Hans De Meyer

A Hitchhikers Guide to Poset Ranking

When ranking objects (like chemicals, geographical sites, river sections, etc.) by multicriteria analysis, it is in most cases controversial and difficult to find a common scale among the criteria of concern. Therefore, ideally, one should not resort to such artificial additional constraints. The theory of partially ordered sets (or posets for short) provides a solid formal framework for the ranking of objects without assigning a common scale and/or weights to the criteria, and therefore constitutes a valuable alternative to traditional approaches. In this paper, we aim to give a comprehensive literature review on the topic. First we formalize the problem of ranking objects according to some predefined criteria. In this theoretical framework, we focus on several algorithms and illustrate them on a toy example. To conclude, a more realistic real-world application shows the power of some of the algorithms considered in this paper.

OPPOSITE DIAGONAL SECTIONS OF QUASI-COPULAS AND COPULAS

In this paper, we study opposite diagonal sections of quasi-copulas and copulas. The best-possible upper bound for the set of copulas with a given opposite diagonal section being known, we focus on the best-possible lower bound, which in general is a quasi-copula. Moreover, it exhibits an interesting type of bivariate symmetry called opposite symmetry.

Constructing copulas with given diagonal and opposite diagonal sections

A method for constructing copulas with given diagonal and opposite diagonal sections is presented. It makes use of a recently developed method for constructing cross-copulas with given horizontal and vertical sections. Conditions guaranteeing the existence of a cross-copula with the given diagonal and opposite diagonal sections are derived. It is shown how the new method facilitates the construction of families of copulas that simultaneously model tail dependences of upper-upper, upper-lower, lower-lower and lower-upper type.

A fuzzy approach to stochastic dominance of random variables

The discrete dice model essentially amounts to comparing discrete uniform probability distributions and generates reciprocal relations that exhibit a particular type of transitivity called dice-transitivity. In this contribution, this comparison method is extended and applied to general probability distributions and the generated reciprocal relations are shown to generalize the concept of stochastic dominance. For a variety of parametrized probability distributions, we analyse the transitivity properties of these reciprocal relations within the framework of cycle-transitivity. The relationship between normal probability distributions and the different types of stochastic transitivity is emphasized.

New Operations for Informative Combination of Two Partial Order Relations with Illustrations on Pollution Data

We discuss various ways in which to construct and process partial order relations or partially ordered sets (posets) in the context of ranking objects on the basis of multiple criteria. Oftentimes, it is undesirable or even impossible to devise a weighting scheme to compute a final score on the basis of the criteria. An alternative is then to restrict oneself to the information contained in the partial ordering of all objects implied by the criteria. We will consider some ways in which one can exploit partial order relations to determine a ranking of a collection of objects. More exactly, we will examine how to combine information coming from two sources, both for the case in which the sources are considered to be equally important, as well as for the case in which one source of information should take priority. We illustrate the concepts on pollution data coming from 59 regions in Baden-Württemberg.

A top-down algorithm for generating the Hasse tree of a fuzzy preorder closure

This paper describes a new top-down algorithm for the stepwise generation of the different levels or Hasse diagrams of the Hasse tree associated to the fuzzy preorder closure (min-transitive closure) of a given reflexive binary fuzzy relation. The algorithm is based upon a recently established weight-driven method for computing the min-transitive closure of a reflexive binary fuzzy relation. The way in which this method gradually establishes the fuzzy preorder closure implies that for the generation of a specific level of the Hasse tree, the newly proposed algorithm does not require the complete computation of this closure.

Toward Graded and Nongraded Variants of Stochastic Dominance

We establish a pairwise comparison method for random variables. This comparison results in a probabilistic relation on a given set of random variables. The transitivity of this probabilistic relation is investigated in the case of independent random variables, as well as when these random variables are pairwisely coupled by means of a copula, more in particular the minimum operator or the Łukasiewicz t-norm. A deeper understanding of this transitivity, which can be captured only in the framework of cycle-transitivity, allows to identify appropriate strict or weak thresholds, depending upon the copula involved, turning the probabilistic relation into a strict order relation. Using 1/2 as a fixed weak threshold does not guarantee an acyclic relation, but is always one-way compatible with the classical concept of stochastic dominance. The proposed method can therefore also be seen as a way of generating graded as well as nongraded variants of that popular concept.

Transitive comparison of random variables

A general framework for describing the transitivity of probabilistic relations is presented. A procedure is established for expressing the pairwise comparison of independent random variables in terms of a probabilistic relation. The transitivity of this relation is studied both in general and within popular parameterized families of distributions. It is shown how a copula can be entered into this comparison procedure and how this affects the transitivity of the probabilistic relation.

Characterization of copulas with given diagonal and opposite diagonal sections

In recent years special attention has been devoted to the problem of finding a copula, the diagonal section and opposite diagonal section of which are known. For given diagonal function and opposite diagonal functions, we provide necessary and sufficient conditions for the existence of a copula to have these functions as diagonal and opposite diagonal sections. We make use of techniques related to interpolation between the diagonals, the construction of checkerboard copulas, and linear programming. This result allows us to solve two open problems: to characterize the class of copulas where the knowledge of diagonal and opposite diagonal sections determines the copula in a unique way, and to formulate necessary and sufficient conditions for each pair of such functions to be the diagonal and opposite diagonal sections of a unique copula.

An algorithm for generating consistent and transitive approximations of reciprocal preference relations

We establish an iterative algorithm to generate for any given reciprocal relation and any given type of transitivity fitting into the framework of cycle-transitivity, a unique reciprocal relation that approximates the given reciprocal relation and possesses the given transitivity property. In the context of decision making, the algorithm can be used to generate a consistent approximation of a non-consistent reciprocal preference relation.