Solve weighted least squares (WLS) problem for a single lambda value

Solves the weighted least squares (WLS) problem for a single lambda value. Internal function that users should not call directly.

Usage

elnet.fit(
  x,
  y,
  weights,
  lambda,
  alpha = 1,
  intercept = TRUE,
  penalty.factor = rep(1, nvars),
  exclude = c(),
  lower.limits = -Inf,
  upper.limits = Inf,
  warm = NULL,
  from.glmnet.fit = FALSE,
  save.fit = FALSE,
  control = glmnet.control()
)

Arguments

x: Input matrix, of dimension nobs x nvars; each row is an observation vector. If it is a sparse matrix, it is assumed to be unstandardized. It should have attributes xm and xs, where xm(j) and xs(j) are the centering and scaling factors for variable j respsectively. If it is not a sparse matrix, it is assumed that any standardization needed has already been done.
y: Quantitative response variable.
weights: Observation weights. elnet.fit does NOT standardize these weights.
lambda: A single value for the lambda hyperparameter.
alpha: The elasticnet mixing parameter, with $0 \le \alpha \le 1$. The penalty is defined as $$(1-\alpha)/2||\beta||_2^2+\alpha||\beta||_1.$$ alpha=1 is the lasso penalty, and alpha=0 the ridge penalty.
intercept: Should intercept be fitted (default=TRUE) or set to zero (FALSE)?
penalty.factor: Separate penalty factors can be applied to each coefficient. This is a number that multiplies lambda to allow differential shrinkage. Can be 0 for some variables, which implies no shrinkage, and that variable is always included in the model. Default is 1 for all variables (and implicitly infinity for variables listed in exclude). Note: the penalty factors are internally rescaled to sum to nvars.
exclude: Indices of variables to be excluded from the model. Default is none. Equivalent to an infinite penalty factor.
lower.limits: Vector of lower limits for each coefficient; default -Inf. Each of these must be non-positive. Can be presented as a single value (which will then be replicated), else a vector of length nvars.
upper.limits: Vector of upper limits for each coefficient; default Inf. See lower.limits.
warm: Either a glmnetfit object or a list (with names beta and a0 containing coefficients and intercept respectively) which can be used as a warm start. Default is NULL, indicating no warm start. For internal use only.
from.glmnet.fit: Was elnet.fit() called from glmnet.fit()? Default is FALSE.This has implications for computation of the penalty factors.
save.fit: Return the warm start object? Default is FALSE.
control: A fully resolved 17-key control list of the form returned by glmnet.control(). Default is glmnet.control() – current session state. This function does not resolve or validate the list; keys (thresh, maxit, big, etc.) are read directly. See ?glmnet.path for the same contract.

Value

An object with class "glmnetfit" and "glmnet". The list returned has the same keys as that of a glmnet object, except that it might have an additional warm_fit key.

a0: Intercept value.
beta: A nvars x 1 matrix of coefficients, stored in sparse matrix format.
df: The number of nonzero coefficients.
dim: Dimension of coefficient matrix.
lambda: Lambda value used.
dev.ratio: The fraction of (null) deviance explained. The deviance calculations incorporate weights if present in the model. The deviance is defined to be 2*(loglike_sat - loglike), where loglike_sat is the log-likelihood for the saturated model (a model with a free parameter per observation). Hence dev.ratio=1-dev/nulldev.
nulldev: Null deviance (per observation). This is defined to be 2*(loglike_sat -loglike(Null)). The null model refers to the intercept model.
npasses: Total passes over the data.
jerr: Error flag, for warnings and errors (largely for internal debugging).
offset: Always FALSE, since offsets do not appear in the WLS problem. Included for compability with glmnet output.
call: The call that produced this object.
nobs: Number of observations.
warm_fit: If save.fit=TRUE, output of C++ routine, used for warm starts. For internal use only.

Details

WARNING: Users should not call elnet.fit directly. Higher-level functions in this package call elnet.fit as a subroutine. If a warm start object is provided, some of the other arguments in the function may be overriden.

elnet.fit is essentially a wrapper around a C++ subroutine which minimizes

$$1/2 \sum w_i (y_i - X_i^T \beta)^2 + \sum \lambda \gamma_j [(1-\alpha)/2 \beta^2+\alpha|\beta|],$$

over $\beta$, where $\gamma_j$ is the relative penalty factor on the jth variable. If intercept = TRUE, then the term in the first sum is $w_i (y_i - \beta_0 - X_i^T \beta)^2$, and we are minimizing over both $\beta_0$ and $\beta$.

None of the inputs are standardized except for penalty.factor, which is standardized so that they sum up to nvars.