Collaborative Research: Bayesian Residual Learning and Random Recursive Partitioning Methods for Gaussian Process Modeling

罕见的自然灾害（例如，风暴潮和飓风）可能造成生命损失，并对社会和环境造成毁灭性破坏。例如，卡特里娜飓风（2005年）由于风暴潮在新奥尔良地区和密西西比沿岸造成1 500多人死亡，估计总损失达750亿美元。不确定性量化（UQ）已被广泛用于了解、监测和预测这些罕见的自然灾害。高斯过程（GP）建模框架是解决此类UQ应用的最广泛使用的工具之一，已在多个领域进行了研究，包括空间统计、计算机实验的设计和分析以及机器学习。随着测量技术的进步和计算能力的提高，在现代应用中经常收集大量分辨率越来越高的测量和大规模数值模拟，并在预测具有相关不确定性的现实世界过程中产生了几个关键挑战。虽然GP为沿海洪水灾害研究等现代新兴应用提供了一种很有前途的方法来执行UQ任务，但现有的GP方法不足以解决几个值得注意的问题，例如大数据集导致的计算瓶颈和多维域复杂结构导致的空间异质性。该项目将开发新的贝叶斯GP方法，以允许可扩展的计算和捕获空间异质性。新的方法、算法、理论和软件有望改进GP建模，以解决包括物理科学、工程、医学、公共卫生和商业科学在内的广泛领域的数据分析问题。该项目将开发和分发用户友好的开源软件，并为本科生和研究生提供跨学科的研究培训机会。本项目旨在开发一个新的贝叶斯多尺度残差学习框架，该框架具有强大的理论支持，允许GP建模的可扩展计算和空间非平稳性。该框架集成并扩展了GP和多尺度建模文献中出现的几种强大技术，包括预测过程逼近、块收缩和域上的随机递归划分。该框架将GP分解为级联的残差过程，这些残差过程在不同分辨率下表征底层协方差结构，并且可以以各种方式在空间上异构。新的框架允许采用块收缩来推断剩余过程的协方差，并结合随机分割先验，以适应多维域的各种空间结构。受小波收缩和状态空间模型启发的新递归算法将被开发出来，以实现观测数方面的线性计算复杂性和线性存储复杂性。由此产生的GP方法将保证串行计算环境下的线性计算复杂度，并且易于并行化。这种贝叶斯多尺度残差学习方法为解决空间统计、计算机实验设计与分析、机器学习和非参数回归等领域的GP建模问题提供了新的途径。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。

Rare natural hazards (for example, storm surge and hurricanes) can cause loss of lives and devastating damage to society and the environment. For instance, Hurricane Katrina (2005) caused over 1,500 deaths and total estimated damages of $75 billion in the New Orleans area and along the Mississippi coast as a result of storm surge. Uncertainty quantification (UQ) has been used widely to understand, monitor, and predict these rare natural hazards. The Gaussian process (GP) modeling framework is one of the most widely used tools to address such UQ applications and has been studied across several areas, including spatial statistics, design and analysis of computer experiments, and machine learning. With the advance of measurement technology and increasing computing power, large numbers of measurements and large-scale numerical simulations at increasing resolutions are routinely collected in modern applications and have given rise to several critical challenges in predicting real-world processes with associated uncertainty. While GP presents a promising route to carrying out UQ tasks for modern emerging applications such as coastal flood hazard studies, existing GP methods are inadequate in addressing several notable issues such as computational bottleneck due to big datasets and spatial heterogeneity due to complex structures in multi-dimensional domains. This project will develop new Bayesian GP methods to allow scalable computation and to capture spatial heterogeneity. The new methods, algorithms, theory, and software are expected to improve GP modeling for addressing data analytical issues across a wide range of fields, including physical science, engineering, medical science, public health, and business science. The project will develop and distribute user-friendly open-source software and provide interdisciplinary research training opportunities for undergraduate and graduate students.This project aims to develop a new Bayesian multi-scale residual learning framework with strong theoretical support that allows scalable computation and spatial nonstationarity for GP modeling. This framework integrates and extends several powerful techniques respectively arising in the literature on GP and that on multi-scale modeling, including predictive process approximation, blockwise shrinkage, and random recursive partitioning on the domain. This framework decomposes the GP into a cascade of residual processes that characterize the underlying covariance structures at different resolutions and that can be spatially heterogeneous in a variety of ways. The new framework allows for adoption of blockwise shrinkage to infer the covariance of the residual processes and incorporates random partition priors to enable adaptivity to various spatial structures in multi-dimensional domains. New recursive algorithms inspired by wavelet shrinkage and state-space models will be developed to achieve linear computational complexity and linear storage complexity in terms of the number of observations. The resulting GP method will guarantee linear computational complexity in a serial computing environment and also be easily parallelizable. This Bayesian multi-scale residual learning method provides a new approach to addressing GP modeling issues among spatial statistics, design and analysis of computer experiments, machine learning, and nonparametric regression.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.