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Estimation of the Diffusion Constant from Intermittent Trajectories with Variable Position Uncertainties

The movement of a particle described by Brownian motion is quantified by a single parameter, $D$, the diffusion constant. The estimation of $D$ from a discrete sequence of noisy observations is a fundamental problem in biological single particle tracking experiments since it can report on the environment and/or the state of the particle itself via hydrodynamic radius. Here we present a method to estimate $D$ that takes into account several effects that occur in practice, that are important for correct estimation of $D$, and that have hitherto not been combined together for estimation of $D$. These effects are motion blur from finite integration time of the camera, intermittent trajectories, and time-dependent localization uncertainty. Our estimation procedure, a maximum likelihood estimation, follows directly from the likelihood expression for a discretely observed Brownian trajectory that explicitly includes these effects. The manuscript begins with the formulation of the likelihood expression and then presents three methods to find the exact solution. Each method has its own advantages in either computational robustness, theoretical insight, or the estimation of hidden variables. We then compare our estimator to previously published estimators using a squared log loss function to demonstrate the benefit of including these effects.

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