[Paper Review] Modeling noisy time series: Physiological tremor

Jens Timmer|arXiv (Cornell University)|May 8, 1998

Control Systems and IdentificationEngineering21 citations

TL;DR

This paper investigates the impact of observational noise on modeling physiological tremor time series using linear and nonlinear models. It compares autoregressive (AR) models that ignore noise to linear state space models (LSSM) that explicitly account for it, demonstrating that LSSM provides more accurate parameter estimation and better spectral fitting, especially with high noise-to-signal ratios (~0.93), and extends the approach to nonlinear deterministic systems using Bock’s algorithm.

ABSTRACT

Empirical time series often contain observational noise. We investigate the effect of this noise on the estimated parameters of models fitted to the data. For data of physiological tremor, i.e. a small amplitude oscillation of the outstretched hand of healthy subjects, we compare the results for a linear model that explicitly includes additional observational noise to one that ignores this noise. We discuss problems and possible solutions for nonlinear deterministic as well as nonlinear stochastic processes. Especially we discuss the state space model applicable for modeling noisy stochastic systems and Bock's algorithm capable for modeling noisy deterministic systems.

Motivation & Objective

• To assess how observational noise affects parameter estimation in time series models of physiological tremor.
• To compare the performance of standard autoregressive (AR) models with linear state space models (LSSM) that explicitly model observational noise.
• To evaluate the applicability of Bock’s algorithm for modeling noisy deterministic systems in the context of nonlinear dynamics.
• To investigate whether noise-informed modeling improves the accuracy of functional relationships and spectral reconstruction in physiological time series.
• To explore the challenges in assigning physiological meaning to fitted parameters in nonlinear models when noise is not properly accounted for.

Proposed method

• Uses a 35-second physiological tremor time series sampled at 300 Hz (10,240 points) with normalized zero mean and unit variance.
• Applies autoregressive (AR) models of order p to fit the data, estimating parameters via regression and assessing significance using residual variance and whiteness tests.
• Employs linear state space models (LSSM) with state transition matrix A, observation matrix C, process noise Q, and observational noise R to explicitly model measurement noise.
• Utilizes Bock’s algorithm to fit nonlinear deterministic maps by minimizing the sum of squared residuals over a continuous trajectory, incorporating temporal continuity.
• Compares model performance using spectral estimation (periodogram and model-simulated spectra), residual whiteness (Kolmogorov-Smirnov test), and residual variance reduction.
• Applies polynomial-based nonlinear modeling with embedding delay τ = 10 and regression order up to 4, selecting models based on residual variance knee and spectral match.

Experimental results

Research questions

• RQ1 How does observational noise affect parameter estimation in linear and nonlinear time series models of physiological tremor?
• RQ2 Can a linear state space model (LSSM) that explicitly accounts for observational noise outperform a standard AR model that ignores it in terms of spectral fit and parameter accuracy?
• RQ3 To what extent can Bock’s algorithm improve modeling of deterministic nonlinear dynamics when observational noise is present?
• RQ4 Why do fitted nonlinear models fail to assign physiological meaning to parameters, even when they fit the data well?
• RQ5 What are the implications of noise misestimation for distinguishing between deterministic chaos and stochastic processes in time series?

Key findings

• The signal-to-noise ratio in the physiological tremor data is approximately 0.93 in relative amplitude, indicating substantial observational noise.
• The linear state space model (LSSM) provides a significantly better fit to the data than the standard AR model, particularly in capturing the broad spectral peak at ~7.5 Hz.
• The LSSM explicitly models observational noise η(t) with variance R, leading to more accurate parameter estimation compared to AR models that conflate process and measurement noise.
• A nonlinear model based on 4th-order polynomials with embedding delay τ = 10 and regression order 3 achieved a good fit, with a clear knee in residual variance and white residuals, though no physiological interpretation was assigned.
• The model-simulated spectrum closely matches the empirical periodogram, including the aliased peak near 13.5 Hz due to the τ = 10 sampling.
• Despite good fit, the parameters of the nonlinear model could not be meaningfully interpreted in physiological or physical terms, highlighting a key limitation in nonlinear modeling of complex systems.