Regression Modeling for Size-and-Shape Data Based on a Gaussian Model for Landmarks

Abstract

In this article, 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 labeled landmarks; these landmarks are used to represent the random objects under study. The regression structure, assumed in this article 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 modeling framework is demonstrated through real-data examples. Supplementary materials for this article are available online.

keyword:

• EM algorithm
• Offset-normal shape distributions
• Mean shape
• Shape of the mean
• Size-and-shape analysis

Acknowledgments

The authors are extremely grateful to the associate editor and the reviewers for their constructive suggestions.

Supplementary Materials

Part A contains results for reflection size-and-shape which complement the results in Section 2 for size-and-shape, while Part B contains the proofs of all the theoretical results. Part C contains formulas for the observed information matrix which enables the calculation of standard errors. Finally, part D contains further numerical results which complement those in Section 4.

Additional information

Funding

This work was supported by the Engineering and Physical Sciences Research Council [grant number EP/K022547/1].