Tangent Coordinates in Statistical Shape Analysis: Properties and Two-Sample tests

Abstract

The concept of shape space is widely used for statistical inference of landmarks data. However, shape spaces are non-linear manifolds and that means standard multivariate statistics cannot be applied. Therefore, that gives rise to the concepts of tangent spaces. This dissertation aims to review the idea behind tangent coordinates on the landmark space and examine the performance of commonly used two-sample tests. Distribution-based and bootstrap Goodall’s tests depend only on the overall shape variability and similarly for Hotelling’s tests which rely on the choice of the tangent coordinates as well as the overall shape variability. Simulation study results show that the performance of all suggested tests is at different levels heavily affected by the variability measured using full Procrustes distance in the shape space. In addition, all suggested tests perform quite similarly in the case of highly concentrated data sets. For Hotelling’s tests, the bootstrap test seems to be more powerful when it is applied to inverse exponential map tangent coordinates than to partial Procrustes tangent coordinates with low concentrated data set, but they behave the same for high concentrated data sets. On the contrary, distribution based Hotelling’s test in the situation considered seems to not be affected by the choice of the tangent coordinates