# R/WhiteNoise.R In weakARMA: Tools for the Analysis of Weak ARMA Models

#### Documented in simGARCH

#' Weak white noise
#' @description Simulates an uncorrelated but dependant noise process.
#' @param n Number of observations.
#' @param sigma Standard deviation.
#' @param k Integer \eqn{\neq 0} to prevent a zero denominator.
#' @param ninit Length of 'burn-in' period.
#'
#' @importFrom stats rnorm
#'
#' @export
#'
#' @return Vector of size \code{n} containing  a nonlinear sequence \eqn{X_{i}} such as
#'   \eqn{X_i = \frac{Z_{i}}{|Z_{i+1}| + k}} , where \eqn{Z_{i}} is a sequence of iid
#'     random variables mean-zero random variable with variance \eqn{\sigma^2}.
#'
#' @references Romano, J. and Thombs, L. 1996, Inference for autocorrelation under weak assumptions,
#'  \emph{Journal of the American Statistical Association}, vol. 91, no. 434, pp. 590-600
#'
#' @examples
#' wnRT(100)
#' wnRT(100, sigma = 1)
wnRT <- function (n, sigma = 1, k = 1, ninit = 100)
{
eps <- rep(0, (n + ninit))
eta <- rnorm((n + ninit + 1), sd = sigma)
for (t in 1:(n + ninit)) {
eps[t] <- eta[t + 1] / (abs(eta[t]) + k)}
return(eps[(ninit + 1): (n + ninit)])
}

#' Weak white noise
#' @description Simulates an uncorrelated but dependant noise process.
#' @param n Number of observations.
#' @param sigma Standard deviation.
#' @param k Integer corresponding to the number of past observation will be used.
#' @param ninit Length of 'burn-in' period.
#'
#' @importFrom stats rnorm
#'
#' @export
#'
#' @return Vector of size \code{n} containing  a nonlinear sequence \eqn{X_{i}} such as
#'   \eqn{X_{i} = Z_{i}Z_{i-1}...Z_{i-k}} , where \eqn{Z_{i}} is a sequence of iid
#'     random variables mean-zero random variable with variance \eqn{\sigma^2}.
#'
#' @references Romano, J. and Thombs, L. 1996, Inference for autocorrelation under weak assumptions,
#'  \emph{Journal of the American Statistical Association}, vol. 91, no. 434, pp. 590-600
#'
#' @examples
#' wnPT(100)
#' wnPT(100, sigma = 1, k = 1)
#' wnPT(100, k = 0) #strong noise
wnPT <- function (n, sigma = 1, k = 1, ninit = 100)
{
eps <- rep(0, (n + ninit))
eta <- rnorm((n + ninit) , sd = sigma)
for(t in (k + 1):(n + ninit))
eps[t] <- prod(eta[t:(t - k)])
return(eps[(ninit + 1) : (n + ninit)])
}

#' Weak white noise
#' @description Simulates an uncorrelated but dependant noise process.
#' @param n Number of observations.
#' @param sigma Standard deviation.
#' @param k Integer corresponding to the number of past observation will be used.
#' @param ninit Length of 'burn-in' period.
#'
#' @importFrom stats rnorm
#' @export
#' @return Vector of size \code{n} containing  a nonlinear sequence \eqn{X_{i}} such as
#'   \eqn{X_{i} = Z^{2}_iZ_{i-1}...Z_{i-k}} , where \eqn{Z_{i}} is a sequence of iid
#'     random variables mean-zero random variable with variance \eqn{\sigma^2}.
#'
#' @references Romano, J. and Thombs, L. 1996, Inference for autocorrelation under weak assumptions,
#'  \emph{Journal of the American Statistical Association}, vol. 91, no. 434, pp. 590-600
#'
#' @examples
#' wnPT_SQ(100)
#' wnPT_SQ(100, sigma = 1, k = 1)
wnPT_SQ <- function (n, sigma = 1, k = 1, ninit=100)
{
eps <- rep(0, (n + ninit))
eta <- rnorm(n + ninit, sd = sigma)
for(t in (k + 1):(n + ninit))
eps[t] <- eta[t]*prod(eta[t:(t - k)])
return(eps[(ninit + 1) : (n + ninit)])
}

#' GARCH process
#' @description Simulates a GARCH process which is an example of a weak white noise.
#'
#' @param n Number of observations.
#' @param c Positive number.
#' @param A Vector of ARCH coefficients >=0.
#' @param B Vector of GARCH coefficients >=0. If \code{NULL}, the
#'   simulation is a ARCH process.
#' @param ninit Length of 'burn-in' period.
#'
#'
#' @importFrom stats rnorm
#' @export
#'
#' @return Vector of size \code{n} containing  a nonlinear sequence \eqn{\epsilon_t} such as
#'   \deqn{\epsilon_{t} = H_{t}^{1 / 2}  \eta_{t}} where \deqn{H_{t} =  c +
#'   a_{1}\epsilon_{t - 1}^ {2}+...+a_{q}\epsilon_{t - q} ^{2} + b_{1}H_{t-1}+...+ b_{p}H_{t-p}}
#'
#' @references Francq C. and Zakoïan J.M., 2010, \emph{GARCH models: structure, statistical inference and financial applications}
#'
#'
#'
#' @examples
#' simGARCH(100, c = 1, A = 0.25)
#' simGARCH(100, c = 1, A = 0.1,  B = 0.88)

simGARCH <- function(n, c, A, B = NULL, ninit = 100)
{
q <- length(A)
p <- length(B)

if (missing(B)) {B <- 0 ; p <- 0}

eps <- rep(0 , n + ninit)
eta <- rnorm(n + ninit, sd = 1)
H <- rep(0, n + ninit)
H[1] <- c
eps[1] <- H[1] * (eta[1])

for (t in 2:(n + ninit)){
H[t] <- c + sum(A[1:min((t-1),q)] * (eps[(t-1):max(1,(t-q))])^2) + sum(B[1:min((t-1),p)] * H[(t-1):max(1,(t-p))])
eps[t] <- sqrt(H[t]) * eta[t]
}

eps <- eps[(1 + ninit):(n + ninit)]
return(eps)
}

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weakARMA documentation built on April 5, 2022, 1:16 a.m.