Fitting a function to time-dependent ensemble averaged data

Karl Fogelmark,Anders Irbäck,Michael A Lomholt,Tobias Ambjörnsson

doi:10.1038/s41598-018-24983-y

Karl Fogelmark, Anders Irbäck + Show 2 more

Open Access

https://doi.org/10.1038/s41598-018-24983-y

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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.

Highlights

Time-dependent ensemble averages, i.e., trajectory-based averages of some observable, are of importance in many fields of science
We demonstrate that such neglect leads to unreliable error estimation for parameters and can in some cases lead to underestimated errors for fitted parameters by more than one order of magnitude for our prototype systems
Our first test of the fitting methods involve comparing histograms of fitted parameters for our four prototype systems. For both correlated chi-square method (CCM) and weighted least squares (WLS) the S fitted values of a given parameter were binned to a histogram, see Fig. 1, and compared to a Gaussian centered on the mean of the estimated parameters with a variance from the average of the error estimates, using either the WLS-ECE or WLS-ICE method

Summary

Methods

The random walk’s guide to anomalous diffusion: a fractional dynamics approach. Journal of Physics A: Mathematical and General 37, R161 (2004). Improved estimation of anomalous diffusion exponents in single-particle tracking experiments. Anomalous diffusion models and their properties: non-stationarity, nonergodicity, and ageing at the centenary of single particle tracking. The Journal of Chemical Physics 91, 461–466 (1989). G. Stochastic processes in physics and chemistry, vol 1 (Elsevier, 1992). S. Static and dynamic errors in particle tracking microrheology. A. Apparent subdiffusion inherent to single particle tracking. P. Motion blur filtering: A statistical approach for extracting confinement forces and diffusivity from a single blurred trajectory.

Results

Method