Distance-based tests for planar shape

Abstract

Statistical shape analysis (SSA) may be understood like a field of Multivariate Analysis and has assumed a prominent position due to its applicability in various areas; such as in imagery processing, biology, anatomy, among others. Biological hypotheses have been often formulating in the SSA framework, which is based on the concept of “shape” whose related data are in a non Euclidian space. Thus, tailored tests are sought to take accurate decision in this area. To that end, we first assume the complex Bingham (CB) distribution to describe preshape data which are extracted from two-dimensional objects, called planar shape. Adopting adequate distances is crucial in SSA; particularly at hypothesis tests, discriminant analysis and clustering processes. In this paper, we derive the Rényi, Kullback–Leibler (KL), Bhattacharyya and Hellinger distances for the CB distribution. These quantities are then used as distance-based hypothesis tests to assess if two planar shape samples come from the same distribution. Even though useful for many models having Euclidean supports, we prove that the CB KL discrepancy is invariant in rotation, what restricts its use. Four new homogeneity hypothesis tests are proposed, three among them may also be used like tests involving mean shape. The performance of the proposed tests is quantified and compared (in terms of both test size and power and robustness) to that due to other four mean shape tests of the SSA literature: Hotelling T2, Goodall, James and lambda. Finally, an application in evolutionary biology is done to assess sexual dimorphism in the Pan troglodytes (chimpanzee) species. Results point out our proposals may offer meaningfully advantages comparatively to considered literature tests.