Regression modelling for size-and-shape data based on a Gaussian model for landmarks

In this paper we propose a regression model for size-and-shape response data. So far as we are aware, few such models have been explored in the literature to date. We assume a Gaussian model for labelled landmarks; these landmarks are used to represent the random objects under study. The regression structure, assumed in this paper to be linear in the ambient space, enters through the landmark means. Two approaches to parameter estimation are considered. The first approach is based directly on the marginal likelihood for the landmark-based shapes. In the second approach we treat the orientations of the landmarks as missing data, and we set up a model-consistent estimation procedure for the parameters using the EM algorithm. Both approaches raise challenging computational issues which we explain how to deal with. The usefulness of this regression modelling framework is demonstrated through real-data examples.