A comparison of inferential methods for highly non-linear state space models in ecology and epidemiology

Highly non-linear, chaotic or near chaotic, dynamic models are important in fields such as ecology and epidemiology: for example, pest species and diseases often display highly non-linear dynamics. However, such models are problematic from the point of view of statistical inference. The defining feature of chaotic and near chaotic systems is extreme sensitivity to small changes in system states and parameters, and this can interfere with inference. There are two main classes of methods for circumventing these difficulties: information reduction approaches, such as Approximate Bayesian Computation or Synthetic Likelihood and state space methods, such as Particle Markov chain Monte Carlo, Iterated Filtering or Parameter Cascading. The purpose of this article is to compare the methods, in order to reach conclusions about how to approach inference with such models in practice. We show that neither class of methods is universally superior to the other. We show that state space methods can suffer multimodality problems in settings with low process noise or model mis-specification, leading to bias toward stable dynamics and high process noise. Information reduction methods avoid this problem but, under the correct model and with sufficient process noise, state space methods lead to substantially sharper inference than information reduction methods. More practically, there are also differences in the tuning requirements of different methods. Our overall conclusion is that model development and checking should probably be performed using an information reduction method with low tuning requirements, while for final inference it is likely to be better to switch to a state space method, checking results against the information reduction approach

Data-Driven Forecasting of High-Dimensional Chaotic Systems with Long Short-Term Memory Networks

We introduce a data-driven forecasting method for high-dimensional chaotic systems using long short-term memory (LSTM) recurrent neural networks. The proposed LSTM neural networks perform inference of high-dimensional dynamical systems in their reduced order space and are shown to be an effective set of nonlinear approximators of their attractor. We demonstrate the forecasting performance of the LSTM and compare it with Gaussian processes (GPs) in time series obtained from the Lorenz 96 system, the Kuramoto-Sivashinsky equation and a prototype climate model. The LSTM networks outperform the GPs in short-term forecasting accuracy in all applications considered. A hybrid architecture, extending the LSTM with a mean stochastic model (MSM-LSTM), is proposed to ensure convergence to the invariant measure. This novel hybrid method is fully data-driven and extends the forecasting capabilities of LSTM networks.Comment: 31 page