Inference for nonlinear dynamical systems

TLDR

Nonlinear stochastic dynamical systems are widely used across sciences and engineering, but inferring unknown parameters from time‑series data remains challenging. This work introduces a new method that enables maximum‑likelihood estimation for partially observed nonlinear stochastic dynamical systems where it was previously infeasible. The method employs a sequence of filtering operations that converge to a maximum‑likelihood estimate, leveraging recent advances in nonlinear filtering and applied to cholera dynamics in Bangladesh with confidence‑interval construction and diagnostic checks. The analysis uncovers nonlinear effects in cholera transmission that earlier studies missed.

Abstract

Nonlinear stochastic dynamical systems are widely used to model systems across the sciences and engineering. Such models are natural to formulate and can be analyzed mathematically and numerically. However, difficulties associated with inference from time-series data about unknown parameters in these models have been a constraint on their application. We present a new method that makes maximum likelihood estimation feasible for partially-observed nonlinear stochastic dynamical systems (also known as state-space models) where this was not previously the case. The method is based on a sequence of filtering operations which are shown to converge to a maximum likelihood parameter estimate. We make use of recent advances in nonlinear filtering in the implementation of the algorithm. We apply the method to the study of cholera in Bangladesh. We construct confidence intervals, perform residual analysis, and apply other diagnostics. Our analysis, based upon a model capturing the intrinsic nonlinear dynamics of the system, reveals some effects overlooked by previous studies.