Statistical analysis of extremes motivated by weather and climate studies: Applied and theoretical advances.

摘要

The statistical study of extreme values has seen much growth since the field's beginnings about eighty years ago. Extreme value theory (EVT) is used in a wide range of applications from hydrology to engineering to economics. In this thesis, three EVT investigations related to weather and climate problems are presented. Two are novel applications of extremes techniques to weather or climate studies, and the other is an advancement of extremes theory motivated by issues arising in weather and climate studies.; The first investigation is a unique application of extreme value theory to lichenometry. The aim is to estimate the ages of paleo-climate events by employing lichen measurements on glacial features formed by the events. This application of extremes differs from most in that although the lichen data are extremes, a hidden covariate of the data is the main quantity of interest. Presented are two different models for estimating the ages of the climate events. The first model relies on the EM-algorithm to estimate the ages, while the second builds a hierarchical model which estimates the ages through a Bayesian framework. In addition to providing age estimates and confidence intervals, the models provide a comprehensive statistical model for lichenometry based on EVT, which had not been done previously.; Often in weather and climate studies, the data are recorded at specific locations, and the second and third investigations deal with spatial data. The second investigation advances the theory of extremes on spatial fields. One of the primary issues with spatial observations is quantifying their spatial dependence. The field of geostatistics is typically used to model spatial data, but it has its roots in studying the central tendencies of the distribution rather than the distribution's tails. The madogram, a new dependence measure for extremes, is introduced. The madogram has its roots in traditional geostatistics and is therefore well-suited for measuring spatial dependence. Estimators for the madogram are also presented and are compared those of other extremes dependence measures.; The goal of the third project is to produce a map describing potential extreme precipitation and uncertainty measures for a study region in Colorado. Like the previous project, it is necessary to characterize spatial dependence. However, here the goal is to model climatological precipitation, and thus it is necessary to model the spatial dependence between the distributions which characterize the extremes rather than the observations themselves. The methodology constructs a Bayesian hierarchical model to characterize the extreme precipitation for the region. The spatial analysis is performed in a space defined by climatological coordinates rather than in the traditional latitude/longitude space. Using this methodology, the precipitation map has strong ties to the orography of the region.