Non-symmetrical factorial discriminant analysis for symbolic objects (Q2711711)

A generalization of the factorial discriminant analysis (FDA) to complex data structures named Symbolic Objects (SOs) is proposed. SOs are complex structured data described on the basis of their own properties and defined, similarly to classical statistical units, on the basis of a set of monovalued variables, but also by set-valued variables and by the definition of logical relations. A SO can also be referred to as an a priori concept and then is called intention SO. Discriminant analysis refers to the statistical methods aimed at ``distinguishing'' \(K\) groups belonging to the same population on the basis of \(p\) explicative variables. Many authors distinguish between two aspects of discriminant analysis on the basis of the goals: the first deals with the selection of a subset of predictors in order to represent the a priori groups as distinct as possible from each other; the second looks for the definition of a decision rule in order to classify a new statistical unit, belonging to the same population, with respect to one of the \(K\) groups.NEWLINENEWLINENEWLINEThe present paper mainly deals with the first aspect. This leads to tackle the problem from a geometrical point of view. The geometrical approach looks for a set of new variables (Canonical Variables), obtained by means of linear transformations of the \(p\) predictors, such that the \(K\) groups are well separated. These new variables are called discriminant factors. As a final result, the FDA furnishes the images of the statistical units on the discriminant factorial axes. The paper proposes a three-step discrimination procedure. Symbolic data are coded in suitable numerical matrices, coded variables are transformed into canonical variables, and symbolic objects are visualized building maximum covering area rectangles, with respect to the canonical variables. Referring to the graphical representation, geometrical rules are proposed in order to assign new objects to a priori classes on the basis of proximity measures.