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January 24, 2020 @ 4:00 pm - 5:00 pm
Title: Kriging: Beyond Matérn
Presenter: Pulong Ma
Abstract: Satellite instruments and computer models that simulate physical processes of interest often lead to massive amount of data with complicated structures. Statistical analysis of such data needs to deal with a wide range of challenging problems such as high-dimensionality and nonstationarity. To understand and predict real-world processes, kriging, originated in geostatistics in the 1960s, has been widely used for prediction in spatial statistics and uncertainty quantification (UQ). In the first part of my talk, I shall give a brief overview of my research related to kriging or Gaussian process regression to tackle these challenging issues in various real-world applications. In the second part of my talk, I shall introduce a new family of covariance functions to perform kriging. Over the past several decades, the Matérn covariance function has been a popular choice to model dependence structures. A key benefit of the Matérn class is that it is possible to get precise control over the degree of differentiability of the process realizations. However, the Matérn class possesses exponentially decaying tails, and thus may not be suitable for modeling long range dependence. This problem can be remedied using polynomial covariances; however, one loses control over the degree of differentiability of the process realizations, in that the realizations using polynomial covariances are either infinitely differentiable or not differentiable at all. To overcome this dilemma, a new family of covariance functions is constructed using a scale mixture representation of the Matérn class where one obtains the benefits of both Matérn and polynomial covariances. The resultant covariance contains two parameters: one controls the degree of differentiability near the origin and the other controls the tail heaviness, independently of each other. This new covariance function also enjoys nice theoretical properties under infill asymptotics including equivalence measures, asymptotic behavior of the maximum likelihood estimators, and asymptotically efficient prediction under misspecified models. The improved theoretical properties in predictive performance of this new covariance class are verified via extensive simulations. Application using NASA’s Orbiting Carbon Observatory-2 satellite data confirms the advantage of this new covariance class over the Matérn class, especially in extrapolative settings. This talk concludes with discussions on extrapolation in UQ studies.