Calculate the autocorrelation of the fSD model.

Compute the autocorrelation of the Savin & Doyle (2005) localization error model with fBM increments (see 'Details').

fsd_acf(alpha, tau, sigma2, dt, N)

Arguments

alpha

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

tau

The ratio between camera shutter open time and interobservation time dt (see 'Details'). A scalar between 0 and 1.

sigma2

The magnitude of the static error (see 'Details'). A positive scalar.

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 the position of an fBM process at time t. The Savin-Doyle localization error model describes the observed position Y_n at time t = n * dt as

Y_n = sigma * eps_n + 1/tau * int_0^tau X_{n*dt + s} ds,

where eps_n ~iid N(0,1) is a Gaussian white noise process.

This function returns the autocorrelation of the stationary process dY_n = Y_{n+1} - Y_n.

References

Savin, T., and Doyle, P.S. "Static and dynamic errors in particle tracking microrheology." Biophysical Journal 88.1 (2005): 623-638.