MultipleCorrespondence Class

A multiple correspondence examines the relationships existing among the variables observed in a categorical data set. Both individuals and categories are represented as weighted points in a multidimensional space, having coordinates w.r.t. specific bases, forming what are referred to as clouds (For a definition, see the Cloud documentation). The aim of a multiple correspondence analysis is to project those clouds in a space having a lower dimensionality, in such a way that distances in space relate to dissimilarities among categories or among individuals.

Let be a categorical data set, where individuals are assigned to rows and variables to columns. Let be, for , the domain of the -th categorical variable, i.e. the set of its observed categories in . Given the definition



one can map the overall observed categories to indexes , so that a disjunctive form of can be represented as a matrix, say , such that, for and , if and only if the -th category has been observed at the -th individual, otherwise. The number of individuals at which the -th category has been observed is thus given as .

Notice that matrix is a contingency table, so that the cloud of individuals and that of categories can be interpreted, respectively, as the clouds of row and column profiles taken into account while analyzing a correspondence. That is, a multiple correspondence analysis of consists precisely in the correspondence analysis of . As a consequence, in what follows is exploited the same notation used in the documentation of class Correspondence.

Cloud of individuals

Individuals are points in that correspond to the row profiles based on matrix . Their cloud can thus be obtained by specializing the cloud , as defined in the remarks about class Correspondence.

More thoroughly, the marginal relative frequency of the -th category is equal to , hence



so that basis is representable through the matrix



Furthermore, since variables are observed at each individual, one also has, for ,



so that



and



where .

Cloud of categories

Categories are the column profiles in represented by the cloud , as discussed in the documentation of Correspondence.

In the current context, such cloud can be expressed as follows.

Basis has a representation matrix evaluating to



while the point coordinates and weights are given by, respectively,



and, for ,

Principal projections

Information about the principal projections of categories and individuals are exposed through properties Categories and Individuals, respectively. These properties return objects of class PrincipalProjections. Check its documentation for a thorough explanation of the underlying statistical methods, and for a discussion about how to exploit the principal information of the clouds, or to locate supplementary points.

Reference

Other Resources