Document Detail

Power spectrum analysis with least-squares fitting: amplitude bias and its elimination, with application to optical tweezers and atomic force microscope cantilevers.
MedLine Citation:
PMID:  20687755     Owner:  NLM     Status:  MEDLINE    
Optical tweezers and atomic force microscope (AFM) cantilevers are often calibrated by fitting their experimental power spectra of Brownian motion. We demonstrate here that if this is done with typical weighted least-squares methods, the result is a bias of relative size between -2/n and +1/n on the value of the fitted diffusion coefficient. Here, n is the number of power spectra averaged over, so typical calibrations contain 10%-20% bias. Both the sign and the size of the bias depend on the weighting scheme applied. Hence, so do length-scale calibrations based on the diffusion coefficient. The fitted value for the characteristic frequency is not affected by this bias. For the AFM then, force measurements are not affected provided an independent length-scale calibration is available. For optical tweezers there is no such luck, since the spring constant is found as the ratio of the characteristic frequency and the diffusion coefficient. We give analytical results for the weight-dependent bias for the wide class of systems whose dynamics is described by a linear (integro)differential equation with additive noise, white or colored. Examples are optical tweezers with hydrodynamic self-interaction and aliasing, calibration of Ornstein-Uhlenbeck models in finance, models for cell migration in biology, etc. Because the bias takes the form of a simple multiplicative factor on the fitted amplitude (e.g. the diffusion coefficient), it is straightforward to remove and the user will need minimal modifications to his or her favorite least-squares fitting programs. Results are demonstrated and illustrated using synthetic data, so we can compare fits with known true values. We also fit some commonly occurring power spectra once-and-for-all in the sense that we give their parameter values and associated error bars as explicit functions of experimental power-spectral values.
Simon F Nørrelykke; Henrik Flyvbjerg
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Publication Detail:
Type:  Journal Article; Research Support, Non-U.S. Gov't    
Journal Detail:
Title:  The Review of scientific instruments     Volume:  81     ISSN:  1089-7623     ISO Abbreviation:  Rev Sci Instrum     Publication Date:  2010 Jul 
Date Detail:
Created Date:  2010-08-06     Completed Date:  2010-11-22     Revised Date:  -    
Medline Journal Info:
Nlm Unique ID:  0405571     Medline TA:  Rev Sci Instrum     Country:  United States    
Other Details:
Languages:  eng     Pagination:  075103     Citation Subset:  IM    
Max Planck Institute for the Physics of Complex Systems, 01187 Dresden, Germany.
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MeSH Terms
Least-Squares Analysis
Microscopy, Atomic Force / instrumentation,  methods*
Optical Tweezers*
Spectrum Analysis / methods*

From MEDLINE®/PubMed®, a database of the U.S. National Library of Medicine

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