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

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.

💡 Research Summary

The paper addresses a subtle but practically important source of systematic error that arises when calibrating optical tweezers or atomic‑force‑microscope (AFM) cantilevers by fitting the experimentally measured power spectrum of Brownian motion with a least‑squares (LS) algorithm. The authors show that, for the usual practice of averaging n independent spectra and then applying a weighted LS fit, the estimated diffusion coefficient (or, more generally, any amplitude parameter) is biased by a factor β whose magnitude lies between –2/n and +1/n. The sign and size of β depend entirely on the weighting scheme: weighting by the inverse standard deviation yields a negative bias (≈ –2/n), whereas weighting by the inverse variance gives a positive bias (≈ +1/n). Because n is typically 10–20 in routine calibrations, the bias can be as large as 10–20 % if left uncorrected.

Crucially, the bias affects only the amplitude‑type parameters; the characteristic frequency (or corner frequency) that determines the shape of the spectrum is unbiased. Consequently, AFM force measurements are not compromised provided an independent length‑scale calibration is available, but optical‑tweezer stiffness determinations are, because the spring constant is obtained as the ratio of the corner frequency to the diffusion coefficient.

The authors derive the bias analytically for any linear (integro‑differential) dynamical system driven by additive white or coloured noise. The derivation starts from the fact that each spectral bin follows an exponential (χ² with two degrees of freedom) distribution, whose mean equals the true spectrum and whose variance equals the square of the mean. When a weighted LS cost function is minimized, the expectation value of the fitted amplitude can be written as

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