Fitting a function to time-dependent ensemble averaged data

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...

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Bibliographic Details
Published in:Scientific reports 2018-05, Vol.8 (1), p.6984-11, Article 6984
Main Authors: Fogelmark, Karl, Lomholt, Michael A., Irbäck, Anders, Ambjörnsson, Tobias
Format: Article
Language:English
Subjects:
631/57/2265
639/766/530
639/766/747
Annan fysik
Biofysiks
Biologi
Biological Sciences
Biophysics
Brownian motion
Fysik
Humanities and Social Sciences
Least squares method
Matematik
Mathematics
multidisciplinary
Natural Sciences
Naturvetenskap
Other Physics Topics
Parameter estimation
Physical Sciences
Probability Theory and Statistics
Sannolikhetsteori och statistik
Science
Science (multidisciplinary)
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Summary: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.
ISSN:2045-2322
2045-2322
DOI:10.1038/s41598-018-24983-y