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Supplementary Material to Publications

Research Statement

My research focuses on statistical models and methods for spatial-temporal processes and their applications to environmental problems. The key problem, in my view, is to find models that allow for interesting spatial-temporal interactions and methods for estimating and assessing the fit of such models. My present focus is on models for Gaussian processes (or processes that can be made Gaussian after a transformation), so that it suffices to specify the mean and covariance structure. The following are the main topics of my personal research;

  1. Simply stated, the ultimate goal of any model for space-time covariance functions is to capture the variance of any linear combination of observations of a process accurately. To do this, it is essential to consider the nature of space-time interactions of the process and not just the spatial and temporal variations separately. I have been studying how these interactions relate to the behavior of space-time covariance functions away from the origin and, in turn, how this behavior depends on the space-time spectral density function.
  2. Most atmospheric spatial-temporal processes are not time reversible. In particular, the space-time covariance functions are generally not fully symmetric, in that the covariance between site x at time s and site y at time t is not the same as that between x at time t and y at s. Some of the goals of this work are to have flexible and interpretable asymmetries, explicit expressions for the resulting covariance functions, and useful diagnostics for assessing asymmetry.
  3. A continuing interest of mine is the development of covariance structures for space-time processes when the spatial domain is a sphere, which has obvious relevance to the atmospheric sciences and climatology. In joint work with a number of students over the years, including Mikyoung Jun, Marcin Hitczenko, Stefano Castruccio and currently Michael Horrell, I have been developing approaches to generate explicit covariance functions for this setting, including ways of producing rich classes of space-time asymmetries. Specific challenges that arise when modeling atmospheric processes on a global scale include the variation of dependence structures with latitude and on whether one is over land or water.
  4. While new and better models are critical to advancing spatial-temporal statistics, one also needs computational and graphical techniques for analyzing and summarizing large space-time datasets. One setting I have been working on for some time is regularly collected monitoring data, for which a combination of ideas from multiple time series and spatial statistics is appropriate. Models and analyses that work in the spectral domain in time and the spatial domain in space are natural for regular monitoring data and lead to interesting classes of partially nonparametric models that can be fitted reasonably easily using spectral methods. With Joseph Guinness, I have worked on extending these ideas to processes that are nonstationary in time, which is critical for modeling high frequency meteorological data. More recently, I have been focusing on computational methods that can be applied when the observations are not regularly spaced in space or time, which is commonly the case for many environmental datasets and for which spectral methods are not so useful. This research makes heavy use of iterative methods for solving large systems of linear equations. Collaborators on this work include Mihai Anitescu, Jie Chen and Ying Sun.
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