In spatial data, especially in geostatistics data where measurements are often provided by satellite scanning, some parts of data may get missed. Due to spatial dependence in the data, these missing values probably are caused by some latent spatial random fields. In this case, ignoring missingness is not logical and may lead to invalid inferences. Thus incorporating the missingness process model into the inferences could improve the results. There are several approaches to take into account the non-ignorable missingness, one of them is the shared parameter model method. In this paper, we extend it for spatial data so that we will have a joint spatial Bayesian shared parameter model. Then the missingness process will be jointly modeled with the measurement process and one or more latent spatial random fields as shared parameters would describe their association. Bayesian inference is implemented by Integrated nested Laplace approximation. A computationally effective approach is applied via a stochastic partial differential equation for approximating latent Gaussian random field. In a simulation study, the proposed spatial joint model is compared with a model that assumes data are missing at random. Based on these two models, the lake surface water temperature data for lake Vänern in Sweden are analyzed. The results of estimation and prediction confirm the efficiency of the spatial joint model.

中文翻译：

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具有不可忽略缺失的空间数据的贝叶斯预测

在空间数据中，尤其是在通常由卫星扫描提供测量值的地质统计数据中，可能会遗漏某些部分的数据。由于数据的空间依赖性，这些缺失值可能是由一些潜在的空间随机场引起的。在这种情况下，忽略缺失是不合逻辑的，可能会导致无效的推论。因此，将缺失过程模型纳入推理可以改善结果。有几种方法可以考虑不可忽略的缺失，其中之一是共享参数模型方法。在本文中，我们将其扩展到空间数据，以便我们将拥有一个联合空间贝叶斯共享参数模型。然后缺失过程将与测量过程和一个或多个潜在空间随机场联合建模，因为共享参数将描述它们的关联。贝叶斯推理是通过集成嵌套拉普拉斯近似实现的。通过随机偏微分方程应用计算上有效的方法来逼近潜在高斯随机场。在模拟研究中，将提议的空间联合模型与假设数据随机丢失的模型进行比较。基于这两个模型，对瑞典Vänern湖的湖面水温数据进行了分析。估计和预测的结果证实了空间联合模型的有效性。通过随机偏微分方程应用计算上有效的方法来逼近潜在高斯随机场。在模拟研究中，将提议的空间联合模型与假设数据随机丢失的模型进行比较。基于这两个模型，对瑞典Vänern湖的湖面水温数据进行了分析。估计和预测的结果证实了空间联合模型的有效性。通过随机偏微分方程应用计算上有效的方法来逼近潜在高斯随机场。在模拟研究中，将提议的空间联合模型与假设数据随机丢失的模型进行比较。基于这两个模型，对瑞典Vänern湖的湖面水温数据进行了分析。估计和预测的结果证实了空间联合模型的有效性。