Geometry gives naturally invariant information and insight via its quantifications and pictures. Long-established statistical examples include analysis of variance decompositions and the least-squares optimality of dimension reduction via principal component analysis (PCA). Such global Euclidean geometric methods need corresponding statistical conditions to hold – notably, constant dispersion. Depending on context, other geometries and associated methodologies can be appropriate. Generic statistical problems where this arises – and current geometries for them – include these instances:
Overall, a variety of geometries – affine, convex, differential, algebraic, … – have been emerging to meet requirement in a growing range of important statistical problems arising across the sciences and industry.