Default Bayesian Analysis of Spatial Data

空间数据的收集和分析在生态学、流行病学、地质学和水文学等自然科学和地球科学中无处不在。该研究项目将开发统计方法，用于高斯模型的自动贝叶斯分析，不需要任何主观输入。这些模型发挥了突出的作用，因为它们具有多功能性，可以模拟空间变化的现象，并且可以作为构建更复杂模型的基础。当主要目标是空间插值时，贝叶斯方法特别吸引人，但是在这样的模型中实现它面临着两大挑战：(i)自动制定适应所调查数据规模的合理先验分布；（ii）分析当今标准的大量数据集。该项目旨在发展理论和实践来克服这两个挑战，这将使大型空间数据集的自动贝叶斯分析具有实际可行性。此外，该项目将培养研究生空间统计的一般知识，特别是本项目的主题。该项目的结果将在不同的渠道传播，并将向公众提供执行该方法的软件。通过对贝叶斯模型的两个部分进行创新，该研究项目将使大型空间数据集的默认贝叶斯分析变得切实可行。首先，将开发协方差模型参数的近似参考先验，允许以自动方式对这些模型进行贝叶斯分析，而不需要主观启发。这些将基于平稳随机场的谱近似。其次，对大型空间数据集可行的似然近似将通过制定策略来调整最近提出的对平稳协方差函数的近似来阐述。调整近似的目的是在精度和计算量之间取得平衡。这两种近似都依赖于模型的谱密度，而不是协方差函数。同时，参考先验和似然近似将使包括模型选择和评估在内的默认贝叶斯分析成为可能。此外，该项目将严格评估将随机场平滑度固定在与分析数据无关的值（由惯例或传统选择）的常见做法。该项目将研究关于平滑参数的空间数据中量化信息内容的方法，并揭示这如何依赖于样本设计。该方法将在地球科学的各种数据集上进行测试，特别侧重于空间降雨数据。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

The collection and analysis of spatial data are ubiquitous in most natural and earth sciences, such as ecology, epidemiology, geology and hydrology. This research project will develop statistical methodology for automatic Bayesian analysis of Gaussian models that does not require any subjective input. These models play a prominent role due to their versatility to model spatially varying phenomena, and because they serve as building blocks for the construction of more elaborate models. The Bayesian approach is especially appealing when the main goal is spatial interpolation, but implementing it for such models faces two big challenges: (i) Automatic formulation of sensible prior distributions that are adapted to the scale of the data under investigation and (ii) Analysis of massive data sets that are the norm nowadays. The project aims at developing theory and practice to overcome both challenges, which will make practicably feasible the automatic Bayesian analysis of large spatial data sets. In addition, the project will train graduate students in spatial statistics in general, and the topics of this project in particular. The results derived from the project will be disseminated in diverse outlets, and software to implement the methodology will be made publicly available.The research project will make practicably feasible default Bayesian analyses of large spatial data sets, by contributing innovations to the two parts of the Bayesian model. First, approximate reference priors for the parameters of covariance models will be developed that allow carrying out Bayesian analyses for these models in an automatic fashion, not requiring subjective elicitation. These will be based on the spectral approximation of stationary random fields. Second, likelihood approximations feasible for large spatial data sets will be elaborated by developing strategies to tune a recently proposed approximation for stationary covariance functions. The tuning of the approximation aims at striking a balance between accuracy and computational effort. Both approximations rely on the spectral density, rather than the covariance function, of the model. Together, the reference prior and likelihood approximations will make possible carrying out default Bayesian analyses that include model selection and assessment. In addition, the project will critically assess the common practice of fixing the smoothness of the random field at a value, chosen by convention or tradition, that bears no relation to the data under analysis. The project will investigate methods to quantify the information content in spatial data about smoothness parameters, and uncover how this depends on the sample design. The methodology will be tested on diverse data sets from the earth sciences, with special focus on spatial rainfall data.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.