Inverse Correspondence Analysis

In correspondence analysis, 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 chisquared distance between the original profiles and their approximations, see or example, Greenacre (1984). In this paper, we will study the inverse correspondence analysis problem, that is, the possibilities of retrieving one or more data matrices from a low dimensional correspondence analysis solution. We will show that there exists a nonempty closed and bounded polyhedron of such matrices. We also present an algorithm to find the vertices of the polyhedron. 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 correspondence analysis problem. As a special case, we present a method for imposing integer restrictions on the data matrix as well. The approach to inverse correspondence analysis followed here is similar to the one employed by De Leeuw and Groenen (1997) in their inverse multidimensional scaling problem.