Abstract
Symbolic objects (Diday (1987, 1992), Brito, Diday (1990), Brito (1991)) allow to model data on the form of descriptions by intension, thus generalizing the usual tabular model of data analysis. This modelisation allows to take into account variability within a set. The formalism of symbolic objects has some notions in common with VL1, proposed by Michalski (1980); however VL1 is mainly based on prepositional and predicate calculus, while the formalism of symbolic objects allows for an explicit interpretation within its framework, by considering the duality intension-extension. That is, given a set of observations, we consider the couple (symbolic object — extension in the given set). This results from the wish to keep a statistics point of view. The need to represent non-deterministic knowledge, that is, data for which the values for the different variables are assigned a weight, led to considering an extension of assertion objects to probabilist objects (Diday 1992). In this case, data are represented by probability distributions on the variables observation sets. The notions previously defined for assertion objects are the generalized to this new kind of symbolic objects. Other extensions can be found in Diday (1992). © Springer-Verlag Berlin Heidelberg 1993.

Abstract
This paper deals with hierarchical or pyramidal conceptual clustering methods, where each formed cluster corresponds to a concept, i.e., a pair (extent, intent).We consider data presenting real or interval-valued numerical values, ordered values and/or probability/frequency distributions on a set of categories. Concepts are obtained by a Galois connection with generalisation by intervals, which allows dealing with different variable types on a common framework. In the case of distribution data, the obtained concepts are more homogeneous and more easily interpretable than those obtained by using the maximum and minimum operators previously proposed. A measure of generality of a concept is defined similarly for all these variable types. An example illustrates the proposed method.

Abstract

Abstract
This work comes within the field of data analysis using Galois lattices. We consider ordinal, numerical single or interval data as well as data that consist on frequency/probability distributions on a finite set of categories. Data are represented and dealt with on a common framework, by defining a generalization operator that determines intents by intervals. In the case of distribution data, the obtained concepts are more homogeneous and more easily interpretable than those obtained by using the maximum and minimum operators previously proposed. The number of obtained concepts being often rather large, and to limit the influence of atypical elements, we propose to identify stable concepts using interval distances in a cross validation-like approach.

Abstract
In this paper we propose a divisive top-down clustering method designed for interval and histogram-valued data. The method provides a hierarchy on a set of objects together with a monothetic characterization of each formed cluster. At each step, a cluster is split so as to minimize intra-cluster dispersion, which is measured using a distance suitable for the considered variable types. The criterion is minimized across the bipartitions induced by a set of binary questions. Since interval-valued variables may be considered a special case of histogram-valued variables, the method applies to data described by either kind of variables, or by variables of both types. An example illustrates the proposed approach.

Abstract