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Zbl 1091.62054
Efron, Bradley; Hastie, Trevor; Johnstone, Iain; Tibshirani, Robert
Least angle regression. (With discussion). (English)
[J] Ann. Stat. 32, No. 2, 407-499 (2004). ISSN 0090-5364

Summary: The purpose of model selection algorithms such as All Subsets, Forward Selection and Backward Elimination is to choose a linear model on the basis of the same set of data to which the model will be applied. Typically we have available a large collection of possible covariates from which we hope to select a parsimonious set for the efficient prediction of a response variable. Least Angle Regression (LARS), a new model selection algorithm, is a useful and less greedy version of traditional forward selection methods. Three main properties are derived: (1) A simple modification of the LARS algorithm implements the Lasso, an attractive version of ordinary least squares that constrains the sum of the absolute regression coefficients; the LARS modification calculates all possible Lasso estimates for a given problem, using an order of magnitude less computer time than previous methods. (2) A different LARS modification efficiently implements Forward Stagewise linear regression, another promising new model selection method; this connection explains the similar numerical results previously observed for the Lasso and Stagewise, and helps us understand the properties of both methods, which are seen as constrained versions of the simpler LARS algorithm. (3) A simple approximation for the degrees of freedom of a LARS estimate is available, from which we derive a Cp estimate of prediction error; this allows a principle choice among the range of possible LARS estimates. LARS and its variants are computationally efficient: the paper describes a publicly available algorithm that requires only the same order of magnitude of computational effort as ordinary least squares applied to the full set of covariates.

MSC 2000:: *62J05 Linear regression
62J07 Ridge regression
65C60 Computational problems in statistics

Keywords: Lasso; boosting; coefficient paths; variable selection

Cited in: Zbl 1235.62097 Zbl 1180.62081 Zbl 1194.62094 Zbl 1222.68290

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