Estimation and Inference for Massive Multivariate Spatial Data

卫星对地球大气和海洋的观测有可能改进对飓风和其他极端天气事件的预报。对井水中存在的化学成分进行大量采样可以减少绘制地下水中危险物质地图的不确定性。对亚微米级化学反应的观察可能会对有毒微量元素在土壤中的行为产生新的见解。然而，如果分析数据的统计技术不跟上步伐，这些昂贵的收集海量数据的努力的价值将无法充分实现。目前可用的技术不足以灵活地对一个地区收集的由许多变量组成的海量数据集进行建模和提取信息。这一研究项目旨在开发计算高效的方法来解决分析海量多变量空间数据的主要挑战：(1)对多变量之间的关系得出合理的结论；(2)在绘制数据时充分和适当地利用所有变量。解决第一个挑战对于将观测和实验数据转化为科学知识至关重要。解决第二个问题对于提供对潜在有害结果的预测至关重要，而解决这两个挑战的关键是将多变量和空间数据分析整合到一个统一的框架中。时间序列和空间数据的内在相关性是使内插和预测成为可能的特征，但它也使多变量关系的估计复杂化。因此，时间序列数据的分析通常从数据到谱域的变换开始，在谱域中变换的数据近似不相关。虽然谱域在发展空间数据模型的理论方面发挥了核心作用，但有几个问题阻碍了实用的空间数据谱域方法的实施。该项目旨在开发方法创新以克服这些障碍，并为从业人员提供一套灵活的工具，以同时从数十个空间变量中提取信息，并使用所有可用数据预测未采样位置的变量。该方法利用计算效率高的周期性数据增加来简化分析，显著提高表征不确定性的能力，并得到新的理论结果的支持。

Satellite observations of the Earth's atmosphere and oceans have the potential to improve forecasting of hurricanes and other extreme weather events. Massive efforts to sample the chemical constituents present in well water can reduce uncertainty in mapping of hazardous materials in groundwater. Observations of chemical reactions at the sub-micron scale may lead to new insights about the behavior of toxic trace elements in soils. However, the value of these expensive efforts to collect massive amounts of data will not be fully realized if the statistical techniques for analyzing them do not keep pace. The current techniques available are inadequate to flexibly model and extract information from massive datasets consisting of many variables collected across a region. This research project aims to develop computationally efficient methods for addressing the central challenges for analyzing massive multivariate spatial data: (1) drawing justifiable conclusions about the relationships among the multiple variables, and (2) making full and appropriate use of all variables when mapping the data. Addressing the first challenge is essential to translating observational and experimental data into scientific knowledge. Addressing the second is crucially important for providing predictions of potentially harmful outcomes, and the key to solving both challenges is integrating the multivariate and spatial data analysis into a unified framework.The inherent correlation in time series and spatial data is the feature that makes interpolation and forecasting possible, but it also complicates estimation of multivariate relationships. As a result, analyses of time series data often start with a transformation of the data into the spectral domain, in which the transformed data are approximately uncorrelated. Although the spectral domain has played a central role in developing theory for models for spatial data, several issues have hindered the implementation of practical spectral domain methods for spatial data. This project aims to develop methodological innovations to overcome those barriers and provide practitioners with a flexible set of tools to extract information from dozens of spatial variables simultaneously, and predict variables at unsampled locations using all of the available data. The methods employ computationally efficient periodic data augmentations to simplify analyses, dramatically improve the ability to characterize uncertainty, and are supported by novel theoretical results.