Catálogo de publicaciones - libros

The set of Euclidean distance matrices has a well-known representation as a convex cone. The problems of representing the group averages of distance matrices are discussed, but not fully resolved, in the context of SMACOF, Generalized Orthogonal Procrustes Analysis and Individual Differences Scaling. The polar (or dual) cone representation, corresponding to inner-products around a centroid, is also discussed. Some new characterisations of distance cones in terms of circumhyperspheres are presented.

The weakly indexed paired-hierarchies (shortly, p-hierarchies) provide a clustering model that allows overlapping clusters and extends the hierarchical model. There exists a bijection between weakly indexed p-hierarchies and the so-called paired-ultrametrics (shortly, p-ultrametrics), this correspondence being a restriction of the bijection between weakly indexed pyramids and Robinsonian dissimilarities. This paper proposes a generalization of the well-known HAC clustering method to compute a weakly indexed p-hierarchy from a given dissimilarity . Moreover, the p-ultrametric associated to such a weakly indexed p-hierarchy is proved to be lower-maximal for and larger than the sub-dominant ultrametric of .

This note is devoted to three-way dissimilarities defined on unordered triples. Some of them are derived from two-way dissimilarities via an -transformation (1 ≤ ≤ ∞). For < ∞, a six-point condition of Menger type is established. Based on the definitions of Joly-Le Calvé and Heiser-Bennani Dosse, the concepts of three-way distances are also discussed. A particular attention is paid to three-way ultrametrics and three-way tree distances.

We place ourselves in a setting where singletons are not all required to be clusters, and we show that the resulting cluster structures and their corresponding closure under finite nonempty intersections still have the same minimal members. Moreover, we show that indexed cluster structures and weakly indexed closed cluster structures correspond in a one-to-one way.

DNA microarray technology allows to monitor simultaneously the expression levels of thousands of genes during important biological processes and across collections of related experiments. Clustering and classification techniques have proved to be helpful to understand gene function, gene regulation, and cellular processes. However the conventional proximity measures between genes expression data, used for clustering or classification purpose, do not fit gene expression specifications as they are based on the closeness of the expression magnitudes regardless of the overall gene expression profile (shape). We propose in this paper an adaptive dissimilarity index which would cover both values and behavior proximity. The effectiveness of the adaptive dissimilarity index is illustrated through a classification process for identification of genes cell cycle phases.

Edwin Diday, some two decades ago, was among the first few individuals to recognize the importance of the (anti-)Robinson form for representing a proximity matrix, and was the leader in suggesting how such matrices might be depicted graphically (as pyramids). We characterize the notions of an anti-Robinson (AR) and strongly anti-Robinson (SAR) matrix, and provide open-source M-files within a MATLAB environment to effect additive decompositions of a given proximity matrix into sums of AR (or SAR) matrices. We briefly introduce how the AR (or SAR) rank of a matrix might be specified.

The aim of this paper is twofold. First it is shown that taking densities between objects into account to define proximities between them is intuitively a right way to process. Secondly, some new distances based on density estimates are defined and some properties are presented. Many algorithms in clustering or discriminant analysis require the choice of a dissimilarity: two applications are presented, one in clustering and the other in discriminant analysis, and illustrate the benefits of using these new distances.

A square similarity matrix is called a Robinson matrix if the highest entries within each row and column are on the main diagonal and if, when moving away from this diagonal, the entries never increase. This paper formulates Robinson cubes as three-way generalizations of Robinson matrices. The first definition involves only those entries that are in a row, column or tube with an entry of the main diagonal. A stronger definition, called a regular Robinson cube, involves all entries. Several examples of the definitions are presented.

Strata generalisation by symbolic objects is presented when there is a class variable to be explained simultaneously in all strata. This is attained by a generalised recursive tree-building algorithm for populations partitioned into strata and described by symbolic data, that is, more complex data structures than classical data. Symbolic objects describe decisional nodes and strata. This paper presents some measures to interpret strata and nodes. The method is integrated into the SODAS Software (Symbolic Official Data Analysis System), partially supported by ESPRIT-20821 SODAS and IST-25161 ASSO.

In previous papers, we propose a generalized principal component analysis (GPCA) aimed to display salient features of a multidimensional data set, in particular the existence of clusters. In the light of an example, this article evidences how GPCA and clustering methods are complementary. The projections provided by GPCA and the sequence of eigenvalues give useful indications on the number and the type of clusters to be expected; submitting GPCA principal components to a clustering algorithm instead of the raw data can improve the classification. The use of a convenient robustification of GPCA is also evoked.