R/fbm-msd.R

Defines functions fbm.msd

Documented in fbm.msd

#' Mean square displacement of fractional Brownian motion.
#'
#' @param tseq Length-\code{N} vector of timepoints.
#' @param H Hurst parameter (between 0 and 1).
#' @return Length-\code{N} vector of mean square displacements.
#' @details The mean squared displacement (MSD) of a stochastic process \eqn{X_t} is defined as
#' \deqn{
#' \mathrm{\scriptsize MSD}_X(t) = E[(X_t - X_0)^2].
#' }{
#' MSD_X(t) = E[(X_t - X_0)^2].
#' }
#' Fractional Brownian motion (fBM) is a continuous Gaussian process with stationary increments, such that its covariance function is entirely defined the MSD, which in this case is \eqn{\textrm{\small MSD}_X(t) = |t|^{2H}}{MSD_X(t) = |t|^(2H)}.
#' @examples
#' fbm.msd(tseq = 1:10, H = 0.4)
#' @export
fbm.msd <- function(tseq, H) {
  abs(tseq)^(2*H)
}

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SuperGauss documentation built on May 1, 2019, 7:58 p.m.