On ANOVA-Like Matrix Decompositions

The analysis of variance plays a fundamental role in statistical theory and practice, the standard Euclidean geometric form being particularly well-established. The geometry and associated linear algebra underlying such standard analysis of variance methods permit, essentially direct, generalisation to other settings. Specifically, as jointly developed here: (a) to minimum distance estimation problems associated with subsets of pairwise orthogonal subspaces; (b) to matrix, rather than vector, contexts; and (c) to general, not just standard Euclidean, inner products, and their induced distance functions. To this end, we characterise inner products rendering pairwise orthogonal a given set of nontrivial subspaces of a linear space any two of which meet only at the origin. Applications in a variety of areas are highlighted, including: (i) the analysis of asymmetry, and (ii) asymptotic comparisons in Invariant Coordinate Selection and Independent Component Analysis. A variety of possible further generalisations and applications are noted.