Calculate the mean square displacement of the fSD model.

Compute the MSD of the Savin & Doyle (2005) localization error model with fBM increments (see fsd_acf()).

fsd_msd(t, dt, alpha, tau, sigma2)

Arguments

t

Vector of timepoints at which to calculate the MSD.

dt

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

alpha

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

tau

The ratio between camera shutter open time and the interobservation time dt. A scalar between 0 and 1.

sigma2

The magnitude of the static error. A positive scalar.

Value

Vector of MSD values the same length as t.

Details

Let X_t denote the true position of an fBM process at time t. The Savin-Doyle localization error model describes the measured position Y_n at time t = n dt as

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

where eps_n ~iid N(0,1) is a Gaussian white noise process. The resulting MSD as a function of tau and sigma2 = sigma^2 is what gets passed to the csi_model base class to construct the fsd_model derived class.

References

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