Correspondence Class

A Correspondence usually examines the relationships existing between two categorical variables whose observations have been summarized in a contingency table, i.e. a table showing the joint frequency distribution of the variables, in which the rows are assigned to the categories of a first variable, and the columns to those of another one. However, the current implementation of Correspondence does not control that the analyzed data strictly represent counts: what is checked is only that the marginal data sums, i.e. the sums computed separately on each row and on each column, are all positive values.

Both rows and columns are represented as weighted points in a multidimensional space, forming what are referred to as the clouds of row or column profiles (See the Cloud documentation for a formal definition of the cloud concept). The aim of a correspondence analysis is to project such clouds in a space having a lower dimensionality, so that distances in such space relate to dissimilarities among the categories of the two variables. Check the PrincipalProjections documentation for a thorough explanation of the statistical methods underlying such projections.

Note
DIrections along which to project the points are added until the corresponding projected variance is greater than 1e-6.

Contingency tables

Let be a contingency table. The number of overall observations is



where and are column vectors having lengths and , respectively, so that the relative joint frequency distribution can be represented by the matrix



The relative marginal frequency distribution of the row variable is



and that of the column one is



Given vectors and , one can also define the following diagonal matrices:



and

Cloud of row profiles

Row profiles are points in , say . The points, if measured w.r.t. the basis , with



have coordinates



and weights



forming the cloud



Such cloud and its projections can be inspected via property RowProfiles.

Cloud of column profiles

Column profiles are points in , say . Such points are measured w.r.t. the basis , with



having coordinates



and weights



forming the cloud



Such cloud and its projections can be inspected via property ColumnProfiles.

Reference

Other Resources