Bayesian and Regularization Methods for Spatial Homogeneity Pursuit with Large Datasets

空间数据来源于农业、地质、经济和社会科学等不同学科的研究。在许多应用问题中，从业者对研究空间响应和一组解释变量之间的关系感兴趣。随着大空间数据的日益可用性，有必要研究这些关联的空间变化模式。特别是，检测空间关系中的聚类模式是需要的，因为它允许从业者对局部关联有直接的解释。在这个项目中，PI将开发新的统计模型和高效的计算算法来追求空间均匀性，既有很强的理论味道，也有现实的现实考虑。总体方法本质上是跨学科的。它整合了统计学、机器学习、计算和地球科学的进步。在这个项目中，PI将考虑一个变系数回归模型来研究响应和协变量之间的聚类关系。特别是，将开发基于树的正则化方法，以鼓励邻近位置的回归系数之间的空间均匀性。PI将设计惩罚优化和贝叶斯MCMC算法来实现所提出的模型。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。PI还将结合分段常数函数的近似理论、组合和代数图论以及高维渐近理论，研究正则化方法的行为的理论性质。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。

Spatial data arises from the research of diverse disciplines such as agricultural, geological, economic and social sciences. In many application problems, practitioners are interested in studying the associations between spatial responses and a set of explanatory variables. With the increasing availability of big spatial data, there is a great need to investigate the spatially varying patterns in such associations. In particular, detecting clustering patterns in spatial relations is desired since it allows practitioners to have straightforward interpretations of local associations. In this project, the PI will develop new statistical models and efficient computation algorithms for spatial homogeneity pursuit with both strong theoretical flavor and realistic practical considerations. The overall approach is interdisciplinary in nature. It integrates the advancements in statistics, machine learning, computation, and geosciences. In this project, the PI will consider a varying coefficient regression model to study the clustered relationship between responses and covariates. In particular, tree-based regularization methods will be developed to encourage spatial homogeneity between regression coefficients at neighboring locations. The PI will design both penalized optimizations and Bayesian MCMC algorithms to implement the proposed models. The performance of the proposed methods will be tested with simulation studies and applied to real-life applications. The PI will also study theoretical properties concerning the behavior of the regularization methods by combining the approximation theory of piecewise constant functions, combinatorial and algebraic graph theory, and high dimensional asymptotic theories.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.