Procrustes Analysis and Shape Alignment

Named for the mythological figure who stretched or trimmed travelers to fit his bed exactly, Procrustes analysis solves a very concrete problem: given two configurations of points (say, factor loadings estimated on two different data samples, or two versions of the same shape measured differently), find the rotation, reflection, and optionally rescaling that best superimposes one onto the other, minimizing the total squared distance between corresponding points. Because factor loadings are only identified up to rotation, two runs of the same factor model on different subsamples can produce loadings that look completely different even though they represent the same underlying factors, just rotated arbitrarily — Procrustes alignment removes that arbitrary rotation so the two solutions can be compared point for point.

The orthogonal Procrustes problem — allowing only rotation and reflection, no stretching — has a closed-form solution via the singular value decomposition of the cross-product matrix between the two configurations, making it fast to compute even for many variables. Adding an allowed scaling factor extends this to comparing shapes of genuinely different sizes.

If a five-factor model is estimated separately on 2010–2015 and 2016–2020 data, the two sets of loadings might look unrelated at first glance simply because each run picked a different arbitrary rotation; a Procrustes rotation of one set onto the other typically reveals that the underlying factors — value, momentum, quality, and so on — actually line up closely across the two periods.