Smoothing Windows for the Synthesis of Gaussian Stationary Random Fields Using Circulant Matrix Embedding

When generating Gaussian stationary random fields, a standard method based on circulant matrix embedding usually fails because some of the associated eigenvalues are negative. The eigenvalues can be shown to be nonnegative in the limit of increasing sample size. Computationally feasible large sample sizes, however, rarely lead to nonnegative eigenvalues. Another solution is to extend suitably the covariance function of interest so that the eigenvalues of the embedded circulant matrix become nonnegative in theory. Though such extensions have been found for a number of examples of stationary fields, the method depends on nontrivial constructions in specific cases.In this work, the embedded circulant matrix is smoothed at the boundary by using a cutoff window or overlapping windows over a transition region. The windows are not specific to particular examples of stationary fields. The resulting method modifies the standard circulant embedding, and is easy to use. It is shown that this straightforward approach works for many examples of interest, with the overlapping windows performing consistently better. The method even outperforms in the cases where extending the covariance leads to nonnegative eigenvalues in theory, in the sense that the transition region is considerably smaller. The Matlab code implementing the method is included in the online supplementary materials and also publicly available at www.hermir.org.

在生成高斯平稳随机场（Gaussian stationary random fields）时，基于循环矩阵嵌入（circulant matrix embedding）的标准方法常会失效，原因是其部分关联特征值为负值。研究表明，当样本量趋于无穷大时，特征值可变为非负；但实际计算中可行的大样本规模，却极少能让特征值保持非负。另一种解决方案是对目标协方差函数（covariance function）进行适度拓展，使得嵌入的循环矩阵的特征值在理论上变为非负。尽管针对诸多平稳随机场的示例已找到此类拓展方式，但该方法需针对特定场景开展非平凡的构造工作。
本研究通过在边界处使用过渡区域上的截断窗（cutoff window）或重叠窗（overlapping windows）对嵌入循环矩阵进行平滑处理。所用窗口不局限于特定的平稳随机场示例，所得到的方法对标准循环矩阵嵌入流程进行了改进，且易于使用。研究表明，这一直截了当的方法可适用于诸多目标场景，其中重叠窗的表现始终更优。在理论上通过拓展协方差函数即可获得非负特征值的场景中，该方法的表现甚至更为出色——这体现为其过渡区域的规模显著更小。实现该方法的Matlab代码已包含于在线补充材料中，同时可在www.hermir.org公开获取。