Generalized cross-validation as a method for choosing a good ridge parameter
Consider the ridge estimate (λ) for β in the model unknown,(λ)=(XTX+ n λ I)− 1 XT y. We
study the method of generalized cross-validation (GCV) for choosing a good value for λ from
the data. The estimate is the minimizer of V (λ) given by where A (λ)= X (XT X+ n λ I)− 1 XT.
This estimate is a rotation-invariant version of Allen's PRESS, or ordinary cross-validation.
This estimate behaves like a risk improvement estimator, but does not require an estimate of
σ2, so can be used when n− p is small, or even if p≥ 2 n in certain cases. The GCV method …
study the method of generalized cross-validation (GCV) for choosing a good value for λ from
the data. The estimate is the minimizer of V (λ) given by where A (λ)= X (XT X+ n λ I)− 1 XT.
This estimate is a rotation-invariant version of Allen's PRESS, or ordinary cross-validation.
This estimate behaves like a risk improvement estimator, but does not require an estimate of
σ2, so can be used when n− p is small, or even if p≥ 2 n in certain cases. The GCV method …
Consider the ridge estimate (λ) for β in the model unknown, (λ) = (X T X + nλI)−1 X T y. We study the method of generalized cross-validation (GCV) for choosing a good value for λ from the data. The estimate is the minimizer of V(λ) given by
where A(λ) = X(X T X + nλI)−1 X T . This estimate is a rotation-invariant version of Allen's PRESS, or ordinary cross-validation. This estimate behaves like a risk improvement estimator, but does not require an estimate of σ2, so can be used when n − p is small, or even if p ≥ 2 n in certain cases. The GCV method can also be used in subset selection and singular value truncation methods for regression, and even to choose from among mixtures of these methods.