Estimating covariance parameters for multivariate spatial Gaussian random fields is computationally challenging, as the number of parameters grows rapidly with the number of variables, and likelihood evaluation requires operations of order $\mathcal{O}((np)^3)$. In many applications, however, not all cross-dependencies between variables are relevant, suggesting that sparse covariance structures may be both statistically advantageous and practically necessary. We propose a LASSO-penalized estimation framework that induces sparsity in the Cholesky factor of the multivariate Matérn correlation matrix, enabling automatic identification of uncorrelated variable pairs while preserving positive semidefiniteness. Estimation is carried out via a projected block coordinate descent algorithm that decomposes the optimization into tractable subproblems, with constraints enforced at each iteration through appropriate projections. Regularization parameter selection is discussed for both the likelihood and composite likelihood approaches. We conduct a simulation study demonstrating the ability of the method to recover sparse correlation structures and reduce estimation error relative to unpenalized approaches. We illustrate our procedure through an application to a geochemical dataset with $p = 36$ variables and $n = 3998$ spatial locations, showing the practical impact of the method and making spatial prediction feasible in a setting where standard approaches fail entirely.

翻译：多元空间高斯随机场的协方差参数估计在计算上极具挑战性，因为参数数量随变量个数急剧增长，且似然函数求值需要计算阶数为$\mathcal{O}((np)^3)$的运算。然而在许多应用中，并非所有变量间的交叉依赖关系都相关，这表明稀疏协方差结构在统计上既有优势又在实际中必不可少。我们提出一种基于LASSO惩罚的估计框架，该框架在多变量马特恩相关矩阵的Cholesky因子中引入稀疏性，能够在保持半正定性的同时自动识别不相关的变量对。估计过程通过投影块坐标下降算法实现，该算法将优化问题分解为易于处理的子问题，并在每次迭代中通过适当投影施加约束。针对似然函数和复合似然方法分别讨论了正则化参数的选择问题。我们通过模拟研究证明了该方法能够恢复稀疏相关结构，且相比非惩罚方法能降低估计误差。通过一个包含$p=36$个变量和$n=3998$个空间点位的地球化学数据集应用，我们展示了该方法的实用效果，并在标准方法完全失效的情境下实现了可行的空间预测。