the Lorentz space of weak type Lp,∞(μ), the space L∞(μ) of all μ-essentially bounded measurable functions, and their quotient spaces are defined together with suitable prenorms on them. Among those function spaces, the Choquet-Lorentz space is defined by the Choquet integral, while the Lorentz space of weak type is defined by the Shilkret integral. Then the completeness and separability of those spaces are investigated. A new characteristic of nonadditive measures, called property (C), is introduced to establish the Cauchy criterion for convergence in μ-measure of measurable functions. This criterion and suitable convergence theorems of the Choquet and Shilkret integrals provide instruments for carrying out our investigation.

A new class of decomposition integrals on finite spaces

A new type of decomposition integral is introduced by using a family of decomposition integrals based on the collections relating to partitions and maximal chains of sets. This new integral extends the Lebesgue integral, and it is different from those well-known decomposition integrals, such as the Choquet, concave, pan-, Shilkret integrals and PC-integral. In the structure of a lattice on the class of decomposition integrals, the introduced decomposition integral is between the Choquet integral and the concave integral, and also between the pan-integral and the concave integral, and it is a lower bound of PC-integral. The coincidences among several well-known integrals and this new integral are also shown.

On quadruplets of nonadditive integrals

The definition of the well-known Choquet integral of a nonnegative random variable is based on a survival (decumulative distribution) function. A quadruplet of Choquet integrals is obtained by extending this idea also to a (cumulative) distribution function for real random variables. The concept includes the asymmetric and symmetric Choquet integrals as well as their complementary versions. These four integrals coincide for any additive measure. For a finite measure, the complementary integrals are related to the original asymmetric and symmetric Choquet integrals via dual measure. Duality also plays a key role in the study of basic integral properties. An alternative way to look at Choquet integral goes via the area of a survival function subgraph. Substituting the Lebesgue planar measure with an arbitrary planar (fuzzy) measure we generalize the introduced quadruplet concept in which the Sugeno and the Shilkret integral quadruplets are involved. This gives a new insight into a common generalization of the three most known nonadditive integrals of Choquet, Sugeno and Shilkret as well as their symmetric, asymmetric and complementary versions.

Scaled aggregation functions

We introduce and discuss a transformation of aggregation functions yielding interval scale invariant aggregation or weak aggregation functions. Several examples illustrating our approach are included. In particular, we discuss the results for integrals based on semicopulas. We focus our attention on transformations of the Sugeno and Shilkret integrals, and clarify their connection with OWA operators.

Interval-valued seminormed fuzzy operators based on admissible orders

The fuzzy integral is a well-known class of aggregation operators, which includes the Sugeno integral and Shilkret integral. When performing fuzzy integration over vectors of interval values, recent literature showed that using a simplistic method to independently deal with the lower and upper bounds of interval-valued inputs is sometimes not reasonable in practice. This motivated us to conduct a necessary and thorough study of the possible structures and properties of interval-valued fuzzy operators. This study investigated concepts and revealed some related properties of admissible orders and cones such that interval-valued seminormed fuzzy operator (ISFO) is then well defined. We introduce the relevant set and systematically examine some of its main properties, which forms the basis of the fundamental structural analysis of the ISFO. Furthermore, relationships between the proposed concepts are discussed, and several Jensen-type inequalities for the ISFO are examined.

Semicopula based integrals

We define the notion of weak universal integral based on a semicopula, and introduce and discuss two particular classes of weak universal integrals based on semicopulas, which generalize the well-known Sugeno and Shilkret integrals. In special cases, when the considered semicopulas are bounded from above by the Łukasiewicz t-norm, all introduced integrals reduce to the corresponding smallest semicopula based universal integrals. Remarkably, when the product semicopula is considered, the proposed integrals generalizing the Shilkret integral belong to the class of aggregation functions, which is not the case of the minimum semicopula when the introduced integrals generalize the Sugeno integral.

Greedy decomposition integrals

In this contribution we define a new class of non-linear integrals based on decomposition integrals. These integrals are motivated by greediness of many real-life situations. Another view on this new class of integrals is that it is a generalization of both the Shilkret and PAN integrals. Moreover, it can be seen as an iterated Shilkret integral. Also, an example in time-series analysis is provided.

Collection integral vs. Choquet integral

In this paper we discuss integral inequalities for collection integrals that are a special subclass of decomposition integrals introduced as a general framework for many non-linear integrals, including the Choquet integral, the Shilkret integral, the PAN integral, and the concave integral. We give a full characterization of collection integrals that are comonotone additive and for which Chebyshev's, Jensen's, Cauchy's, and Hölder's integral inequalities hold. Interestingly, all these classes of collection integrals coincide and thus we introduce a special subclass of collection integrals, called PCC integrals. The paper is complemented with some examples and remarks for collection and decomposition integrals.

On the (M)-property of monotone measures and integrals on atoms

We further investigate (M)-property, an important structure characteristic of monotone measures. Some necessary and/or sufficient conditions of (M)-property are shown and some characteristics of (M)-property are described. It is shown that the restriction of a monotone measure to an atom of its own possesses (M)-property on this atom. By means of the (M)-property we characterize the equivalence among the Choquet integral, the pan-integral, the concave integral, and the Shilkret integral on atoms of monotone measures and obtain some special results. We also show a necessary and sufficient condition for the concave integral and the pan-integral to coincide on atoms of monotone measures.

(1907-5331) Greedy decomposition integrals

In this contribution we define a new class of non-linear integrals based on decomposition integrals. These integrals are motivated by greediness of many real-life situations. Another view on this new class of integrals is that it is a generalization of both the Shilkret and PAN integrals. Moreover, it can be seen as an iterated Shilkret integral. Also, an example in time-series analysis is provided.

An Application of Non-additive Measures and Corresponding Integrals in Tourism Management

Non-additive measures and corresponding integrals originally have been introduced by Choquet in 1953 (1) and independently defined by Sugeno in 1974 (2) in order to extend the classical measure by replacing the additivity property to non-additive property. An important feature of non –additive measures and fuzzy integrals is that they can represent the importance of individual information sources and interactions among them. There are many applications of non-additive measures and fuzzy integrals such as image processing, multi-criteria decision making, information fusion, classification, and pattern recognition. This paper presents a mathematical model for discussing an application of non-additive measures and corresponding integrals in tourism management. First, the problem of tourism management is described for one of the tourism companies in Iraq. Then, fuzzy integrals (Sugeno integral, Choquet integral, and Shilkret integral) are applied with respect to non-additive measures to evaluate the grade of the gratification of the tourist of staying in a particular town for determining the best evaluation.

The Vitali convergence in measure theorem of nonlinear integrals

The Vitali convergence in measure theorem for the abstract Lebesgue integral is fundamental in Lebesgue integration theory and yields the bounded convergence theorem and the dominated convergence theorem as its applications. In this paper, in a unified way using the perturbation method the Vitali convergence in measure theorem is established for nonlinear integrals such as the Choquet, Šipoš, Sugeno, and Shilkret integrals, and their symmetric and asymmetric extensions. It is derived from the Fatou and the reverse Fatou type lemmas for perturbative nonlinear integral functionals.

K-maxitive aggregation functions

Inspired by the idea of k-maxitive measures, we introduce and study k-maxitive aggregation functions. In particular, 1-maxitive aggregation functions are shown to be just maxitive aggregation functions, i.e., maxima of distorted inputs. We introduce, among others, a representation of symmetric k-maxitive aggregation functions. Moreover, we show that for any k-maxitive capacity and the smallest universal integral based on an arbitrary fixed semicopula, including the Sugeno and Shilkret integrals, the resulting aggregation function is k-maxitive.

Monotone Measures Defined by Pan-Integral

Given a pan-space and a nonnegative measurable function f on measurable space (X, A), the pan-integral of f with respect to monotone measure μ and pan-operation determines a new monotone measure on (X, A). Such the new monotone measure is absolutely continuous with respect to the monotone measure μ. We show that the new monotone measure preserves some important structural characteristics of the original monotone measures, such as continuity from below, subadditivity, null-additivity, weak null-additivity and (S) property. Since the pan-integral based on a pair of pan-operations covers the Sugeno integral (based on ) and the Shilkret integral (based on ), therefore, the previous related results for the Sugeno integral are covered by the results presented here, in the meantime, some special results related the Shilkret integral are also obtained.

A unified approach to the monotone convergence theorem for nonlinear integrals

We present a unified approach to the monotone convergence theorem for nonlinear integrals such as the Choquet, the Šipoš, the Sugeno, and the Shilkret integral. A nonlinear integral may be viewed as a nonlinear functional defined on a set of pairs of a nonadditive measure and a measurable function. We thus formulate our general type of monotone convergence theorem for such a functional. The key tool is a perturbation of functional that manages not only the monotonicity of the functional but also the small change of the functional value arising as a result of adding small amounts to a measure and a function in the domain of the functional. Our approach is also applicable to the Lebesgue integral when a nonadditive measure is σ-additive.

Superadditive and subadditive transformations of integrals and aggregation functions

We propose the concepts of superadditive and of subadditive transformations of aggregation functions acting on non-negative reals, in particular of integrals with respect to monotone measures. We discuss special properties of the proposed transforms and links between some distinguished integrals. Superadditive transformation of the Choquet integral, as well as of the Shilkret integral, is shown to coincide with the corresponding concave integral recently introduced by Lehrer. Similarly the transformation of the Sugeno integral is studied. Moreover, subadditive transformation of distinguished integrals is also discussed.

Integrals based on monotone set functions

An overview of various integrals is given which can be defined on arbitrary monotone set functions vanishing in the empty set (called here monotone measures). Our survey includes not only the Choquet integral (1954) [10], the Shilkret integral (1971) [66] and the Sugeno integral (1974) [71] and some of their properties, but also some more general and more recent concepts as universal integrals Klement et al. (2010) [27] and decomposition integrals Even (2014) [13], together with some of their properties, such as integral inequalities and convergence theorems.

Shilkret Integral Based on Binary-Element Sets and its Application in The Area of Synthetic Evaluation

Shilkret Integral Based on Binary-Element Sets and its Application in The Area of Synthetic Evaluation