Statistics on Stiefel manifolds

When the orientation of an object lies in a space of non-zero curvature usual distributions of probability cannot be used to describe its directions. One of such spaces is the Stiefel manifold. We focus on a probability distribution defined on that space, the matrix Langevin distribution. Classical and Bayesian methods of estimation of the parameter of the distribution are discussed. As the dimension of the Stiefel manifold increases, the more complicated the estimation process becomes given the complexity of the functions to be evaluated. A method is given that efficiently parameterizes the elements of the singular value decomposition of the parameter of the matrix Langevin distribution in terms of generalized Euler angles. How to implement that parameterization in the context of Bayesian estimation is shown. The methodology is illustrated with a dataset on trace element concentrations in bullet tips from the Federal Bureau of Investigations