Generalized Procrustes Analysis

Abstract

Suppose P i ( i ) ( i = 1, 2, ..., m, j = 1, 2, ..., n ) give the locations of mn points in p -dimensional space. Collectively these may be regarded as m configurations, or scalings, each of n points in p -dimensions. The problem is investigated of translating, rotating, reflecting and scaling the m configurations to minimize the goodness-of-fit criterion Σ i=1 m Σ i=1 n Δ 2 ( P j ( i ) G i ), where G i is the centroid of the m points P i ( i ) ( i = 1, 2, ..., m ). The rotated positions of each configuration may be regarded as individual analyses with the centroid configuration representing a consensus, and this relationship with individual scaling analysis is discussed. A computational technique is given, the results of which can be summarized in analysis of variance form. The special case m = 2 corresponds to Classical Procrustes analysis but the choice of criterion that fits each configuration to the common centroid configuration avoids difficulties that arise when one set is fitted to the other, regarded as fixed.