SuperGauss (version 1.0.2)

msd2acf: Convert mean square displacement to autocorrelations.

Description

Converts the mean squared displacement (MSD) of positions to the autocorrelation (ACF) of the corresponding increments.

Usage

msd2acf(msd)

Arguments

msd

Length-N vector of MSDs at regular timepoints dt, 2*dt, ..., N*dt.

Value

Length N vector of ACFs.

Details

For a stationary increments process \(X_t\), converts a sequence \(\eta_1, \ldots, \eta_N\) of regularly spaced MSDs, $$ \eta_i = E[(X_{i\Delta t} - X_0)^2], $$ into \(\gamma_1, \ldots, \gamma_N\), a sequence of regularly spaced ACFs, $$ \gamma_i = \mathrm{cov}\{X_{(i+1)\Delta t} - X_{i \Delta_i}, X_{\Delta t} - X_{0}\}. $$ This only produces correct results when msd corresponds to equally-spaced observations.

Examples

# NOT RUN {
# autocorrelation of fBM increments
msd2acf(msd = fbm.msd(tseq = 0:10, H = .3))
# }