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#' @title Fits the SparseStep model
#'
#' @description Fits the SparseStep model for a single value of the
#' regularization parameter.
#'
#' @param x matrix of predictors
#' @param y response
#' @param lambda regularization parameter
#' @param gamma0 starting value of the gamma parameter
#' @param gammastop stopping value of the gamma parameter
#' @param IMsteps number of steps of the majorization algorithm to perform for
#' each value of gamma
#' @param gammastep factor to decrease gamma with at each step
#' @param normalize if TRUE, each variable is standardized to have unit L2
#' norm, otherwise it is left alone.
#' @param intercept if TRUE, an intercept is included in the model (and not
#' penalized), otherwise no intercept is included
#' @param force.zero if TRUE, absolute coefficients smaller than the provided
#' threshold value are set to absolute zero as a post-processing step,
#' otherwise no thresholding is performed
#' @param threshold threshold value to use for setting coefficients to
#' absolute zero
#' @param XX The X'X matrix; useful for repeated runs where X'X stays the same
#' @param Xy The X'y matrix; useful for repeated runs where X'y stays the same
#' @param use.XX whether or not to compute X'X and return it
#' @param use.Xy whether or not to compute X'y and return it
#'
#' @return A "sparsestep" object is returned, for which predict, coef, methods
#' exist.
#'
#' @export
#'
#' @examples
#' data(diabetes)
#' attach(diabetes)
#' object <- sparsestep(x, y)
#' plot(object)
#' detach(diabetes)
#'
sparsestep <- function(x, y, lambda=1.0, gamma0=1e6,
gammastop=1e-8, IMsteps=2, gammastep=2.0, normalize=TRUE,
intercept=TRUE, force.zero=TRUE, threshold=1e-7,
XX=NULL, Xy=NULL, use.XX = TRUE, use.Xy = TRUE)
{
call <- match.call()
nm <- dim(x)
n <- nm[1]
m <- nm[2]
one <- rep(1, n)
if (intercept) {
meanx <- drop(one %*% x)/n
x <- scale(x, meanx, FALSE)
mu <- mean(y)
y <- drop(y - mu)
} else {
meanx <- rep(0, m)
mu <- 0
y <- drop(y)
}
if (normalize) {
normx <- sqrt(drop(one %*% (x^2)))
names(normx) <- NULL
x <- scale(x, FALSE, normx)
cat("Normalizing in sparsestep\n")
} else {
normx <- rep(1, m)
}
if (use.XX & is.null(XX)) {
XX <- t(x) %*% x
}
if (use.Xy & is.null(Xy)) {
Xy <- t(x) %*% y
}
# Start solving SparseStep
gamma <- gamma0
beta <- matrix(0.0, m, 1)
it <- 0
while (gamma > gammastop)
{
for (i in 1:IMsteps) {
alpha <- beta
omega <- gamma/(alpha^2 + gamma)^2
Omega <- diag(as.vector(omega), m, m)
beta <- solve(XX + lambda * Omega, Xy)
}
it <- it + 1
gamma <- gamma / gammastep
}
# perform IM until convergence
epsilon <- 1e-14
loss <- get.loss(x, y, gamma, beta, lambda)
lbar <- (1.0 + 2.0 * epsilon) * loss
while ((lbar - loss)/loss > epsilon)
{
alpha <- beta
omega <- gamma/(alpha^2 + gamma)^2
Omega <- diag(as.vector(omega), m, m)
beta <- solve(XX + lambda * Omega, Xy)
lbar <- loss
loss <- get.loss(x, y, gamma, beta, lambda)
}
# postprocessing
if (force.zero) {
beta[which(abs(beta) < threshold)] <- 0
}
residuals <- y - x %*% beta
beta <- scale(t(beta), FALSE, normx)
RSS <- apply(residuals^2, 2, sum)
R2 <- 1 - RSS/RSS[1]
object <- list(call = call, lambda = lambda, R2 = R2, RSS = RSS,
gamma0 = gamma0, gammastop = gammastop,
IMsteps = IMsteps, gammastep = gammastep,
intercept = intercept, force.zero = force.zero,
threshold = threshold, beta = beta, mu = mu,
normx = normx, meanx = meanx,
XX = if(use.XX) XX else NULL,
Xy = if(use.Xy) Xy else NULL)
class(object) <- "sparsestep"
return(object)
}
get.loss <- function(x, y, gamma, beta, lambda)
{
Xb <- x %*% beta
diff <- y - Xb
b2 <- beta^2
binv <- 1/(b2 + gamma)
loss <- t(diff) %*% diff + lambda * t(b2) %*% binv
return(loss)
}
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