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#' @title Path algorithm for the SparseStep model
#'
#' @description Fits the entire regularization path for SparseStep using a
#' Golden Section search.
#'
#' @param x matrix of predictors
#' @param y response
#' @param max.depth maximum recursion depth
#' @param ... further arguments to sparsestep()
#'
#' @export
#'
sparsestep.path <- function(x, y, max.depth=10, intercept=TRUE,
normalize=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 path.sparsestep\n")
} else {
normx <- rep(1, m)
}
XX <- t(x) %*% x
Xy <- t(x) %*% y
iter <- 0
# First find the smallest lambda for which all coefficients are zero
lambda.max <- 2^25
fit <- NULL
while (1) {
last.fit <- fit
fit <- sparsestep(x, y, lambda=lambda.max, normalize=FALSE,
XX=XX, Xy=Xy, ...)
iter <- iter + 1
if (all(fit$beta == 0)) {
lambda.max <- lambda.max / 2
} else {
lambda.max <- lambda.max * 2
break
}
}
cat("Found maximum value of lambda: 2^(", log(lambda.max)/log(2), ")\n")
iter <- iter + 1
if (is.null(last.fit)) {
betas.max <- fit$beta
} else {
betas.max <- last.fit$beta
}
# Now find the largest lambda for which no coefficients are zero
lambda.min <- 2^-12
fit <- NULL
while (1) {
last.fit <- fit
fit <- sparsestep(x, y, lambda=lambda.min, normalize=FALSE,
XX=XX, Xy=Xy, ...)
iter <- iter + 1
if (all(fit$beta != 0)) {
lambda.min <- lambda.min * 2
} else {
lambda.min <- lambda.min / 2
break
}
}
cat("Found minimum value of lambda: 2^(", log(lambda.min)/log(2), ")\n")
iter <- iter + 1
if (is.null(last.fit)) {
betas.min <- fit$beta
} else {
betas.min <- last.fit$beta
}
# Run binary section search
have.zeros <- as.vector(matrix(FALSE, 1, m+1))
have.zeros[1] <- TRUE
have.zeros[m+1] <- TRUE
left <- log(lambda.min)/log(2)
right <- log(lambda.max)/log(2)
l <- lambda.search(x, y, 0, max.depth, have.zeros, left, right, 1,
m+1, XX, Xy, ...)
have.zeros <- have.zeros | l$have.zeros
lambdas <- c(lambda.min, l$lambdas, lambda.max)
betas <- rbind(betas.min, l$betas, betas.max)
iter <- iter + l$iter
ord <- order(lambdas)
lambdas <- lambdas[ord]
betas <- betas[ord, ]
betas <- scale(betas, FALSE, normx)
object <- list(call=call, lambdas=lambdas, betas=betas,
iterations=iter)
}
lambda.search <- function(x, y, depth, max.depth, have.zeros, left, right,
lidx, ridx, XX, Xy, ...)
{
cat("Running search in interval [", left, ",", right, "] ... \n")
nm <- dim(x)
m <- nm[2]
betas <- NULL
lambdas <- NULL
middle <- left + (right - left)/2
lambda <- 2^middle
fit <- sparsestep(x, y, lambda=lambda, normalize=FALSE, XX=XX, Xy=Xy,
...)
iter <- 1
num.zero <- length(which(fit$beta == 0))
cidx <- num.zero + 1
if (have.zeros[cidx] == FALSE) {
have.zeros[cidx] = TRUE
betas <- rbind(betas, as.vector(fit$beta))
lambdas <- c(lambdas, lambda)
}
idx <- rbind(c(lidx, cidx), c(cidx, ridx))
bnd <- rbind(c(left, middle), c(middle, right))
for (r in 1:2) {
i1 <- idx[r, 1]
i2 <- idx[r, 2]
b1 <- bnd[r, 1]
b2 <- bnd[r, 2]
if (depth < max.depth && any(have.zeros[i1:i2] == F)) {
ds <- lambda.search(x, y, depth+1, max.depth,
have.zeros, b1, b2, i1, i2, XX, Xy,
...)
have.zeros <- have.zeros | ds$have.zeros
lambdas <- c(lambdas, ds$lambdas)
betas <- rbind(betas, ds$betas)
iter <- iter + ds$iter
}
}
out <- list(have.zeros=have.zeros, lambdas=lambdas, betas=betas,
iter=iter)
return(out)
}
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