Space–Time Covariance Functions
Space–Time Covariance Functions
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DOI：
10.1198/016214504000000854
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发表时间：
2005-03
影响因子：
3.7
通讯作者：
M. Stein
中科院分区：
文献类型：
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作者：
M. Stein
这项工作考虑了一些时空协方差函数的属性，以及这些属性如何与过程的时空相互作用有关。首先，它研究了空间-时间协方差函数远离原点的平滑度如何影响空间差异的时间相关性。远离原点的模型并不比在原点处更平滑，例如可分离模型，在某些情况下可能希望避免某些相关性的不连续性。平滑远离原点的协方差函数示出遵循从相应的谱密度具有有限的时刻的衍生物。这些结果被用来获得一个参数类的谱密度，其相应的时空协方差函数是无限可微远离原点，并允许基本上是任意的，并可能不同程度的平滑的过程中的空间和时间。其次，这项工作考虑了时空不对称的模型;时间t的站点x和时间s的站点y之间的协方差不同于时间s的站点x和时间t的站点y之间的协方差。一种通用的方法来产生非对称模型的对称模型，通过采取衍生物。最后，在时间上的马尔可夫假设高斯过程的空间-时间协方差函数的影响进行了检查，并给出了一个明确的表征所有这样的连续协方差函数。在这项工作中描述的几个新的模型被应用到风数据从爱尔兰。
This work considers a number of properties of space–time covariance functions and how these relate to the spatial-temporal interactions of the process. First, it examines how the smoothness away from the origin of a space–time covariance function affects, for example, temporal correlations of spatial differences. Models that are not smoother away from the origin than they are at the origin, such as separable models, have a kind of discontinuity to certain correlations that one might wish to avoid in some circumstances. Smoothness away from the origin of a covariance function is shown to follow from the corresponding spectral density having derivatives with finite moments. These results are used to obtain a parametric class of spectral densities whose corresponding space–time covariance functions are infinitely differentiable away from the origin and that allows for essentially arbitrary and possibly different degrees of smoothness for the process in space and time. Second, this work considers models that are asymmetric in space–time; the covariance between site x at time t and site y at time s is different than the covariance between site x at time s and site y at time t. A general approach is described for generating asymmetric models from symmetric models by taking derivatives. Finally, the implications of a Markov assumption in time on space–time covariance functions for Gaussian processes are examined, and an explicit characterization of all such continuous covariance functions is given. Several of the new models described in this work are applied to wind data from Ireland.