For multivariate spatial Gaussian process (GP) models, customary specifications of cross-covariance functions do not exploit relational inter-variable graphs to ensure process-level conditional independence among the variables. This is undesirable, especially for highly multivariate settings, where popular cross-covariance functions such as the multivariate Mat\'ern suffer from a "curse of dimensionality" as the number of parameters and floating point operations scale up in quadratic and cubic order, respectively, in the number of variables. We propose a class of multivariate "Graphical Gaussian Processes" using a general construction called "stitching" that crafts cross-covariance functions from graphs and ensures process-level conditional independence among variables. For the Mat\'ern family of functions, stitching yields a multivariate GP whose univariate components are Mat\'ern GPs, and conforms to process-level conditional independence as specified by the graphical model. For highly multivariate settings and decomposable graphical models, stitching offers massive computational gains and parameter dimension reduction. We demonstrate the utility of the graphical Mat\'ern GP to jointly model highly multivariate spatial data using simulation examples and an application to air-pollution modelling.
翻译:对于多变空间高斯进程(GP) 模型来说,交叉变量函数的习惯性规格并不利用关系间可变图形来确保变量之间的进程性有条件独立。 这不可取, 特别是对于高度多变环境来说。 在这种高度多变环境中, 诸如多变 Mat\'ern 等流行的交叉变量性函数会因参数和浮动点操作数量在变数数量中以二次和立方顺序分别扩大而受“ 维度的诅咒” 的“ 维度” 作用。 我们建议使用一个名为“ 静态” 的一般构造, 即从图形中工艺性跨变量功能的跨变量性功能和确保变量之间的进程性有条件独立性。 对于功能的 Mat\'ern 组, 缝合产生一个多变式GP, 其单数组成部分是 Mat\'ern GPs, 符合图形模型规定的进程级条件性独立性。 对于高多变式设置和可调化的图形模型, 我们用一个巨大的计算模型和参数度模型进行缝配制, 并用高度模型化的模型和高空位数模型化模型化模型模拟应用。