发表机构
Iowa State University; University of Chicago（爱荷华州立大学; 芝加哥大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
针对高维域中异质结构大型空间数据，提出基于二叉树的残差分解高斯过程模型ResTGP，实现多尺度协方差刻画与分而治之计算，并证明后验一致性，在数值和风暴潮应用中表现优异。
AI 中文摘要
随着测量技术的进步和计算能力的提升，人们常常在高维域上收集具有异质结构的大型空间数据。现有的高斯过程（GP）模型和计算策略往往不足以分析多维域中的此类数据集。为应对这些挑战，我们开发了一种名为ResTGP的贝叶斯残差树高斯过程方法，用于处理多维域中可能具有异质结构的大型空间数据。其关键思想是通过迭代计算预测过程和残差过程，沿二叉树的级联分辨率分解高斯过程，使得每个树节点（包括内部节点和叶节点）上的残差过程足以捕捉该节点内的更细粒度依赖。这使得能够以灵活、多尺度的方式刻画底层协方差结构，同时在数据域上实现分而治之，从而带来计算效率。为了支持高效的树推断，我们引入了一种基于递归消息传递的贝叶斯推断计算策略，在给定树的情况下，其计算复杂度随样本量线性增长。本文还证明了在非参数回归框架下，该模型对连续函数估计的后验一致性。大量的数值示例和风暴潮应用验证了所提方法的优势。
英文摘要
With the advance of measurement technologies and increasing computing power, large spatial data with heterogeneous structures are often collected over high-dimensional domains. Existing Gaussian process (GP) models and computational strategies are often inadequate for analyzing such datasets in multi-dimensional domains. To address these challenges, we develop a Bayesian residual tree GP methodology called ResTGP for large spatial data with potentially heterogeneous structures in multi-dimensional domains. The key idea is to decompose a Gaussian process at a cascade of resolutions along a dyadic tree through iteratively computing predictive and residual processes so that the residual process on each tree node, both interior and leaf, becomes sufficient for the finer-level dependency within that node. This allows characterization of the underlying covariance structure in a flexible, multi-scale manner while achieving divide-and-conquer on the data domain, which leads to computational efficiency. To allow efficient tree inference, we introduce a computational strategy for Bayesian inference based on recursive message passing, which scales linearly with the sample size given the tree. This paper also proves posterior consistency of the model for estimating continuous functions in a nonparametric regression framework. Extensive numerical examples and the storm surge application confirm the advantages of the proposed method.
