Default Bayesian Analysis of Spatial Data

在生态学、流行病学、地质学和水文学等大多数自然科学和地球科学中，空间数据的收集和分析无处不在。该研究项目将开发用于高斯模型自动贝叶斯分析的统计方法，不需要任何主观输入。这些模型发挥了突出的作用，因为它们的多功能性，以模拟空间变化的现象，因为它们作为构建更精细的模型的积木。当主要目标是空间插值时，贝叶斯方法特别有吸引力，但将其用于此类模型面临两大挑战：（i）自动制定适应调查数据规模的合理先验分布，以及（ii）分析当今规范的海量数据集。该项目旨在发展理论和实践，以克服这两个挑战，这将使大型空间数据集的自动贝叶斯分析切实可行。此外，该项目还将对研究生进行一般空间统计培训，特别是本项目的专题培训。该项目的成果将在各种渠道传播，并将公开提供实施该方法的软件，该研究项目将通过对贝叶斯模型的两个部分作出创新，对大型空间数据集进行切实可行的缺省贝叶斯分析。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。第二，可能性近似可行的大型空间数据集将制定战略，调整最近提出的近似平稳协方差函数。近似值的调整旨在精确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进行默认贝叶斯分析，包括模型选择和评估成为可能。此外，该项目将严格评估将随机场的平滑度固定在一个值的常见做法，该值是根据惯例或传统选择的，与所分析的数据无关。该项目将研究量化空间数据中关于平滑参数的信息内容的方法，并揭示这如何取决于样本设计。该方法将在来自地球科学的各种数据集上进行测试，特别关注空间降雨数据。该奖项反映了NSF的法定使命，并被认为值得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。

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.