195 research outputs found
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
The Symmetric Sugeno Integral
We propose an extension of the Sugeno integral for negative numbers, in the spirit of the symmetric extension of Choquet integral, also called \Sipos\ integral. Our framework is purely ordinal, since the Sugeno integral has its interest when the underlying structure is ordinal. We begin by defining negative numbers on a linearly ordered set, and we endow this new structure with a suitable algebra, very close to the ring of real numbers. In a second step, we introduce the Möbius transform on this new structure. Lastly, we define the symmetric Sugeno integral, and show its similarity with the symmetric Choquet integral.
Axiomatizations of quasi-polynomial functions on bounded chains
Two emergent properties in aggregation theory are investigated, namely
horizontal maxitivity and comonotonic maxitivity (as well as their dual
counterparts) which are commonly defined by means of certain functional
equations. We completely describe the function classes axiomatized by each of
these properties, up to weak versions of monotonicity in the cases of
horizontal maxitivity and minitivity. While studying the classes axiomatized by
combinations of these properties, we introduce the concept of quasi-polynomial
function which appears as a natural extension of the well-established notion of
polynomial function. We give further axiomatizations for this class both in
terms of functional equations and natural relaxations of homogeneity and median
decomposability. As noteworthy particular cases, we investigate those
subclasses of quasi-term functions and quasi-weighted maximum and minimum
functions, and provide characterizations accordingly
Weighted lattice polynomials
We define the concept of weighted lattice polynomial functions as lattice
polynomial functions constructed from both variables and parameters. We provide
equivalent forms of these functions in an arbitrary bounded distributive
lattice. We also show that these functions include the class of discrete Sugeno
integrals and that they are characterized by a median based decomposition
formula.Comment: Revised version (minor changes
Aggregation on bipolar scales
The paper addresses the problem of extending aggregation operators typically defined on to the symmetric interval , where the ``0'' value plays a particular role (neutral value). We distinguish the cases where aggregation operators are associative or not. In the former case, the ``0'' value may play the role of neutral or absorbant element, leading to pseudo-addition and pseudo-multiplication. We address also in this category the special case of minimum and maximum defined on some finite ordinal scale. In the latter case, we find that a general class of extended operators can be defined using an interpolation approach, supposing the value of the aggregation to be known for ternary vectors.bipolar scale; bi-capacity; aggregation
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.
Invariant functionals on completely distributive lattices
In this paper we are interested in functionals defined on completely
distributive lattices and which are invariant under mappings preserving
{arbitrary} joins and meets. We prove that the class of nondecreasing invariant
functionals coincides with the class of Sugeno integrals associated with
-valued capacities, the so-called term functionals, thus extending
previous results both to the infinitary case as well as to the realm of
completely distributive lattices. Furthermore, we show that, in the case of
functionals over complete chains, the nondecreasing condition is redundant.
Characterizations of the class of Sugeno integrals, as well as its superclass
comprising all polynomial functionals, are provided by showing that the
axiomatizations (given in terms of homogeneity) of their restriction to
finitary functionals still hold over completely distributive lattices. We also
present canonical normal form representations of polynomial functionals on
completely distributive lattices, which appear as the natural extensions to
their finitary counterparts, and as a by-product we obtain an axiomatization of
complete distributivity in the case of bounded lattices
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
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