Statistical models for spatial processes play a central role in analyses of spatial data. Yet, it is the simple, interpretable, and well understood models that are routinely employed even though, as is revealed through prior and posterior predictive checks, these can poorly characterise the spatial heterogeneity in the underlying process of interest. Here, we propose a new, flexible class of spatial-process models, which we refer to as spatial Bayesian neural networks (SBNNs). An SBNN leverages the representational capacity of a Bayesian neural network; it is tailored to a spatial setting by incorporating a spatial “embedding layer” into the network and, possibly, spatially-varying network parameters. An SBNN is calibrated by matching its finite-dimensional distribution at locations on a fine gridding of space to that of a target process of interest. That process could be easy to simulate from or we may have many realisations from it. We propose several variants of SBNNs, most of which are able to match the finite-dimensional distribution of the target process at the selected grid better than conventional BNNs of similar complexity. We also show that an SBNN can be used to represent a variety of spatial processes often used in practice, such as Gaussian processes, lognormal processes, and max-stable processes. We briefly discuss the tools that could be used to make inference with SBNNs, and we conclude with a discussion of their advantages and limitations.

中文翻译：

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空间贝叶斯神经网络

空间过程的统计模型在空间数据分析中发挥着核心作用。然而，常规使用的是简单、可解释且易于理解的模型，尽管通过先验和后验预测检查显示，这些模型不能很好地表征感兴趣的潜在过程中的空间异质性。在这里，我们提出了一类新的、灵活的空间过程模型，我们将其称为空间贝叶斯神经网络（SBNN）。 SBNN 利用贝叶斯神经网络的表征能力；它通过将空间“嵌入层”并入网络以及可能的空间变化的网络参数来针对空间设置进行定制。 SBNN 通过将空间精细网格上的有限维分布与感兴趣的目标过程的位置进行匹配来进行校准。这个过程很容易模拟，或者我们可以从中得到很多认识。我们提出了 SBNN 的几种变体，其中大多数能够比类似复杂度的传统 BNN 更好地匹配所选网格处目标过程的有限维分布。我们还表明，SBNN 可用于表示实践中经常使用的各种空间过程，例如高斯过程、对数正态过程和最大稳定过程。我们简要讨论了可用于通过 SBNN 进行推理的工具，最后讨论了它们的优点和局限性。