Canonical Separated Hierarchical Decomposition of the Choquet Integral Over a Finite Set

1998 ◽  
Vol 06 (03) ◽  
pp. 257-272 ◽  
Author(s):  
Toshiaki Murofushi ◽  
Katsushige Fujimoto ◽  
Michio Sugeno
Keyword(s):  
Choquet Integral ◽  
Hierarchical Decomposition ◽  
Finite Set ◽  
Choquet Integrals ◽  
Hierarchical Decompositions ◽  
Disjoint Domains

The paper shows the existence of a canonical separated hierarchical decomposition of the Choquet integral over a finite set. The decomposed system is a hierarchical combination of Choquet integrals with mutually disjoint domains, and equivalent to the original Choquet integral. The paper also gives canonical overlapped hierarchical decompositions of the Choquet integral over a semiatom.

Separated Hierarchical Decomposition of the Choquet Integral

1997 ◽  
Vol 05 (05) ◽  
pp. 563-585 ◽  
Author(s):  
Toshiaki Murofushi ◽  
Michio Sugeno ◽  
Katsushige Fujimoto
Keyword(s):  
Choquet Integral ◽  
Hierarchical Decomposition ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Integral System ◽  
Choquet Integrals ◽  
Necessary And Sufficient ◽  
Disjoint Domains

The paper gives a necessary and sufficient condition for a Choquet integral to be decomposable into an equivalent separated hierarchical Choquet-integral system, which is a hierarchical combination of ordinary Choquet integrals with mutually disjoint domains.

Canonical Hierarchical Decomposition of Choquet Integral Over Finite Set with Respect to Null Additive Fuzzy Measure

1998 ◽  
Vol 06 (04) ◽  
pp. 345-363 ◽  
Author(s):  
Katsushige Fujimoto ◽  
Toshiaki Murofushi ◽  
Michio Sugeno
Keyword(s):  
Hierarchical Structure ◽  
Choquet Integral ◽  
Fuzzy Measure ◽  
Integral Model ◽  
Hierarchical Decomposition ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Finite Set ◽  
Necessary And Sufficient

In this paper, we provide the necessary and sufficient condition for a Choquet integral model to be decomposable into a canonical hierarchical Choquet integral model constructed by hierarchical combinations of some ordinary Choquet integral models. This condition is characterized by the pre-Znclusion-Exclusion Covering (pre-IEC). Moreover, we show that the pre-IEC is the subdivision of an IEC and that the additive hierarchical structure is the most fundamental one on considering a hierarchical decomposition of the Choquet integral model.

On a Choquet-Stieltjes type integral on intervals

2019 ◽  
Vol 69 (4) ◽  
pp. 801-814 ◽  
Author(s):  
Sorin G. Gal
Keyword(s):  
Lebesgue Measure ◽  
Choquet Integral ◽  
Stieltjes Integral ◽  
Line Integral ◽  
Type Integral ◽  
Choquet Integrals ◽  
Lebesgue Measures ◽  
Stieltjes Type

Abstract In this paper we introduce a new concept of Choquet-Stieltjes integral of f with respect to g on intervals, as a limit of Choquet integrals with respect to a capacity μ. For g(t) = t, one reduces to the usual Choquet integral and unlike the old known concept of Choquet-Stieltjes integral, for μ the Lebesgue measure, one reduces to the usual Riemann-Stieltjes integral. In the case of distorted Lebesgue measures, several properties of this new integral are obtained. As an application, the concept of Choquet line integral of second kind is introduced and some of its properties are obtained.

scholarly journals Neural Representation and Learning of Hierarchical 2-additive Choquet Integrals

2020 ◽  
Author(s):  
Roman Bresson ◽  
Johanne Cohen ◽  
Eyke Hüllermeier ◽  
Christophe Labreuche ◽  
Michèle Sebag
Keyword(s):  
Real World ◽  
Choquet Integral ◽  
Marginal Utility ◽  
Decision Makers ◽  
Neural Representation ◽  
Automatic Identification ◽  
Neural Architecture ◽  
Real World Applications ◽  
Choquet Integrals ◽  
The Cost

Multi-Criteria Decision Making (MCDM) aims at modelling expert preferences and assisting decision makers in identifying options best accommodating expert criteria. An instance of MCDM model, the Choquet integral is widely used in real-world applications, due to its ability to capture interactions between criteria while retaining interpretability. Aimed at a better scalability and modularity, hierarchical Choquet integrals involve intermediate aggregations of the interacting criteria, at the cost of a more complex elicitation. The paper presents a machine learning-based approach for the automatic identification of hierarchical MCDM models, composed of 2-additive Choquet integral aggregators and of marginal utility functions on the raw features from data reflecting expert preferences. The proposed NEUR-HCI framework relies on a specific neural architecture, enforcing by design the Choquet model constraints and supporting its end-to-end training. The empirical validation of NEUR-HCI on real-world and artificial benchmarks demonstrates the merits of the approach compared to state-of-art baselines.

Cardinal-Probabilistic Interaction Indices and their Applications: A Survey

2003 ◽  
Vol 7 (2) ◽  
pp. 79-85 ◽  
Author(s):  
Katsushige Fujimoto ◽  
Keyword(s):  
Choquet Integral ◽  
Axiomatic Characterization ◽  
Probability Measures ◽  
Choice Probability ◽  
Hierarchical Decomposition ◽  
Möbius Transform

The class of cardinal probabilistic interaction indices obtained as expected marginal interactions includes the Shapley, Banzhaf, and chaining interaction indices and the Möbius and co-Möbius transform so. We will survey cardinal-probabilistic interaction indices and their applications, focusing on axiomatic characterization of the class of cardinal-probabilistic interaction indices. We show that these typical cardinal-probabilistic interaction indices can be represented as the Stieltjes integrals with respect to choice-probability measures on [0,1]. We introduce a method for hierarchical decomposition of systems represented by the Choquet integral using interaction indices.

Characterization, Robustness, and Aggregation of Signed Choquet Integrals

2020 ◽  
Vol 45 (3) ◽  
pp. 993-1015 ◽  
Author(s):  
Ruodu Wang ◽  
Yunran Wei ◽  
Gordon E. Willmot
Keyword(s):  
Risk Management ◽  
General Class ◽  
Choquet Integral ◽  
Risk Measures ◽  
Functional Characterization ◽  
Risk Aggregation ◽  
Choquet Integrals ◽  
Equivalent Conditions ◽  
Comonotonic Additivity

This article contains various results on a class of nonmonotone, law-invariant risk functionals called the signed Choquet integrals. A functional characterization via comonotonic additivity is established along with some theoretical properties, including six equivalent conditions for a signed Choquet integral to be convex. We proceed to address two practical issues currently popular in risk management, namely robustness (continuity) issues and risk aggregation with dependence uncertainty, for signed Choquet integrals. Our results generalize in several directions those in the literature of risk functionals. From the results obtained in this paper, we see that many profound and elegant mathematical results in the theory of risk measures hold for the general class of signed Choquet integrals; thus, they do not rely on the assumption of monotonicity.

scholarly journals Choquet Integral of Fuzzy-Number-Valued Functions: The Differentiability of the Primitive with respect to Fuzzy Measures and Choquet Integral Equations

2014 ◽  
Vol 2014 ◽  
pp. 1-11 ◽  
Author(s):  
Zengtai Gong ◽  
Li Chen ◽  
Gang Duan
Keyword(s):  
Integral Equations ◽  
Real Line ◽  
Fuzzy Number ◽  
Laplace Transformation ◽  
Choquet Integral ◽  
Fuzzy Measures ◽  
Nonadditive Measure ◽  
Choquet Integrals ◽  
Lebesgue Measures ◽  
Interval Valued

This paper deals with the Choquet integral of fuzzy-number-valued functions based on the nonnegative real line. We firstly give the definitions and the characterizations of the Choquet integrals of interval-valued functions and fuzzy-number-valued functions based on the nonadditive measure. Furthermore, the operational schemes of above several classes of integrals on a discrete set are investigated which enable us to calculate Choquet integrals in some applications. Secondly, we give a representation of the Choquet integral of a nonnegative, continuous, and increasing fuzzy-number-valued function with respect to a fuzzy measure. In addition, in order to solve Choquet integral equations of fuzzy-number-valued functions, a concept of the Laplace transformation for the fuzzy-number-valued functions in the sense of Choquet integral is introduced. For distorted Lebesgue measures, it is shown that Choquet integral equations of fuzzy-number-valued functions can be solved by the Laplace transformation. Finally, an example is given to illustrate the main results at the end of the paper.

A Method for Multi-Attribute Group Decision Making Based on Generalized Interval-Valued Intuitionistic Fuzzy Choquet Integral Operators

2017 ◽  
Vol 25 (05) ◽  
pp. 821-849 ◽  
Author(s):  
Fanyong Meng ◽  
Chunqiao Tan
Keyword(s):  
Decision Making ◽  
Group Decision Making ◽  
Group Decision ◽  
Choquet Integral ◽  
Intuitionistic Fuzzy ◽  
Fuzzy Environment ◽  
Operational Laws ◽  
Choquet Integrals ◽  
The Given ◽  
Interval Valued

As an extension of the classical averaging operators, Choquet integral has been shown a powerful tool for decision theory. In this paper, a method based on the generalized interval-valued intuitionistic fuzzy Choquet integrals w.r.t. the generalized interaction indices is proposed for multiattribute group decision making problems, where the importance of the elements is considered, and their interactions are reflected. Based on the given operational laws on interval-valued intuitionistic fuzzy sets, the interval-valued intuitionistic fuzzy Choquet integrals with respect to the generalized Shapley and Banzhaf indices are defined. Moreover, some of their properties are studied, such as idempotency, boundary, comonotonic linearity and μ–linearity. Furthermore, a decision procedure based on the proposed operators is developed for solving multi-attribute group decision making under interval-valued intuitionistic fuzzy environment. Finally, a numerical example is provided to illustrate the developed procedure.

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