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，在给定附近气象站网络的情况下，对一个人家里的温度进行内插。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，并提供了不确定性的自然量化。然而，对于大数据集，直接应用高斯过程的计算代价太高。这个项目用新的算法解决了计算挑战，并弥合了统计和机器学习方法之间的差距。随着大数据现在几乎出现在科学和社会的每个领域，提供强大、可扩展和自由的软件来分析这些数据集可以产生革命性的影响。这项工作将取代目前针对海量空间数据的做法和近似，这些做法和近似往往由于计算限制而过于简单。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、机械臂的校准以及叛乱活动的建模和预测。因此，所开发的方法和软件将成为计算和数据支持的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演示分享项目成果。该项目的目标是开发一个几乎通用的工具箱，用于可伸缩高斯过程(GP)建模。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概念，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。该项目将导致新的、高精度的OCA方法，具有保证的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，还将开发扩展，允许这些新的空间统计方法用于各种机器学习应用程序，在这些应用程序中，OCA类型的方法迄今尚未受到太多关注。对于新的方法，计算成本在数据大小上保证是线性的，通过并行化可以进一步加速。所有方法都将在易于使用的开源软件中实施。这将允许用户将全科医生的能力应用于现代数据集，实现空间预测、校准、参数学习和大数据的非参数回归。该奖项反映了NSF的法定使命，并通过使用基金会的智力优势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.