Wavelet methods for locally stationary time series

McGonigle, Euan and Killick, Rebecca and Nunes, Matthew (2020) Wavelet methods for locally stationary time series. PhD thesis, Lancaster University.

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Abstract

Time series data can often possess complex and dynamic characteristics. Two key statistical properties of time series -- the mean (first-order) and autocovariance (second-order) -- commonly change over time. Modelling this evolution of so-called nonstationary time series is crucial to making informed inference on the data. This thesis focuses on wavelet-based methodology for the simultaneous modelling of first and second-order nonstationary time series, for which we provide three main contributions. First, we propose a method using differencing to jointly estimate the time-varying trend and second-order structure of a time series, within the locally stationary wavelet processes framework. We discuss a wavelet-based estimator of the second-order structure of the original time series by employing differencing, and show how this can be incorporated into the estimation of the trend of the time series. Second, we propose a framework for modelling series with simultaneous time-varying first and second-order structure by removing the restrictive zero-mean assumption of locally stationary wavelet (LSW) processes and extending the applicability of the locally stationary wavelet model to include a trend component. We develop associated estimation theory for both first and second-order time series quantities and show that our estimators achieve good properties in isolation of each other by making appropriate assumptions on the series trend. Last, we consider simultaneous modelling of first and second-order structure in the scenario where the mean function is piecewise constant. We propose a likelihood-based method using wavelets to detect changes in mean in time series that exhibit time-varying autocovariance. This allows for a more flexible model for mean changepoint detection, since commonly the second-order structure is assumed to be independent and identically distributed. The performance of the method is investigated via simulation, and is shown to perform well in a variety of time series scenarios.