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:高斯进程被广泛用作空间统计、功能数据分析、计算机建模和机器学习的多种应用的多功能建模和预测工具,在欧几里德空间进行了广泛研究,使用共变函数或共变图来模拟复杂的依赖性。关于高斯进程在里曼尼主义多元体上的文献越来越多,以开发非欧几里德主义数据的更丰富、更灵活的推论框架。虽然通过图形表达式的数值近似值已经为马特尔恩正变方形图和热内核进行了深入研究,但对共变图参数的无常推论行为却受到相对较少的注意。我们侧重于在紧凑里曼主义多元体中构建的高斯进程无损推论,以开发较丰富、更灵活的非欧数框架。我们采用正式的概念和条件来对等两个马特尔尼调调调调调调调调调调调调调调调调调调调调调调调调调调调调调调调调调调调调调调调调调调调调调调调调调调调调调调调调调调调调调调调调调调调调调调调调调调调调调调调调调调调调调调调调调调调调调调调调调调调调调调调调调调调调调调调调调调调调调调调调调调调调调调调调的参数,以得出,以得出调调调调调调调调调调调调调,以得出调调调调调和调调调调调调调调调调调调调调调调调和调调调调调调调调和调调调调调调调调调调调调调调调调调调调调调调调调调,以得出调调调调调调调调调调调调调调调调,以得出调调调调调调调调调调调调调调调调调调调调调调调调调调调调调调调调调调调调调调调调调调调调调调调调调调调调调调调调调调调调调调调调调调调调调调调调调调调调调调调调调调调调调调调调调调调调,以确定</s>