Statistical inference based on information theory for pre-shape data

Abstract

An important branch at Multivariate Analysis is Statistical Shape Analysis (SSA). A common demand in SSA is to study the shape property over objects in images, called planar-shape. To quantify difference in planar-shape between distinct groups is crucial in several areas – such as biology, medical image analysis, among others – and we provide advances in this sense. This thesis assumes pre-shapes which are obtained from twodimensional objects follow the complex Bingham (CB) distribution, having like the most important particular case the complex Watson (CW) model. From numerical evidence we present in this thesis, statistical tests which are well-defined in the SSA literature may provide low empirical test power curves. In order to obtain new alternatives to overcome this issue, we use information theory measures; in particular, stochastic entropies and distances. These measures play an important role in statistical theory, specifically into estimation and hypothesis inference procedures under large samples. First, we propose new distance-based two-sample hypothesis tests for triangle mean shapes. Closed-form expressions for the Rényi, Kullback-Leibler (KL), Bhattacharyya and Hellinger distances for the CW distribution are derived. The performance of proposed tests is quantified and compared with that due to the F2 test (analysis-of-variance tailored to the SSA literature). Furthermore, we perform an application to real data. Second, we extend the first topic proposing new distance-based two-sample hypothesis tests (for both homogeneity and mean shape) for the CB distribution and landmarks number higher than three. We derive the Rényi and KL divergences and the Bhattacharyya and Hellinger distances for the CB distribution. Three from among them may also be used like tests between two mean shapes or as discrepancy measures between the CB models. We prove also that the KL discrepancy for the CB model is rotation invariant. In order to evaluate and compare our proposals with other four SSA mean shape tests, a simulation study is also made to evaluate asymptotic and robustness properties. Finally an application in evolutionary biology is made. Third we tackle the proposal of new entropy-based multi-sample tests for variability in planar-shape. We develop closed-form expressions for the Rényi and Shannon entropies at the CB and CW models. From these quantities, hypothesis tests are obtained to assess if multiple spherical samples have the same degree of disorder.