59 research outputs found
On maxitive integration
A functional is said to be maxitive if it commutes with the (pointwise) supremum operation. Such functionals find application in particular in decision theory and related fields. In the present paper, maxitive functionals are characterized as integrals with respect to maxitive measures (also known as possibility measures or idempotent measures). These maxitive integrals are then compared with the usual additive and nonadditive integrals on the basis of some important properties, such as convexity, subadditivity, and the law of iterated expectations
Representation of maxitive measures: an overview
Idempotent integration is an analogue of Lebesgue integration where
-maxitive measures replace -additive measures. In addition to
reviewing and unifying several Radon--Nikodym like theorems proven in the
literature for the idempotent integral, we also prove new results of the same
kind.Comment: 40 page
On maxitive integration
The Shilkret integral is maxitive (i.e., the integral of a pointwise supremum of functions is the supremum of their integrals), but defined only for nonnegative functions. In the present paper, some properties of this integral (such as subadditivity and a law of iterated expectations) are studied, in comparison with the additive and Choquet integrals. Furthermore, the definition of a maxitive integral for all real functions is discussed. In particular, a convex, maxitive integral is introduced and some of its properties are derived
How regular can maxitive measures be?
We examine domain-valued maxitive measures defined on the Borel subsets of a
topological space. Several characterizations of regularity of maxitive measures
are proved, depending on the structure of the topological space. Since every
regular maxitive measure is completely maxitive, this yields sufficient
conditions for the existence of a cardinal density. We also show that every
outer-continuous maxitive measure can be decomposed as the supremum of a
regular maxitive measure and a maxitive measure that vanishes on compact
subsets under appropriate conditions.Comment: 24 page
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
Measure and integral with purely ordinal scales
We develop a purely ordinal model for aggregation functionals for lattice valued functions, comprising as special cases quantiles, the Ky Fan metric and the Sugeno integral. For modeling findings of psychological experiments like the reflection effect in decision behaviour under risk or uncertainty, we introduce reflection lattices. These are complete linear lattices endowed with an order reversing bijection like the reflection at on the real interval . Mathematically we investigate the lattice of non-void intervals in a complete linear lattice, then the class of monotone interval-valued functions and
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.
Basic generated universal fuzzy measures
AbstractThe concept of basic generated universal fuzzy measures is introduced. Special classes and properties of basic generated universal fuzzy measures are discussed, especially the additive, the symmetric and the maxitive case. Additive (symmetric) basic universal fuzzy measures are shown to correspond to the Yager quantifier-based approach to additive (symmetric) fuzzy measures. The corresponding fuzzy integral-based aggregation operators are introduced, including the generated OWA operators
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