Bayesian singular spectrum analysis with state dependent models.

Abstract

The analysis of time series using Singular Spectrum Analysis has become an important area of statistics with application in a variety of fields such as economics, geophysics, engineering, medicine and many others. In fact, in this method there is no need to make any statistical assumptions such as the stationarity of the series or normality of the residuals. Therefore, SSA is recognised as an extremely practical tool which can be used to solve problems without considering any parametric model. At its core SSA depends on an eigenvector decomposition of the covariance matrix of a time series which may be utilised for forecasting via a linear recurrent formula. However, many time series exhibit structural breaks which interfere with a linear continuation of the time series although the underlying data generating process may not have changed. In addition, in a multivariate setting there is the added complication of combining time series. In this case the linear recurrence relationships of each time series may either reinforce each other or alternatively may lead to degraded forecasts. In this thesis a state dependent model is proposed under the assumption that if a system moves from one homogeneous state to another rapidly that this transition may be tracked using a Bayesian model in which the state transitions are state dependent. In addition, it is proven that for basic SSA the linear recurrent coe cients are biased and that this bias decays linearly with the samples. Empirically, the state dependent model shows far superior performance over two multivariate data sets. In the second part of the thesis component matching is examined. The core issue is how to identify which time series to group together without testing every possible combination. Geographical information resulted in superior forecasts on USA unemployment time series via a spatial SSA model. Subsequent research into data driven methods to group the time series concludes that a novel variant of the self organising map leads to a signi cant improvement over methods based on standard techniques like tensor analysis and joint diagonalisation.