Fast and scalable computations for Gaussian hierarchical models with intrinsic conditional autoregressive spatial random effects
Fast and scalable computations for Gaussian hierarchical models with intrinsic conditional autoregressive spatial random effects
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DOI：
10.1016/j.csda.2021.107264
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发表时间：
2021-04
期刊：
影响因子：
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通讯作者：
Marco A. R. Ferreira;Erica M. Porter;C. Franck
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文献类型：
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作者：
Marco A. R. Ferreira;Erica M. Porter;C. Franck
提出了具有内禀条件自回归（ICAR）空间随机效应的高斯层次模型贝叶斯分析的快速算法。为了实现计算加速，首先证明了使用以苍蝇为中心的不适当的CAR先验与使用和零约束ICAR先验之间的等价性。这个等价的结果为算法提供了关键的洞察力，这些算法是基于在谱域中重写层次模型的。这两种新算法分别是谱吉布斯采样器（SGS）和谱后验最大化器（SPM）。这两种算法都是基于单一矩阵谱分解计算。经过此计算，SGS和SPM算法随样本量线性扩展。SGS算法更适合较小的样本量，而SPM算法更适合样本量大到足以进行渐近计算以提供良好的近似。由于矩阵谱分解只需要计算一次，因此在需要拟合多个模型的情况下，SPM算法比基于稀疏矩阵分解的算法（需要对随机效应方差参数的每个值进行计算）具有计算优势。进行了三次仿真研究：第一次仿真研究表明，与使用精度矩阵谱分解的算法相比，SGS算法在估计计算速度方面的性能有所提高；第二项模拟研究表明，对于10个回归量和样本大小从49到3600不等的模型选择计算，与R包INLA中实现的当前最快的最先进算法相比，SPM计算速度快550到1825倍；第三个模拟研究表明，与默认INLA设置相比，SGS和SPM结合参考先验提供了更充分的不确定性量化。最后，通过对2017年美国3108个县的县级家庭收入中位数的空间回归研究，说明了新型SGS和SPM算法的应用。
Fast algorithms are developed for Bayesian analysis of Gaussian hierarchical models with intrinsic conditional autoregressive (ICAR) spatial random effects. To achieve computational speed-ups, first a result is proved on the equivalence between the use of an improper CAR prior with centering on the fly and the use of a sum-zero constrained ICAR prior. This equivalence result then provides the key insight for the algorithms, which are based on rewriting the hierarchical model in the spectral domain. The two novel algorithms are the Spectral Gibbs Sampler (SGS) and the Spectral Posterior Maximizer (SPM). Both algorithms are based on one single matrix spectral decomposition computation. After this computation, the SGS and SPM algorithms scale linearly with the sample size. The SGS algorithm is preferable for smaller sample sizes, whereas the SPM algorithm is preferable for sample sizes large enough for asymptotic calculations to provide good approximations. Because the matrix spectral decomposition needs to be computed only once, the SPM algorithm has computational advantages over algorithms based on sparse matrix factorizations (which need to be computed for each value of the random effects variance parameter) in situations when many models need to be fitted. Three simulation studies are performed: the first simulation study shows improved performance in computational speed in estimation of the SGS algorithm compared to an algorithm that uses the spectral decomposition of the precision matrix; the second simulation study shows that for model selection computations with 10 regressors and sample sizes varying from 49 to 3600, when compared to the current fastest state-of-the-art algorithm implemented in the R package INLA, SPM computations are 550 to 1825 times faster; the third simulation study shows that, when compared to default INLA settings, SGS and SPM combined with reference priors provide much more adequate uncertainty quantification. Finally, the application of the novel SGS and SPM algorithms is illustrated with a spatial regression study of county-level median household income for 3108 counties in the contiguous United States in 2017.