Optimal estimation of drift and diffusion coefficients in the presence of static localization error

We consider the inference of the drift velocity and the diffusion coefficient of a particle undergoing a directed random walk in the presence of static localization error. A weighted least-squares fit to mean-square displacement (MSD) data is used to infer the parameters of the assumed drift-diffusion model. For experiments which cannot be repeated we show that the quality of the inferred parameters depends on the number of MSD points used in the fitting. An optimal number of fitting points p_{opt} is shown to exist which depends on the time interval between frames Δt and the unknown parameters. We therefore also present a simple iterative algorithm which converges rapidly toward p_{opt}. For repeatable experiments the quality depends crucially on the measurement time interval over which measurements are made, reflecting the different timescales associated with drift and diffusion. An optimal measurement time interval T_{opt} exists, which depends on the number of measurement points and the unknown parameters, and so again we present an iterative algorithm which converges quickly toward T_{opt} and is shown to be robust to initial parameter guesses.