A brief introduction to geometrical statistics

When data are geometrical objects, classical statistical tools cannot be employed directly because the object space is no longer Euclidean. Even worse, often one is interested in object descriptors that are different from the original objects, e.g. principal components, which live on real projective spaces. Analogs of such descriptors usually live in even more non-Euclidean spaces. Extending the concept of the Euclidean expected value to generalized Fréchet means we provide for a framework, for instance for dimension reduction and hypothesis testing. Further, we sketch some of the pitfalls coming along with the non-Euclidean geometry, e.g. finite sample smeariness, and how to avoid them. We conclude with a list of open problems.