99 research outputs found
A note on the Sobol' indices and interactive criteria
The Choquet integral and the Owen extension (or multilinear extension) are
the most popular tools in multicriteria decision making to take into account
the interaction between criteria. It is known that the interaction transform
and the Banzhaf interaction transform arise as the average total variation of
the Choquet integral and multilinear extension respectively. We consider in
this note another approach to define interaction, by using the Sobol' indices
which are related to the analysis of variance of a multivariate model. We prove
that the Sobol' indices of the multilinear extension gives the square of the
Fourier transform, a well-known concept in computer sciences. We also relate
the latter to the Banzhaf interaction transform and compute the Sobol' indices
for the 2-additive Choquet integral
Multicriteria sustainability evaluation of transport networks for selected European countries
As an essential economic activity, transportation has complex interactions with the environment and society. Since the concept of sustainable development has become one of the top priorities for nations, there has been a growing interest in evaluating the performance of transport systems with respect to sustainability issues. The main purpose of this study is to introduce a decision making framework to assess the sustainability of the transport networks in a multidimensional setting and a technique to identify non-compromise alternatives. We also propose an elucidation technique to identify according to which criteria a system needs to be improved and how much improvement is required to attain a certain level of sustainability. The proposed methods are applied to a set of selected European countries within a case study
Approximations of Lovasz extensions and their induced interaction index
The Lovasz extension of a pseudo-Boolean function is
defined on each simplex of the standard triangulation of as the
unique affine function that interpolates at the
vertices of the simplex. Its degree is that of the unique multilinear
polynomial that expresses . In this paper we investigate the least squares
approximation problem of an arbitrary Lovasz extension by Lovasz
extensions of (at most) a specified degree. We derive explicit expressions of
these approximations. The corresponding approximation problem for
pseudo-Boolean functions was investigated by Hammer and Holzman (1992) and then
solved explicitly by Grabisch, Marichal, and Roubens (2000), giving rise to an
alternative definition of Banzhaf interaction index. Similarly we introduce a
new interaction index from approximations of and we present some of
its properties. It turns out that its corresponding power index identifies with
the power index introduced by Grabisch and Labreuche (2001).Comment: 19 page
The Importance and Interaction Indices of Bi-Capacities Based on Ternary-Element Sets
Grabisch and Labreuche have recently proposed a generalization of capacities, called the bi-capacities. Recently, a new approach for studying bi-capacities through introducing a notion of ternary-element sets proposed by the author. In this paper, we propose many results such as bipolar Mobius transform, importance index, and interaction index of bi-capacities based on our approach
On the set of imputations induced by the k-additive core
An extension to the classical notion of core is the notion of -additive core, that is, the set of -additive games which dominate a given game, where a -additive game has its Möbius transform (or Harsanyi dividends) vanishing for subsets of more than elements. Therefore, the 1-additive core coincides with the classical core. The advantages of the -additive core is that it is never empty once , and that it preserves the idea of coalitional rationality. However, it produces -imputations, that is, imputations on individuals and coalitions of at most individuals, instead of a classical imputation. Therefore one needs to derive a classical imputation from a -order imputation by a so-called sharing rule. The paper investigates what set of imputations the -additive core can produce from a given sharing rule.
Games on lattices, multichoice games and the Shapley value: a new approach
Multichoice games have been introduced by Hsiao and Raghavan as a generalization of classical cooperative games. An important notion in cooperative game theory is the core of the game, as it contains the rational imputations for players. We propose two definitions for the core of a multichoice game, the first one is called the precore and is a direct generalization of the classical definition. We show that the precore coincides with the definition proposed by Faigle, and that it contains unbounded imputations, which makes its application questionable. A second definition is proposed, imposing normalization at each level, causing the core to be a convex closed set. We study its properties, introducing balancedness and marginal worth vectors, and defining the Weber set and the pre-Weber set. We show that the classical properties of inclusion of the (pre)core into the (pre)-Weber set as well as their equality remain valid. A last section makes a comparison with the core defined by van den Nouweland et al.multichoice game ; lattice ; core
p-symmetric fuzzy measures
In this paper we propose a generalization of the concept of symmetric fuzzy measure based in a decomposition of the universal set in what we have called subsets of indifference. Some properties of these measures are studied, as well as their Choquet integral. Finally, a degree of interaction between the subsets of indifference is defined.
Enabling Explainable Fusion in Deep Learning with Fuzzy Integral Neural Networks
Information fusion is an essential part of numerous engineering systems and
biological functions, e.g., human cognition. Fusion occurs at many levels,
ranging from the low-level combination of signals to the high-level aggregation
of heterogeneous decision-making processes. While the last decade has witnessed
an explosion of research in deep learning, fusion in neural networks has not
observed the same revolution. Specifically, most neural fusion approaches are
ad hoc, are not understood, are distributed versus localized, and/or
explainability is low (if present at all). Herein, we prove that the fuzzy
Choquet integral (ChI), a powerful nonlinear aggregation function, can be
represented as a multi-layer network, referred to hereafter as ChIMP. We also
put forth an improved ChIMP (iChIMP) that leads to a stochastic gradient
descent-based optimization in light of the exponential number of ChI inequality
constraints. An additional benefit of ChIMP/iChIMP is that it enables
eXplainable AI (XAI). Synthetic validation experiments are provided and iChIMP
is applied to the fusion of a set of heterogeneous architecture deep models in
remote sensing. We show an improvement in model accuracy and our previously
established XAI indices shed light on the quality of our data, model, and its
decisions.Comment: IEEE Transactions on Fuzzy System
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