PRESS statistic

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.

Leave-one-out cross-validation: Illustration of fitting a model and finding the PRESS statistic for n=8 observations

Procedure

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]

${\displaystyle \operatorname {PRESS} =\sum _{i=1}^{n}(y_{i}-{\hat {y}}_{i,-i})^{2}}$

Usage

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]

See also

• Model selection
• Stepwise regression
• Cross-validation (statistics)