| Summary: | Glaciers have been melting and reforming on the Earth for millions of years. Over the last several decades, Geologists have created δ 18O proxy records that measure the amount of ice on the Earth’s surface over the last 5 million years The proxy records provide evidence for two major changes in the Earth’s ice sheet dynamics. The first is an increase in the magnitude of glacial events around 2.7 Million years ago (Ma), which coincides with the intensification of glaciations in the Northern Hemisphere; the second, around 1Ma, is known as the Mid-Pleistocene Transition and represents not only a change in the magnitude of glacial events, but their frequency as well. Thus, two of the most important questions that can be asked about the δ18O proxy record are: (1) When (exactly) have changes occurred? (2) Which mechanisms are operating in each of the different glacial regimes? This dissertation is concerned with statistical inference in discrete high dimensional space, in particular on the δ18O proxy record. We begin with the simplest of models, the Least Squares Change Point algorithm, which aims to optimally partition a time series into k regimes, fitting each with a different regression model. However, there is no way to characterize the uncertainty surrounding the optimal solution. To deal with this limitation, we introduce the Bayesian Change Point algorithm, which creates a probabilistic model for the δ18O record, yielding uncertainty estimates on the number of change points, their locations and the regression parameters. In addition, to address the wide range of mechanisms proposed for glacial dynamics after the Mid-Pleistocene Transition, we introduce a novel variable selection technique (EBMA or Exact Bayesian Model Averaging) that has a smaller time complexity than existing algorithms. By combining EBMA with the Bayesian Change Point algorithm, we produce a highly flexible statistical model that can search through an enormously high dimensional space in a practical amount of time.
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