On Computing the Prediction Sum of Squares Statistic in Linear Least Squares Problems with Multiple Parameter or Measurement Sets

The prediction sum of squares is a useful statistic for comparing different models. It is based on the principle of leave-one-out or ordinary cross-validation, whereby every measurement is considered in turn as a test set, for the model parameters trained on all but the held out measurement. As for linear least squares problems, there is a simple well-known non-iterative formula to compute the prediction sum of squares without having to refit the model as many times as the number of measurements. We extend this formula to cases where the problem has multiple parameter or measurement sets. We report experimental results on the fitting of a warp between two images, for which the number of deformation centres is automatically selected, based on one of the proposed non-iterative formulae.