R/tools_wntest.R

In HDTSA: High Dimensional Time Series Analysis Tools

#' @name WN_test
#' @title Testing for white noise hypothesis in high dimension
#' @description \code{WN_test()} is the test proposed in Chang, Yao and Zhou
#' (2017) for the following hypothesis testing problems: \deqn{H_0:\{{\bf x}_t
#' \}_{t=1}^n\mathrm{\ is\ white\ noise\ \ versus\ \ }H_1:\{{\bf x}_t
#' \}_{t=1}^n\mathrm{\ is\ not\ white\ noise.} }
#'
#' @param X \eqn{{\bf X} = \{{\bf x}_1, \dots , {\bf x}_n \}'}, an \eqn{n\times
#'   p} sample matrix, where \eqn{n} is the sample size and \eqn{p} is the
#'   dimension of \eqn{{\bf x}_t}.
#' @param lag.k Time lag \eqn{K}, a positive integer, used to calculate the test
#'   statistic [See (4) in Chang, Yao and Zhou (2017)]. Default is \code{lag.k}
#'   \eqn{=2}.
#' @param B Bootstrap times for generating multivariate normal distributed
#'   random vectors in calculating the critical value. Default is \code{B}
#'   \eqn{=2000}.
#' @param pre Logical value which determines whether to performs preprocessing
#'   procedure on data matrix \code{X} or not, see Remark 1 in Chang, Yao and
#'   Zhou (2017) for more information. If \code{TRUE}, then the segment
#'   procedure will be performed to data \code{X} first. The three additional
#'   options including \code{thresh}, \code{tuning.vec} and \code{cv.num} are
#'   the same as those in \code{\link{PCA4_TS}}.
#' @param alpha The prescribed significance level. Default is 0.05.
#' @param kernel.type String, an option for choosing the symmetric kernel used
#'   in the estimation of long-run covariance matrix, for example, \code{'QS'}
#'   (Quadratic spectral kernel), \code{'Par'} (Parzen kernel) and \code{'Bart'}
#'   (Bartlett kernel), see Andrews (1991) for more information. Default option
#'   is\code{kernel.type = 'QS'}.
#' @param k0 A positive integer specified to calculate \eqn{\widehat{{\bf
#'   W}}_y}. See parameter \code{lag.k} in \code{\link{PCA4_TS}} for more
#'   information.
#' @param thresh Logical. It determines whether to perform the threshold method
#'   to estimate \eqn{\widehat{{\bf W}}_y} or not. See parameter \code{thresh}
#'   in \code{\link{PCA4_TS}} for more information.
#' @param tuning.vec The value of thresholding tuning parameter \eqn{\lambda}.
#'   See parameter \code{tuning.vec} in \code{\link{PCA4_TS}} for more
#'   information.
#' @seealso \code{\link{PCA4_TS}}
#'
#' @return An object of class "WN_test" is a list containing the following
#'   components:
#'
#'   \item{reject}{Logical value. If \code{TRUE}, it means rejecting the null
#'   hypothesis, otherwise it means not rejecting the null hypothesis.}
#'   \item{p.value}{Numerical value which represents the p-value of the test
#'   based on the observed data \eqn{\{{\bf x}_t\}_{t=1}^n}.}
#' @references Chang, J., Yao, Q. & Zhou, W. (2017). \emph{Testing for
#'   high-dimensional white noise using maximum cross-correlations}, Biometrika,
#'   Vol. 104, pp. 111–127.
#'
#'   Chang, J., Guo, B. & Yao, Q. (2018). \emph{Principal component analysis for
#'   second-order stationary vector time series}, The Annals of Statistics, Vol.
#'   46, pp. 2094–2124.
#'
#'   Cai, T. and Liu, W. (2011). \emph{Adaptive thresholding for sparse
#'   covariance matrix estimation},  Journal of the American Statistical
#'   Association, Vol. 106, pp. 672--684.
#'
#' @examples
#' n <- 200
#' p <- 10
#' X <- matrix(rnorm(n*p),n,p)
#' res <- WN_test(X)
#' Pvalue <- res$p.value
#' rej <- res$reject
#' @useDynLib HDTSA
#' @importFrom Rcpp sourceCpp
#' @importFrom Rcpp evalCpp
#' @import Rcpp
#' @export

WN_test = function(X, lag.k=2, B=2000, 
                   kernel.type=c('QS','Par','Bart'), 
                   pre=FALSE,alpha=0.05,k0=5,thresh=FALSE,
                   tuning.vec=NULL){
  
  kernel.type <- match.arg(kernel.type)
  ken_type <- switch(kernel.type,
                     "QS" = 1,
                     "Par" = 2,
                     "Bart" = 3)
  
  if (pre==T){
    X_pre = PCA4_TS(X, lag.k=k0, 
                    thresh=thresh, just4pre = pre, 
                    tuning.vec = tuning.vec)
    X = X_pre$X
    Bx = X_pre$B
  }
  n = nrow(X)
  p = ncol(X)
  
  Tn_list = WN_teststatC(X,n,p,lag.k)
  Tn = Tn_list$Tn
  sigma_zero = Tn_list$sigma_zero
  X_mean = Tn_list$X_mean
  
  ft = WN_ftC(n,lag.k,p,X,X_mean)
  bn = WN_bandwith(ft,lag.k,p,ken_type)
  
  Gnstar = WN_bootc(n,lag.k,p,B,bn,ken_type,ft,X,sigma_zero) # critical value
  p.value = mean(Gnstar>Tn)
  Results = list(reject = (p.value<0.05), p.value = p.value)
  
  return(Results)
}