cdfPen {penalizedcdf}

Fit a Linear Model with with CDF regularization

Description

Uses the CDF penalty to estimate a linear model with the maximum penalized likelihood. The path of coefficients is computed for a grid of values for the lambda regularization parameter.

Usage

cdfPen(X,
       y,
       nu,
       lmb,
       nlmb = 100L,
       e = 1E-3,
       rho = 2,
       algorithm = c("lla", "opt"),
       nstep = 1E+5,
       eps = 1E-6,
       eps.lla = 1E-6,
       nstep.lla = 1E+5)

Arguments

X

Matrix of covariates, each row is a vector of observations. The matrix must not contain the intercept.

y

Vector of response variable.

nu

Shape parameter of the penalty. It affects the degree of the non-convexity of the penalty. If no value is specified, the smallest value that ensures a single solution will be used.

lmb

A user-supplied tuning parameter sequence.

nlmb

number of lambda values; 100 is the default value.

e

The smallest lambda value, expressed as a percentage of maximum lambda. Default value is .001.

rho

Parameter of the optimization algorithm. Default is 2.

algorithm

Approximation to be used to obtain the sparse solution.

nstep

Maximum number of iterations of the global algorithm.

eps

Convergence threshold of the global algorithm.

eps.lla

Convergence threshold of the LLA-algorithm (if used).

nstep.lla

Maximum number of iterations of the LLA-algorithm (if used).

Details

We consider a local quadratic approximation of the likelihood to treat the problem as a weighted linear model.

The choice of value assigned to \nu is of fundamental importance: it affects both computational and estimation aspects. It affects the ”degree of non-convexity” of the penalty and determines which of the good and bad properties of convex and non-convex penalties are obtained. Using a high value of \nu ensures the uniqueness of solution, but the estimates will be biased. Conversely, a small value of \nu guarantees negligible bias in the estimates. The parameter \nu has the role of determining the convergence rate of non-null estimates$: the lower the value, the higher the convergence rate. Using lower values of \nu, the objective function will have local minima.

Value

coefficients

The coefficients fit matrix. The number of columns is equal to nlmb, and the number of rows is equal to the number of coefficients.

lmb

The vector of lambda used.

e

The smallest lambda value, expressed as a percentage of maximum lambda. Default value is .001.

rho

The parameter of the optimization algorithm used

nu

The shape parameters of the penalty used.

X

The design matrix.

y

The response.

algorithm

Approximation used

Author(s)

Daniele Cuntrera, Luigi Augugliaro, Vito Muggeo

References

Aggiungere Arxiv

Examples

p <- 10
n <- 100
X <- cbind(1, matrix(rnorm(n * p), n , p))
b.s <- c(1, rep(0, p))
b.s[sample(2:p, 3)] <- 1
y <- drop(crossprod(t(X), b.s))
out <- cdfPen(X = X, y = y)