In statistics, the predicted residual error sum of squares (PRESS) is a form of cross-validation used in regression analysis to provide a summary measure of the fit of a model to a sample of observations that were not themselves used to estimate the model. It is calculated as the sum of squares of the prediction residuals for those observations.[1] [2] [3] Specifically, the PRESS statistic is an exhaustive form of cross-validation, as it tests all the possible ways that the original data can be divided into a training and a validation set.
Instead of fitting only one model on all data, leave-one-out cross-validation is used to fit N models (on N observations) where for each model one data point is left out from the training set. The out-of-sample predicted value is calculated for the omitted observation in each case, and the PRESS statistic is calculated as the sum of the squares of all the resulting prediction errors:[4]
\operatorname{PRESS}
| n | |
| =\sum | |
| i=1 |
(yi-\hat{y}i,)2
Given this procedure, the PRESS statistic can be calculated for a number of candidate model structures for the same dataset, with the lowest values of PRESS indicating the best structures.Models that are over-parameterised (over-fitted) would tend to give small residuals for observations included in the model-fitting but large residuals for observations that are excluded.The PRESS statistic has been extensively used in lazy learning and locally linear learning to speed-up the assessment and the selection of the neighbourhood size.[5] [6]