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My research focuses on applying ideas from mathematical statistics and probability theory to questions about the large-scale spatial distributions of species. I am particularly interested in understanding the spatial distributions and biogeography of microorganisms as part of the iSEEM metagenomics project. My research has two interrelated strands: Geometric probability and species distributions. It remains unclear what processes generate many patterns of species distributions -- for instance, the species area, distance decay, and endemics area relationships. We have shown that ideas based on geometric probability can be used to understand the causes and interrelatedness of these relationships. Essentially, we propose that these relationships are related to problems in geometric probability, particularly covering problems which ask the probability of randomly paced regions covering other, fixed regions. Through the application of this approach, we have developed a theory for the distance-decay relationship that successfully accounts for many observed patterns. In addition to having excellent predictive power, this theory allows the shapes of the underlying species ranges to be inferred from the distance-decay relationship. Thus, it provides a powerful tool for inferring and understanding the spatial distributions of species. We are currently working to extend the geometric probability approach to understand distance-decay relationships in other spaces, for instance niche spaces, and to understand other species distribution patterns. Optimal null models tests. Many critical questions about the processes that shape communities at large scales cannot be addressed experimentally. Instead, to investigate these questions, inferences must be made from observational data. Null model testing comprises a key tool in making these inferences, allowing investigators to check whether observations of species ranges, community phylogenetic patterns, and other traits are consistent with effects of environmental heterogeneity, competition, and other processes. However, null model tests are usually developed intuitively, rather than via a principled mathematical framework. In this strand of my research, we have implemented ideas from mathematical statistics to develop optimal null model tests. These tests can be proven to have the greatest possible power subject to a controlled false positive rate. In addition, we have developed principled null model tests by imposing robustness constraints. Collectively, these null model tests allow stronger inferences about large-scale effects of ecological processes from observational data. Software related to this research is available here. Collaborators
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