Effect of multiplicative noise on least-squares parameter estimation with applications to the atomic force microscope

Abstract

Measurement of the power spectral density of (stochastic) Brownian fluctuations of micro- and nano-devices is used frequently to gain insight into their mechanistic properties. Noise is always present in these measurements and can directly influence any parameter estimation obtained through a least-squares analysis. Importantly, measurements of the spectral density of stationary random signals, such as Brownian motion, inherently contain multiplicative noise. In this article, we theoretically analyze the impact of multiplicative noise on fit parameters extracted using a least-squares analysis. A general analysis is presented that is valid for any fit function with any number of fit parameters. This yields closed-form expressions for the expected value and variance in the fit parameters and provides a rigorous theoretical framework for a priori determination of the effect of measurement uncertainty. The theory is demonstrated and validated through Monte Carlo simulation of synthetic data and by comparison to power spectral density measurements of the Brownian fluctuations of an atomic force microscope cantilever - analytical formulas for the uncertainty in the fitted resonant frequency and quality factor are presented. The results of this study demonstrate that precise measurements of fit parameters in the presence of noise are inherently problematic - individual measurements of the power spectral density are capable of yielding fit parameters that are many standard deviations away from the mean, with finite probability. This is of direct relevance to a host of applications in measurement science, including those connected with the atomic force microscope.

Spring constant calibration of atomic force microscope cantilevers of arbitrary shape

“…The measured resonant frequency and quality factor in air (1 atm) for each cantilever are also given, and possess smaller uncertainty: between ±0.0006% and ±0.02% for the resonant frequency, and ±0.3% and ±1% for the quality factor. 53 The spring constants of two test cantilevers, BL-RC150VB(S) and TR400(L), could not be measured; also see Appendix A.…”

“…Scatter in measurements of the resonant frequency and quality factor, as obtained from fits to the PSD of Brownian motion, was due to sampling noise inherent in periodogram estimates of the PSD. 53 …”

Estimating Damping in Microresonators by Measuring Thermomechanical Noise Using Laser Doppler Vibrometry

“…The data, fitting curve, and residual errors were plotted to check that the fit captured the baseline, peak shape, and peak height, and that the residuals were uncorrelated [40]- [42]. Figure 4 shows the resonance peak and fitting curves for one representative measurement performed on Specimen 4.…”

“…The final version of record is available athttp://dx.doi.org/10.1109/JMEMS.2013.2286199 the following format: mean value ( ± relative uncertainty), where the relative uncertainty is the standard deviation reported as a percentage of the mean. The precision of our measurements is influenced by: (i) intrinsic errors associated with estimating the PSD using finite time-series of stochastic fluctuations and (ii) uncertainties associated with fit parameters obtained using least-squares analysis [37]- [40]. Sader et al [40] recommended statistical analysis of multiple independent measurements because the goodness-of-fit of a single measurement may not provide an estimate of uncertainty even when the analysis results in an apparently excellent fit to measurements.…”

“…The precision of our measurements is influenced by: (i) intrinsic errors associated with estimating the PSD using finite time-series of stochastic fluctuations and (ii) uncertainties associated with fit parameters obtained using least-squares analysis [37]- [40]. Sader et al [40] recommended statistical analysis of multiple independent measurements because the goodness-of-fit of a single measurement may not provide an estimate of uncertainty even when the analysis results in an apparently excellent fit to measurements. Accordingly, we use the relative uncertainty, which ranges from 6% to 26%, as the measure of precision.…”