Model Criticism for Regression on the Grassmannian

Reliable estimation of model parameters from data requires a suitable model. In this work, we investigate and extend a recent model criticism approach to evaluate regression models on the Grassmann manifold. Model criticism allows us to check if a model fits and if the underlying model assumptions are justified by the observed data. This is a critical step to check model validity which is often neglected in practice. Using synthetic data we demonstrate that the proposed model criticism approach can indeed reject models that are improper for observed data and that the approach can guide the model selection process. We study two real applications: degeneration of corpus callosum shapes during aging and developmental shape changes in the rat calvarium. Our experimental results suggest that the three tested regression models on the Grassmannian (equivalent to linear, time-warped, and cubic-spline regression in R, respectively) can all capture changes of the corpus callosum, but only the cubic-spline model is appropriate for shape changes of the rat calvarium. While our approach is developed for the Grassmannian, the principles are applicable to smooth manifolds in general.