Power spectrum analysis with least-squares fitting: Amplitude bias and its elimination, with application to optical tweezers and atomic force microscope cantilevers

TL;DR

This paper identifies a bias in least-squares fitting of power spectra for optical tweezers and AFM cantilevers, quantifies it analytically, and provides a simple correction method applicable to various systems with linear dynamics.

Contribution

It analytically characterizes the bias in weighted least-squares power spectrum fitting and offers a straightforward correction applicable across multiple scientific fields.

Findings

Bias ranges from -2/n to +1/n depending on weighting scheme

Characteristic frequency estimates are unaffected by the bias

Bias can be easily corrected with minimal modifications to fitting programs

Abstract

Optical tweezers and AFM cantilevers are often calibrated by fitting their experimental powerspectra 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 depends 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-square 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.