This work considers Gaussian process interpolation with a periodized version of the Mat{\'e}rn covariance function introduced by Stein (22, Section 6.7). Convergence rates are studied for the joint maximum likelihood estimation of the regularity and the amplitude parameters when the data is sampled according to the model. The mean integrated squared error is also analyzed with fixed and estimated parameters, showing that maximum likelihood estimation yields asymptotically the same error as if the ground truth was known. Finally, the case where the observed function is a fixed deterministic element of a Sobolev space of continuous functions is also considered, suggesting that bounding assumptions on some parameters can lead to different estimates.

翻译：这项工作考虑了Gaussian过程的内插和Stein(22,第6.7节)引入的马特伊恩共变功能的周期性版本(Mat_'e}rn Covention),在根据模型对数据进行抽样抽样时,为了对规律性和振幅参数进行联合最大可能性估计,研究了趋同率,还用固定和估计参数分析了平均集成的方差,表明最大可能性估计产生与已知地面真相相同的误差。最后,还考虑了所观察到的函数是连续函数Sobolev空间的固定确定要素的情况,表明对某些参数的套装假设可能导致不同的估计。