The uncertainty of changepoints in time series

Abstract

Analysis concerning time series exhibiting changepoints have predominantly focused on detection and estimation. However, changepoint estimates such as their number and location are subject to uncertainty which is often not captured explicitly, or requires sampling long latent vectors in existing methods. This thesis proposes efficient, flexible methodologies in quantifying the uncertainty of changepoints. The core proposed methodology of this thesis models time series and changepoints under a Hidden Markov Model framework. This methodology combines existing work on exact changepoint distributions conditional on model parameters with Sequential Monte Carlo samplers to account for parameter uncertainty. The combination of the two provides posterior distributions of changepoint characteristics in light of parameter uncertainty. This thesis also presents a methodology in approximating the posterior of the number of underlying states in a Hidden Markov Model. Consequently, model selection for Hidden Markov Models is possible. This methodology employs the use of Sequential Monte Carlo samplers, such that no additional computational costs are incurred from the existing use of these samplers. The final part of this thesis considers time series in the wavelet domain, as opposed to the time domain. The motivation for this transformation is the occurrence of autocovariance changepoints in time series. Time domain modelling approaches are somewhat limited for such types of changes, with approximations often taking place. The wavelet domain relaxes these modelling limitations, such that autocovariance changepoints can be considered more readily. The proposed methodology develops a joint density for multiple processes in the wavelet domain which can then be embedded within a Hidden Markov Model framework. Quantifying the uncertainty of autocovariance changepoints is thus possible. These methodologies will be motivated by datasets from Econometrics, Neuroimaging and Oceanography