Fitting a function to time-dependent ensemble averaged data

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 d...

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Autores principales:	Karl Fogelmark, Michael A. Lomholt, Anders Irbäck, Tobias Ambjörnsson
Formato:	article
Lenguaje:	EN
Publicado:	Nature Portfolio 2018
Materias:	Medicine R Science Q
Acceso en línea:	https://doaj.org/article/4bf7b352ae46409b8ae46a9f37d8019e
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spelling	oai:doaj.org-article:4bf7b352ae46409b8ae46a9f37d8019e2021-12-02T15:07:52ZFitting a function to time-dependent ensemble averaged data10.1038/s41598-018-24983-y2045-2322https://doaj.org/article/4bf7b352ae46409b8ae46a9f37d8019e2018-05-01T00:00:00Zhttps://doi.org/10.1038/s41598-018-24983-yhttps://doaj.org/toc/2045-2322Abstract 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.Karl FogelmarkMichael A. LomholtAnders IrbäckTobias AmbjörnssonNature PortfolioarticleMedicineRScienceQENScientific Reports, Vol 8, Iss 1, Pp 1-11 (2018)
institution	DOAJ
collection	DOAJ
language	EN
topic	Medicine R Science Q
spellingShingle	Medicine R Science Q Karl Fogelmark Michael A. Lomholt Anders Irbäck Tobias Ambjörnsson Fitting a function to time-dependent ensemble averaged data
description	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.
format	article
author	Karl Fogelmark Michael A. Lomholt Anders Irbäck Tobias Ambjörnsson
author_facet	Karl Fogelmark Michael A. Lomholt Anders Irbäck Tobias Ambjörnsson
author_sort	Karl Fogelmark
title	Fitting a function to time-dependent ensemble averaged data
title_short	Fitting a function to time-dependent ensemble averaged data
title_full	Fitting a function to time-dependent ensemble averaged data
title_fullStr	Fitting a function to time-dependent ensemble averaged data
title_full_unstemmed	Fitting a function to time-dependent ensemble averaged data
title_sort	fitting a function to time-dependent ensemble averaged data
publisher	Nature Portfolio
publishDate	2018
url	https://doaj.org/article/4bf7b352ae46409b8ae46a9f37d8019e
work_keys_str_mv	AT karlfogelmark fittingafunctiontotimedependentensembleaverageddata AT michaelalomholt fittingafunctiontotimedependentensembleaverageddata AT andersirback fittingafunctiontotimedependentensembleaverageddata AT tobiasambjornsson fittingafunctiontotimedependentensembleaverageddata
_version_	1718388382224089088

Fitting a function to time-dependent ensemble averaged data

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