  Traditionally, time-development of the mean square displacement has been
employed to determine the diffusion coefficient from the trajectories of single
particles. However, this approach is sensitive to the noise and the motion blur
upon image acquisition. Recently, Vestergaard et al. has proposed a novel
method based on the covariance between the shifted displacement series. This
approach gives a more robust estimator of the diffusion coefficient of
one-dimensional diffusion without bias, i.e., when mean velocity is zero. Here,
we extend this approach to a potentially biased random walk on a
two-dimensional lattice. First, we describe the relationship between the
hopping rates to the eight adjacent sites and the time development of the
higher-order moments of the stochastic two-dimensional displacements. Then, we
derive the covariance-based estimators for these higher-order moments.
Numerical simulations confirmed that the procedure presented here allows
inference of the stochastic hopping rates from two-dimensional trajectory data
with location error and motion blur.
