Entropy-based pivotal statistics for multi-sample problems in planar shape

Abstract

Morphometric data come from several natural and man-made phenomena; e.g., biological processes and medical image processing. The analysis of these data—known as statistical shape analysis (SSA)—requires tailored methods because the majority of multivariate techniques are for the Euclidean space. An important branch at the SSA consists in using landmark data in two dimensions, called planar shape. Hypothesis tests in the last context have been proposed to check difference of mean shapes in multiple samples. A common assumption in these tests is that involved samples have the same covariance matrix, but few works are devoted to assess the former hypothesis. This paper addresses the proposal of new entropy-based multi-samples tests for variability. To formulate these tests, we assume that pre-shape data are well described by the complex Bingham ( $${\mathbb {C}} B$$ ) distribution. We derive expressions for the Renyi and Shannon entropies for the $${\mathbb {C}} B$$ and complex Watson models. Moreover, these quantities are understood as entropy-based tests to assess if multiple spherical samples have the same degree of disorder, a kind of variability. To quantify the performance of proposed tests, we perform a Monte Carlo study, adopting empirical test sizes and powers as comparison criteria. An application to real data using the second thoracic vertebra T2 of mouses is done to assess possible effects of body weight on the shape of mouse vertebra. Numerical results indicate that the test based on the optimized Renyi entropy may consist in a good tool to recognize variability similarity on planar-shape data.