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Canonical Analysis of two Convex Polyhedral Cones and Applications

Published online by Cambridge University Press:  01 January 2025

Michel Tenenhaus
Michel Tenenhaus*
Affiliation:
Centre HEC-ISA
*
Requests for reprints should be sent to Michel Tenenhaus, Département S.LA.D., Centre HEC-ISA, 78350 Jouy-en-Josas, FRANCE.
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Abstract

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 properties of the optimal solution to this problem, and also discusses in detail the convergence of an alternating least squares algorithm. The set of scalings of an ordinal variable is a convex polyhedral 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.

Keywords

convex polyhedral conealternating least squares algorithmoptimal scalingmonotone analysis of varianceoptimal multiple regression
Type
Original Paper
Information
Psychometrika , Volume 53 , Issue 4 , December 1988 , pp. 503 - 524
Copyright
Copyright © 1988 The Psychometric Society

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Footnotes

This paper is dedicated to the memory of Patrice Bertier who suggested its subject to me in 1974. This paper also owes much to the very valuable suggestions of J. P. Benzécri, P. Cazes and J. de Leeuw, and I am very grateful to them. I also want to thank J. Yellin for his help in improving the English of the manuscript.

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