On the Estimation of Parameters from Time Traces originating from an Ornstein-Uhlenbeck Process

Abstract

In this article, we develop a Bayesian approach to estimate parameters from time traces that originate from an overdamped Brownian particle in a harmonic potential, or Ornstein-Uhlenbeck process (OU). We show that least-square fitting the autocorrelation function, which is often the standard way of analyzing such data, is significantly underestimating the confidence intervals of the fitted parameters. Here, we develop a rigorous maximum likelihood theory that properly captures the underlying statistics. From the analytic solution, we found that there exists an optimal measurement spacing ($\Delta t = 0.7968 \tau$) that maximizes the statistical accuracy of the estimate for the decay-time $\tau$ of the process for a fixed number of samples $N$, which plays a similar role than the Nyquist-Shannon theorem for the OU-process. In summary, our results have strong implications for parameter estimation for processes that result in a single exponential decay in the autocorrelation function. Our analysis can directly be applied to single-component dynamic light scattering experiments or optical trap calibration experiments.