Abstract

Abstract
We recall a formalism based on the notion of symbolic object (Diday [15], Brito and Diday [8]), which allows to generalize the classical tabular model of Data Analysis. We study assertion objects, a particular class of symbolic objects which is endowed with a partial order and a quasi-order. Operations are then defined on symbolic objects. We study the property of completeness, already considered in Brito and Diday [8], which expresses the duality extension/intension. We formalize this notion in the framework of the theory of Galois connections and study the order structure of complete assertion objects. We introduce the notion of c-connection, as being a pair of mappings (f, g) between two partially ordered sets which should fulfil given conditions. A complete assertion object is then defined as a fixed point of the composed f o g; this mapping is called a ''completeness operator'' for it ''completes'' a given assertion object. The set of complete assertion objects forms a lattice and we state how suprema and infima are obtained. The lattice structure being too complex to allow a clustering study of a data set, we have proposed a pyramidal clustering approach [8]. The symbolic pyramidal clustering method builds a pyramid bottom-up, each cluster being described by a complete assertion object whose extension is the cluster itself. We thus obtain an inheritance structure on the data set. The inheritance structure then leads to the generation of rules.

Abstract
This paper is concerned with the problem of characterization of classification algorithms. The aim is to determine under what circumstances a particular classification algorithm is applicable. The method used involves generation of different kinds of models. These include regression and rule models, piecewise linear models (model trees) and instance based models. These are generated automatically on the basis of dataset characteristics and given test results. The lack of data is compensated for by various types of preprocessing. The models obtained are characterized by quantifying their predictive capability and the best models are identified. © Springer-Verlag Berlin Heidelberg 1995.

Abstract

Abstract
Sullivan's scaling function provides a complete description of the smooth conjugacy classes of cookie-cutters. However, for smooth conjugacy classes of Markov maps on a train track, such as expanding circle maps and train track mappings induced by pseudo-Anosov systems, the generalisation of the scaling function suffers from a deficiency. It is difficult to characterise the structure of the set of those scaling functions which correspond to smooth mappings. We introduce a new invariant for Markov maps called the solenoid function. We prove that for any prescribed topological structure, there is a one-to-one correspondence between smooth conjugacy classes of smooth Markov maps and pseudo-Hölder solenoid functions. This gives a characterisation of the moduli space for smooth conjugacy classes of smooth Markov maps. For smooth expanding maps of the circle with degree d this moduli space is the space of Hölder continuous functions on the space {0,…, d - 1} satisfying the matching condition.

Abstract
We classify the C1+a structures on embedded trees. This extends the results of Sullivan on embeddings of the binary tree to trees with arbitrary topology and to embeddings without bounded geometry and with contact points. We used these results in an earlier paper to describe the moduli spaces of smooth conjugacy classes of expanding maps and Markov maps on train tracks. In later papers we will use those results to do the same for pseudo-Anosov diffeomorphisms of surfaces. These results are also used in the classification of renormalisation limits of C1+a diffeomorphisms of the circle.