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

TL;DR

This paper introduces a Bayesian maximum likelihood method for accurately estimating parameters of an Ornstein-Uhlenbeck process from time traces, revealing an optimal sampling interval and improving confidence interval estimation over traditional least-squares fitting.

Contribution

The authors develop a rigorous Bayesian maximum likelihood approach for parameter estimation in OU processes, identifying an optimal measurement spacing that enhances statistical accuracy.

Findings

Optimal measurement spacing is approximately 0.8 times the decay time τ.

Traditional least-squares fitting underestimates confidence intervals.

The method improves parameter estimation in single exponential decay processes.

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.