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

Area of Science:

• Geometric morphometrics
• Statistical shape analysis
• Multivariate statistics

Background:

• Procrustes analysis aligns shapes by minimizing distances between corresponding points.
• Traditional least-squares methods fail with heterogeneous variances or correlated errors.
• Real-world data often exhibit heteroscedasticity and correlation, compromising LS accuracy.

Purpose of the Study:

• To provide a comprehensive solution for the nonisotropic maximum likelihood (ML) Procrustes problem.
• To generalize and simplify existing ML Procrustes methodologies.
• To address limitations of ordinary least-squares (LS) in Procrustes analysis.

Main Methods:

• Utilizing a matrix Gaussian distribution with factored covariances.
• Developing a complete solution for the nonisotropic ML Procrustes problem.
• Implementing an iterative algorithm for simultaneous numerical determination of ML solutions.

Main Results:

• The proposed ML approach provides accurate and consistent estimates under heteroscedasticity and correlation.
• The analysis extends and simplifies previous work on ML Procrustes problems.
• A novel iterative algorithm facilitates the computation of ML solutions.

Conclusions:

• Maximum likelihood estimation is superior to least-squares for Procrustes analysis with complex error structures.
• The presented method offers a robust framework for shape comparison in the presence of data heterogeneity.
• The iterative algorithm enables practical application of advanced ML Procrustes techniques.