欧几里得空间中二阶平稳随机场模型的高斯超几何协方差核：紧支撑、性质及谱表示
The Gauss Hypergeometric Covariance Kernel for Modeling Second-Order Stationary Random Fields in Euclidean Spaces: its Compact Support, Properties and Spectral Representation
摘要：
本文提出了一个紧正半定核的参数族，旨在模拟定义在$d$维欧几里得空间中的二阶平稳各向同性随机场的协方差结构。协方差和它的谱密度都有一个解析表达式，分别涉及超几何函数${}_2F_1$和${}_1F_2$，以及与协方差的相关范围，光滑度和形状有关的四个实值参数。所提出的超几何核家庭包含，作为特殊情况下，球形，立方，五，Askey，广义Wendland和截断功率协方差，并作为渐近的情况下，Matérn，Laguerre，Tricomi，不完全伽马和高斯协方差，等等。的参数空间的单变量超几何内核的确定和其功能特性-连续性，平滑性，传递放大（montée）和缩小（decente）-进行了检查。几组充分条件也推导出获得有效的平稳二元和多元协方差核，其特征在于由四个矩阵值参数。这样的内核是通用的，只要直接和互协方差不一定具有相同的形状，相关范围或行为在短尺度，从而与矢量随机场，其组件是互相关的，但具有不同的空间结构。
英文摘要：
This paper presents a parametric family of compactly-supported positive semidefinite kernels aimed to model the covariance structure of second-order stationary isotropic random fields defined in the $d$-dimensional Euclidean space. Both the covariance and its spectral density have an analytic expression involving the hypergeometric functions ${}_2F_1$ and ${}_1F_2$, respectively, and four real-valued parameters related to the correlation range, smoothness and shape of the covariance. The presented hypergeometric kernel family contains, as special cases, the spherical, cubic, penta, Askey, generalized Wendland and truncated power covariances and, as asymptotic cases, the Matérn, Laguerre, Tricomi, incomplete gamma and Gaussian covariances, among others. The parameter space of the univariate hypergeometric kernel is identified and its functional properties -- continuity, smoothness, transitive upscaling (montée) and downscaling (descente) -- are examined. Several sets of sufficient conditions are also derived to obtain valid stationary bivariate and multivariate covariance kernels, characterized by four matrix-valued parameters. Such kernels turn out to be versatile, insofar as the direct and cross-covariances do not necessarily have the same shapes, correlation ranges or behaviors at short scale, thus associated with vector random fields whose components are cross-correlated but have different spatial structures.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
