We conduct a study of the aliased spectral densities of Mat\'ern covariance functions on a regular grid of points, providing clarity on the properties of a popular approximation based on stochastic partial differential equations; while others have shown that it can approximate the covariance function well, we find that it assigns too much power at high frequencies and does not provide increasingly accurate approximations to the inverse as the grid spacing goes to zero, except in the one-dimensional exponential covariance case. We provide numerical results to support our theory, and in a simulation study, we investigate the implications for parameter estimation, finding that the SPDE approximation tends to overestimate spatial range parameters.

翻译：我们对固定的点网格上的 Mat\'ern 共变函数的别名光谱密度进行了一项研究,澄清了基于随机偏差部分方程式的流行近似值的特性;而其他人则表明,它能够很好地接近共变函数,我们发现,它给高频率分配了太多的功率,并且没有提供在网格间距变为零时,越来越准确的反向近似值,但单维指数共变情况除外。我们提供了数字结果来支持我们的理论,在模拟研究中,我们调查了参数估计的影响,发现SPDE的近似值往往高估了空间范围参数。