  Predictive modelling of multivariate data where both the covariates and
responses are high-dimensional is becoming an increasingly popular task in many
data mining applications. Partial Least Squares (PLS) regression often turns
out to be a useful model in these situations since it performs dimensionality
reduction by assuming the existence of a small number of latent factors that
may explain the linear dependence between input and output. In practice, the
number of latent factors to be retained, which controls the complexity of the
model and its predictive ability, has to be carefully selected. Typically this
is done by cross validating a performance measure, such as the predictive
error. Although cross validation works well in many practical settings, it can
be computationally expensive. Various extensions to PLS have also been proposed
for regularising the PLS solution and performing simultaneous dimensionality
reduction and variable selection, but these come at the expense of additional
complexity parameters that also need to be tuned by cross-validation. In this
paper we derive a computationally efficient alternative to leave-one-out cross
validation (LOOCV), a predicted sum of squares (PRESS) statistic for two-block
PLS. We show that the PRESS is nearly identical to LOOCV but has the
computational expense of only a single PLS model fit. Examples of the PRESS for
selecting the number of latent factors and regularisation parameters are
provided.
