Forecasting Complex Systems Using Stochastic Models for Low Dimensional Approximations

Computer simulators are often used to model complex systems, which by their very nature are in general expensive to run and hard to condition. Statistical methods can be used to simplify and help analyse these simulator outputs, with the aim of increasing understanding of the underlying complex system and producing simplified forecasts of this system. This work investigates some statistical methods that can be used, and applies these methods to some output from the HadCM3 climate simulator. In Part I of this thesis, the statistical framework is introduced, and can be split broadly into the areas of dimension reduction, stochastic modelling, and forecasting. Within the dimension reduction section, the fastICA independent component analysis method is examined with some definciencies highlighted. A novel independent component analysis method is introduced, called clusterICA, which uses clustering in the projective space and Householder reflections to obtain independent directions. Modelling of the components found using dimension reduction is then examined. This includes using a block-average Ornstein-Uhlenbeck process, where the pointwise Ornstein-Uhlenbeck process is integrated over disjoint time intervals. As the first step of modelling involves removing seasonality, a novel spline function is introduced that can be used to ensure seasonal means between pointwise and block-average Ornstein-Uhlenbeck processes remain equal. Forecasts using the distributional properties of the block-average Ornstein-Uhlenbeck are also examined. In Part II of this thesis, methods introduced in Part~I are applied to output obtained from the HadCM3 climate simulator. Two different forecasting methods are used to obtain forecasts of the sets of low dimensional components, and these forecasts are used to reconstruct the full high dimensional simulator output. These forecasted reconstructions are compared to the reconstructions using the true low dimensional components, and the full HadCM3 simulator output.