Calculate a solution path of the natural lasso estimate (of error standard deviation) with a list of tuning parameter values. In particular, this function solves the lasso problems and returns the lasso objective function values as estimates of the error variance: \(\hat{\sigma}^2_{\lambda} = \min_{\beta} ||y - X \beta||_2^2 / n + 2 \lambda ||\beta||_1.\) The output also includes a path of naive estimates and a path of degree of freedom adjusted estimates of the error standard deviation.
nlasso_path(x, y, lambda = NULL, nlam = 100, flmin = 0.01,
thresh = 1e-08, intercept = TRUE, glmnet_output = NULL)
An n
by p
design matrix. Each row is an observation of p
features.
A response vector of size n
.
A user specified list of tuning parameter. Default to be NULL, and the program will compute its own lambda
path based on nlam
and flmin
.
The number of lambda
values. Default value is 100
.
The ratio of the smallest and the largest values in lambda
. The largest value in lambda
is usually the smallest value for which all coefficients are set to zero. Default to be 1e-2
.
Threshold value for the underlying optimization algorithm to claim convergence. Default to be 1e-8
.
Indicator of whether intercept should be fitted. Default to be TRUE
.
Should the estimate be computed using a user-specified output from glmnet
. If not NULL
, it should be the output from glmnet
call with standardize = TRUE
, and then the arguments lambda
, nlam
, flmin
, thresh
, and intercept
will be ignored. Default to be NULL
, in which case the function will call glmnet
internally.
A list object containing:
n
and p
: The dimension of the problem.
lambda
: The path of tuning parameters used.
beta
: Matrix of estimates of the regression coefficients, in the original scale. The matrix is of size p
by nlam
. The j
-th column represents the estimate of coefficient corresponding to the j
-th tuning parameter in lambda
.
a0
: Estimate of intercept. A vector of length nlam
.
sig_obj_path
: Natural lasso estimates of the error standard deviation. A vector of length nlam
.
sig_naive_path
: Naive estimates of the error standard deviation based on lasso regression, i.e., \(||y - X \hat{\beta}||_2 / \sqrt n\). A vector of length nlam
.
sig_df_path
: Degree-of-freedom adjusted estimate of standard deviation of the error. A vector of length nlam
. See Reid, et, al (2016).
type
: whether the output is of a natural or an organic lasso.
# NOT RUN {
set.seed(123)
sim <- make_sparse_model(n = 50, p = 200, alpha = 0.6, rho = 0.6, snr = 2, nsim = 1)
nl_path <- nlasso_path(x = sim$x, y = sim$y[, 1])
# }
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