Gaussian processes are widely employed as versatile modelling and predictive tools in spatial statistics, functional data analysis, computer modelling and diverse applications of machine learning. They have been widely studied over Euclidean spaces, where they are specified using covariance functions or covariograms for modelling complex dependencies. There is a growing literature on Gaussian processes over Riemannian manifolds in order to develop richer and more flexible inferential frameworks for non-Euclidean data. While numerical approximations through graph representations have been well studied for the Mat\'ern covariogram and heat kernel, the behaviour of asymptotic inference on the parameters of the covariogram has received relatively scant attention. We focus on the asymptotic inference for Gaussian processes constructed over compact Riemannian manifolds. Building upon the recently introduced Mat\'ern covariogram on a compact Riemannian manifold, we employ formal notions and conditions for the equivalence of two Mat\'ern Gaussian random measures on compact manifolds to derive the parameter that is identifiable, also known as the microergodic parameter, and formally establish the consistency of the maximum likelihood estimate and the asymptotic optimality of the best linear unbiased predictor. The circle is studied as a specific example of compact Riemannian manifolds with numerical experiments to illustrate and corroborate the theory.
翻译:高斯过程被广泛应用于空间统计学、函数数据分析、计算机建模和各种机器学习应用作为多功能建模和预测工具。它们在欧几里德空间上广泛研究,其中使用协方差函数或协方差函数来建模复杂的依赖关系。 随着大量非欧几里德数据的出现,关于在紧致里曼流形上的高斯过程的研究越来越多,以便针对非欧几里德数据开发更丰富和灵活的推理框架。虽然通过图形表示的数值近似方法已经对Matérn协方差函数和热核进行了研究,但协方差函数参数的渐近推理行为仍然受到相对较少的关注。我们专注于构建在紧致里曼流形上的高斯过程的渐近推理。基于最近在紧致里曼流形上引入的Matérn协方差函数,我们采用正式的概念和条件来等价于两个Matérn高斯随机测度,以得出可识别的参数,也被称为微观马尔可夫参数,并正式建立最大似然估计的一致性和最优线性无偏估计器的渐近最优性。特定于研究圆作为紧致里曼流形的示例,并进行数值实验证明和证实理论。