Parameter estimation in sparse state-space models

Abstract

State-space models are a flexible framework for modelling sequential data in the presence of noise or incomplete observations, within which we model a system via a hidden state process and a related observation process. These processes are described via a pair of distributions encoding the state dynamics and the observation process, with the distribution of the current state depending only on the previous state, and the distribution of the current observation depending only on the current state. In general, the parameters of these distributions are unknown, and are challenging to estimate, with conventional estimation schemes failing due to the temporal dependence of the time series, and the resultant concentration of the likelihood function. In this work we present several methods to estimate the parameters of state-space models, as well as some methods for estimating the form of the model itself when this is unknown. In particular, we focus on methods that admit interpretable estimates via promoting sparsity in the parameters, thereby shrinking many parameter values to zero.

In the first contributing chapter of this work, Chapter 3, we propose a method to obtain sparse Bayesian estimates of the transition matrix of a linear-Gaussian statespace model by utilising reversible jump Markov chain Monte Carlo. We discuss the construction of the reversible jump kernel, and how to interpret the sampled sparsity in terms of a Bayesian causality. We demonstrate our method on several synthetic datasets, where we have the ground truth of causality, and on real-world weather data where we do not, comparing the performance to the existing state-of-the-art.

In Chapter 4, we propose a method to promote graphical clusters in the transition parameters of a linear-Gaussian state-space model by utilising a sparsity promoting estimation scheme in conjunction with a dynamically adaptive penalty. We design a general framework to construct state clustering methods within state-space models, and then construct a representative method as a case of this general framework, wherein we apply ideas from network analysis to design an iteratively applied cluster promoting penalty function. We test our method on a series of synthetic datasets, and compare the performance to the existing state of the art.

In Chapter 5, we propose a method to construct a polynomial representation of a general state-space model, whereby we learn a sparse approximation of the transition function from a basis of polynomial terms. This allows us to infer the connectivity of the hidden states, thereby providing insight into the unknown underlying dynamics.

In the final main chapter, Chapter 6, we propose a method to approximate the intractable optimal proposal of a particle filter utilising a shallow neural network which parametrises a Gaussian mixture distribution. We compare this proposal to several standard proposals, and extend the work to simultaneous estimation of the transition and proposal distributions.

Finally, we provide some concluding remarks on the techniques developed, and present a number of potential avenues for future research.