Symbolic objects: Order structure and pyramidal clustering

We recall a formalism based on the notion of symbolic object, 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, 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 fℝoslash;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. The symbolic pyramidal clustering method builds a pyramid bottom-up, each closter 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 than leads to the generation of rules.