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

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.马尔特南协方差函数（Matérn covariance function）是空间统计学（spatial statistics）与不确定性量化（uncertainty quantification）领域中用于预测的常用选择。该协方差族的核心优势在于，可对随机过程（random process）的均方可微性（mean-square differentiability）程度实现精准控制。然而，马尔特南协方差族具有指数衰减尾部（exponentially decaying tails），因此不适用于建模多项式衰减相依性（polynomially decaying dependence）。此类问题可通过多项式协方差（polynomial covariances）解决，但此时会丧失对对应随机过程均方可微程度的控制——现有多项式协方差对应的随机过程，要么是无限均方可微（infinitely mean-square differentiable）的，要么处处均不可微（nowhere mean-square differentiable）。我们基于马尔特南协方差族的尺度混合表示（scale mixture representation），构建了一类全新的协方差函数族，命名为合流超几何（Confluent Hypergeometric, CH）族，该族兼具马尔特南协方差与多项式协方差的双重优势。所得协方差函数包含两个独立参数：其一用于控制原点附近的均方可微程度，其二用于控制尾重性（tail heaviness），二者相互独立。我们通过谱表示（spectral representation）推导了该新型协方差的多项理论性质，包括等价测度（equivalent measures）以及填充渐近（infill asymptotics）框架下极大似然估计量（maximum likelihood estimators）的渐近行为。通过大量模拟实验验证了CH族相较于现有方法的优良理论性质。利用美国国家航空航天局（National Aeronautics and Space Administration, NASA）的轨道碳观测站-2（Orbiting Carbon Observatory-2, OCO-2）卫星数据开展的实证应用表明，CH族相较于马尔特南协方差族具有显著优势，尤其在外推场景中。本文的补充材料可在线获取。