R/AVNS_theo_covmat.R

In avar: Allan Variance

#' @title Calculate Theoretical Covariance Matrix of AR(1) Blocks Process
#' @description
#' This function allows us to calculate the theoretical covariance matrix of
#' a non-stationary AR(1) blocks process.
#' @export
#' @usage covmat_ar1blocks(n_total, n_block, phi, sigma2)
#' @param n_total An \code{integer} indicating the length of the whole AR(1) blocks process.
#' @param n_block An \code{integer} indicating the length of each block of the AR(1) blocks process.
#' @param phi A \code{double} value for the autocorrection parameter \eqn{\phi}{phi}.
#' @param sigma2 A \code{double} value for the variance parameter \eqn{\sigma ^2}{sigma^2}.
#' @return The theoretical covariance \code{matrix} of the AR(1) blocks process.
#' @note This function helps calculate the theoretical covariance matrix of a non-stationary
#' process, AR(1) blocks. It is helpful to calculate the theoretical allan variance of
#' non-stationary processes, which can be used to compare with the theoretical allan variance
#' of stationary processes as shown in "A Study of the Allan Variance for Constant-Mean Non-Stationary
#' Processes" by Xu et al., 2017, IEEE Signal Processing Letters, 24(8): 1257–1260.
#' @author Yuming Zhang
#' @examples
#' covmat1 = covmat_ar1blocks(n_total = 1000, n_block = 10,
#' phi = 0.9, sigma2 = 1)
#' covmat2 = covmat_ar1blocks(n_total = 800, n_block = 20,
#' phi = 0.5, sigma2 = 2)
covmat_ar1blocks = function(n_total, n_block, phi, sigma2){
  a = matrix(rep(0, n_block^2), nrow = n_block)
  for (i in 1:n_block){
    a[,i] = seq(from = 1 - i, length.out = n_block)
  }
  a.diag = diag(a)
  a[upper.tri(a,diag=T)] = 0
  a = a + t(a) + diag(a.diag)
  covmat_single = phi^a

  covmat = sigma2/(1- phi^2)*kronecker(diag(rep(1,floor(n_total/n_block))), covmat_single)
  return(covmat)
}


#' @title Calculate Theoretical Covariance Matrix of Non-Stationary White Noise Process
#' @description
#' This function allows us to calculate the theoretical covariance matrix of a non-stationary
#' white noise process.
#' @export
#' @usage covmat_nswn(sigma2, n_total)
#' @param sigma2 A \code{double} value for the variance parameter \eqn{\sigma ^2}{sigma^2}.
#' @param n_total An \code{integer} indicating the length of the whole non-stationary white noise
#' process.
#' @return The theoretical covariance \code{matrix} of the non-stationary white noise process.
#' @note This function helps calculate the theoretical covariance matrix of a non-stationary
#' process, non-stationary white noise. It is helpful to calculate the theoretical allan variance of
#' non-stationary processes, which can be used to compare with the theoretical allan variance
#' of stationary processes as shown in "A Study of the Allan Variance for Constant-Mean Non-Stationary
#' Processes" by Xu et al., 2017, IEEE Signal Processing Letters, 24(8): 1257–1260.
#' @author Yuming Zhang
#' @examples
#' covmat1 = covmat_nswn(sigma2 = 1, n_total = 1000)
#' covmat2 = covmat_nswn(sigma2 = 2, n_total = 800)
covmat_nswn = function(sigma2, n_total){
  covmat = diag(sigma2, n_total, n_total)
  return(covmat)
}


#' @title Calculate Theoretical Covariance Matrix of Bias-Instability Process
#' @description
#' This function allows us to calculate the theoretical covariance matrix of a bias-instability
#' process.
#' @export
#' @usage covmat_bi(sigma2, n_total, n_block)
#' @param sigma2 A \code{double} value for the variance parameter \eqn{\sigma ^2}{sigma^2}.
#' @param n_total An \code{integer} indicating the length of the whole bias-instability process.
#' @param n_block An \code{integer} indicating the length of each block of the bias-instability
#' process.
#' @return The theoretical covariance \code{matrix} of the bias-instability process.
#' @note This function helps calculate the theoretical covariance matrix of a non-stationary
#' process, bias-instability. It is helpful to calculate the theoretical allan variance of
#' non-stationary processes, which can be used to compare with the theoretical allan variance
#' of stationary processes as shown in "A Study of the Allan Variance for Constant-Mean Non-Stationary
#' Processes" by Xu et al., 2017, IEEE Signal Processing Letters, 24(8): 1257–1260.
#' @author Yuming Zhang
#' @examples
#' covmat1 = covmat_bi(sigma2 = 1, n_total = 1000, n_block = 10)
#' covmat2 = covmat_bi(sigma2 = 2, n_total = 800, n_block = 20)
covmat_bi = function(sigma2, n_total, n_block){
  covmat = kronecker(diag(rep(sigma2,floor(n_total/n_block))),
                     matrix(sigma2, n_block, n_block))
  return(covmat)
}