Journal
ANNALS OF STATISTICS
Volume 39, Issue 1, Pages 48-81Publisher
INST MATHEMATICAL STATISTICS
DOI: 10.1214/10-AOS823
Keywords
Nonparametric regression; manifold; collinearity; model selection; regularization
Categories
Funding
- NSF [CCR-0225610, DMS-06-05236]
- Direct For Computer & Info Scie & Enginr
- Division Of Computer and Network Systems [0931843] Funding Source: National Science Foundation
- Direct For Mathematical & Physical Scien [0906808] Funding Source: National Science Foundation
- Division Of Mathematical Sciences [0906808] Funding Source: National Science Foundation
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Collinearity and near-collinearity of predictors cause difficulties when doing regression. In these cases, variable selection becomes untenable because of mathematical issues concerning the existence and numerical stability of the regression coefficients, and interpretation of the coefficients is ambiguous because gradients are not defined. Using a differential geometric interpretation, in which the regression coefficients are interpreted as estimates of the exterior derivative of a function, we develop a new method to do regression in the presence of collinearities. Our regularization scheme can improve estimation error, and it can be easily modified to include lasso-type regularization. These estimators also have simple extensions to the large p, small n context.
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