On the Correspondence Between Procrustes Analysis and Bidimensional Regression

Abstract

Procrustes analysis is defined as the problem of fitting a matrix of data to a target matrix as closely as possible (Gower and Dijksterhuis, 2004). The problem can take many forms, but the most common form, orthogonal Procrustes analysis, has as allowable transformations, a translation, a scaling, an orthogonal rotation, and a reflection. Procrustes analysis and other rotation methods have a long history in quantitative psychology, as well as in other fields, such as biology (Siegel and Benson, 1982) and shape analysis (Kendall, 1984). In the field of quantitative geography, the use of bidimensional regression (Tobler, 1965) has recently become popular. Tobler (1994) defines bidimensional regression as "an extension of ordinary regression to the case in which both the independent and dependent variables are two-dimensional." In this paper, it is established that orthogonal Procrustes analysis (without reflection) and Euclidean bidimensional regression are the same. As such, both areas of development can borrow from the other, allowing for a richer landscape of possibilities.