Generalized procrustes analysis

Abstract

SupposeP ⁽ⁱ⁾ _i (i = 1, 2, ...,m, j = 1, 2, ...,n) give the locations ofmn points inp-dimensional space. Collectively these may be regarded asm configurations, or scalings, each ofn points inp-dimensions. The problem is investigated of translating, rotating, reflecting and scaling them configurations to minimize the goodness-of-fit criterion Σ ^m _i=1 Σ ⁿ _i=1 Δ²(P ⁽ⁱ⁾ _j G _i ), whereG _i is the centroid of them pointsP ⁽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 casem = 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.