Parameter estimation from an Ornstein-Uhlenbeck process with measurement noise

TL;DR

This work develops a probabilistic framework for estimating Ornstein-Uhlenbeck parameters in the presence of measurement noise, including additive (thermal) and multiplicative noise. It introduces an EM algorithm that efficiently estimates OU parameters under additive noise, while demonstrating that multiplicative noise poses fundamental identifiability challenges for standard MCMC methods unless the ratio of noise types is known and the data sampling rate is sufficiently high. The authors show that the second-order power spectrum and latent-variable inference can reveal and mitigate multiplicative noise effects, but that accurate separation hinges on noise ratio knowledge and sampling cadence; in practice, EM yields fast parameter estimates but underestimates uncertainty, whereas HMC/NUTS provides more reliable uncertainty quantification at substantial computational cost. The Appendix offers detailed Gaussian algebra and spectral derivations that underpin the noise-separation results, including products/convolutions of Gaussians and OU spectral shapes, with implications for fMRI and other noisy time-series applications.

Abstract

This article aims to investigate the impact of noise on parameter fitting for an Ornstein-Uhlenbeck process, focusing on the effects of multiplicative and thermal noise on the accuracy of signal separation. To address these issues, we propose algorithms and methods that can effectively distinguish between thermal and multiplicative noise and improve the precision of parameter estimation for optimal data analysis. Specifically, we explore the impact of both multiplicative and thermal noise on the obfuscation of the actual signal and propose methods to resolve them. First, we present an algorithm that can effectively separate thermal noise with comparable performance to Hamilton Monte Carlo (HMC) but with significantly improved speed. We then analyze multiplicative noise and demonstrate that HMC is insufficient for isolating thermal and multiplicative noise. However, we show that, with additional knowledge of the ratio between thermal and multiplicative noise, we can accurately distinguish between the two types of noise when provided with a sufficiently large sampling rate or an amplitude of multiplicative noise smaller than thermal noise. Thus, we demonstrate the mechanism underlying an otherwise counterintuitive phenomenon: when multiplicative noise dominates the noise spectrum, one can successfully estimate the parameters for such systems after adding additional white noise to shift the noise balance.

Figures (7)

• Figure 1: Graph representation of a chain of N values $\{x_{i}\}$ that originate from an OU process with corresponding $\{y_{i}\}$ that represent the measured values containing noise
• Figure 2: The parameter estimation as a function of the sample rate. The left graph was generated using Hamilton-Monte-Carlo with the NUTS sampler, and the right one using our EM algorithm. The ribbon around each is the standard deviation of the estimate, and for both parameters, the true values are equal to 1 (dashed line)
• Figure 3: The Expectation Maximization (EM) fit for the first 100 points of an OU process obfuscated by thermal noise. The OU time series was created with parameters $A=1$ and $\tau =1$ at $\Delta t =0.1$ over a time interval of 100s. We then added thermal noise with variance 1. The Pearson correlation between the OU time series and the OU time series with added noise was 0.72. The average of the posterior of the latent OU time series correlated with the original OU time series resulted in an improved Pearson correlation of 0.89.
• Figure 4: Power spectra of the Ornstein-Uhlenbeck process with different types of noise added. Fig. A is the baseline Ornstein-Uhlenbeck signal. Thermal noise (Fig. B) adds a constant in the power spectrum. Multiplicative noise (Fig. C) closely resembles the Ornstein-Uhlenbeck signal in the second-order power spectrum, while it looks thermal in the first order. For the power spectrum, the curves have been normalized such that the area under the curve is equal to the variance of the signal. For the second-order power spectrum, this is equivalent to the variance of the signal squared.
• Figure 5: Parameter estimation for pure multiplicative noise using HMC. We cannot estimate the true values for the amplitude and tau of the OU signal for any sample rate tested.