The Living Thing / Notebooks :

Cross Validation

On substituting simulation for analysis in model selection, in e.g. choosing the “right” regularisation parameter for sparse regression.

Asymptotically equivalent to generalised Akaike information criteria. (e.g. Ston77) Related to bootstrap in various ways.

The computationally expensive default option when your model doesn’t have any obvious short cuts for complexity regularization. For example, I think that the AIC for penalised reqgression requires penalties twice-differentiable at the optimum. I’m not sure they couldn’t be made to work, however. should investigate.

Present alternatives, especially outside n-fold cross-validation, especially computationally tractable ones. Methods based on statistical learning theory or concentration inequalities win gratitude.

Basic Cross Validation

TBD

Generalised Cross Validation

Why the name? It’s specialised cross-validation, AFAICT.

TBD. Hat matrix, smoother matrix. Note comparative computational efficiency. Define hat matrix.

Refs

Andr91
Andrews, D. W. K.(1991) Asymptotic optimality of generalized CL, cross-validation, and generalized cross-validation in regression with heteroskedastic errors. Journal of Econometrics, 47(2), 359–377. DOI.
GoHW79
Golub, G. H., Heath, M., & Wahba, G. (1979) Generalized Cross-Validation as a Method for Choosing a Good Ridge Parameter. Technometrics, 21(2), 215–223. DOI.
HaRL04
Hall, P., Racine, J., & Li, Q. (2004) Cross-validation and the estimation of conditional probability densities. Journal of the American Statistical Association, 99(468), 1015–1026. DOI.
Li87
Li, K.-C. (1987) Asymptotic Optimality for $C_p, C_L$, Cross-Validation and Generalized Cross-Validation: Discrete Index Set. The Annals of Statistics, 15(3), 958–975. DOI.
Ston77
Stone, M. (1977) An Asymptotic Equivalence of Choice of Model by Cross-Validation and Akaike’s Criterion. Journal of the Royal Statistical Society. Series B (Methodological), 39(1), 44–47.
Wood94
Wood, S. (1994) Monotonic Smoothing Splines Fitted by Cross Validation. SIAM Journal on Scientific Computing, 15(5), 1126–1133. DOI.