cdfPen: 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)

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

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).

Author

Daniele Cuntrera, Luigi Augugliaro, Vito Muggeo

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.

References

Aggiungere Arxiv

Examples

Run this code


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)

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