Two algorithms are proposed to simulate space-time Gaussian random fields with a covariance function belonging to an extended Gneiting class, the definition of which depends on a completely monotone function associated with the spatial structure and a conditionally negative definite function associated with the temporal structure. In both cases, the simulated random field is constructed as a weighted sum of cosine waves, with a Gaussian spatial frequency vector and a uniform phase. The difference lies in the way to handle the temporal component. The first algorithm relies on a spectral decomposition in order to simulate a temporal frequency conditional upon the spatial one, while in the second algorithm the temporal frequency is replaced by an intrinsic random field whose variogram is proportional to the conditionally negative definite function associated with the temporal structure. Both algorithms are scalable as their computational cost is proportional to the number of space-time locations that may be irregular in space and time. They are illustrated and validated through synthetic examples.

中文翻译：

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用不可分的Gneiting型协方差函数模拟时空随机场

提出了两种算法来模拟时空高斯随机场，它们具有属于扩展Gneiting类的协方差函数，其定义取决于与空间结构相关的完全单调函数和与时间结构相关的条件负定函数。在这两种情况下，模拟的随机场都被构造为余弦波的加权和，具有高斯空间频率矢量和均匀相位。区别在于处理时间分量的方式。第一种算法依靠频谱分解来模拟以空间频率为条件的时间频率，而在第二种算法中，时间频率由内在随机场代替，后者的变异函数与与时间结构相关的条件负定函数成比例。两种算法都是可扩展的，因为它们的计算成本与时空位置的数量成正比，而时空位置的时空可能不规则。通过综合示例对它们进行了说明和验证。