Beyond Matérn: On A Class of Interpretable Confluent Hypergeometric Covariance Functions
针对Matérn协方差函数尾部指数衰减、无法建模多项式衰减依赖的问题，提出合流超几何协方差函数，通过两个独立参数分别控制均方可微性和尾部重性，在NASA卫星数据中验证了其外推优势。
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.