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Heuristics of instability and stabilization in model selection. (English) Zbl 0867.62055

Summary: In model selection, usually a “best” predictor is chosen from a collection \(\{\widehat{\mu} (\cdot,s)\}\) of predictors where \(\widehat{\mu} (\cdot,s)\) is the minimum least-squares predictor in a collection \({\mathcal U}_s\) of predictors. Here, \(s\) is a complexity parameter; that is, the smaller \(s\), the lower dimensional/smoother the models in \({\mathcal U}_s\). If \({\mathcal L}\) is the data used to derive the sequence \(\{\widehat{\mu} (\cdot,s)\}\), the procedure is called unstable if a small change in \({\mathcal L}\) can cause large changes in \(\{\widehat{\mu} (\cdot,s)\}\). With a crystal ball, one could pick the predictor in \(\{\widehat{\mu} (\cdot,s)\}\) having minimum prediction error. Without prescience, one uses test sets, cross-validation and so forth. The difference in prediction error between the crystal ball seletion and the statistician’s choice we call predictive loss. For an unstable procedure the predictive loss is large.
This is shown by some analytics in a simple case and by simulation results in a more complex comparison of four different linear regression methods. Unstable procedures can be stabilized by perturbing the data, getting a new predictor sequence \(\{\widehat{\mu}' (\cdot,s)\}\) and then averaging over many such predictor sequences.

MSC:

62H99 Multivariate analysis
62J05 Linear regression; mixed models
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References:

[1] BREIMAN, L. 1992. The little bootstrap and other methods for dimensionality selection in regression: x-fixed prediction error. J. Amer. Statist. Assoc. 87 738 754. Z. JSTOR: · Zbl 0850.62518 · doi:10.2307/2290212
[2] BREIMAN, L. 1995. Better subset selection using the non-negative garotte. Technometrics 37 373 384. Z. JSTOR: · Zbl 0862.62059 · doi:10.2307/1269730
[3] BREIMAN, L. 1996a. Stacked regressions. Machine Learning 24 41 64. Z. · Zbl 0849.68104
[4] BREIMAN, L. 1996b. Bagging predictors. Machine Learning 26 123 140. Z. · Zbl 0858.68080
[5] BREIMAN, L. 1996c. Bias, variance and arcing classifiers. Report 460, Dept. Statistics, Univ. California. Z.
[6] BREIMAN, L. and SPECTOR, P. 1992. Submodel selection and evaluation in regression. The random X case. Internat. Statist. Rev. 60 291 319. Z.
[7] WOLPERT, D. 1992. Stacked generalization. Neural Networks 5 241 259.
[8] BERKELEY, CALIFORNIA 94720-3860
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