Calculate the autocorrelation of the fBM model.

Compute the autocorrelation of fractional Brownian motion (fBM) increments at equally-spaced time points (see 'Details').

fbm_acf(alpha, dt, N)

Arguments

alpha

Subdiffusion exponent. A scalar between 0 and 2.

dt

Interobservation time \(\Delta t\) = 1/fps (positive scalar).

N

Number of observations (positive integer).

Value

A vector of N autocorrelation values.

Details

Let X_t denote an fBM process and dX_n = X_{dt * (n+1)} - X_{dt * n} denote the nth increment of X_t with interobservation time dt. The autocorrelation of the fBM increment process dX_n is given by

acf_dX(n) = 0.5 * dt^alpha * (|n+1|^alpha + |n-1|^alpha - 2|n|^\alpha).