R/data_gen_AR.R

In zejiang-unsw/WASP: Wavelet System Prediction

# An autoregressive model of order p, AR(p), can be written as
# yt=c+<U+03D5>1y_t-1+<U+03D5>2y_t-2+<U+22EF>+<U+03D5>py_t-p+et,
# where et is white noise.
# For an AR(1) model:
# •	when <U+03D5>1=0, yt is equivalent to white noise;
# •	when <U+03D5>1=1 and c=0, yt is equivalent to a random walk;
# •	when <U+03D5>1=1 and c<U+2260>0, yt is equivalent to a random walk with drift;
# •	when <U+03D5>1<0, yt tends to oscillate around the mean.
# We normally restrict autoregressive models to stationary data, in which case some constraints on the values of the parameters are required.
# •	For an AR(1) model: -1<<U+03D5>1<1.
# •	For an AR(2) model: -1<<U+03D5>2<1, <U+03D5>1+<U+03D5>2<1, <U+03D5>2-<U+03D5>1<1.
# When p=3, the restrictions are much more complicated. R takes care of these restrictions when estimating a model.

#' Generate predictor and response data from AR1 model.
#'
#' @param nobs The data length to be generated.
#' @param ndim The number of potential predictors (default is 9).
#'
#' @return A list of 2 elements: a vector of response (x), and a matrix of potential predictors (dp) with each column containing one potential predictor.
#' @export
#'
#' @examples
#' # AR1 model from paper with 9 dummy variables
#' data.ar1 <- data.gen.ar1(500)
#' plot.ts(cbind(data.ar1$x, data.ar1$dp))
data.gen.ar1 <- function(nobs, ndim = 9) {
  nwarm1 <- nwarm2 <- 50
  n <- nobs + nwarm1 + nwarm2
  x <- matrix(0, n, 1)
  for (i in 1:nwarm1) {
    x[i] <- rnorm(1, mean = 0, sd = 1)
  }
  dp <- matrix(0, (nobs), ndim)
  for (i in (nwarm1 + 1):n) {
    eps <- rnorm(1, mean = 0, sd = 1)
    x[i] <- 0.9 * x[i - 1] + 0.866 * eps
  }
  for (i in 1:ndim) dp[, i] <- x[(n - i - nobs + 1):(n - i)]
  x <- x[(n - nobs + 1):n]
  data_generated <- list(x = x, dp = dp, true.cpy = c(1))
  return(data_generated)
}

#' Generate predictor and response data from AR4 model.
#'
#' @param nobs The data length to be generated.
#' @param ndim The number of potential predictors (default is 9).
#'
#' @return A list of 2 elements: a vector of response (x), and a matrix of potential predictors (dp) with each column containing one potential predictor.
#' @export
#'
#' @examples
#' # AR4 model from paper with total 9 dimensions
#' data.ar4 <- data.gen.ar4(500)
#' plot.ts(cbind(data.ar4$x, data.ar4$dp))
data.gen.ar4 <- function(nobs, ndim = 9) {
  nwarm1 <- nwarm2 <- 50
  n <- nobs + nwarm1 + nwarm2
  x <- matrix(0, n, 1)
  for (i in 1:nwarm1) {
    x[i] <- rnorm(1, mean = 0, sd = 1)
  }
  dp <- matrix(0, (nobs), ndim)
  for (i in (nwarm1 + 1):n) {
    eps <- rnorm(1, mean = 0, sd = 1)
    x[i] <- 0.6 * x[i - 1] - 0.4 * x[i - 4] + eps
  }
  for (i in 1:ndim) dp[, i] <- x[(n - i - nobs + 1):(n - i)]
  x <- x[(n - nobs + 1):n]
  data_generated <- list(x = x, dp = dp, true.cpy = c(1, 4))
  return(data_generated)
}

#' Generate predictor and response data from AR9 model.
#'
#' @param nobs The data length to be generated.
#' @param ndim The number of potential predictors (default is 9).
#'
#' @return A list of 2 elements: a vector of response (x), and a matrix of potential predictors (dp) with each column containing one potential predictor.
#' @export
#'
#' @examples
#' # AR9 model from paper with total 9 dimensions
#' data.ar9 <- data.gen.ar9(500)
#' plot.ts(cbind(data.ar9$x, data.ar9$dp))
data.gen.ar9 <- function(nobs, ndim = 9) {
  nwarm1 <- nwarm2 <- 50
  n <- nobs + nwarm1 + nwarm2
  x <- matrix(0, n, 1)
  for (i in 1:nwarm1) {
    x[i] <- rnorm(1, mean = 0, sd = 1)
  }
  dp <- matrix(0, (nobs), ndim)
  for (i in (nwarm1 + 1):n) {
    eps <- rnorm(1, mean = 0, sd = 1)
    x[i] <- 0.3 * x[i - 1] - 0.6 * x[i - 4] - 0.5 * x[i - 9] + eps
  }
  for (i in 1:ndim) dp[, i] <- x[(n - i - nobs + 1):(n - i)]
  x <- x[(n - nobs + 1):n]
  data_generated <- list(x = x, dp = dp, true.cpy = c(4, 9, 1))
  return(data_generated)
}

#' Generate predictor and response data from TAR1 model.
#'
#' @param nobs  The data length to be generated.
#' @param ndim  The number of potential predictors (default is 9).
#' @param noise The white noise in the data
#'
#' @return A list of 2 elements: a vector of response (x), and a matrix of potential predictors (dp) with each column containing one potential predictor.
#' @export
#'
#' @references Sharma, A. (2000). Seasonal to interannual rainfall probabilistic forecasts for improved water supply management: Part 1—A strategy for system predictor identification. Journal of Hydrology, 239(1-4), 232-239.
#'
#' @examples
#' # TAR1 model from paper with total 9 dimensions
#' data.tar1 <- data.gen.tar1(500)
#' plot.ts(cbind(data.tar1$x, data.tar1$dp))
data.gen.tar1 <- function(nobs, ndim = 9, noise = 0.1) {
  nwarm1 <- nwarm2 <- 50
  n <- nobs + nwarm1 + nwarm2
  x <- matrix(0, n, 1)
  for (i in 1:nwarm1) {
    x[i] <- rnorm(1, mean = 0, sd = 1)
  }
  dp <- matrix(0, (nobs), ndim)
  for (i in (nwarm1 + 1):n) {
    eps <- rnorm(1, mean = 0, sd = 1)
    xi3 <- x[i - 3]
    if (xi3 <= 0) x[i] <- -0.9 * x[i - 3] + noise * eps else x[i] <- 0.4 * x[i - 3] + noise * eps
  }
  for (i in 1:ndim) dp[, i] <- x[(n - i - nobs + 1):(n - i)]
  x <- x[(n - nobs + 1):n]
  data_generated <- list(x = x, dp = dp, true.cpy = c(3))
  return(data_generated)
}

#' Generate predictor and response data from TAR2 model.
#'
#' @param nobs  The data length to be generated.
#' @param ndim  The number of potential predictors (default is 9).
#' @param noise The white noise in the data
#'
#' @return A list of 2 elements: a vector of response (x), and a matrix of potential predictors (dp) with each column containing one potential predictor.
#' @export
#'
#' @references Sharma, A. (2000). Seasonal to interannual rainfall probabilistic forecasts for improved water supply management: Part 1—A strategy for system predictor identification. Journal of Hydrology, 239(1-4), 232-239.
#'
#' @examples
#' # TAR2 model from paper with total 9 dimensions
#' data.tar2 <- data.gen.tar2(500)
#' plot.ts(cbind(data.tar2$x, data.tar2$dp))
data.gen.tar2 <- function(nobs, ndim = 9, noise = 0.1) {
  nwarm1 <- nwarm2 <- 50
  n <- nobs + nwarm1 + nwarm2
  x <- matrix(0, n, 1)
  for (i in 1:nwarm1) {
    x[i] <- rnorm(1, mean = 0, sd = 1)
  }
  dp <- matrix(0, (nobs), ndim)
  for (i in (nwarm1 + 1):n) {
    eps <- rnorm(1, mean = 0, sd = 1)
    xi6 <- x[i - 6]
    if (xi6 <= 0) x[i] <- -0.5 * x[i - 6] + 0.5 * x[i - 10] + noise * eps else x[i] <- 0.8 * x[i - 10] + noise * eps
  }
  for (i in 1:ndim) dp[, i] <- x[(n - i - nobs + 1):(n - i)]
  x <- x[(n - nobs + 1):n]
  data_generated <- list(x = x, dp = dp, true.cpy = c(10, 6))
  return(data_generated)
}