Fit a GLM with elastic net regularization for a path of lambda values

Fit a generalized linear model via penalized maximum likelihood for a path of lambda values. Can deal with any GLM family.

Usage

glmnet.path(
  x,
  y,
  weights = NULL,
  lambda = NULL,
  nlambda = 100,
  lambda.min.ratio = ifelse(nobs < nvars, 0.01, 1e-04),
  alpha = 1,
  offset = NULL,
  family = gaussian(),
  standardize = TRUE,
  intercept = TRUE,
  thresh = 1e-10,
  maxit = 1e+05,
  penalty.factor = rep(1, nvars),
  exclude = integer(0),
  lower.limits = -Inf,
  upper.limits = Inf,
  trace.it = 0
)

Arguments

x: Input matrix, of dimension nobs x nvars; each row is an observation vector. Can be a sparse matrix.
y: Quantitative response variable.
weights: Observation weights. Default is 1 for each observation.
lambda: A user supplied lambda sequence. Typical usage is to have the program compute its own lambda sequence based on nlambda and lambda.min.ratio. Supplying a value of lambda overrides this.
nlambda: The number of lambda values, default is 100.
lambda.min.ratio: Smallest value for lambda as a fraction of lambda.max, the (data derived) entry value (i.e. the smallest value for which all coefficients are zero). The default depends on the sample size nobs relative to the number of variables nvars. If nobs >= nvars, the default is 0.0001, close to zero. If nobs < nvars, the default is 0.01. A very small value of lambda.min.ratio will lead to a saturated fit in the nobs < nvars case. This is undefined for some families of models, and the function will exit gracefully when the percentage deviance explained is almost 1.
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.
offset: A vector of length nobs that is included in the linear predictor. Useful for the "poisson" family (e.g. log of exposure time), or for refining a model by starting at a current fit. Default is NULL. If supplied, then values must also be supplied to the predict function.
family: A description of the error distribution and link function to be used in the model. This is the result of a call to a family function. Default is gaussian(). (See family for details on family functions.)
standardize: Logical flag for x variable standardization, prior to fitting the model sequence. The coefficients are always returned on the original scale. Default is standardize=TRUE. If variables are in the same units already, you might not wish to standardize.
intercept: Should intercept be fitted (default=TRUE) or set to zero (FALSE)?
thresh: Convergence threshold for coordinate descent. Each inner coordinate-descent loop continues until the maximum change in the objective after any coefficient update is less than thresh times the null deviance. Default value is 1e-10.
maxit: Maximum number of passes over the data; default is 10^5.
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.
trace.it: Controls how much information is printed to screen. Default is trace.it=0 (no information printed). If trace.it=1, a progress bar is displayed. If trace.it=2, some information about the fitting procedure is printed to the console as the model is being fitted.

Value

An object with class "glmnetfit" and "glmnet".

a0: Intercept sequence of length length(lambda).
beta: A nvars x length(lambda) matrix of coefficients, stored in sparse matrix format.
df: The number of nonzero coefficients for each value of lambda.
dim: Dimension of coefficient matrix.
lambda: The actual sequence of lambda values used. When alpha=0, the largest lambda reported does not quite give the zero coefficients reported (lambda=inf would in principle). Instead, the largest lambda for alpha=0.001 is used, and the sequence of lambda values is derived from this.
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 summed over all lambda values.
jerr: Error flag, for warnings and errors (largely for internal debugging).
offset: A logical variable indicating whether an offset was included in the model.
call: The call that produced this object.
family: Family used for the model.
nobs: Number of observations.

Details

glmnet.path solves the elastic net problem for a path of lambda values. It generalizes glmnet::glmnet in that it works for any GLM family.

Sometimes the sequence is truncated before nlambda values of lambda have been used. This happens when glmnet.path detects that the decrease in deviance is marginal (i.e. we are near a saturated fit).

Examples

set.seed(1)
x <- matrix(rnorm(100 * 20), nrow = 100)
y <- ifelse(rnorm(100) > 0, 1, 0)

# binomial with probit link
fit1 <- glmnet:::glmnet.path(x, y, family = binomial(link = "probit"))