Within this very nice piece, Rob drops this bomb of mathematical knowledge:

**It is not necessary to actually fit**

**separate models when computing the CV statistic for linear models.**

Say what?

Here is a broader excerpt and the method itself (after the jump).

While cross-validation can be computationally expensive in general, it is very easy and fast to compute LOOCV for linear models. A linear model can be written as

Then

and the fitted values can be calculated using

where is known as the “hat-matrix” because it is used to compute (“Y-hat”).

If the diagonal values of are denoted by , then the cross-validation statistic can be computed using

where is the residual obtained from fitting the model to all observations. See Christensen’s book Plane Answers to Complex Questions for a proof. Thus, it is not necessary to actually fit separate models when computing the CV statistic for linear models. This remarkable result allows cross-validation to be used while only fitting the model once to all available observations.

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