The Matérn family of covariance functions is currently the most popularly used model in spatial statistics, geostatistics, and machine learning to specify the correlation between two geographical locations based on spatial distance. Compared to existing covariance functions, the Matérn family has more flexibility in data fitting because it allows the control of the field smoothness through a dedicated parameter. Moreover, it generalizes other popular covariance functions. However, fitting the smoothness parameter is computationally challenging since it complicates the optimization process. As a result, some practitioners set the smoothness parameter at an arbitrary value to reduce the optimization convergence time. In the literature, studies have used various parameterizations of the Matérn covariance function, assuming they are equivalent. This work aims at studying the effectiveness of different parameterizations under various settings. We demonstrate the feasibility of inferring all parameters simultaneously and quantifying their uncertainties on large-scale data using the ExaGeoStat parallel software. We also highlight the importance of the smoothness parameter by analyzing the Fisher information of the statistical parameters. We show that the various parameterizations have different properties and differ from several perspectives. In particular, we study the three most popular parameterizations in terms of parameter estimation accuracy, modeling accuracy and efficiency, prediction efficiency, uncertainty quantification, and asymptotic properties. We further demonstrate their differing performances under nugget effects and approximated covariance. Lastly, we give recommendations for parameterization selection based on our experimental results.

Matérn 协方差函数系列是目前空间统计、地统计学和机器学习中最常用的模型，用于根据空间距离指定两个地理位置之间的相关性。与现有的协方差函数相比，Matérn 系列在数据拟合方面具有更大的灵活性，因为它允许通过专用参数控制场平滑度。此外，它还推广了其他流行的协方差函数。然而，拟合平滑参数在计算上具有挑战性，因为它使优化过程变得复杂。因此，一些从业者将平滑参数设置为任意值，以减少优化收敛时间。在文献中，研究使用了 Matérn 协方差函数的各种参数化，假设它们是等价的。这项工作旨在研究不同设置下不同参数化的有效性。我们证明了使用ExaGeoStat并行软件同时推断所有参数并量化大规模数据的不确定性的可行性。我们还通过分析统计参数的 Fisher 信息来强调平滑参数的重要性。我们表明，各种参数化具有不同的属性，并且从多个角度来看都有所不同。特别是，我们在参数估计精度、建模精度和效率、预测效率、不确定性量化和渐近性质方面研究了三种最流行的参数化。我们进一步证明了它们在金块效应和近似协方差下的不同性能。最后，我们根据实验结果给出了参数化选择的建议。