SymScal: symbolic multidimensional scaling of interval dissimilarities

Multidimensional scaling aims at reconstructing dissimilaritiesbetween pairs of objects by distances in a low dimensional space.However, in some cases the dissimilarity itself is unknown, but therange of the dissimilarity is given. Such fuzzy data fall in thewider class of symbolic data (Bock and Diday, 2000).Denoeux and Masson (2000) have proposed to model an intervaldissimilarity by a range of the distance defined as the minimum andmaximum distance between two rectangles representing the objects. Inthis paper, we provide a new algorithm called SymScal that is basedon iterative majorization. The advantage is that each iteration isguaranteed to improve the solution until no improvement is possible.In a simulation study, we investigate the quality of thisalgorithm. We discuss the use of SymScal on empirical dissimilarityintervals of sounds.iterative majorization;multidimensional scaling;symbolic data analysis;distance smoothing

An Introduction to symbolic data analysis

Projet CLORECThe main aim of the symbolic approach in data analysis is to extend problems, methods and algorithms used on standard data to more complex data called "symbolic objects" in order to distinguish them from objects (described by numerical or categorical variables) treated by standard data nalysis methods. Symbolic objects extend classical objects of data analysis in two ways : first in case of individuals by giving the possibility of introducing in their definition, structured information, second, in case of sets or classes, by being intentionally defined. In both cases in order to represent uncertainty knowledge, it may be useful to use probabilities, possibilities (in case of vagueness and imprecision for instance) belief (in case of probabilities only known on parts and to express ignorance) that why, we introduce several kinds of symbolic objetcs : boolean, possibilist, probabilist and belief. We briefly present some of their qualities and properties, three theorems, show how probability, possibility and evidences theories may be extended on these objects. Some mixture decomposition problems on these objects are settled. We show that in some cases, fractals are well adapted to represent duality between symbolic objects. Sets of symbolic objects are represented by categories of different kinds (hierarchies, pyramids and lattices). Four kinds of data analysis problems including the symbolic extension are illustrated by several algorithms which induce knowledge from classical data or from a set of symbolic objects. Finally, important steps of a symbolic data analysis are described and illustrate by an example concerning road accidents

Quelques aspects de l'analyse des donnees symboliques

Projet CLORECSavoir representer nos connaissances par des expressions a la fois symboliques et numeriques, savoir manipuler et utiliser ces expressions dans le but d'aider a decider, de mieux analyser, synthetiser et organiser notre experience et nos observations, tel est l'objectif que s'assigne l'analyse des donnees symboliques. On presente d'abord les "objets symboliques" (sortes d'atomes de connaissances et ce qui les distingue des objets classiques de l'analyse des donnees usuelles. Ces objets qui constituent les individus de l'analyse des donnees symboliques, permettent de representer des individus complexes ou des classes d'invidus par des conjonctions de proprietes ou des descripteurs peuvent prendre des valeurs multiples et ponderees (selon differentes semantiques) et sont parfois relies entre eux par des relations d'ordre logique. On introduit des outils pour manipuler ces objets : union, intersection, generalisation, extension, etc. On s'interesse en particulier aux objets symboliques dits probabilistes et l'on enonce un resultat permettant d'etendre les probabilites a ce type d'objet, on construit ainsi un espace d'objets symboliques probabiliste dual ou les individus sont des objets definis en intension, dans cet espace on pose des problemes de decomposition en lois de lois. On s'interesse ensuite a la representation graphique de ces objets par differentes categories (hierarchies, pyramides, treillis, etc. d'objets symboliques). En utilisant la dualite, on peut construire des suites d'objets symboliques (devenant individus dans l'espace dual suivant), ces suites definissent des fractals dans certains cas que nous preciserons. On decrit differents types d'analyse des donnees symboliques ainsi que les principales etapes d'une telle analyse. On illustre enfin par une application concernant la construction et l'etude de scenarios d'accidents de la route