On physical units in multivariate analysis

Abstract: comparatively stringent mathematical frame has been developed, where the behaviour of physical dimensions (like ``length'', ``mass'', and ``time'') and physical units (like 1 m [1 metre], 1 kg [1 kilogramme], and 1 s [1 second]) is modelled. Easy-to-apply conditions for additivity and multiplicativity of physcial matrices are presented. Precise meanings of physically dimensional random variables, expectations, and covariance matrices are proposed. Physically dimensional counterparts of linear transformations and the multivariate normal distribution are introduced. Four everyday statistical tools are investigated from the point of view of physical dimensions, namely 1) point estimations in linear models, 2) principal components, 3) Fisher's discriminant, and 4) canonical correlations. The governing idea is the way a mathematically constructed entity transforms when we convert from one collection of physical units to another collection of physical units, e.g. 1 m; 1 kg; 1 s) curvearrowright (1 in; 1 lbavdp; 1 min). When the entity is shown to be indifferent under such a conversion, it will be considered a physical entity, having an intrinsic meaning independent of how we happen to represent it in terms of physical units.