Efficient methods for fitting nonlinear non-Gaussian state space models of wildlife population dynamics

Abstract

State-space models (SSMs) are a popular and flexible framework for modelling time series due to their ability to separate changes in the underlying state of a system from the noisy observations made on these states. This thesis explores methods for estimating states and model parameters in non-linear and non-Gaussian Bayesian SSMs. We focus on models of wildlife population dynamics, in particular a case study of the UK grey seal population.

Calculation of the likelihood is fundamental to Bayesian analysis, but direct calculation is typically intractable for non-linear non-Gaussian SSMs. We use a class of simulation-based methods, Sequential Monte Carlo (SMC), which build on repeated importance sampling of simulated states to deliver an unbiased estimate of the likelihood. We find that variance of the estimated likelihood can be high and explore techniques for variance reduction.

For parameter inference, we use particle marginal Metropolis-Hastings (PMMH), which embeds the SMC likelihood within a Markov chain Monte Carlo (MCMC) algorithm. Careful balance is needed between computational effort expended on the SMC step and the number of MCMC samples.

A much faster alternative is the Kalman filter, designed for linear and Gaussian SSMs. We applied the Kalman filter to an approximation of the seal model. The posterior distribution obtained was often close to the true posterior, while reducing computation time by a factor of 1790.

We show the seal model suffers from identifiability issues which cannot be resolved by increasing the accuracy of the observations or allowing more flexibility in the underlying biological process with random effects. However, estimation of underlying states (i.e., population sizes) is unaffected by these issues.

A reduction in PMMH computation time can be achieved by exploiting the structure of the state model: separately estimating likelihood components in each of the 4 seal regions led to a 5-fold increase in speed.