Empirical Bayes hierarchical models for regularizing maximum likelihood estimation in the matrix Gaussian Procrustes problem

Abstract

Procrustes analysis involves finding the optimal superposition of two or more "forms" via rotations, translations, and scalings. Procrustes problems arise in a wide range of scientific disciplines, especially when the geometrical shapes of objects are compared, contrasted, and analyzed. Classically, the optimal transformations are found by minimizing the sum of the squared distances between corresponding points in the forms. Despite its widespread use, the ordinary unweighted least-squares (LS) criterion can give erroneous solutions when the errors have heterogeneous variances (heteroscedasticity) or the errors are correlated, both common occurrences with real data. In contrast, maximum likelihood (ML) estimation can provide accurate and consistent statistical estimates in the presence of both heteroscedasticity and correlation. Here we provide a complete solution to the nonisotropic ML Procrustes problem assuming a matrix Gaussian distribution with factored covariances. Our analysis generalizes, simplifies, and extends results from previous discussions of the ML Procrustes problem. An iterative algorithm is presented for the simultaneous, numerical determination of the ML solutions.

Fig. 1.

ML and LS superposition of simulated protein structure data. (A) The “true” superposition, generated from a known mean structure with known covariance matrices, before arbitrary translations and rotations have been applied. In generating the simulated data, the set of alpha carbon atoms from model 1 of Protein Data Bank entry 2SDF (www.pdb.org) was used as the mean form (67 atoms/landmarks, squared radius of gyration = 152 Ų). The nondiagonal 67 × 67 landmark covariance matrix was based on values calculated from the superposition given in 2SDF, with variances ranging from 0.01 to 80 Ų and correlations ranging from 0 to 0.99 (see Data Sets 1 and 2). The known dimensional covariance matrix had eigenvalues of 0.16667, 0.33333, and 0.5 corresponding to the x, y, and z axes of 2SDF model 1, respectively. (B) An ordinary LS superposition of the simulated data. (C) A ML superposition of the simulated data, assuming a diagonal landmark covariance matrix Σ (i.e., no correlations), Ξ = 1, and inverse gamma distributed variances.