CANONICAL ANALYSIS OF TWO CONVEX POLYHEDRAL CONES AND APPLICATIONS

Canonical analysis of two convex polyhedral cones consists in looking for two vectors (one in each cone) whose square cosine is a maximum. This paper presents new results about the proper-ties of the optimal solution to this problem, and also discusses in detail the convergence of an alternating least squares algorithm. The set of sealings of an ordinal variable is a convex poly-hedral cone, which thus plays an important role in optimal scaling methods for the analysis of ordinal data. Monotone analysis of variance, and correspondence analysis subject to an ordinal constraint on one of the factors are both canonical analyses of a convex polyhedral cone and a subspace. Optimal multiple regression of a dependent ordinal variable on a set of independent ordinal variables is a canonical analysis of two convex polyhedral cones as long as the signs of the regression coefficients are given. We discuss these three situations and illustrate them by examples. Key words: convex polyhedral cone, alternating least squares algorithm, optimal scaling, mono-tone analysis of variance, optimal multiple regression