Beyond Matérn: On A Class of Interpretable Confluent Hypergeometric Covariance Functions

The Mat\'ern covariance function is a popular choice for prediction in spatial statistics and uncertainty quantification literature. A key benefit of the Mat\'ern class is that it is possible to get precise control over the degree of differentiability of the process realizations. However, the Mat\'ern 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 the random processes with 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 \emph{Confluent Hypergeometric} (CH) class using a scale mixture representation of the Mat\'ern class where one obtains the benefits of both Mat\'ern 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 Mat\'ern class, especially in extrapolative settings.