Fit an MCP- or SCAD-penalized regression path
Fit coefficients paths for MCP- or SCAD-penalized regression models over a grid of values for the regularization parameter lambda. Fits linear and logistic regression models, with option for an additional L2 penalty.
ncvreg(X, y, family=c("gaussian", "binomial", "poisson"),
penalty=c("MCP", "SCAD", "lasso"), gamma=switch(penalty, SCAD=3.7, 3),
alpha=1, lambda.min=ifelse(n>p,.001,.05), nlambda=100, lambda, eps=1e-4,
max.iter=10000, convex=TRUE, dfmax=p+1, penalty.factor=rep(1, ncol(X)),
warn=TRUE, returnX, ...)X |
The design matrix, without an intercept. |
y |
The response vector. |
family |
Either "gaussian", "binomial", or "poisson", depending on the response. |
penalty |
The penalty to be applied to the model. Either "MCP" (the default), "SCAD", or "lasso". |
gamma |
The tuning parameter of the MCP/SCAD penalty (see details). Default is 3 for MCP and 3.7 for SCAD. |
alpha |
Tuning parameter for the Mnet estimator which controls
the relative contributions from the MCP/SCAD penalty and the ridge,
or L2 penalty. |
lambda.min |
The smallest value for lambda, as a fraction of lambda.max. Default is .001 if the number of observations is larger than the number of covariates and .05 otherwise. |
nlambda |
The number of lambda values. Default is 100. |
lambda |
A user-specified sequence of lambda values. By default,
a sequence of values of length |
eps |
Convergence threshhold. The algorithm iterates until the
RMSD for the change in linear predictors for each coefficient is
less than |
max.iter |
Maximum number of iterations (total across entire path). Default is 10000. |
convex |
Calculate index for which objective function ceases to be locally convex? Default is TRUE. |
dfmax |
Upper bound for the number of nonzero coefficients. Default is no upper bound. However, for large data sets, computational burden may be heavy for models with a large number of nonzero coefficients. |
penalty.factor |
A multiplicative factor for the penalty applied
to each coefficient. If supplied, |
warn |
Return warning messages for failures to converge and model saturation? Default is TRUE. |
returnX |
Return the standardized design matrix along with the
fit? By default, this option is turned on if X is under 100 MB, but
turned off for larger matrices to preserve memory. Note that
certain methods, such as |
... |
Not used. |
The sequence of models indexed by the regularization parameter
lambda is fit using a coordinate descent algorithm. For
logistic regression models, some care is taken to avoid model
saturation; the algorithm may exit early in this setting. The
objective function is defined to be
Q(β|X, y) = (1/n)*L(β|X, y) + P(β, λ),
where the loss function L is the deviance (-2 times the log likelihood) for the specified outcome distribution (gaussian/binomial/poisson). See here for more details.
This algorithm is stable, very efficient, and generally converges quite rapidly to the solution. For GLMs, adaptive rescaling is used.
An object with S3 class "ncvreg" containing:
The fitted matrix of coefficients. The number of rows
is equal to the number of coefficients, and the number of columns
is equal to nlambda.
A vector of length nlambda containing the number
of iterations until convergence at each value of lambda.
The sequence of regularization parameter values in the path.
Same as above.
Same as above.
Same as above.
Same as above.
The last index for which the objective function is
locally convex. The smallest value of lambda for which the
objective function is convex is therefore
lambda[convex.min], with corresponding coefficients
beta[,convex.min].
A vector containing the deviance (i.e., the loss) at
each value of lambda. Note that for gaussian
models, the loss is simply the residual sum of squares.
Same as above.
Sample size.
Additionally, if returnX=TRUE, the object will also contain
The standardized design matrix.
The response, centered if family='gaussian'.
Patrick Breheny
Breheny P and Huang J. (2011) Coordinate descentalgorithms for nonconvex penalized regression, with applications to biological feature selection. Annals of Applied Statistics, 5: 232-253. doi: 10.1214/10-AOAS388
# Linear regression --------------------------------------------------
data(Prostate)
X <- Prostate$X
y <- Prostate$y
op <- par(mfrow=c(2,2))
fit <- ncvreg(X, y)
plot(fit, main=expression(paste(gamma,"=",3)))
fit <- ncvreg(X, y, gamma=10)
plot(fit, main=expression(paste(gamma,"=",10)))
fit <- ncvreg(X, y, gamma=1.5)
plot(fit, main=expression(paste(gamma,"=",1.5)))
fit <- ncvreg(X, y, penalty="SCAD")
plot(fit, main=expression(paste("SCAD, ",gamma,"=",3)))
par(op)
op <- par(mfrow=c(2,2))
fit <- ncvreg(X, y)
plot(fit, main=expression(paste(alpha,"=",1)))
fit <- ncvreg(X, y, alpha=0.9)
plot(fit, main=expression(paste(alpha,"=",0.9)))
fit <- ncvreg(X, y, alpha=0.5)
plot(fit, main=expression(paste(alpha,"=",0.5)))
fit <- ncvreg(X, y, alpha=0.1)
plot(fit, main=expression(paste(alpha,"=",0.1)))
par(op)
op <- par(mfrow=c(2,2))
fit <- ncvreg(X, y)
plot(mfdr(fit)) # Independence approximation
plot(mfdr(fit), type="EF") # Independence approximation
perm.fit <- perm.ncvreg(X, y)
plot(perm.fit)
plot(perm.fit, type="EF")
par(op)
# Logistic regression ------------------------------------------------
data(Heart)
X <- Heart$X
y <- Heart$y
op <- par(mfrow=c(2,2))
fit <- ncvreg(X, y, family="binomial")
plot(fit, main=expression(paste(gamma,"=",3)))
fit <- ncvreg(X, y, family="binomial", gamma=10)
plot(fit, main=expression(paste(gamma,"=",10)))
fit <- ncvreg(X, y, family="binomial", gamma=1.5)
plot(fit, main=expression(paste(gamma,"=",1.5)))
fit <- ncvreg(X, y, family="binomial", penalty="SCAD")
plot(fit, main=expression(paste("SCAD, ",gamma,"=",3)))
par(op)
op <- par(mfrow=c(2,2))
fit <- ncvreg(X, y, family="binomial")
plot(fit, main=expression(paste(alpha,"=",1)))
fit <- ncvreg(X, y, family="binomial", alpha=0.9)
plot(fit, main=expression(paste(alpha,"=",0.9)))
fit <- ncvreg(X, y, family="binomial", alpha=0.5)
plot(fit, main=expression(paste(alpha,"=",0.5)))
fit <- ncvreg(X, y, family="binomial", alpha=0.1)
plot(fit, main=expression(paste(alpha,"=",0.1)))
par(op)Please choose more modern alternatives, such as Google Chrome or Mozilla Firefox.