Extreme value theory finds wide applications in areas such as environmental science, financial strategy of risk management and biomedical data processing. In this thesis, we present two spatial extreme value studies related to weather and climate events observed in space and in time, one of which motivates a novel methodology in constructing continuous spatial process for extreme values. Motivated by finding multiscale spatial variations in extreme climate studies, we offer a new Bayesian analysis tool for large scale spatial variations and microscale variations. The last chapter presents a novel application of space-time models to synoptic climatology.
The first investigation is a development of hierarchical modeling approach for explaining a collection of spatially-referenced time series of extreme values. We assume that the observations follow Generalized Extreme Value(GEV) distributions whose locations and scales are jointly spatially dependent where the dependence is captured using multivariate Markov random field models specified through coregionalization. We fit the models to a set of gridded interpolated precipitation data collected over a 50 year period for the Cape Floristic Region in South Africa, summarizing results for what appears to be the best choice of model.
We extend the hierarchical modeling approach in Chapter 3 for explaining a collection of point-referenced time series of extreme values. Here, we relax the conditionally independency assumption previously imposed in the first stage hierarchical models for annual maxima. Instead, a continuous spatial process model is proposed to account for spatial dependence which is unexplained by the latent spatio-temporal specifications for the GEV parameters. In addition, we offer an approach to make spatial interpolation for extreme values based on this hierarchical models with smoothed residuals across space. A simulation study is illustrated to investigate the model fitting behavior.
Motivated by the findings in extreme climate studies, which is, large scale spatial variations and small scale spatial variations coexist some extreme climate phenomena, we present a Bayesian spatial modeling approach to account for both large scale and small scale spatial patterns.
In the last Chapter, the application we focus on is to synoptic climatology where the goal is to develop an array of atmospheric states to capture a collection of distinct circulations. In particular, Self Organizing Maps (SOMs) are one of the widely used techniques in the meteorology community with regard to developing synoptic weather states. Little discussion about this technique has been found in the statistics literature. We introduce the stochasticity in the form of a space-time process model aiming to illuminate and interpret its performance in the context of application to daily data collection. That is, the observed daily state vectors are viewed as a time series of multivariate process realizations which we try to understand under the dimension reduction achieved by the SOM procedure.