Statistical Inference for Nonlinear Dynamical Systems.

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. A new method is introduced which 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. A key element in the implementation of this new method as presented here is simulation from the proposed model, taking advantage of recent advances in simulation based nonlinear filtering. Analytical calculations using the model, such as transition densities and their derivatives, are not required for the inference. This allows statistical inference for models where analytical properties are hard to derive, as is likely the case when models are based on scientifically proposed mechanisms. A framework for carrying out inference is developed for a novel class of models for dynamical systems composed of interacting populations of individuals for which analytical properties are not readily available. These models are obtained by introducing stochastic rates in continuous time Markov counting processes. Unlike previous models with stochastic rates, these new models retain the Markov property and exhibit over-dispersion. The relationship between adding noise to the rates and over-dispersed continuous time Markov counting processes is studied, and some analytic results, such as infinitesimal moments and generators, are derived for simple population models. The theory developed is applied in an analysis of cholera mortality dynamics in Dhaka, Bangladesh during the historical period 1891-1940. Specifically, the role of a cholera reservoir in the environment and the role of the El Nino Southern Oscillation index are investigated. Another application investigates the structure and interaction between two competing strains of the pathogen Vibrio cholerae, as well as the role of cross-immunity in the dynamics of the disease in the more recent period 1975-2005 in Matlab, Bangladesh.