Interval-valued Choquet Integral Research Articles

The interval-valued Choquet-Sugeno-like operator as a tool for aggregation of interval-valued functions

We propose a novel mapping from a space of interval-valued functions defined on an arbitrary set to the family of closed subintervals of nonnegative extended real numbers, called the interval-valued Choquet-Sugeno-like operator. It is based on the notion of the Choquet-Sugeno-like operator and the interval-valued seminormed fuzzy operator introduced by Boczek et al. (2021) and generalizes many interval-valued operators known in the literature, such as n-ary interval-valued aggregation operators, interval-valued ordered weighting average operators, interval-valued seminormed fuzzy operators, and discrete interval-valued Choquet integral. We study its properties and show that it can be used to aggregate interval-valued data.

Discrete IV dG-Choquet integrals with respect to admissible orders

In this work, we introduce the notion of dG-Choquet integral, which generalizes the discrete Choquet integral replacing, in the first place, the difference between inputs represented by closed subintervals of the unit interval [0,1] by a dissimilarity function; and we also replace the sum by more general appropriate functions. We show that particular cases of dG-Choquet integral are both the discrete Choquet integral and the d-Choquet integral. We define interval-valued fuzzy measures and we show how they can be used with dG-Choquet integrals to define an interval-valued discrete Choquet integral which is monotone with respect to admissible orders. We finally study the validity of this interval-valued Choquet integral by means of an illustrative example in a classification problem.

Interval-valued Choquet integral for set-valued mappings: definitions, integral representations and primitive characteristics

In this paper, a new kind of real-valued major Choquet integral, real-valued minor Choquet integral and interval-valued Choquet integrals for set-valued functions is introduced and investigated. The representations of the Choquet integral of set-valued functions with respect to a fuzzy measure are given. In particular, we focus on the case of the distorted Lebesgue measure as a fuzzy measure. Furthermore, the characteristics of the primitive of Choquet integral for set-valued functions are given as Radon-Nikodym property in some sense.

The Interval-Valued Choquet Integral Based on Admissible Permutations

Aggregation or fusion of interval data is not a trivial task, since the necessity of arranging data arises in many aggregation functions, such as OWA operators or the Choquet integral. Some arranging procedures have been given to solve this problem, but they need certain parameters to be set. In order to solve this problem, we propose the concept of an admissible permutation of intervals. Based on this concept, which avoids any parameter selection, we propose a new approach for the interval-valued Choquet integral that takes into account every possible permutation fitting to the considered ordinal structure of data. Finally, a consensus among all the permutations is constructed.

Interval-Valued Model Level Fuzzy Aggregation-Based Background Subtraction.

In a recent work, the effectiveness of neighborhood supported model level fuzzy aggregation was shown under dynamic background conditions. The multi-feature fuzzy aggregation used in that approach uses real fuzzy similarity values, and is robust for low and medium-scale dynamic background conditions such as swaying vegetation, sprinkling water, etc. The technique, however, exhibited some limitations under heavily dynamic background conditions, as features have high uncertainty under such noisy conditions and these uncertainties were not captured by real fuzzy similarity values. Our proposed algorithm is particularly focused toward improving the detection under heavy dynamic background conditions by modeling uncertainties in the data by interval-valued fuzzy set. In this paper, real-valued fuzzy aggregation has been extended to interval-valued fuzzy aggregation by considering uncertainties over real similarity values. We build up a procedure to calculate the uncertainty that varies for each feature, at each pixel, and at each time instant. We adaptively determine membership values at each pixel by the Gaussian of uncertainty value instead of fixed membership values used in recent fuzzy approaches, thereby, giving importance to a feature based on its uncertainty. Interval-valued Choquet integral is evaluated using interval similarity values and the membership values in order to calculate interval-valued fuzzy similarity between model and current. Adequate qualitative and quantitative studies are carried out to illustrate the effectiveness of the proposed method in mitigating heavily dynamic background situations as compared to state-of-the-art.

A New Approach to Interval-Valued Choquet Integrals and the Problem of Ordering in Interval-Valued Fuzzy Set Applications

We consider the problem of choosing a total order between intervals in multiexpert decision making problems. To do so, we first start researching the additivity of interval-valued aggregation functions. Next, we briefly treat the problem of preserving admissible orders by linear transformations. We study the construction of interval-valued ordered weighted aggregation operators by means of admissible orders and discuss their properties. In this setting, we present the definition of an interval-valued Choquet integral with respect to an admissible order based on an admissible pair of aggregation functions. The importance of the definition of the Choquet integral, which is introduced by us here, lies in the fact that if the considered data are pointwise (i.e., if they are not proper intervals), then it recovers the classical concept of this aggregation. Next, we show that if we make use of intervals in multiexpert decision making problems, then the solution at which we arrive may depend on the total order between intervals that has been chosen. For this reason, we conclude the paper by proposing two new algorithms such that the second one allows us, by means of the Shapley value, to pick up the best alternative from a set of winning alternatives provided by the first algorithm.

On Choquet Integrals with Respect to a Fuzzy Complex Valued Fuzzy Measure of Fuzzy Complex Valued Functions

In this paper, using fuzzy complex valued functions and fuzzy complex valued fuzzy measures ([11]) and interval-valued Choquet integrals ([2-6]), we define Choquet integral with respect to a fuzzy complex valued fuzzy measure of a fuzzy complex valued function and investigate some basic properties of them.

구간치 쇼케이적분에 의해 정의된 단조 구간치 집합함수의 구조적 성질에 관한 연구

We introduce nonnegative interval-valued set functions and nonnegative measurable interval-valued Junctions. Then the interval-valued Choquet integral determines a new nonnegative monotone interval-valued set function which is a generalized concept of monotone set function defined by Choquet integral in [17]. We also obtained absolutely continuity of them in [9]. In this paper, we investigate some characterizations of the monotone interval-valued set function defined by the interval-valued Choquet integral, and such as subadditivity, superadditivity, null-additivity, converse-null-additivity.

THE AUTOCONTINUITY OF MONOTONE INTERVAL-VALUED SET FUNCTIONS DEFINED BY THE INTERVAL-VALUED CHOQUET INTEGRAL

In a previous work [18], the authors investigated autocontinuity, converse-autocontinuity, uniformly autocontinuity, uniformly converse-autocontinuity, and fuzzy multiplicativity of monotone set function defined by Choquet integral([3,4,13,14,15]) instead of fuzzy integral([16,17]). We consider nonnegative monotone interval-valued set functions and nonnegative measurable interval-valued functions. Then the interval-valued Choquet integral determines a new nonnegative monotone interval-valued set function which is a generalized concept of monotone set function defined by Choquet integral in [18]. These integrals, which can be regarded as interval-valued aggregation operators, have been used in [10,11,12,19,20]. In this paper, we investigate some characterizations of monotone interval-valued set functions defined by the interval-valued Choquet integral such as autocontinuity, converse-autocontinuity, uniform autocontinuity, uniform converse-autocontinuity, and fuzzy multiplicativity.

단조집합함수에 의해 정의된 구간치 쇼케이적분에 대한 르베그형태 정리에 관한 연구

In this paper, we consider Lebesgue-type theorems in non-additive measure theory and then investigate interval valued Choquet integrals and interval-valued fuzzy integral with respect to a additive monotone set function. Furthermore, we discuss the equivalence among the Lebesgue`s theorems, the monotone convergence theorems of interval-valued fuzzy integrals with respect to a monotone set function and find some sufficient condition that the monotone convergence theorem of interval-valued Choquet integrals with respect to a monotone set function holds.

구간치 쇼케이적분과 위험률 가격 측정에서의 응용

Non-additive measures and their corresponding Choquet integrals are very useful tools which are used in both insurance and financial markets. In both markets, it is important to update prices to account for additional information. The update price is represented by the Choquet integral with respect to the conditioned non-additive measure. In this paper, we consider a price functional H on interval-valued risks defined by interval-valued Choquet integral with respect to a non-additive measure. In particular, we prove that if an interval-valued pricing functional H satisfies the properties of monotonicity, comonotonic additivity, and continuity, then there exists an two non-additive measures <TEX>${\mu}1,\;{\mu}2$</TEX> such that it is represented by interval-valued choquet integral on interval-valued risks.

A NOTE ON THE MONOTONE INTERVAL-VALUED SET FUNCTION DEFINED BY THE INTERVAL-VALUED CHOQUET INTEGRAL

At first, we consider nonnegative monotone interval-valued set functions and nonnegative measurable interval-valued functions. In this paper we investigate some properties and structural characteristics of the monotone interval-valued set function defined by an interval-valued Choquet integral.

Some characterizations of a mapping defined by interval-valued Choquet integrals

Note that Choquet integral is a generalized concept of Lebesgue integral, because two definitions of Choquet integral and Lebesgue integral are equal if a fuzzy measure is a classical measure. In this paper, we consider interval-valued Choquet integrals with respect to fuzzy measures(see [4,5,6,7]). Using these Choquet integrals, we define a mappings on the classes of Choquet integrable functions and give an example of a mapping defined by interval-valued Choquet integrals. And we will investigate some relations between m-convex mappings <TEX>${\phi}$</TEX> on the class of Choquet integrable functions and m-convex mappings <TEX>$T_{\phi}$</TEX>, defined by the class of closed set-valued Choquet integrals with respect to fuzzy measures.

쇼케이 거리측도와 응용에 관한 연구

구간치 퍼지집합은 Gorzalczang(1983)에 의해 처음 제의되었다. 이를 토대로 Wang과 Li는구간치 퍼지수에 관한 연산으로 일반화하여 연구하였다. 최근에 홍(2002)는 왕과 리의 이론을 리만적분에 의해 구간치 퍼지수 상의 거리측도에 관한 연구를 하였다. 우리는 일반측도와 관련된 리만적분 대신에 퍼지측도와 관련된 쇼케이적분을 이용한 구간치 퍼지수 상의 쇼케이 거리측도를 연구하였다(2005). 본 논문에서는 퍼지수치 퍼지수 상의 쇼케이 거리측도를 정의하고 이와 관련된 성질들을 조사하였다. Internal-valued fuzzy sets were suggested for the first time by Gorzalczang(1983). Based on this, Wang and Li extended their operations on interval-valued fuzzy numbers. Recently, Hong(2002) generalized results of Wang and Li and extended to interval-valued fuzzy numbers with Riemann integral. By using interval-valued Choquet integrals with respect to a fuzzy measure instead of Riemann integrals with respect to a classical measure, we studied some characterizations of interval-valued Choquet distance(2005). In this paper, we define Choquet distance measure for fuzzy number-valued fuzzy numbers and investigate some properties of them.

보단조 가법 구간치 범함수와 구간치 쇼케이적분에 관한 연구(II)

이 논문에서는 Schmeidler[14]와 Narukawa[12]에 나오는 보단조 가법 실수치 범함수 개념의 일반화인 보단조 가법 구간치 범함수를 정의하고 그들의 성질을 연구한다. 또한 보단조 가법 구간치 범함수와 구간치 쇼케이적분이 적당한 함수공간 상에서 서로간의 관계를 조사한다. 수의 값을 갖는 함수들의 쇼케이적분을 생각하고자 한다. 이러한 구간 수의 값을 갖는 함수들의 성질들을 조사한다. In this paper, we will define comonotonically additive interval-valued functionals which are generalized comonotonically additive real-valued functionals in Schmeidler[14] and Narukawa[12], and prove some properties of them. And we also investigate some relations between comonotonically additive interval-valued functionals and interval-valued Choquet integrals on a suitable function space, cf.[9,10,11,13].