3,209 research outputs found
An equivalent condition to the Jensen inequality for the generalized Sugeno integral.
For the classical Jensen inequality of convex functions, i.e., [Formula: see text] an equivalent condition is proved in the framework of the generalized Sugeno integral. Also, the necessary and sufficient conditions for the validity of the discrete form of the Jensen inequality for the generalized Sugeno integral are given
Characterizations of discrete Sugeno integrals as polynomial functions over distributive lattices
We give several characterizations of discrete Sugeno integrals over bounded
distributive lattices, as particular cases of lattice polynomial functions,
that is, functions which can be represented in the language of bounded lattices
using variables and constants. We also consider the subclass of term functions
as well as the classes of symmetric polynomial functions and weighted minimum
and maximum functions, and present their characterizations, accordingly.
Moreover, we discuss normal form representations of these functions
Synergy Modelling and Financial Valuation : the contribution of Fuzzy Integrals.
Les méthodes d’évaluation financière utilisent des opérateurs d’agrégation reposant sur les propriétés d’additivité (sommations, intégrales de Lebesgue). De ce fait, elles occultent les phénomènes de renforcement et de synergie (ou de redondance) qui peuvent exister entre les éléments d’un ensemble organisé. C’est particulièrement le cas en ce qui concerne le problème d’évaluation financière du patrimoine d’une entreprise : en effet, en pratique, il est souvent mis en évidence une importante différence de valorisation entre l’approche « valeur de la somme des éléments » (privilégiant le point de vue financier) et l’approche « somme de la valeur des différents éléments » (privilégiant le point de vue comptable). Les possibilités offertes par des opérateurs d’agrégation comme les intégrales floues (Sugeno, Grabisch, Choquet) permettent, au plan théorique, de modéliser l’effet de synergie. La présente étude se propose de valider empiriquement les modalités d’implémentation opérationnelle de ce modèle à partir d’un échantillon d’entreprises cotées ayant fait l’objet d’une évaluation lors d’une OPA.Financial valuation methods use additive aggregation operators. But a patrimony should be regarded as an organized set, and additivity makes it impossible for these aggregation operators to formalize such phenomena as synergy or mutual inhibition between the patrimony’s components. This paper considers the application of fuzzy measure and fuzzy integrals (Sugeno, Grabisch, Choquet) to financial valuation. More specifically, we show how integration with respect to a non additive measure can be used to handle positive or negative synergy in value construction.Fuzzy measure; Fuzzy integral; Aggregation operator; Synergy; Financial valuation;
Intangibles mismeasurements, synergy, and accounting numbers : a note.
For the last two decades, authors (e.g. Ohlson, 1995; Lev, 2000, 2001) have regularly pointed out the enforcement of limitations by traditional accounting frameworks on financial reporting informativeness. Consistent with this claim, it has been then argued that accounting finds one of its major limits in not allowing for direct recognition of synergy occurring amongst the firm intangible and tangible items (Casta, 1994; Casta & Lesage, 2001). Although the firm synergy phenomenon has been widely documented in the recent accounting literature (see for instance, Hand & Lev, 2004; Lev, 2001) research hitherto has failed to provide a clear approach to assess directly and account for such a henceforth fundamental corporate factor. The objective of this paper is to raise and examine, but not address exhaustively, the specific issues induced by modelling the synergy occurring amongst the firm assets whilst pointing out the limits of traditional accounting valuation tools. Since financial accounting valuation methods are mostly based on the mathematical property of additivity, and consequently may occult the perspective of regarding the firm as an organized set of assets, we propose an alternative valuation approach based on non-additive measures issued from the Choquet's (1953) and Sugeno's (1997) framework. More precisely, we show how this integration technique with respect to a non-additive measure can be used to cope with either positive or negative synergy in a firm value-building process and then discuss its potential future implications for financial reporting.Financial reporting; accounting goodwill; assets synergy; non-additive measures; Choquet’s framework;
Aggregation functions: Means
The two-parts state-of-art overview of aggregation theory summarizes the essential information concerning aggregation issues. Overview of aggregation properties is given, including the basic classification of aggregation functions. In this first part, the stress is put on means, i.e., averaging aggregation functions, both with fixed arity (n-ary means) and with open arity (extended means).
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
A Discrete Choquet Integral for Ordered Systems
A model for a Choquet integral for arbitrary finite set systems is presented.
The model includes in particular the classical model on the system of all
subsets of a finite set. The general model associates canonical non-negative
and positively homogeneous superadditive functionals with generalized belief
functions relative to an ordered system, which are then extended to arbitrary
valuations on the set system. It is shown that the general Choquet integral can
be computed by a simple Monge-type algorithm for so-called intersection
systems, which include as a special case weakly union-closed families.
Generalizing Lov\'asz' classical characterization, we give a characterization
of the superadditivity of the Choquet integral relative to a capacity on a
union-closed system in terms of an appropriate model of supermodularity of such
capacities
A decade of application of the Choquet and Sugeno integrals in multi-criteria decision aid
The main advances regarding the use of the Choquet and Sugeno integrals in multi-criteria decision aid over the last decade are reviewed. They concern mainly a bipolar extension of both the Choquet integral and the Sugeno integral, interesting particular submodels, new learning techniques, a better interpretation of the models and a better use of the Choquet integral in multi-criteria decision aid. Parallel to these theoretical works, the Choquet integral has been applied to many new fields, and several softwares and libraries dedicated to this model have been developed.Choquet integral, Sugeno integral, capacity, bipolarity, preferences
An Ordinal Approach to Risk Measurement
In this short note, we aim at a qualitative framework for modeling multivariate risk. To this extent, we consider completely distributive lattices as underlying universes, and make use of lattice functions to formalize the notion of risk measure. Several properties of risk measures are translated into this general setting, and used to provide axiomatic characterizations. Moreover, a notion of quantile of a lattice-valued random variable is proposed, which shown to retain several desirable properties of its real-valued counterpart.lattice; risk measure; Sugeno integral; quantile.
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