Non-additive Measures Research Articles

Approximating Choquet integral in generalized measure theory: Choquet-midpoint rule

In generalized measure theory, Choquet integral is a generalization of Lebesgue integral and mathematical expectation. Approximating Choquet integral in the continuous case on real line is not very easy. So, we need mainly to estimate Choquet integral with respect to non-additive measures. There are few studies on the approximating Choquet integral in the continuous case on real line. In approximation theory, there are many interesting properties of midpoint rule. As a subject for research, there are no results on the midpoint rule for Choquet integral. The main objective of this paper is to propose some applications of midpoint rule for approximating continuous Choquet integral. The choquet-midpoint rule helps us to numerically solve Choquet integrals, in particular, the singular and unbounded integrals. Several numerical examples are considered to illustrate the application of our proposed methodology.

A Hybrid Interactive Decision-making Model Based on Multi-objective Optimization: An Application to Sustainable Supplier Selection

Sustainable development is crucial for conserving resources and ensuring smooth supply chain operations. In the context of the complex task of supplier selection, the traditional assumptions of attribute independencies are frequently found to be inadequate. The effective interaction between attributes and their weights plays a crucial role in navigating the decision outcomes. This paper presents a Choquet integral-based hybrid interactive decision-making approach to address the two important issues regarding the weights and interaction of criteria. To achieve this aim, first a hybrid weight determination method is proposed, combining the merits of multi-objective optimization and clustering techniques. To comprehensively assess the interactions among attributes, a non-additive measure determination method is introduced, leveraging concepts such as standard deviation and correlation. This approach effectively captures the interdependencies among attributes, leading to a more accurate decision-making process. Then, the Choquet integral incorporates a logarithmic function, which serves to enhance the modeling of uncertainty and effectively address the complexities present in intricate decision-making situations. Furthermore, a comprehensive interactive MCDM model is constructed, serving as a framework for supplier selection. Finally, experimental and sensitivity analysis demonstrate the effectiveness and stability of the proposed approach.

Research on a Three-Way Decision-Making Approach, Based on Non-Additive Measurement and Prospect Theory, and Its Application in Aviation Equipment Risk Analysis.

Due to the information non-independence of attributes, combined with a complex and changeable environment, the analysis of risks faces great difficulties. In view of this problem, this paper proposes a new three-way decision-making (3WD) method, combined with prospect theory and a non-additive measure, to cope with multi-source and incomplete risk information systems. Prospect theory improves the loss function of the original 3WD model, and the combination of non-additive measurement and probability measurement provides a new perspective to understand the meaning of decision-making, which could measure the relative degree by considering expert knowledge and objective data. The theoretical basis and framework of this model are illustrated, and this model is applied to a real in-service aviation equipment structures risk evaluation problem involving multiple incomplete risk information sources. When the simulation analysis is carried out, the results show that the availability of this method is verified. This method can also evaluate and rank key risk factors in equipment structures, which provides a reliable basis for decisions in aviation safety management.

Non-additive Measures, Envelopes and Extensions to Quasi-Measures

Non-additive Measures, Envelopes and Extensions to Quasi-Measures

A nonadditive rough set model for long-term clinical efficacy evaluation of chronic diseases in real-world settings

The pursuit of clinical effectiveness in real-world settings is at the core of clinical practice progression. In this study, we address a long-term clinical efficacy evaluation decision-making problem with temporal correlation hybrid attribute characteristics. To address this problem, we propose a novel approach that combines a temporal correlation feature rough set model with machine learning techniques and nonadditive measures. Our proposed approach involves several steps. First, over the framework of granular computing, we construct a temporal correlation hybrid information system, the gradient method is employed to characterize the temporal attributes and the similarity between objects is measured using cosine similarity. Second, based on the similarity of gradient and cosine, we construct a composite binary relation of temporal correlation hybrid information, enabling effective classification of this information. Third, we develop a rough set decision model based on the Choquet integral, which describes temporal correlation decision process. We provide the ranking results of decision schemes with temporal correlation features. To demonstrate the practical applications of our approach, we conduct empirical research using an unlabeled dataset consisting of 3094 patients with chronic renal failure (CRF) and 80,139 EHRs from various clinical encounters. These findings offer valuable support for clinical decision-making. Two main innovations are obtained from this study. First, it establishes general theoretical principles and decision-making methods for temporal correlation and hybrid rough sets. Second, it integrates data-driven clinical decision paradigms with traditional medical research paradigms, laying the groundwork for exploring the feasibility of data-driven clinical decision-making in the field.

Inequalities in Riemann–Lebesgue Integrability

In this paper, we prove some inequalities for Riemann–Lebesgue integrable functions when the considered integration is obtained via a non-additive measure, including the reverse Hölder inequality and the reverse Minkowski inequality. Then, we generalize these inequalities to the framework of a multivalued case, in particular for Riemann–Lebesgue integrable interval-valued multifunctions, and obtain some inequalities, such as a Minkowski-type inequality, a Beckenbach-type inequality and some generalizations of Hölder inequalities.

How Quantum Mechanics Requires Non-Additive Measures.

Measure theory is used in physics, not just to capture classical probability, but also to quantify the number of states. In previous works, we found that state quantification plays a foundational role in classical mechanics, and, therefore, we set ourselves to construct the quantum equivalent of the Liouville measure. Unlike the classical counterpart, this quantized measure is non-additive and has a unitary lower bound (i.e., no set of states can have less than one state). Conversely, requiring that state quantification is finite for finite continuous regions and that each state counts as one already implies non-additivity, which in turn implies the failure of classical theory. In this article we show these preliminary results and outline a new line of inquiry that may provide a different insight into the foundations of quantum theory. Additionally, this new approach may prove to be useful to those interested in a quantized theory of space-time, as we believe this requires a quantized measure for the quantification of the independent degrees of freedom.

A Basic Set of Cancellation Violating Sequences for Finite Two-Dimensional Non-Additive Measurement

Abstract Cancellation conditions play a central role in the representation theory of measurement for a weak order on a finite two-dimensional Cartesian product set X. A weak order has an additive representation if and only if it violates no cancellation conditions. Given X, a longstanding open problem is to determine the simplest set of cancellation conditions that is violated by every linear order that is not additively representable. Here, we report that the simplest set of cancellation conditions on a 5 by 5 product X is obtained.

Fuzzy neighborhood Choquet integrals with overlap functions and their applications in attribute reduction

In fuzzy rough sets, fuzzy neighborhood operators and fuzzy rough approximate operators are mainly used by traditional methods to deal with the problem of attribute reduction (i.e., feature selection). However, they are parametric linear, while the corresponding weights are additive measures. For complex data, parametric linear operators and additive measures do not reflect correlations among data (or attributes). Therefore, it is important to develop non-additive measures and nonlinear operators for attribute reduction, which can effectively find those attributes that are not needed to reach a decision from the viewpoint of correlations. Along these lines, in this work, OI-fuzzy β-neighborhood measures (i.e., non-additive measures) and generalized Choquet integrals (i.e., nonlinear integrals) are presented by using overlap functions to deal with the problem of attribute reduction under the fuzzy β-covering approximation space. First, four pairs of OI-fuzzy β-neighborhood operators are constructed based on the implementation of overlap functions and their residual implications in a fuzzy β-covering approximation space. Furthermore, four pairs of OI-fuzzy β-neighborhood measures, as non-additive measures, are introduced to instead of rough approximate operators in data mining. Then, a novel method with respect to OI-fuzzy β-neighborhood measures is presented to deal with the attribute reduction in a fuzzy β-covering information decision table. Third, four pairs of generalized Choquet integrals based on the employment of OI-fuzzy β-neighborhood measures are constructed. On top of that, another method with respect to these generalized Choquet integrals is proposed to deal with the attribute reduction in a fuzzy β-covering information decision table. Finally, the presented methods are used to deal with the problem of classification. Several public data sets are applied to illustrate the feasibility and effectiveness of the above-mentioned proposed methods.

Bounds on Choquet risk measures in finite product spaces with ambiguous marginals

Abstract We investigate the problem of finding upper and lower bounds for a Choquet risk measure of a nonlinear function of two risk factors, when the marginal distributions of the risk factors are ambiguous and represented by nonadditive measures on the marginal spaces and the joint nonadditive distribution on the product space is unknown. We treat this problem as a generalization of the optimal transport problem to the setting of nonadditive measures. We provide explicit characterizations of the optimal solutions for finite marginal spaces, and we investigate some of their properties. We further discuss the connections with linear programming, showing that the optimal transport problems for capacities are linear programs, and we also characterize their duals explicitly. Finally, we investigate a series of numerical examples, including a comparison with the classical optimal transport problem, and applications to counterparty credit risk.

Spaces of non-additive measures generated by triangular norms

We consider non-additive measures on the compact Hausdorff spaces, which are generalizations of the idempotent measures and max-min measures. These measures are related to the continuous triangular norms and they are defined as functionals on the spaces of continuous functions from a compact Hausdorff space into the unit segment.The obtained space of measures (called ∗-measures, where ∗ is a triangular norm) are endowed with the weak* topology. This construction determines a functor in the category of compact Hausdorff spaces. It is proved, in particular, that the ∗-measures of finite support are dense in the spaces of ∗-measures. One of the main results of the paper provides an alternative description of ∗-measures on a compact Hausdorff space X, namely as hyperspaces of certain subsets in X × [0, 1]. This is an analog of a theorem for max-min measures proved by Brydun and Zarichnyi.

On the definition of probabilistic metric spaces by means of fuzzy measures

Metric spaces are defined in terms of a space and a metric, or distance. Probabilistic metric spaces are a useful extension of metric spaces where the distance is a distribution instead of a number. In this way, we can take into account uncertainty. Then, the triangle inequality is replaced by a condition based on triangle functions on the distributions. In this paper we introduce F-spaces. This is a new type of probabilistic metric spaces which is based on fuzzy measures (also known as non-additive measures and capacities). We prove some properties that describe which families of fuzzy measures are compatible with which type of triangle functions. Then, we show how we can use Sugeno, Choquet integrals, and, in general, any other fuzzy integral as a tool for building these spaces. We show how these results can be used to compute distances between functions. We illustrate the example comparing three types of means when applied to a set of databases. The example uses Sugeno λ-measures to illustrate the theoretical results presented in the paper.

An Approximation Method for Variational Inequality with Uncertain Variables

In this paper, a Stieltjes integral approximation method for uncertain variational inequality problem (UVIP) is studied. Firstly, uncertain variables are introduced on the basis of variational inequality. Since the uncertain variables are based on nonadditive measures, there is usually no density function. Secondly, the expected value model of UVIP is established after the expected value is discretized by the Stieltjes integral. Furthermore, a gap function is constructed to transform UVIP into an uncertain constraint optimization problem, and the optimal value of the constraint problem is proved to be the solution of UVIP. Finally, the convergence of solutions of the Stieltjes integral discretization approximation problem is proved.

Topological and topological linear properties of the Sugeno-Lorentz spaces

The properties of spaces of Sugeno integrable functions are quite different from those of ordinary spaces of Lebesgue integrable functions. In our previous research, given a nonadditive measure μ and constants 0<p<∞ and 0<q<∞, the completeness and separability of the Sugeno-Lorentz spaces were discussed with respect to the Sugeno-Lorentz prenorm (⋅)p,q. The purpose of the paper is to further advance our study of the Sugeno-Lorentz spaces from the perspective of topology. To this end, the Sugeno-Lorentz topologies are defined as the topologies generated by the prenorms (⋅)p,q. Some topological and topological linear properties of the Sugeno-Lorentz spaces are discussed. Of particular interest is that the Sugeno-Lorentz spaces and the Sugeno-Lorentz topologies are unique regardless of p and q.

The transport problem for non-additive measures

Non-additive measures, also known as fuzzy measures, capacities, and monotonic games, are increasingly used in different fields. Applications have been built within computer science and artificial intelligence related to e.g., decision making, image processing, machine learning for both classification, and regression. Tools for measure identification have been built. In short, as non-additive measures are more general than additive ones (i.e., than probabilities), they have better modeling capabilities allowing to model situations and problems that cannot be modeled by the latter. See e.g., the application of non-additive measures and the Choquet integral to model both Ellsberg paradox and Allais paradox. Because of that, there is an increasing need to analyze non-additive measures.The optimal transport problem is a useful tool for probabilities. It is used to define the Wasserstein distance, and also to solve problems as data matching. In this work with tackle the problem of defining the optimal transport problem for non-additive measures. We consider that it is necessary to provide appropriate definitions with a flavour similar to the one for probabilities, but for non-additive measures. The solutions should generalize the classical ones.We provide definitions based on the Möbius transform, but also based on the (max,+)-transform that we consider that has some advantages. We will discuss in this paper the problems that arise to define the transport problem for non-additive measures, and provide ways to solve them. We include the definitions of the optimal transport problem, and prove some properties. We also conclude providing a definition of the Wasserstein distance for non-additive measures.

Results and applications of a new inaccuracy measure based on cumulative Tsallis entropy

Non-additive measures are important for many applications. In this paper we propose an extension of the non-additive measure using the concept of Kerridge inaccuracy and cumulative Tsallis entropy. Several properties including non-linear transformation, bounds and characterizations are obtained using this measure. We also study different properties of the quantile version of the proposed measure. Finally, we numerically illustrate the usefulness of the measure and its quantile form using kernel density estimators.

Меры доверия и правдоподобия при оценке рисков информационной безопасности

Purpose of the research: to develop methods for assessing information security risks under uncertainty, to describe the mechanism of propagating belief and plausibility in the attack graph.Research method: application of soft computing techniques, including the combination of Dempster-Shafer evidence theory, integration with respect to non-additive measures of belief and plausibility.Research result: risk assessment methods and methods for assessing expected losses have been developed in the case when risk factors are characterized by high uncertainty and do not allow sufficiently justified applying objective (probabilistic) assessment methods. The initial information is the upper and lower estimates of the probability of risk realization. Using the methods of the Dempster-Shafer evidence theory, belief and plausibility measures are built on the attack graph. An approach is described that allows building belief and plausibility measures in the space of attack scenarios based on probabilistic estimates of typical information security events. It is shown how the expected damage (severity) can be estimated by the expectation of damage with respect to to these measures using the Choquet integral. Scientific novelty: a method of propagation belief along the attack graph has been developed. The method is based on an original approach to evaluating logical combinations of evidence given on binary frames and represented by disjunctive normal forms.

RFI-GAN: A reference-guided fuzzy integral network for ultrasound image augmentation

The Generative Adversarial Network (GAN) is commonly used for medical image augmentation, a method to alleviate the data shortage for downstream tasks. However, traditional GANs is not effective in mining the intrinsic association between samples and data space. Also, the multiple sections and variable scaling ratios of ultrasound images lead to inconsistency among images. The current GAN-based image generation methods cannot obtain the overall anatomical semantic constraints of inconsistent images, thus illogical images are generated. Therefore, a Reference-guided Fuzzy Integral GAN (RFI-GAN) is proposed. First, RFI-GAN implicitly constrains the logicality of the generated images through feature fusion between samples and reference set. These references, a kind of prior knowledge, cover most of the anatomical structures in the dataset. Second, as the general linear fusion takes features as additive and cannot fit the inconsistency of features, we generalize the additivity of features to non-additivity through the Fuzzy Integral, which is used to fusion non-additive measures. Two Fuzzy Integral Modules are proposed to nonlinearly fusion the texture and structure features of ultrasound images under the guidance of references. Finally, we perform experiments on the thyroid ultrasound image dataset, demonstrating the effectiveness of RFI-GAN in terms of generation and downstream classification tasks.

Coincidences of nonlinear integrals related to ordered pair of non-additive measures

This note studies the coincidences among the Choquet, concave and pan-integral in the setting of an ordered pair of monotone measures. We further investigate some properties of the minimal atoms and (M)-property of monotone measures, and optimal measures. By using these properties we present several sufficient and/or necessary conditions of coincidences among these three integrals, in the context relating to two monotone measures. These generalized versions, as special cases, cover respectively the coincidences in the usual sense among the pan-integral, the Choquet integral, and the concave integral on finite spaces.

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.