In the last two decades, the linear model of coregionalization (LMC) has been widely used to model multivariate spatial processes. From a computational standpoint, the LMC is a substantially easier model to work with than other multidimensional alternatives. Up to now, this fact has been largely overlooked in the literature. Starting from an analogy with matrix normal models, we propose a reformulation of the LMC likelihood that highlights the linear, rather than cubic, computational complexity as a function of the dimension of the response vector. Further, we describe in detail how those simplifications can be included in a Gaussian hierarchical model. In addition, we demonstrate in two examples how the disentangled version of the likelihood we derive can be exploited to improve Markov chain Monte Carlo (MCMC) based computations when conducting Bayesian inference. The first is an interwoven approach that combines samples from centered and whitened parametrizations of the latent LMC distributed random fields. The second is a sparsity-inducing method that introduces structural zeros in the coregionalization matrix in an attempt to reduce the number of parameters in a principled way. It also provides a new way to investigate the strength of the correlation among the components of the outcome vector. Both approaches come at virtually no additional cost and are shown to significantly improve MCMC performance and predictive performance respectively. We apply our methodology to a dataset comprised of air pollutant measurements in the state of California.

翻译：在过去的二十年中，协同区域化线性模型（LMC）已被广泛用于多元空间过程的建模。从计算角度来看，LMC相较于其他多维替代模型在易用性上具有显著优势，然而这一事实至今在文献中尚未得到充分重视。基于矩阵正态模型的类比，我们提出了一种LMC似然函数的重新表述方法，该方法突显了其计算复杂度随响应向量维度呈线性增长（而非三次方增长）的特性。进一步地，我们详细描述了如何将这些简化方法整合到高斯层次模型中。此外，通过两个实例展示了如何利用所推导的似然解耦形式来改进基于马尔可夫链蒙特卡洛（MCMC）的贝叶斯推理计算：第一个实例采用交织方法，结合了潜在LMC分布随机场在中心化和白化参数化下的样本；第二个实例则是一种稀疏诱导方法，通过在协同区域化矩阵中引入结构零值，以原理性方式减少参数数量，同时为探究结果向量分量间相关强度提供了新途径。两种方法几乎不增加额外计算成本，并分别显著提升了MCMC性能和预测性能。我们将该方法应用于加利福尼亚州空气污染物测量数据集。