R/standardize.R

In SIS: Sure Independence Screening

#' Standardization of High-Dimensional Design Matrices
#' 
#' Standardizes the columns of a high-dimensional design matrix to mean zero
#' and unit Euclidean norm.
#' 
#' Performs a location and scale transform to the columns of the original
#' design matrix, so that the resulting design matrix with \eqn{p}-dimensional
#' observations \eqn{\{x_i : i=1,...,n\}} of the form
#' \eqn{x_i=(x_{i1},x_{i2},...,x_{ip})} satisfies \eqn{\sum_{i=1}^{n} x_{ij} =
#' 0} and \eqn{\sum_{i=1}^{n} x_{ij}^{2} = 1} for \eqn{j=1,...,p}.
#' 
#' @export
#' @param X A design matrix to be standardized.
#' @return A design matrix with standardized predictors or columns.
#' @author Jianqing Fan, Yang Feng, Diego Franco Saldana, Richard Samworth, and
#' Yichao Wu
#' @references 
#' Diego Franco Saldana and Yang Feng (2018) SIS: An R package for Sure Independence Screening in
#' Ultrahigh Dimensional Statistical Models, \emph{Journal of Statistical Software}, \bold{83}, 2, 1-25.
#' @keywords models
#' @examples
#' 
#' set.seed(0)
#' n = 400; p = 50; rho = 0.5
#' corrmat = diag(rep(1-rho, p)) + matrix(rho, p, p)
#' corrmat[,4] = sqrt(rho)
#' corrmat[4, ] = sqrt(rho)
#' corrmat[4,4] = 1
#' corrmat[,5] = 0
#' corrmat[5, ] = 0
#' corrmat[5,5] = 1
#' cholmat = chol(corrmat)
#' x = matrix(rnorm(n*p, mean=15, sd=9), n, p)
#' x = x%*%cholmat
#' 
#' x.standard = standardize(x)
#' 
standardize <- function(X) {
    center = colMeans(X)
    X.c = sweep(X, 2, center)
    unit.var = sqrt(apply(X.c, 2, crossprod))
    val = sweep(X.c, 2, unit.var, "/")
    return(val)
}