Taylor & Francis Group
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Beyond Matérn: On A Class of Interpretable Confluent Hypergeometric Covariance Functions

posted on 2022-01-20, 16:40 authored by Pulong Ma, Anindya Bhadra

The Matérn covariance function is a popular choice for prediction in spatial statistics and uncertainty quantification literature. A key benefit of the Matérn class is that it is possible to get precise control over the degree of mean-square differentiability of the random process. However, the Matérn class possesses exponentially decaying tails, and thus, may not be suitable for modeling polynomially decaying dependence. This problem can be remedied using polynomial covariances; however, one loses control over the degree of mean-square differentiability of corresponding processes, in that random processes with existing polynomial covariances are either infinitely mean-square differentiable or nowhere mean-square differentiable at all. We construct a new family of covariance functions called the Confluent Hypergeometric (CH) class 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 mean-square differentiability near the origin and the other controls the tail heaviness, independently of each other. Using a spectral representation, we derive theoretical properties of this new covariance including equivalent measures and asymptotic behavior of the maximum likelihood estimators under infill asymptotics. The improved theoretical properties of the CH class are verified via extensive simulations. Application using NASA’s Orbiting Carbon Observatory-2 satellite data confirms the advantage of the CH class over the Matérn class, especially in extrapolative settings. Supplementary materials for this article are available online.


Ma carried out this work in part as a postdoctoral fellow at the Statistical and Applied Mathematical Sciences Institute (SAMSI) and Duke University and gratefully acknowledges the postdoctoral fellowship at SAMSI that was awarded by the National Science Foundation (NSF) under Grant DMS-1638521 to SAMSI. Bhadra gratefully acknowledges a visiting fellowship at SAMSI where part of this research was conducted. The research of Bhadra is also supported by NSF grant DMS-2014371.