Least Squares Approximation of Matrices by Additive and Multiplicative Models

Summary Reduced rank approximation of matrices by Householder–Young methods is shown to be equivalent to fitting of a bilinear (multiplicative) model and to projection onto an optimally chosen subspace. The relation to linear approximation is made evident. Least squares fitting of a mixed linear (additive) and bilinear (multiplicative) model is proved to be a two-stage process: (1) fit the linear part of the model, then take residuals, and (2) fit the bilinear part to the residuals. Extensions of this result are given as well as some results on distributions of residuals and tests against alternatives of given rank. Statistical applications are shown in principal component analysis, in biplot graphical display and in fitting additive or Mandel-type models to two-way tables.