Stochastic State Space Modelling of Nonlinear systems - With application to Marine Ecosystems

Abstract

This thesis deals with stochastic dynamical systems in discrete and continuous time. Traditionally dynamical systems in continuous time are modelled using Ordinary Differential Equations (ODEs). Even the most complex system of ODEs will not be able to capture every detail of a complex system like a natural ecosystem, and hence residual variation between the model and observations will always remain. In stochastic state-space models the residual variation is separated into observation and system noise and a main theme of the thesis is a proper description of the system noise. Additive Gaussian noise is the standard approach to introduce system noise, but this may lead to undesirable consequences for the state variables. In biological models, where the statespace generally contains positive real numbers only, modelling in the log-domain ensures positive state variables, however, this transformation is likely to conflict with the concept of mass balances. One of the central conclusions of the thesis is that the stochastic formulations should be an integral part of the model formulation. As discrete-time stochastic processes are simpler to handle numerically than continuous-time stochastic processes, I start by considering discrete-time processes. An novel approach combining multiplicative and additive log-normal noise has been developed in discrete time, and used to demonstrate the effect of stochastic forcing in simple discrete-time regime shift models. An approximate maximum likelihood estimation procedure based on the second order moment representation of the multiplicative and additive log-normal noise model was developed and tested in simulation studies. The transition to continuous-time stochastic models (here Stochastic Differential Equations (SDEs)) offers the opportunity of embedding parts of the ODE processes into the stochastic part of the model (the diffusion term). The estimation method we use here (maximum likelihood and the Extended Kalman Filter (EKF)) rely on state-independent diffusion, but for a wide class of SDEs there exist an alternative description (given by the Lamperti transform) of the input-output relation, where the diffusion term is independent of the state. This alternative description is used to develop better parametric descriptions of the diffusion term, while maintaining the opportunity of estimation by standard software. Additionally, the state-space formulation facilitates estimation of unobserved states. Based on estimation of random walk hidden states and examination of simulated distributions and stationarity characteristics, a methodological framework for structural identification based on information embedded in the observations of the system has been developed. The applicability of the methodology is demonstrated using phytoplankton and nitrogen data from a Danish estuary as well as bacterial growth data from a controlled experiment. In summary, the novelty of the work presented here is the introduction of more appropriate stochastic descriptions in non-linear state-space models, which can include combinations of additive and multiplicative noise components under various distributional assumptions. A model identification and estimation framework for working with such models has been developed and tested using data from biological and ecological systems typically characterised by non-linear and non-Gaussian responses.