Search Results (12)

In this paper, we investigate the second Borel–Cantelli lemma for capacity without the assumption of independence for events. We obtain a sufficient condition under which the second Borel–Cantelli lemma for capacity holds. Our results are natural extensions of the classical Borel–Cantelli lemma. However, the proof is different from the existing literature. Full article

Problems of modeling uncertainty and imprecision for the analysis of insufficient expert data (IED) are considered in the environment of interactive multi-group decision-making (MGDM). Based on the Choquet finite integral, a moments’ method for the IED is developed for the evaluation of the associated probabilities class (APC) of Choquet’s second-order capacity based on the informational entropy maximum principle. Based on the IED new approach of the lower and upper Choquet’s second-order capacities, identification is developed. The second pole of insufficient expert data, the data imprecision indicator, is presented in the form of a fuzzy subset and image on the alternatives set. In the environment of the Dempster–Shafer belief structure, connections between an associated possibilities class (APosC), with the APC, and an associated focal probabilities class (AFPC) are constructed. In the approach of A. Kaufman’s theory of expertons, based on the APosC and the AFPC unique fuzzy subset, the IED image on the alternatives set is constructed. Based on Sugeno’s finite integral most typical value (MTV), as a prediction on possible alternatives set, the IED is constructed. In the example, a sensitive and comparative analysis is provided for the evaluation of the new approach’s stability and reliability. Full article

The aim to improve diagnosis decision systems of kidneys led to the concept of many methods of texture characterization of kidneys from ultrasound images. Here, as a first main contribution, we propose a texture characterization method based on the singularity defined by the Hölder exponent, which is multifractal local information. Indeed, the originality of our contribution here is to build a singularity-level matrix corresponding to different levels of regularities, from which we extract new texture features. Finally, as a second main contribution, we will evaluate the potential of our proposed multifractal features to characterize textured ultrasound images of the kidney. Having more reproducibility of the texture features first requires a good choice of Choquet capacity to calculate the irregularities and a selection of a more representative region of interest (ROIs) to analyze by carrying out an adapted virtual puncture in the kidney components. The results of the supervised classification, using three classes of images (young, healthy, and glomerulonephritis), are interesting and promising since the classification accuracy reaches about 80%. This encourages conducting further research to yield better results by overcoming the limitations and taking into account the recommendations made in this article. Full article

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. Full article

From recent studies, the concept of “monotone expectation” (ME) of Interactive Multi-Attribute Decision Making (MADM) is well known, which was developed for the case of different fuzzy sets. This article develops the concept of “monotone expectation” for such statistical parameters as variance, $k$-order moment and covariance. We investigate the problem of the definition of some statistical parameters, when the uncertainty is represented by a monotone measure—a fuzzy measure—instead of an additive measure. The study presents the concept of the definition of monotone statistical parameters based on the Choquet finite integral for the definition of monotone expectation, monotone variance, monotone $k$-order moment and monotone covariance. Associated statistical parameters are also presented—expectation, variance, $k$-order moment and covariance—which are defined in relation to associated probabilities of a fuzzy measure. It is shown that the monotone statistical parameters defined in the study are defined by one particular relevant associated statistical parameter out of the total number $n!$ of such parameters. It is also shown that the aggregations with monotone statistical parameters used in interactive MADM models take into account interactions of the focal elements of only one consonant structure from the $n!$ consonant structures of attributes. In order to take into account the interactions of the focal elements of all $n!$ consonant structures of attributes, the monotone statistical parameters were expanded into the $F$-associated statistical parameters. Expansion correctness implies that if dual second-order Choquet capacities are taken as the fuzzy measures of aggregation of the $F$-associated statistical parameters, then the $F$-associated statistical parameters coincide with the corresponding monotone statistical parameters. A scheme for embedding new aggregation operators, monotone statistical parameters and $F$-associated statistical parameters into the interactive MADM model has been developed. Specific numerical examples are presented to illustrate the obtained results. Full article

Nonadditivity of a fuzzy measure, as an indicator of defectiveness, makes a fuzzy mea-sure less useful in applications compared to additive, probabilistic measures. In order to neutralize this indicator of defectiveness to some degree, it is important to study the representations of fuzzy measures, including, in particular, additive, probabilistic representations. In this paper, we discuss a couple of probability representations of a fuzzy measure: the Campos-Bolanos representation (CBR) and the Murofushi–Sugeno representation (MSR). The CBR is mainly represented by the Associated Probability Class (APC). The APC is well studied and the aspects of its use can be found in many interesting studies. This is especially true for the environment of interactive attributes in their identification and multi-attribute group decision-making (MAGDM) models, related to the attributes’ Shapley values and interaction indexes. The MSR is a less-used tool in practice today. The main motivation of the research presented here was to explore the connections between these two representations, which will help increase the usability of the MSR in practice in the future. In the MSR, we constructed the nonequivalent representation class (NERC) of a fuzzy measure. This probabilistic new representation is somewhat similar to the APC in the CBR environment. The proposition on the existence of the MSR induced by the CBR was proven. The presented formula of the APC by the NERC was obtained. The duality property of fuzzy measures for the CBR is well studied with respect to fuzzy measures—Choquet second-order dual capacities. Significant properties were proven for the representation of a monotone expectation (ME) under the NERC conditions: as is known, the necessary and sufficient conditions for the existence of the second-order Choquet dual capacities are proven in the terms of the APC of a CBR and ME. After establishing the links between the APC of a CBR and the NERC of a MSR, we proved the same in the case of the MSR. A recursive connection formula between the interaction indexes, Shapley values, and the probability distribution of the NERC of a two-order additive fuzzy measure was obtained in the environment of a general MAGDM. A new distance concept was introduced for all fuzzy measures’ classes defined in finite sets in terms of the NERC. The distance between two fuzzy measures was defined as the distance between their NERCs. This distance is equivalent to the distance defined on the same class under the conditions of the APC of a CBR. The correctness proposition on the extension of the distance between fuzzy measures for the NERC was preserved: distances between any two fuzzy measures and between their dual fuzzy measures also coincided in the CBR as the MSR. After parameterization, the calculation formula of the new distance was obtained. An illustrative example was considered in order to easily present the obtained results. The connection schemes between the CBR and MSR and the sequential scheme of key facts and results obtained are presented at the end of this work. Full article

This article discusses the need for standards for the assignment of importance to criteria and the measurement of interaction between them in multiple criteria analyses of complex systems. A strategy for criteria evaluation is considered that is suitable to account for the interaction among a wide variety of imprecisely assessed criteria applied simultaneously. It is based on the results of collecting sample information on preferences according to the specified criteria instead of merely an abstract comparison of the criteria. The comparison of alternatives is based on objectives that determine the formation of preferences. It is facilitated by a rating in terms of preference probabilities. Probabilistic standards grant homogeneity of measurements by different criteria, which is useful for the combination of the criteria. These standards apply to a sampling evaluation conducted via pairwise trichotomic comparison of the alternatives according to each criterion, followed by the combination of these multiple evaluations into a single global score by means of the Choquet Integral with respect to a capacity determined by applying preference concentration to the sets of probabilistic assessments. Examples of practical application are discussed. Full article

In some multi-attribute decision-making (MADM) models studying attributes’ interactive phenomena is very important for the minimizing decision risks. Usually, the Choquet integral type aggregations are considered in such problems. However, the Choquet integral aggregations do not consider all attributes’ interactions; therefore, in many cases, when these interactions are revealed in less degree, they do not perceive these interactions and their utility in MADM problems is less useful. For the decision of this problem, we create the Choquet integral-based new aggregation operators’ family which considers all pair interactions between attributes. The problem under the discrimination q-rung picture linguistic and q-rung orthopair fuzzy environments is considered. Construction of a 2-order additive fuzzy measure (TOAFM) involves pair interaction indices and importance values of attributes of a MADM model. Based on the attributes’ pair interactions for the identification of associated probabilities of a 2-order additive fuzzy measure, the Shapley entropy maximum principle is used. The associated probabilities q-rung picture linguistic weighted averaging (APs-q-RPLWA) and the associated probabilities q-rung picture linguistic weighted geometric (APs-q-RPLWG) aggregation operators are constructed with respect to TOAFM. For an uncertainty pole of experts’ evaluations on attributes regarding the possible alternatives, the associated probabilities of a fuzzy measure are used. The second pole of experts’ evaluations as arguments of the aggregation operators by discrimination q-rung picture linguistic values is presented. Discrimination q-rung picture linguistic evaluations specify the attribute’s dominant, neutral and non-dominant impacts on the selection of concrete alternative from all alternatives. Constructed operators consider the all relatedness between attributes in any consonant attribute structure. Main properties on the rightness of extensions are showed: APs-q-RPLWA and APs-q-RPLWG operators match with q-rung picture linguistic Choquet integral averaging and geometric operators for the lower and upper capacities of order two. The conjugation among the constructed operators is also considered. Connections between the new operators and the compositions of dual triangular norms $\left(T_p,S_p^q\right)$ and $(T_{\min },S_{\max })$ are also constructed. Constructed operators are used in evaluation of a selection reliability index (SRI) of candidate service centers in the facility location selection problem, when small degree interactions are observed between attributes. In example MADM, the difference in optimal solutions is observed between the Choquet integral aggregation operators and their new extensions. The difference, however, is due to the need to use indices of all interactions between attributes. Full article

According to the United Nations report, climate disasters have intensified in the past 20 years, and China has the largest number of disasters in the world. So the study of meteorological disaster governance capacities is critically important for China. We designed a meteorological disaster governance capacity evaluation system to calculate the evaluation values by using the generalized λ-Shapley Choquet integral, a method that considers the interaction between indicators. We used various official statistical yearbooks and internal data of China Meteorological Administration (CMA) and weight intervals set by meteorologists for each level of indicators to calculate the evaluation values of meteorological disaster governance capacity in mainland provinces, from 2014 to 2018. We compared them with other methods (entropy weight method, Technique for Order Preference by Similarity to an Ideal Solution (TOPSIS), and Analytic Hierarchy Process (AHP)), and the results showed that the results calculated by the designed interaction method provided in this paper are more stable and differentiated. The results show that provincial meteorological disaster governance capacities in Mainland China are characterized by uneven development and a pro-slight polarization phenomenon. This leads to policy recommendations: Provinces should strengthen the construction of meteorological disaster information; provinces with outstanding capacity must strengthen the experience sharing with provinces with lower capacity. Full article

The Choquet capacity and integral is an eminent scheme to represent the interaction knowledge among multiple decision criteria and deal with the independent multiple sources preference information. In this paper, we enhance this scheme’s decision pattern learning ability by combining it with another powerful machine learning tool, the random forest of decision trees. We first use the capacity fitting method to train the Choquet capacity and integral-based decision trees and then compose them into the capacity random forest (CRF) to better learn and explain the given decision pattern. The CRF algorithms of solving the correlative multiple criteria based ranking and sorting decision problems are both constructed and discussed. Two illustrative examples are given to show the feasibilities of the proposed algorithms. It is shown that on the one hand, CRF method can provide more detailed explanation information and a more reliable collective prediction result than the main existing capacity fitting methods; on the other hand, CRF extends the applicability of the traditional random forest method into solving the multiple criteria ranking and sorting problems with a relatively small pool of decision learning data. Full article

Coherent lower previsions generalize the expected values and they are defined on the class of all real random variables on a finite non-empty set. Well known construction of coherent lower previsions by means of lower probabilities, or by means of super-modular capacities-based Choquet integrals, do not cover this important class of functionals on real random variables. In this paper, a new approach to the construction of coherent lower previsions acting on a finite space is proposed, exemplified and studied. It is based on special decomposition integrals recently introduced by Even and Lehrer, in our case the considered decomposition systems being single collections and thus called collection integrals. In special case when these integrals, defined for non-negative random variables only, are shift-invariant, we extend them to the class of all real random variables, thus obtaining so called super-additive integrals. Our proposed construction can be seen then as a normalized super-additive integral. We discuss and exemplify several particular cases, for example, when collections determine a coherent lower prevision for any monotone set function. For some particular collections, only particular set functions can be considered for our construction. Conjugated coherent upper previsions are also considered. Full article

Several discrete universal integrals on finite universes are discussed from an axiomatic point of view. We start from the first attempt due to B. Riemann and cover also most recent approaches based on level dependent capacities. Our survey includes, among others, the Choquet and the Sugeno integral and general copula-based integrals. Full article