Invariance, Identifiability, and Morphometrics

Abstract

The form of an object is that characteristic that remains invariant under a group of transformations comprising translation, rotation, and possibly reflection. Group invariance thus naturally plays an important role in the statistical analysis of forms. We examine the existing methods for the statistical analysis of form from the invariance perspective. We begin with a review of the important basic ideas behind invariance and derive a maximal invariant under the group of transformations consisting of rotation, reflection, and translation and its distribution under the Gaussian and elliptically symmetric perturbation models. We first consider the single-sample case and discuss the issue of identifiability of the parameters. We show that method-of-moments estimators based on the distances between landmarks and maximum likelihood estimators (MLEs) based on the size and shape coordinates are invariant and estimate identifiable parameters. However, a number of commonly used methods do not. We compare the statistical and computational efficiencies of the method-ofmoments estimator and MLE and show that the method of moments substantially simplifies the estimation procedure computationally with only a small loss of statistical efficiency. We then extend the discussion of invariance to the comparison of two forms. We discuss the relationship between identifiability and invariance in the two-sample case and again show that many commonly used methods base inference on nonidentifiable parameters and discuss the scientific implications of basing inferences on nonidentifiable parameters using a biological example. We provide a brief summary of a method for shape analysis that is invariant.