Abstract
Least angle regression (LARS) by Efron et al. (2004) is a novel method for constructing the piece-wise linear path of Lasso solutions. For several years, it remained also as the de facto method for computing the Lasso solution before more sophisticated optimization algorithms preceded it. LARS method has recently again increased its popularity due to its ability to find the values of the penalty parameters, called knots, at which a new parameter enters the active set of non-zero coefficients. Significance test for the Lasso by Lockhart et al. (2014), for example, requires solving the knots via the LARS algorithm. Elastic net (EN), on the other hand, is a highly popular extension of Lasso that uses a linear combination of Lasso and ridge regression penalties. In this paper, we propose a new novel algorithm, called pathwise (PW-)LARS-EN, that is able to compute the EN knots over a grid of EN tuning parameter α values. The developed PW-LARS-EN algorithm decreases the EN tuning parameter and exploits the previously found knot values and the original LARS algorithm. A covariance test statistic for the Lasso is then generalized to the EN for testing the significance of the predictors. Our simulation studies validate the fact that the test statistic has an asymptotic Exp(1) distribution.
Highlights
INTRODUCTIONWe propose a pathwise (PW)Least angle regression (LARS)-Elastic net (EN) algorithm that computes the knots of EN over a grid of α values
In this paper, we consider a linear model, where the n-vector y ∈ Rn of observations is modeled as y = Xβ + ε, (1)where X ∈ Rn×p is a known predictor matrix, β ∈ Rp is the unknown vector of regression coefficients and ε ∈ Rn is the noise vector
Elastic net (EN) of [1] is a superset of the popular Lasso (Least absolute shrinkage and selection operator) [2] that is termed as basis pursuit denoising (BPDN) in the literature
Summary
We propose a pathwise (PW)LARS-EN algorithm that computes the knots of EN over a grid of α values. We generalize the covariance test statistic for the EN, which we denote as Tk(α) This test statistic requires computing the knots of EN solution for a given fixed α ∈ [1, 0). It was postulated in [5, Sec. 8] that covariance test statistic for the EN follows standard exponential distribution, Exp(1), when the null hypothesis that all signal variables are in the model holds true.
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