An explicit link between Gaussian fields and Gaussian Markov random fields: the stochastic partial differential equation approach
An explicit link between Gaussian fields and Gaussian Markov random fields: the stochastic partial differential equation approach
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DOI：
10.1111/j.1467-9868.2011.00777.x
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发表时间：
2011-01-01
影响因子：
5.8
通讯作者：
Lindstrom, Johan
中科院分区：
文献类型：
--
作者：
Lindgren, Finn;Rue, Havard;Lindstrom, Johan
连续索引高斯场（GF）是空间统计建模和地质统计学中最重要的成分。通过协方差函数的规范给出了场属性的直观解释。在计算方面，GF受到大n问题的阻碍，因为分解稠密矩阵的成本在维度上是立方的。虽然今天的计算能力是在所有时间高，这一事实似乎仍然是在许多应用程序的计算瓶颈。沿着GF，存在离散索引的高斯马尔可夫随机场（GMRF）类。马尔可夫性质使得所涉及的精度矩阵稀疏，这使得可以使用稀疏矩阵的数值算法，对于R-2中的字段，仅使用一般算法所需时间的平方根。一个GMRF的规格是通过其完整的条件分布，但其边际属性是不透明的，在这样的参数化。我们表明，使用（线性）随机偏微分方程的近似随机弱解，我们可以，为一些GF在Matern类，提供一个明确的链接，任何三角化的R-d，GF和GMRF之间，制定为基函数表示。结果是，我们可以从两个世界中取最好的，并使用GF进行建模，但使用GMRF进行计算。也许更重要的是，我们的方法推广到其他协方差函数产生的SPDE，包括振荡和非平稳的GF，以及GF流形上。我们说明了我们的方法，通过分析全球温度数据的非平稳模型定义在一个球体上。
Continuously indexed Gaussian fields (GFs) are the most important ingredient in spatial statistical modelling and geostatistics. The specification through the covariance function gives an intuitive interpretation of the field properties. On the computational side, GFs are hampered with the big n problem, since the cost of factorizing dense matrices is cubic in the dimension. Although computational power today is at an all time high, this fact seems still to be a computational bottleneck in many applications. Along with GFs, there is the class of Gaussian Markov random fields (GMRFs) which are discretely indexed. The Markov property makes the precision matrix involved sparse, which enables the use of numerical algorithms for sparse matrices, that for fields in R-2 only use the square root of the time required by general algorithms. The specification of a GMRF is through its full conditional distributions but its marginal properties are not transparent in such a parameterization. We show that, using an approximate stochastic weak solution to (linear) stochastic partial differential equations, we can, for some GFs in the Matern class, provide an explicit link, for any triangulation of R-d, between GFs and GMRFs, formulated as a basis function representation. The consequence is that we can take the best from the two worlds and do the modelling by using GFs but do the computations by using GMRFs. Perhaps more importantly, our approach generalizes to other covariance functions generated by SPDEs, including oscillating and non-stationary GFs, as well as GFs on manifolds. We illustrate our approach by analysing global temperature data with a non-stationary model defined on a sphere.