Statistical Shape Analysis, 2nd Edition

9 Tangent space inference

In this chapter we outline some simple approaches to inference based on tangent space approximations, which are valid in datasets with small variability in shape. We discuss one and two sample Hotelling’s T² tests for mean shape based on normality assumptions, and then consider non-parametric alternatives. As a special case we consider isotropic covariance structure, although the model is too simple for most applications. We discuss other multivariate inference techniques which work directly on shape coordinates and conclude with the topic of allometry: the relationship between shape and size. Note that any inference is a function of the landmark selection that the investigator designed at the outset.

Statistical models and parametric inference in the pre-shape, shape and size-and-shape space are discussed later in Chapter 10.

9.1 Tangent space small variability inference for mean shapes

The horizontal tangent space to the pre-shape sphere was defined in Section 4.4. A practical approach to analysis is to use the Procrustes tangent space coordinates if the data are concentrated and then perform standard multivariate analysis in this linear space, where the pole is chosen from the data using a consistent estimator of an overall population mean, which is then treated as fixed. For example the pole could be the full Procrustes mean of Equation (6.11).

After the choice of tangent space has been fixed we carry out inference using any convenient linear ...