R/generate_data_brownian.R

In fChange: Functional Change Point Detection and Analysis

#' Generate a Brownian Motion Process
#'
#' Generate a functional time series according to an iid Brownian Motion process.
#'  Each observation is discretized on the points indicated in \code{v}.
#'
#' @param N Numeric. The number of observations for the generated data.
#' @param v Numeric (vector). Discretization points of the curves.This can be
#'  the evaluated points or the number of evenly spaced points on \[0,1\].
#'  By default it is evenly spaced on \[0,1\] with 30 points.
#' @param sd Numeric. Standard deviation of the Brownian Motion process.
#'  The default is \code{1}.
#'
#' @return Functional time series (dfts) object.
#' @export
#'
#' @examples
#' bmotion <- generate_brownian_motion(
#'   N = 100,
#'   v = c(0, 0.25, 0.4, 0.7, 1, 1.5), sd = 1
#' )
#' bmotion1 <- generate_brownian_motion(
#'   N = 100,
#'   v = 10, sd = 2
#' )
generate_brownian_motion <- function(N, v = 30, sd = 1) {
  # Convert resolution to equally spaced points
  if (length(v) == 1) {
    v <- seq(0, 1, length.out = v)
  }
  res <- length(v)

  # Generate data
  if (N > 1) {
    data <- apply(
      t(sapply(c(0, diff(v)),
        function(d, N, sd) {
          stats::rnorm(N, sd = sd * sqrt(d))
        },
        N = N, sd = sd
      )),
      MARGIN = 2, cumsum
    )
  } else {
    data <- cumsum(sapply(c(0, diff(v)),
      function(d, N, sd) {
        stats::rnorm(N, sd = sd * sqrt(d))
      },
      N = N, sd = sd
    ))
  }

  dfts(X = as.matrix(data), fparam = v)
}


#' Generate a Brownian Bridge Process
#'
#' Generate a functional time series from an iid Brownian Bridge process.
#'  If \eqn{W(t)} is a Wiener process, the Brownian Bridge is
#'  defined as \eqn{W(t) - tW(1)}. Each functional observation is discretized on
#'  the points indicated in \code{v}.
#'
#' @inheritParams generate_brownian_motion
#'
#' @return Functional time series (dfts) object.
#' @export
#'
#' @examples
#' bbridge <- generate_brownian_bridge(N = 100, v = c(0, 0.2, 0.6, 1, 1.3), sd = 2)
#' bbridge <- generate_brownian_bridge(N = 100, v = 10, sd = 1)
generate_brownian_bridge <- function(N, v = 30, sd = 1) {
  # Convert resolution to equally spaced points
  if (length(v) == 1) {
    v <- seq(0, 1, length.out = v)
  }
  res <- length(v)

  # Generate Data
  data <- generate_brownian_motion(N, v = v, sd)$data
  data <- data -
    t(data[res, ] * t(matrix(rep(v, times = N) / max(v), ncol = N, nrow = res)))

  dfts(X = data, fparam = v)
}


#' Second-Order Brownian Bridge
#'
#' @inheritParams generate_brownian_motion
#'
#' @return dfts object
#'
#' @noRd
#' @keywords internal
.generate_brownian_bridge_second_order <- function(
    N, v = seq(from = 0, to = 1, length.out = 100), sd = 1) {
  if (length(v) == 1) {
    v <- seq(0, 1, length.out = v)
  }

  n <- length(v)
  dt <- diff(v)
  t <- seq(v[1], v[n], length = n + 1)

  BB <- sapply(1:N, function(m, v, n) {
    X <- c(0, cumsum(stats::rnorm(n - 1) * sqrt(dt) * sd))
    v[1] + X - (v - v[1]) / (v[n] - v[1]) * (X[n] - v[1] + v[1])
  }, v = v, n = n)

  dfts(X = BB, fparam = v)
}


#' Generate Second-Level Brownian Motion
#'
#' Generates a second-order Brownian Motion.
#'
#' @param M Numeric number of simulations
#' @param v Observation points
#'
#' @return dfts object of the Brownian Motion
#'
#' @references MacNeill, I. B. (1978). Properties of Sequences of Partial Sums
#'  of Polynomial Regression Residuals with Applications to Tests for Change of
#'  Regression at Unknown Times. The Annals of Statistics, 6(2), 422-433.
#'
#' @keywords internal
#' @noRd
.generate_second_level_brownian_bridge <- function(N, v, sd = 1) {
  W <- generate_brownian_motion(N, v = v, sd = sd)
  # W <- generate_data(
  #   general=list('resolution'=v,'burnin'=0),
  #   parameters =list(list('N'=N, 'process'='bmotion', 'sd'=sd))
  # )
  V <- W$data +
    (2 * v - 3 * v^2) %*% t(W$data[nrow(W$data), ]) +
    (-6 * v + 6 * v^2) %*% t(dot_integrate_col(W$data, v))

  dfts(X = V, fparam = v)
}