Bayesian and Regularization Methods for Spatial Homogeneity Pursuit with Large Datasets

空间数据产生于农业、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，从业者感兴趣的是研究空间反应和一组解释变量之间的关联。随着大空间数据可用性的增加，有一个很大的需要，以调查在这种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期望的，因为它允许从业者对本地关联进行直接的解释。在这个项目中，PI将开发新的统计模型和有效的计算算法，用于空间均匀性追求，具有强烈的理论味道和现实的实际考虑。总体方法是跨学科性质的。它集成了统计学，机器学习，计算和地球科学的进步。在本项目中，PI将考虑变系数回归模型，以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，以鼓励相邻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进行测试与模拟研究，并应用到现实生活中的应用。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来研究正则化方法的理论特性。该奖项反映了NSF的法定使命，并通过使用基金会的智力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.