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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.
An illustrative plot of a fit to data (green curve in top panel, data in red) plus a plot of residuals: red points in bottom plot. Dashed curve in bottom panel is a straight line fit to the residuals. If the functional form is correct then there should be little or no trend to the residuals - as seen here.
Residuals = residuals from the full model, ^ = regression coefficient from the i-th independent variable in the full model, X i = the i-th independent variable. Partial residual plots are widely discussed in the regression diagnostics literature (e.g., see the References section below).
Thus to compare residuals at different inputs, one needs to adjust the residuals by the expected variability of residuals, which is called studentizing. This is particularly important in the case of detecting outliers, where the case in question is somehow different from the others in a dataset. For example, a large residual may be expected in ...
The residuals from the least squares linear fit to this plot are identical to the residuals from the least squares fit of the original model (Y against all the independent variables including Xi). The influences of individual data values on the estimation of a coefficient are easy to see in this plot.
Given a sample from a normal distribution, whose parameters are unknown, it is possible to give prediction intervals in the frequentist sense, i.e., an interval [a, b] based on statistics of the sample such that on repeated experiments, X n+1 falls in the interval the desired percentage of the time; one may call these "predictive confidence intervals".
On the other hand, the internally studentized residuals are in the range , where ν = n − m is the number of residual degrees of freedom. If t i represents the internally studentized residual, and again assuming that the errors are independent identically distributed Gaussian variables, then: [ 2 ]
Linear least squares (LLS) is the least squares approximation of linear functions to data. It is a set of formulations for solving statistical problems involved in linear regression, including variants for ordinary (unweighted), weighted, and generalized (correlated) residuals.