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

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.