Wavelet Methods for Multivariate Nonstationary Time Series

This thesis proposes novel methods for the modelling of multivariate time series. The work presented falls into three parts. To begin we introduce a new approach for the modelling of multivariate non-stationary time series. The approach, which is founded on the locally stationary wavelet paradigm, models the second order structure of a multivariate time series with smoothly changing process amplitude. We also define wavelet coherence and partial coherence which quantify the direct and indirect links between components of a multivariate time series. Estimation theory is also developed for this model. The second part of the thesis considers the application of the multivariate locally stationary wavelet framework in a classification setting. Methods for the supervised classification of time series generally aim to assign a series to one class for its entire time span. We instead consider an alternative formulation for multivariate time series where the class membership of a series is permitted to change over time. Our aim therefore changes from classifying the series as a whole to classifying the series at each time point to one of a fixed number of known classes. We also present asymptotic consistency results for this framework. The thesis concludes by introducing a test of coherence between components of a multivariate locally stationary wavelet time series