This study emanates from a simple observation: as specified by Vapnik [37] in his study, an artificial neural network cannot generate a universal approximator if the aggregation function chosen to design the artificial neuron does not include non-linearity. The usual option is to follow a linear aggregation by a non-linear function, or so-called activation function. We wonder if this approach could be replaced by one using a natively non-linear aggregation function. Among all of the available non-linear aggregation functions, here we are interested in aggregations based on weighted minimum and weighted maximum operations [8] . As these operators were originally developed within a possibility theory and fuzzy rule framework, such operators cannot be easily integrated into a neural network because the values that are usually considered belong to [ 0 , 1 ] . For gradient descent based learning, a neuron must be an aggregation function derivable with respect to its inputs and synaptic weights, whose variables (synaptic weights, inputs and outputs) must all be signed real values. We thus propose an extension of weighted maximum based aggregation to enable this learning process. We show that such an aggregation can be seen as a combination of four Sugeno integrals. Finally, we compare this type of approach with the classical one.

In this article, we explore the properties of the two generalized Sugeno integrals that we obtain by substituting in the expression of a classical Sugeno integral, the Kleene-Dienes conjunction and the Kleene-Dienes implication by the Gödel conjunction and the Gödel implication, respectively.A major difference compared to the classical Sugeno integrals is that the implication-based Gödel integral and the conjunction-based Gödel integral do not return the same result. In this paper, we investigate the adaptation of classical results for Sugeno integrals to Gödel integrals. Namely, their elicitation according to a piece of data and the extraction of selection and elimination if-then rules. The selection rules are obtained from the focal sets of the capacity underlying a conjunction-based Gödel integral, while the elimination rules are extracted from the focal sets of the conjugate of the capacity defining an implication-based Gödel integral.To illustrate our results, we apply our constructions to a real data example already used for Sugeno integrals.

In this paper, we investigate qualitative integrals (generalizations of Sugeno integral) acting on recently introduced Dragonfly algebras. These algebras are designed for applications in data analysis (based on fuzzy relational compositions) when some data are unknown (e.g., missing). Unknown data are represented by the additional dummy value. Definitions of operations on Dragonfly algebras follow lower estimation strategy, i.e., results of operations can be interpreted as lower estimations of results obtained when unknown values are replaced by known ones. We define qualitative integrals on Dragonfly algebras and show their monotonicity. We prove results characterizing when these integrals return known (or unknown) values. Illustrative example and directions of further research are also presented.

Qualitative or equivalently fuzzy integrals are used as qualitative aggregation functions or as L-fuzzy quantifiers. In both cases they are generalisations of Sugeno integrals. The definitions of these fuzzy integrals are quite similar and coincide in particular cases, but surprisingly there is no deeper analysis of their relationship. The paper attempts to fill this gap and provides unified definitions of fuzzy quantifiers on the basis of which various links between these fuzzy integrals are studied. In order to make these links more visible and to emphasise their logical structure, we present them using the graded square and modern square of opposition.

Possibility theory and Sugeno integrals may be viewed respectively as qualitative counterparts of probability theory and Choquet integrals, which are well-known tools for decision under uncertainty and multi-criteria evaluation. But what is the qualitative counterpart to Shafer's evidence theory for fusing uncertain pieces of information? There is not yet fully clear and definitive answer to this question, in spite of some sparse attempts at developing elements of such a theory. The paper makes a step in this direction and focuses more particularly on the problems of fusing possibilistic counterparts of the basic probability assigments of evidence theory. Making sense of this qualitative counterpart is not fully straightforward, and not just a matter of replacing the sum by the max operation and the product by the min operation. Indeed, as it turns out, any fuzzy measure can stand as a qualitative belief function and more than one possibilistic mass function can be associated to the same fuzzy measure. The particular role of qualitative support functions (whose focal sets are the universe of discourse and some subset of it, just as in the quantitative case) is emphasized in the fusion process. The practical interest of such a qualitative approach is pointed out.

Convolution kernels are essential in signal processing. They are used to model sensors, to define filters, to ensure the interplay between continuous and discrete domains, etc. A classic shortcoming is the difficulty to define which kernel is suitable for a particular application. In previous articles, we relied on a simple analogy between positive convolution kernels and probability distributions to define the notion of maxitive kernel. A maxitive kernel aims at representing a convex set of positive convolution kernels. It therefore models imprecise information on the suitable kernel to be used. Though, in many applications, such as filtering, it may be necessary to use signed convolution kernels. In this article we propose to extend the notion of maxitive kernel domination over signed convolution kernels. This will lead us towards a concept little used until now that are signed - and thus non-monotonous - set functions.

In decision problems involving two dimensions (like several agents in uncertainty) the properties of expected utility ensure that the result of a two-stepped procedure evaluation does not depend on the order with which the aggregations of local evaluations are performed (e.g., agents first, uncertainty next, or the converse). We say that the aggregations on each dimension commute. In a previous conference paper, Ben Amor, Essghaier and Fargier have shown that this property holds when using pessimistic possibilistic integrals on each dimension, or optimistic ones, while it fails when using a pessimistic possibilistic integral on one dimension and an optimistic one on the other. This paper studies and completely solves this problem when more general Sugeno integrals are used in place of possibilistic integrals, leading to double Sugeno integrals. The results show that there are capacities other than possibility and necessity measures that ensure commutation of Sugeno integrals. Moreover, the relationship between two-dimensional capacities and the commutation property for their projections is investigated.