We propose a Bayesian variable selection approach for ultrahigh dimensional linear regression based on the strategy of split and merge. The approach proposed consists of two stages: split the ultrahigh dimensional data set into a number of lower dimensional subsets and select relevant variables from each of the subsets, and aggregate the variables selected from each subset and then select relevant variables from the aggregated data set. Since the approach proposed has an embarrassingly parallel structure, it can be easily implemented in a parallel architecture and applied to big data problems with millions or more of explanatory variables. Under mild conditions, we show that the approach proposed is consistent, i.e. the true explanatory variables can be correctly identified by the approach as the sample size becomes large. Extensive comparisons of the approach proposed have been made with penalized likelihood approaches, such as the lasso, elastic net, sure independence screening and iterative sure independence screening. The numerical results show that the approach proposed generally outperforms penalized likelihood approaches: the models selected by the approach tend to be more sparse and closer to the true model.
Variable selection plays an important role in high dimensional statistical modelling which nowadays appears in many areas and is key to various scientific discoveries. For problems of large scale or dimensionality p, accuracy of estimation and computational cost are two top concerns. Recently, Candes and Tao have proposed the Dantzig selector using L1-regularization and showed that it achieves the ideal risk up to a logarithmic factor log(p). Their innovative procedure and remarkable result are challenged when the dimensionality is ultrahigh as the factor log(p) can be large and their uniform uncertainty principle can fail. Motivated by these concerns, we introduce the concept of sure screening and propose a sure screening method that is based on correlation learning, called sure independence screening, to reduce dimensionality from high to a moderate scale that is below the sample size. In a fairly general asymptotic framework, correlation learning is shown to have the sure screening property for even exponentially growing dimensionality. As a methodological extension, iterative sure independence screening is also proposed to enhance its finite sample performance. With dimension reduced accurately from high to below sample size, variable selection can be improved on both speed and accuracy, and can then be accomplished by a well-developed method such as smoothly clipped absolute deviation, the Dantzig selector, lasso or adaptive lasso. The connections between these penalized least squares methods are also elucidated.
more » « less- NSF-PAR ID:
- 10405649
- Publisher / Repository:
- Oxford University Press
- Date Published:
- Journal Name:
- Journal of the Royal Statistical Society Series B: Statistical Methodology
- Volume:
- 70
- Issue:
- 5
- ISSN:
- 1369-7412
- Page Range / eLocation ID:
- p. 849-911
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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