An approach to periodic, time-varying parameter estimation using nonlinear filtering

TL;DR

This paper introduces a nonlinear filtering method using an augmented ensemble Kalman filter to estimate periodic, time-varying parameters in biological systems, providing flexibility and improved accuracy over traditional approaches.

Contribution

The paper presents a novel approach that models periodic parameters as piecewise functions with unknown coefficients, enabling flexible and accurate estimation in biological systems.

Findings

Successfully estimated parameters in a synthetic FitzHugh-Nagumo model.

Accurately estimated seasonal transmission in measles data.

Demonstrated flexibility in modeling without predefined functional forms.

Abstract

Many systems arising in biological applications are subject to periodic forcing. In these systems the forcing parameter is not only time-varying but also known to have a periodic structure. We present an approach to estimating periodic, time-varying parameters that imposes periodic structure by treating the time-varying parameter as a piecewise function with unknown coefficients. This method allows the resulting parameter estimate more flexibility in shape than prescribing a specific functional form (e.g., sinusoidal) to model its behavior, while still maintaining periodicity. We employ nonlinear filtering, more specifically, a version of the augmented ensemble Kalman filter (EnKF), to estimate the unknown coefficients comprising the piecewise approximation of the periodic, time-varying parameter. This allows for straightforward comparison of the proposed method with an EnKF-based parameter tracking algorithm, where periodicity is not guaranteed. We demonstrate the effectiveness of the proposed approach on two biological examples: a synthetic example with data generated from the nonlinear FitzHugh-Nagumo system, modeling the excitability of a nerve cell, to estimate the external voltage parameter, and a case study using reported measles incidence data from three locations during the pre-vaccine era to estimate the seasonal transmission parameter. The formulation of the proposed approach also allows for simultaneous estimation of initial conditions and other static system parameters, such as the reporting probability of measles cases, which is vital for predicting under-reported incidence data.