High-Dimensional Nonstationary Processes for Spatial Analysis and Machine Learning

地球科学、气候和环境科学、公共卫生、社会科学和交通统计中的许多应用问题涉及从具有非平凡几何形状的复杂约束域收集的大量空间数据，例如具有尖锐凹陷的不规则边界、由于地理限制而产生的内部孔以及河流或道路网络。从业者对复杂空间依赖建模感兴趣，因为它在空间问题的估计和预测中起着最重要的作用。然而，对于复杂域上空间问题的估计和预测工具却非常有限。该项目旨在开发新的统计模型和算法，以更好地表征大型数据集中潜在的异构空间依赖性，同时尊重数据中的不规则几何形状。该方法将适用于多个跨学科领域的广泛实际问题。拟议的研究计划将为本科生和研究生水平的跨学科研究培训提供大量机会，特别侧重于促进统计科学的多样性和包容性。该项目将引入一类新的非平稳模型，具有灵活的局部平稳依赖结构，用于大空间数据。在尊重域边界不规则形状的前提下，采用具有柔性划分边界的流形划分模型实现了局部静止结构的检测。该项目将进一步发展一个新的框架，以建立一个有效的随机过程模型，将局部模型结合在一起。可以在统一的框架下进行参数估计和预测，同时捕获空间随机场的不连续/突变和平滑。此外，该项目将产生新的可扩展和并行化的分并征服推理工具，利用局部平稳假设的力量。所提出的方法的性能将通过模拟研究进行测试，并应用于实际应用。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。

Numerous application problems in geosciences, climate and environmental sciences, public health, social sciences and traffic statistics involve large amounts of spatial data collected from complex constrained domains with non-trivial geometries, such as irregular boundaries with sharp concavities, interior holes due to geographic constraints, and river or road networks. Practitioners are interested in modeling complex spatial dependence because it plays the most important role in the estimation and prediction of spatial problems. However, there is very limited tool for the estimation and prediction of spatial problems on complex domains. This project aims to develop new statistical models and algorithms to better characterize the potentially much more heterogeneous spatial dependence in large data sets while respecting irregular geometries in the data. The methodology will be applicable to a broad range of real problems in multiple interdisciplinary fields. The proposed research initiatives will offer numerous opportunities for interdisciplinary research training at undergraduate and graduate levels, with a particular focus on advancing diversity and inclusion in statistical sciences.This project will introduce a new class of nonstationary models with flexible locally stationary dependence structures for large spatial data. The detection of locally stationary structures is achieved by a novel manifold partition model with flexible partition boundaries while respecting irregular shapes of domain boundaries. The project will further develop a novel framework to build a valid stochastic process model to knit together local models. Both parameter estimation and prediction can be performed under a unified framework, and both discontinuities/abrupt changes and smoothness in spatial random field can be captured. Moreover, the project will result in new scalable and parallelizable divide-merge-conquer inference tools, harnessing the power of locally stationary assumptions. The performance of the proposed methods will be tested with simulation studies and applied to real-life applications.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.