协同区域化线性模型的计算考量
Computational Considerations for the Linear Model of Coregionalization
AI总结：
针对协同区域化线性模型似然推断的三次方计算瓶颈，本文重构似然使其随响应维度呈线性扩展，并提出结构性零元素稀疏方法，在MCMC中近乎零额外成本且提升预测性能。
摘要：
在过去二十年中，协同区域化线性模型（LMC）已被广泛用于对多元空间过程进行建模。然而，由于高斯似然评估伴随的三次方计算成本，对此类模型进行基于似然的推断可能是一项具有挑战性的任务。从与矩阵正态模型的类比出发，我们提出了一种LMC似然的重构形式，该形式凸显了其作为响应向量维度函数的线性而非三次方计算复杂度。我们描述了如何在高斯层次模型中利用这些简化。此外，我们提出了一种新的LMC稀疏诱导方法，该方法在协同区域化矩阵中引入结构性零元素，试图以一种有原则且数据驱动的方式减少参数数量。我们对LMC似然的重构确保了当将稀疏方法纳入马尔可夫链蒙特卡洛（MCMC）算法时，几乎不会产生额外成本。在合成数据上的结果表明，该方法显著提升了预测性能。我们还将所提方法应用于一个由加利福尼亚州空气污染物测量值组成的数据集。我们通过稀疏方法提供的新见解，考察了这些测量值之间相关性的强度。
英文摘要：
In the last two decades, the linear model of coregionalization (LMC) has been widely used to model multivariate spatial processes. However, it can be a challenging task to conduct likelihood-based inference for such models because of the cubic cost associated with Gaussian likelihood evaluations. 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. We describe how those simplifications can be exploited in Gaussian hierarchical models. In addition, we propose a new sparsity-inducing approach to the LMC that introduces structural zeros in the coregionalization matrix in an attempt to reduce the number of parameters in a principled and data-driven way. Our reformulation of the LMC likelihood ensures that our sparse approach comes at virtually no additional cost when included in a Markov chain Monte Carlo (MCMC) algorithm. It is shown, on synthetic data, to significantly improve predictive performance. We also apply our methodology to a dataset comprised of air pollutant measurements from the state of California. We investigate the strength of the correlation among the measurements by providing new insights from our sparse method.
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