Fitting a function to time-dependent ensemble averaged data

TL;DR

This paper introduces a new method, WLS-ICE, for fitting time-dependent ensemble averaged data that accurately accounts for temporal correlations, improving parameter estimation and error analysis over existing methods.

Contribution

The authors derive a closed-form error estimation formula for weighted least squares fitting that includes temporal correlations, addressing limitations of current methods.

Findings

WLS-ICE accurately estimates parameters in Brownian motion, harmonic oscillation, fractional Brownian motion, and continuous time random walks.

The method outperforms existing approaches in robustness and accuracy.

WLS-ICE is applicable to arbitrary fit functions and is supported by publicly available software.

Abstract

Time-dependent ensemble averages, i.e., trajectory-based averages of some observable, are of importance in many fields of science. A crucial objective when interpreting such data is to fit these averages (for instance, squared displacements) with a function and extract parameters (such as diffusion constants). A commonly overlooked challenge in such function fitting procedures is that fluctuations around mean values, by construction, exhibit temporal correlations. We show that the only available general purpose function fitting methods, correlated chi-square method and the weighted least squares method (which neglects correlation), fail at either robust parameter estimation or accurate error estimation. We remedy this by deriving a new closed-form error estimation formula for weighted least square fitting. The new formula uses the full covariance matrix, i.e., rigorously includes temporal correlations, but is free of the robustness issues, inherent to the correlated chi-square method. We demonstrate its accuracy in four examples of importance in many fields: Brownian motion, damped harmonic oscillation, fractional Brownian motion and continuous time random walks. We also successfully apply our method, weighted least squares including correlation in error estimation (WLS-ICE), to particle tracking data. The WLS-ICE method is applicable to arbitrary fit functions, and we provide a publically available WLS-ICE software.