Gaussian processes are widely used as priors for unknown functions in statistics and machine learning. To achieve computationally feasible inference for large datasets, a popular approach is the Vecchia approximation, which is an ordered conditional approximation of the data vector that implies a sparse Cholesky factor of the precision matrix. The ordering and sparsity pattern are typically determined based on Euclidean distance of the inputs or locations corresponding to the data points. Here, we propose instead to use a correlation-based distance metric, which implicitly applies the Vecchia approximation in a suitable transformed input space. The correlation-based algorithm can be carried out in quasilinear time in the size of the dataset, and so it can be applied even for iterative inference on unknown parameters in the correlation structure. The Euclidean- and correlation-based Vecchia approximations are equivalent for strictly decreasing isotropic covariances, but the correlation-based approach has two advantages for more complex settings: It can result in more accurate approximations, and it offers a simple, automatic strategy that can be applied to any covariance, even when Euclidean distance is not applicable. We demonstrate these advantages in several settings, including anisotropic, nonstationary, multivariate, and spatio-temporal processes. We also illustrate our method on multivariate spatio-temporal temperature fields produced by a regional climate model.
翻译:Gausian 进程被广泛用作统计和机器学习中未知功能的前期。 为了实现大型数据集的计算可行的假设, 流行的方法是 Vecchia 近似值, 即数据矢量的定点有条件近似值, 意味着精确矩阵中一个稀疏的Choolesky 系数。 排序和宽度模式通常根据输入或位置与数据点相对应的Eucliidean距离来决定。 在这里, 我们提议使用基于关联的距离尺度, 在合适的转换输入空间中隐含地应用Vecchia 近似值。 基于关联的算法可以在数据集大小的准线性时间内进行, 因而甚至可以用于相关结构中未知参数的迭接推法。 以 Euclidean 和基于关联的Vecchia 近似值相当于严格减少偏移的偏移调值, 但基于关联的方法对于更复杂的环境有两个优点: 它可以导致更精确的近度, 并且它提供了一种简单、 自动的策略, 可用于任何交替性,, 即使当 Euclimdevocarate strate ro a colate strue (we) ro polate) acolate, roviolate (我们所制作的多位、 rocolver) acolate- proview) acolviolver roviolate- proview) a violate (我们所制作的多位、 violver) acolate- procolver) aviolver 。