Advanced

Fitting a function to time-dependent ensemble averaged data

Fogelmark, Karl LU ; Lomholt, Michael A.; Irbäck, Anders LU and Ambjörnsson, Tobias LU (2018) In Scientific Reports 8(1).
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... (More)

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.

(Less)
Please use this url to cite or link to this publication:
author
organization
publishing date
type
Contribution to journal
publication status
published
subject
in
Scientific Reports
volume
8
issue
1
publisher
Nature Publishing Group
external identifiers
  • scopus:85046427281
ISSN
2045-2322
DOI
10.1038/s41598-018-24983-y
language
English
LU publication?
yes
id
27f1fa2b-911a-4329-85bc-3c307522f55c
date added to LUP
2018-05-15 08:59:54
date last changed
2018-05-16 03:00:08
@article{27f1fa2b-911a-4329-85bc-3c307522f55c,
  abstract     = {<p>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.</p>},
  articleno    = {6984},
  author       = {Fogelmark, Karl and Lomholt, Michael A. and Irbäck, Anders and Ambjörnsson, Tobias},
  issn         = {2045-2322},
  language     = {eng},
  month        = {12},
  number       = {1},
  publisher    = {Nature Publishing Group},
  series       = {Scientific Reports},
  title        = {Fitting a function to time-dependent ensemble averaged data},
  url          = {http://dx.doi.org/10.1038/s41598-018-24983-y},
  volume       = {8},
  year         = {2018},
}