Representation of Symbolic Objects according to the description structure Antonio Irpino 1 , N. Carlo Lauro 1 , and Rosanna Verde 2 1 Dipartimento di Matematica e Statistica, University of Naples ”Federico II”, Monte Sant’Angelo, Via Cinthia, 80126 Naples, Italy 2 Dipartimento di strategie aziendali e metodologie quantitative, Second University of Naples, P.zza Umberto I, 81043 Capua (CE), Italy Abstract. As known, in Symbolic Data Analysis (SDA) a relevant role is played by the structure of symbolic object description, which depends on the types of the symbolic variables (i.e., real, interval, multi nominal, modal variables) and on the presence of the hierarchies and logical rules defined on their domains. In the framework of the symbolic - numerical - symbolic analysis of symbolic data some coherent choices of transformation of symbolic objects descriptors are required in order to compare, represent, visualize symbolic objects. In this work, different kinds of transformations are proposed according to the nature of the symbolic de- scriptors. In particular, for analyzing symbolic objects in a reduced sub-spaces, a Symbolic Factorial Analysis can be performed. A suitable coding phase must pre- cede the core of the procedure. The multi-valued descriptors which characterize the symbolic object to be analyzed, can be homogenized by means of suitable functions (B-spline, monotone, etc). The numerical transformation phase in the Symbolic Factorial Analysis allows to discover linear and non-linear associations between dif- ferent symbolic objects characteristics (symbolic descriptors). The transformation of the symbolic descriptors has, as immediate consequence, the transformation of the geometrical space of representation. In particular in this context, we analyze some topological transformations of the parallelotopos which represents symbolic objects described by interval variables in to more general connected shape (where the convex hulls are a particular case). These different types of representation are generated by non linear transformation function applied on the interval variables. The great advantage deriving by this kind of numerical transformation of symbolic data is to represent geometrically the structure of non linear relationships between symbolic descriptors. Some principles for the right choice of the transformation functions of the descriptors are necessary. The original space must be retrieved from the transformed one without lose information (the transformation of an in- terval variable have to represent all the values belonging to each interval and only them). Moreover, at each multi-nominal descriptor, usually codified according to presence/absence of each category, can be transformed in a modal, real one. Finally, some alternative visualizing tools are proposed according to the different kind of transformation of the symbolic objects descriptors. The transformation of symbolic descriptors has consequence in the visualization of the SO’s especially in those analysis whose aim is the reduction of the description space (for example, factorial methods). This means that the classical Hypercube