Date of Award


Document Type

Campus Access Dissertation



First Advisor

Ian Dryden

Second Advisor

Xianzheng Huang


This dissertation deals with the problem of measurement error in shape analysis, especially in the Procrustes matching of shapes. Shape data are often prone to measurement error due to the wide sources of errors. In traditional shape analysis measurement error is usually not considered, but ignoring measurement error when it does exist can cause many problems. We prove that, in the Procrustes matching for both two and three dimensional shape data, the naive estimators (which are obtained by ignoring the measurement error) for rotation and scale are biased. In this dissertation we propose a measurement error model-based Procrustes approach for the ordinary Procrustes analysis (OPA), matching of two shapes based on the conditional score method. In the measurement error literature the conditional score method is widely applicable for models of real variables and it yields consistent inference. We develop the conditional score method for two and three dimensional shape data using the complex and quaternion linear models. Two different estimators: the conditional score estimator (CSE) and the separate regression conditional score estimator (Separate CSE) are developed here. The estimators obtained by this approach for the rotation and scale in the Procrustes matching of two shapes are consistent. The asymptotic properties of the estimators are given by the theory of M-estimators. We also extend this approach to the general Procrustes analysis (GPA) for the matching of a set of shapes. The methodology is demonstrated in some simulations and applications in some real data sets including face landmark data and landmarks on the skulls of rats.