On physical units in multivariate analysis
Tord Rikte
Centre for Mathematical Sciences
Mathematical Statistics
Lund Institute of Technology,
Lund University,
2002
ISSN 1404028X

Abstract:

A 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. Easytoapply 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
lb_{avdp}; 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.


Keywords:
Physical dimensions, physical units, unit conversion, physical matrices,
physical random variables, principal components, Fisher's discriminant, canonical
correlations.
