Calculate the autocorrelation of the fARMA model.

Compute the autocorrelation of fARMA(p,q) process increments at equally-spaced time points (see 'Details').

farma_acf(alpha, phi = numeric(), rho = numeric(), dt, N, m = 50)

Arguments

alpha

Subdiffusion exponent of the underlying fBM process. A scalar between 0 and 2.

phi

A vector of p >= 0 autoregressive (AR) coefficients.

rho

A vector of q >= 0 moving-average (MA) coefficients (see 'Details').

dt

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

N

Number of observations (positive integer).

m

Number of MA coefficients used for approximating the ARMA filter (see Details).

Value

A vector of N autocorrelation values.

Details

Let X_n denote the position of an fBM process at time t = n * dt. The fARMA(p,q) process Y_n is then defined as

Y_n = rho_0 X_n + sum_{i=1}^p phi_i Y_{n-i} + sum_{j=1}^q + rho_j X_{n-j}.

where rho_0 = 1 - sum(phi) - sum(rho).

This function returns the autocorrelation of the stationary process dY_n = Y_{n+1} - Y_n. The ARMA(p,q) filter is approximated with m moving-average terms (see arma_acf() for details).

References

Ling, Y., Lysy, M., Seim, I., Newby, J.M., Hill, D.B., Cribb, J., and Forest, M.G. "Measurement error correction in particle tracking microrheology" (2019). https://arxiv.org/abs/1911.06451.