Generalized cross-validation ($\mathrm{GCV}=\frac{\Vert Y-SY{\Vert }^2/n}{(1-\mathrm{tr}(S)/n{)}^2}$)

For a fixed linear smoother, generalized cross-validation replaces the individual leverage corrections of Leave-one-out cross-validation by their average. It selects smoothing parameters by minimizing the displayed criterion. It approximates a predictive error criterion; it is not a universal exact estimate of future error and requires care when observation errors are dependent.