In this paper we develop a nonparametric multivariate spatial model that avoids specifying a Gaussian distribution for spatial random effects. Our nonparametric model extends the stick-breaking (SB) prior of Sethuraman (1994), which is frequently used in Bayesian modelling to capture uncertainty in the parametric form of an outcome. The stick-breaking prior is extended here to the spatial setting by assigning each location a different, unknown distribution, and smoothing the distributions in space with a series of space-dependent kernel functions that have a space-varying bandwidth parameter. This results in a flexible non-stationary spatial model, as different kernel functions lead to different relationships between the distributions at nearby locations. This approach is the first to allow both the probabilities and the point mass values of the SB prior to depend on space. Thus, there is no need for replications and we obtain a continuous process in the limit. We extend the model to the multivariate setting by having for each process a different kernel function, but sharing the location of the kernel knots across the different processes. The resulting covariance for the multivariate process is in general nonstationary and nonseparable. The modelling framework proposed here is also computationally efficient because it avoids inverting large matrices and calculating determinants, which often hinders the spatial analysis of large data sets. We study the theoretical properties of the proposed multivariate spatial process. The methods are illustrated using simulated examples and an air pollution application to model components of fine particulate matter.

中文翻译：

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通过核过程混合的多元空间非参数建模

在本文中，我们开发了一个非参数多元空间模型，该模型避免为空间随机效应指定高斯分布。我们的非参数模型扩展了 Sethuraman (1994) 的破棒 (SB) 先验，这在贝叶斯建模中经常使用，以捕捉结果参数形式的不确定性。通过为每个位置分配一个不同的未知分布，并使用一系列具有空间变化带宽参数的空间相关核函数来平滑空间中的分布，打破棒的先验在这里扩展到空间设置。这导致了灵活的非平稳空间模型，因为不同的核函数会导致附近位置的分布之间存在不同的关系。这种方法是第一个允许 SB 的概率和点质量值都依赖于空间的方法。因此，不需要重复，我们在极限中获得了一个连续的过程。我们通过为每个进程使用不同的内核函数，但在不同进程之间共享内核节点的位置，将模型扩展到多变量设置。多元过程的结果协方差通常是非平稳和不可分离的。这里提出的建模框架在计算上也很高效，因为它避免了大矩阵的求逆和行列式的计算，这往往会阻碍大数据集的空间分析。我们研究了所提出的多元空间过程的理论特性。