Inverse correspondence analysis

In correspondence analysis (CA), rows and columns of a data matrix are depicted as points in low-dimensional space. The row and column profiles are approximated by minimizing the so-called weighted chi-squared distance between the original profiles and their approximations, see for example, [Theory and applications of correspondence analysis, Academic Press, New York, 1984]. In this paper, we will study the inverse CA problem, that is, the possibilities for retrieving one or more data matrices from a low-dimensional CA solution. We will show that there exists a nonempty closed and bounded polyhedron of such matrices. We also present two algorithms to find the vertices of the polyhedron: an exact algorithm that finds all vertices and a heuristic approach for larger sized problems that will find some of the vertices. A proof that the maximum of the Pearson chi-squared statistic is attained at one of the vertices is given. In addition, it is discussed how extra equality constraints on some elements of the data matrix can be imposed on the inverse CA problem. As a special case, we present a method for imposing integer restrictions on the data matrix as well. The approach to inverse CA followed here is similar to the one employed by De Leeuw and Groenen [J. Classification 14 (1997) 3] in their inverse multidimensional scaling problem. (C) 2003 Elsevier Inc. All rights reserved.