Collaborative Research: Scalable Gaussian-Process Methods for Spatial Statistics and Machine Learning

高斯过程是一种数学工具，它可以使用不完整的数据来填补空白，例如，根据附近气象站的网络来插入一个人家里的温度。高斯过程被用于许多应用领域，如地理空间分析、机器学习和计算机实验分析。高斯过程是灵活的，可解释的，并提供不确定性的自然量化。然而，对于大型数据集，直接应用高斯过程的计算成本太高。该项目用新颖的算法解决了计算方面的挑战，弥合了统计和机器学习方法之间的差距。由于大数据现在几乎出现在科学和社会的每个领域，提供强大的、可扩展的、免费的软件来分析这些数据集可以产生变革性的影响。这项工作将取代目前由于计算限制而往往过于简单化的大规模空间数据的实践和近似。该项目可以在无数对社会有直接影响的应用中提高准确性和不确定性量化，包括碳监测、可再生能源、降雨预测、机械臂校准以及叛乱活动建模和预测。因此，开发的方法和软件将成为计算和数据支持科学和工程的重要工具。研究人员将指导和培训学生研究人员，并通过期刊出版物和会议报告分享项目发现。这个项目的目标是为可扩展的高斯过程（GP）建模开发一个几乎通用的工具箱。该工具箱基于有序条件近似（OCA），这是一个简单但非常强大的思想，它利用了许多流行的协方差函数所表现出的筛选效应（即条件独立性）。OCA框架统一了来自统计学、机器学习和数值线性代数的许多最先进的GP近似。该项目将产生新的、高度精确的OCA方法，保证可扩展性和广泛的适用性，用于非平稳、多变量、多尺度和其他过程的建模和分析。此外，将开发扩展，允许这些新的空间统计方法用于各种机器学习应用程序，其中oca类型的方法迄今尚未受到太多关注。对于新方法，计算成本保证在数据大小上是线性的，通过并行化可以进一步加快速度。所有的方法都将在易于使用的开源软件中实现。这将允许用户将GPs的力量应用于现代数据集，实现大数据的空间预测、校准、参数学习和非参数回归。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。

The Gaussian process is a mathematical tool that can use incomplete data to fill in gaps, for example to interpolate the temperature at a person’s house given a network of nearby weather stations. Gaussian processes are used in many application areas, such as geospatial analysis, machine learning, and the analysis of computer experiments. Gaussian processes are flexible, interpretable, and provide natural quantification of uncertainty. However, direct application of Gaussian processes is too computationally expensive for large datasets. This project addresses the computational challenges with novel algorithms and bridges the gap between statistical and machine learning approaches. As big data now appear in almost every field of science and society, providing powerful, scalable, and free software to analyze such datasets can have a transformative effect. This work will replace current practices and approximations for massive spatial data that are often simplistic due to computational limitations. This project can lead to improved accuracy and uncertainty quantification in countless applications with direct impact on society, including carbon monitoring, renewable energy, rainfall prediction, calibration of robotic arms, and modeling and prediction of insurgent activities. The developed methods and software will thus be an important tool for computational and data-enabled science and engineering. The investigators will mentor and train student researchers, and share the project findings via journal publications and conference presentations.The goal of this project is to develop a nearly universal toolbox for scalable Gaussian process (GP) modeling. The toolbox is based on the ordered conditional approximation (OCA), a simple but very powerful idea that exploits the screening effect (i.e., conditional independence) exhibited by many popular covariance functions. The OCA framework unifies many state-of-the-art GP approximations from statistics, machine learning, and numerical linear algebra. This project will result in new, highly accurate OCA methods with guaranteed scalability and broad applicability for modeling and analysis of nonstationary, multivariate, multi-scale, and other processes. Also, extensions will be developed that allow these new spatial-statistics methods to be used in a variety of machine-learning applications, where OCA-type approaches have not received much attention so far. For the new methods, the computational cost is guaranteed to be linear in the data size, with further speed-ups possible through parallelization. All approaches will be implemented in easy-to-use open-source software. This will allow users to bring the power of GPs to bear on modern datasets, enabling spatial prediction, calibration, parameter learning, and nonparametric regression with big data.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.