Circulant Embedding of Approximate Covariances for Inference From Gaussian Data on Large Lattices

Recently proposed computationally efficient Markov chain Monte Carlo (MCMC) and Monte Carlo expectation–maximization (EM) methods for estimating covariance parameters from lattice data rely on successive imputations of values on an embedding lattice that is at least two times larger in each dimension. These methods can be considered exact in some sense, but we demonstrate that using such a large number of imputed values leads to slowly converging Markov chains and EM algorithms. We propose instead the use of a discrete spectral approximation to allow for the implementation of these methods on smaller embedding lattices. While our methods are approximate, our examples indicate that the error introduced by this approximation is small compared to the Monte Carlo errors present in long Markov chains or many iterations of Monte Carlo EM algorithms. Our results are demonstrated in simulation studies, as well as in numerical studies that explore both increasing domain and fixed domain asymptotics. We compare the exact methods to our approximate methods on a large satellite dataset, and show that the approximate methods are also faster to compute, especially when the aliased spectral density is modeled directly. Supplementary materials for this article are available online.

近年来提出的、用于从格点数据中估计协方差参数的计算高效型马尔可夫链蒙特卡洛（Markov chain Monte Carlo, MCMC）与蒙特卡洛期望-最大化（Monte Carlo expectation–maximization, EM）方法，均需在每一维尺寸至少为原格点两倍的嵌入格点上开展连续值插补。此类方法在部分意义下可被视为精确方法，但我们的研究表明，使用如此大量的插补值会导致马尔可夫链与EM算法的收敛速度大幅放缓。为此，我们提出采用离散谱近似方法，以实现此类算法在更小尺寸嵌入格点上的运行。尽管我们的方法属于近似方案，但案例研究显示，相较于长马尔可夫链或多次迭代的蒙特卡洛EM算法所引入的蒙特卡洛误差，该近似带来的误差微不足道。我们的结论通过模拟研究与数值研究得以验证，后者同时探讨了增域渐近性与固定域渐近性问题。我们在大型卫星数据集上将精确方法与近似方法进行了对比，结果显示近似方法的计算效率更优，尤其是在直接对混叠谱密度进行建模的场景下。本文的补充材料可在线获取。