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.

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.

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.

A HIERARCHICAL DECOMPOSITION OF CHOQUET INTEGRAL MODEL

1995 ◽  
Vol 03 (01) ◽  
pp. 1-15 ◽  
Author(s):  
MICHIO SUGENO ◽  
KATSUSHIGE FUJIMOTO ◽  
TOSHIAKI MUROFUSHI
Keyword(s):  
Choquet Integral ◽  
Integral Model ◽  
Hierarchical Decomposition ◽  
Sufficient Condition

In this paper we give a nesessary and sufficient condition for a Choquet integral model to be decomposable into an equivalent hierarchical Choquet integral model constructed by hierarchical combinations of some ordinary Choquet integral models. The condition is obtained by Inclution-Exclusion Covering (IEC). Moreover we show some properties on the set of IECs.

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.

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.

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