sqrt_lasso {RPtests}

Square-root Lasso regression

Description

Fits a linear model to potentially high-dimensional data using the square-root Lasso, also known as the scaled Lasso. The Lasso path is computed using the glmnet package.

Usage

sqrt_lasso(x, y, lam0 = NULL, exclude = integer(0), output_all = FALSE, ...)

Arguments

x

Input matrix of dimension nobs by nvars; each row is an observation vector.

y

Response variable; shoud be a numeric vector.

lam0

Tuning parameter for the square-root / scaled Lasso. If left blank (recommended) this is chosen using the method of Sun & Zhang (2013) implemented in the scalreg package.

exclude

Indices of variables to be excluded from the model; default is none.

output_all

In addition to the vector of coefficients, if TRUE, also outputs the intercept, an estimate of the noise standard deviation, and the output of glmnet.

...

Additional arguments to be passed to glmnet.

Details

First the Lasso path is computed using glmnet from glmnet. Next the particular point on the path corresponding to the square-root Lasso solution is found. As the path is only computed on a grid of points, the square-root Lasso solution is approximate.

Value

Either an estimated vector of regression coefficients with nvars components or, if output_all is true, a list with components

beta

the vector of regression coefficents

a0

an intercept term

sigma_hat

an estimate of the noise standard deviation; this is calculated as square-root of the average residual sums of squares

glmnet_obj

the fitted glmnet object, an S3 class “glmnet"

lamda_index

the index of the lambda vector in the glmnet object corresponding to the square-root Lasso solution

References

A. Belloni, V. Chernozhukov, and L. Wang. (2011) Square-root lasso: pivotal recovery of sparse signals via conic programming. Biometrika, 98(4):791-806.

T. Sun and C.-H. Zhang. (2012) Scaled sparse linear regression. Biometrika, 99(4):879-898.

T. Sun and C.-H. Zhang. (2013) Sparse matrix inversion with scaled lasso. The Journal of Machine Learning Research, 14(1):3385-3418.

See Also

glmnet and scalreg.

Examples

x <- matrix(rnorm(100*250), 100, 250)
y <- x[, 1] + x[, 2] + rnorm(100)
out <- sqrt_lasso(x, y)