Correspondence Analysis can be presented in several equivalent ways. In a traditional setting, it is the analysis of a contingency table, i.e., a cross-table between two qualitative variables. Two point clouds are associated with such a table: one where one point is a row of the table, i.e., a profile for one modality of the variable attached to the rows, and a second where one point is a column, i.e., one profile attached to the second variable. Each point cloud is given a series of weights and equipped with a distance, in such a way that both point clouds can be projected simultaneously to reveal the dependency structure between the variables, or rows and columns, hence the name Correspondence Analysis, as a correspondence between both point clouds. In this chapter, we start with the observation that Correspondence Analysis is the analysis of the contingency table with two metrics on row and column space each related to weights chosen as inverse to the margins of the table. It is shown that the norm of the table is the \(\chi ^2\) norm and that Correspondence Analysis is the task of finding the table of low rank as close as possible to the table with \(\chi ^2\) norm. The equivalence between both approaches, and the correspondence between them, is presented.

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Correspondence Analysis

  • Alain Franc

摘要

Correspondence Analysis can be presented in several equivalent ways. In a traditional setting, it is the analysis of a contingency table, i.e., a cross-table between two qualitative variables. Two point clouds are associated with such a table: one where one point is a row of the table, i.e., a profile for one modality of the variable attached to the rows, and a second where one point is a column, i.e., one profile attached to the second variable. Each point cloud is given a series of weights and equipped with a distance, in such a way that both point clouds can be projected simultaneously to reveal the dependency structure between the variables, or rows and columns, hence the name Correspondence Analysis, as a correspondence between both point clouds. In this chapter, we start with the observation that Correspondence Analysis is the analysis of the contingency table with two metrics on row and column space each related to weights chosen as inverse to the margins of the table. It is shown that the norm of the table is the \(\chi ^2\) norm and that Correspondence Analysis is the task of finding the table of low rank as close as possible to the table with \(\chi ^2\) norm. The equivalence between both approaches, and the correspondence between them, is presented.