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High-dimensional regression with potential prior information on variable importance

Published version
Peer-reviewed

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Authors

Stokell, Benjamin G 

Abstract

Abstract: There are a variety of settings where vague prior information may be available on the importance of predictors in high-dimensional regression settings. Examples include the ordering on the variables offered by their empirical variances (which is typically discarded through standardisation), the lag of predictors when fitting autoregressive models in time series settings, or the level of missingness of the variables. Whilst such orderings may not match the true importance of variables, we argue that there is little to be lost, and potentially much to be gained, by using them. We propose a simple scheme involving fitting a sequence of models indicated by the ordering. We show that the computational cost for fitting all models when ridge regression is used is no more than for a single fit of ridge regression, and describe a strategy for Lasso regression that makes use of previous fits to greatly speed up fitting the entire sequence of models. We propose to select a final estimator by cross-validation and provide a general result on the quality of the best performing estimator on a test set selected from among a number M of competing estimators in a high-dimensional linear regression setting. Our result requires no sparsity assumptions and shows that only a logM price is incurred compared to the unknown best estimator. We demonstrate the effectiveness of our approach when applied to missing or corrupted data, and in time series settings. An R package is available on github.

Description

Keywords

Article, High-dimensional data, Low variance filter, Lasso, Ridge regression, Missing data, Corrupted data

Journal Title

Statistics and Computing

Conference Name

Journal ISSN

0960-3174
1573-1375

Volume Title

32

Publisher

Springer US
Sponsorship
Engineering and Physical Sciences Research Council (EP/N031938/1)