Confirmed Talks

Kenneth P. BOGART

Talk: Multidimensional Analogs of Interval Orders

Abstract:   Yet to appear.

 

Edwin DIDAY

Talk: Tesselation of Concepts by Spatial Pyramidal Clustering in Symbolic Data Analysis

Abstract:   First we recall the general framework of Symbolic Data Analysis (see Bock, Diday and al (2000)) and its relation to concept analysis. Then, we present several kinds of compatibility between a tesselation and a dissimilarity which allow for a spatial representation of a clustering pyramid where each level is associated to a concept. In Symbolic Data Analysis, the entries of the input data table (Units x Descriptors) are sets of categories or numbers, intervals or probability distributions. This kind of descriptors (so-called symbolic variables) is better suited than the standard numerical or categorical ones, for characterizing a "concept" (as a town, an insurance company, a species of animals) by its intent. In the SODAS software, the description of the concepts can be "native" (i.e. given by expert knowledge) or induced from their extent. The extent of a concept is a class of standard units called "individuals" (as the inhabitant of a town, a set of insurance companies, a sample of animals of a given species, clusters obtained from a clustering). In addition to their standard description, relationships as logical rules and taxonomies between the standard descriptors of the individuals, can be added as input. Starting from a symbolic data table which describes concepts of a first order, new concepts can be obtained from a Galois lattice built on these data in order to obtain second order concepts (representing for instance, classes of towns or classes of species). Three kinds of compatibility between a tessellation and a dissimilarity are considered: convex, connected and "perfect". Then, we extend standard hierarchies and pyramids with linear support to hierarchies and pyramids with multidimensional support based on a tesselation where a) m edges defining m equal angles, meet at each node and b) the smallest cycles contain k edges of equal lengh. Instead of representing the extent of the concepts on a straight line as in standard hierarchical or pyramidal conceptual clustering, it is then possible to represent them with their symbolic interpretation as a surface or as a volume by using a tessellation of the numerous pixels of a screen.

 

Olivier HUDRY

Talk: Links Between Some Tournament Solutions  

Abstract:         Consider a competition organized in such a way that each competitor plays once against each one of the other competitors. If we assume that ties cannot occur, then the result of this pairwise comparison is a tournament T: the vertices of T are the competitors and there is an arc from x to y if x defeats y.       Then, a "tournament solution" S can be defined as a way to choose, from the set X of vertices of any tournament T = (X, U), a subset S(X) of X whose elements can be considered as better than the others: the "winners" of T according to S.       It may happen that, in the competition, there is a player (a vertex) who defeated all the other ones; such a player is called a "Condorcet winner" and can (must ?) reasonably be considered as the unique winner for any tournament solution S. But it may happen also that there is no Condorcet winner in T. Then answering the question "who are the best players ?" is no longer an easy task.       In this talk, we discuss the properties of and the links between several tournament solutions, namely: - the Copeland solution, based on the out-degrees of the vertices; - a solution based on the maximum eigenvalue of the matrix associated with the tournament; - a Markovian solution; - the Slater solution, based on linear orders at minimum distance; - the Banks solution, based on maximal transitive subtournaments; - the tournament equilibrium set (TEQ); - solutions based on covering relations.  

Robert E. JAMISON

Talk: Trees Invariant Under a Half-Turn

Abstract:   The work to be presented here grew out of some theoretical models proposed by David Halitsky in an attempt to describe structural redundancies in DNA and proteins. Halitsky is a linguist by training but has spent most of his career in computing. Underlying his ideas are two main principles. First, natural language, DNA, amino acid sequences in proteins, and even computer databases are all methods for storing and communicating information, and he posits that there should be certain basic structural similarities among them. Second, he believes that seemingly complex low dimensional phenomena are sometimes the result of a projection of very regular higher dimensional structures. Symmetry is, of course, a way of measuring structure and redundancy, and it is Halitsky's recognition of previously unnoted symmetries in language-theoretic structures which differentiates his bio-molecular application of these constructs from their well-known application to bio-informatics by David Searls. ?This talk will focus on some mathematics that was inspired by Halitsky's ideas and questions. In particular, it will be concerned with trees represented in a discrete torus, namely, $Z_n \times Z_n$. Here $Z_n$ carries the structure of a ring as well as the natural {\bf cyclic} order. In this context, it makes sense to call a mapping of the form ${\bf x} \longrightarrow {\bf c} - {\bf x}$ a {\it half-turn} in analogy with the Euclidean situation. Cutting the torus along two fundamental cycles allows it to be flattened out. The same structure on the torus can give rise to many different trees. One of Halitsky's main problems was to characterize those trees which can arise from a set of points on the torus which is left invariant by some half-turn. A solution will be presented in this talk. Halitsky was originally motivated to ask this question when he noted empirically that three families of trees that are important in linguistics have a matching property which turns out to be equivalent to invariance under a half-turn.

 

Mel JANOWITZ

Talk: A Natural Classification Of Isotone Real Mappings  

Abstract:         Let ‘S’ be a semigroup of mappings defined on an interval ‘I’ of the reals. Assume ‘S’ contains all order automorphisms of ‘I’. The members of ‘S’ can then be classified by how they act on arbitrary ‘k’-tuples from ‘I’. The resulting classifications will be exhibited in concrete cases, and the results applied to define corresponding classifications of cluster functions. New insights into the subject will allow a clearer exposition that will unify and clarify earlier results. These results involve isotone mappings on the reals, 0- preserving isotone mappings on the nonnegative reals, residuated mappings on the nonnegative reals, and residuated mappings on [0,1] having 1 in their image.  

Marc PIRLOT

Talk: Preferences for Multi-Attributed Alternatives: Traces, dominance, and numerical representations (by Denis BOUYSSOU and Marc PIRLOT)

Abstract:  Two distinct traditions underlie most of the work done in the area of decision analysis with multiple attributes. The conjoint measurement tradition has deep roots both in Mathematical Psychology and Decision Theory. Starting with a binary relation defined on a product set, it aims at finding conditions under which it is possible to build a convenient numerical representation of the relation. The model that has been most studied in this framework is the additive utility. Besides their theoretical interest, many conjoint measurement results are constructive in nature and, therefore, give hints on how to devise assessment procedures of utility functions and, thus, preferences. A more pragmatic tradition starts with alternatives evaluated along several attributes. Along each attribute, alternatives are supposed to be compared using a well-behaved preference relation. The central problem is then to build a preference relation between alternatives taking all attributes into account, i.e. a global preference relation, based on the preference relations on each attribute and ''inter-attribute'' information such as weights or trade-offs. The notion of dominance plays a crucial role here. An alternative is said to dominate another if the former is judged ``at least as good as'' the latter on all attributes. It is usually considered that global preference should be compatible with dominance, i.e. that the dominance relation is included in the global preference relation. When a global preference relation is compatible with dominance, it makes sense to limit the search for ``good'' alternatives in the set of efficient alternatives, i.e. alternatives that are undominated. These two lines of thought seem to coexist since the beginning of decision analysis with multiple attributes, in the late '60s. Both have generated important theoretical and practical achievements but the two traditions have remained largely unrelated. Their setting differ significantly. The conjoint measurement tradition starts with a well- behaved preference relation taking all attributes into account. The pragmatic one starts with a well-behaved preference relation defined on each attribute and derives a global preference relation using the notion of dominance and inter-attribute information. The principles used in order to build the global preference relation do not always guarantee that this relation will be transitive or complete, e.g. if a qualified weighted majority of attributes is used. ?In this work, we attempt to establish connections between these two traditions. In order to do so, we adopt a classical conjoint measurement setting, while not requiring transitivity or completeness. We provide a simple axiomatic characterization of preference relations compatible with dominance and show that all such relations admit a nontrivial numerical representation. This extends the traditional scope of conjoint measurement to include binary relations that are not ``well- behaved''. Furthermore this shows that many techniques developed in the pragmatic tradition can usefully be analyzed in a conjoint measurement framework. The key tool for the analysis of such preference relations is the consideration of various kinds of traces on coordinates induced by the original relation.

Michel REGENWETTER

Talk: Aggegation of Probabilistic Ordinal Preferences  

Abstract:   Much of social choice theory is dedicated to the elicitation and
aggregation of ordinal preferences. Voters may experience uncertainty as
to which vote to cast, especially when ballots are complex and when the
number of candidates is large. Pollsters and election researchers may
experience uncertainty about voter preferences. Furthermore, as the
whole world witnessed in the 2000 U.S. presidential elections, election
officials may experience uncertainty as to which ballots actually were
cast. It is therefore very natural to investigate social choice theory
when ballots are viewed as probabilistic. The present talk will provide
an introduction and overview of descriptive probabilistic models of
social choice behavior, as well as their application to real world
election and survey data. A major emphasis will be placed on the new
insights that this "behavioral" approach may yield regarding policy
implications and regarding the wise choice of "good" election methods.

 

Donald SAARI

Talk: Symmetries and Other Data Structures  

Abstract:   For the last half century, a standard approach toward understanding data, voting procedures, decision methods, etc. has been to impose conditions on the procedures. But rather than leading to consensus, debates had arisen about which conditions are better than others, and even impossibilities assertions such as Arrow's Theorem. On the other hand, by emphasizing the underlying symmetry and other structures of data, Arrows' theorem is replaced with a possibility assertion and positive assertions are obtained. The ideas are supported by examples from voting, social choice, engineering, etc.  

Rudolf WILLE

Talk: Formal Concept Analysis as Basis for Ordinal Data Analysis  

Abstract:   Formal Concept Analysis has been invented more than twenty years ago. Just from the beginning its development profited largerly from its applications in data analysis. Since the nature of Formal Concept Analysis is ordinal, it was mainly applied to ordinal data which strongly forced the development of methods and procedures for Ordinal Data Analysis. The lecture will present an overview about essentials of that development.

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