Estimating the parameters of spatial models for large spatial datasets can be computationally challenging, as it involves repeated evaluation of sizable spatial covariance matrices. In this paper, we aim to develop Krylov subspace-based methods that are computationally efficient for large spatial data. Specifically, we approximate the inverse and the log-determinant of the spatial covariance matrix in the log-likelihood function via conjugate gradient and stochastic Lanczos on a Krylov subspace. These methods reduce the computational complexity from $O(N^3)$ to $O(N^2\log N)$ and $O(N\log N)$ for dense and sparse matrices, respectively. Moreover, we quantify the difference between the approximated log-likelihood function and the original log-likelihood function and establish the consistency of parameter estimates. Simulation studies are conducted to examine the computational efficiency as well as the finite-sample properties. For illustration, our methodology is applied to analyze a large dataset comprising LiDAR estimates of forest canopy height in western Alaska.

中文翻译：

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大型空间数据建模和分析：Krylov 子空间方法

估计大型空间数据集的空间模型参数在计算上可能具有挑战性，因为它涉及对大量空间协方差矩阵的重复评估。在本文中，我们旨在开发基于 Krylov 子空间的方法，该方法对大型空间数据具有计算效率。具体来说，我们通过 Krylov 子空间上的共轭梯度和随机 Lanczos 来近似对数似然函数中空间协方差矩阵的逆和对数行列式。这些方法降低了计算复杂度$○({ñ}^3)$至$○({ñ}^2\mathrm{日志}ñ)$和$○(ñ\mathrm{日志}ñ)$分别用于密集和稀疏矩阵。此外，我们量化了近似对数似然函数和原始对数似然函数之间的差异，并建立了参数估计的一致性。进行模拟研究以检查计算效率以及有限样本属性。为了说明，我们的方法用于分析一个大型数据集，其中包括对阿拉斯加西部森林冠层高度的激光雷达估计。