Nonparametric Total Variation Regression for Multivariate Process Data

过程数据经常出现在工程、制造、商业、环境科学和其他领域。例如，水或空气污染水平、钻出的金属部件的结构、药品的化学成分以及计算机网络的操作特性，所有这些都随着时间的推移而变化，这些都是常规的实时监测。中断或偏离稳定、一致的工艺数据流是特殊原因入侵的指示。在特定应用程序的上下文中，这些特殊原因可能对决策制定和流程理解非常有害。由于缺乏合适的非参数回归方法，用于监测多变量过程中平均向量的渐变和突变的无参考统计控制图的发展受到了极大的阻碍。为了应对这一挑战，该项目将解决多元过程数据对非参数估计器的迫切需求，并将开发新的无参考方法用于统计过程监测。该项目的成果将通过加强工业制造、商业、商业、医疗保健和其他具有社会重要性的领域的统计质量保证，使社会受益。该项目的成果将以一种公开可用的软件的形式实施。此外，该项目将涉及各种教育级别的多项研究培训和职业指导倡议，并将提供多种跨学科培训机会，特别侧重于扩大对统计科学的参与。该项目将通过发展个人多元过程数据统计过程控制的新理论和方法，推进非参数多元回归的前沿。在独立亚高斯过程的非参数估计的背景下，目标是研究具有分段光滑过程均值的多元过程数据的非参数全变分（TV）和紧弦（TS）估计，建立相关优化问题的适定性，证明它们的等价性，并研究TV/TS估计在各种实际相关拓扑中的渐近一致性/收敛率。这些理论结果将应用于开发TV/TS估计器的计算效率算法实现，研究这些算法的收敛性和复杂性，并基于合成和真实数据展示它们的性能。随后，TV/TS估计器的算法实现将用于设计一类新的无参考统计控制图，用于多变量过程均值的非参数监测，并在各种实际相关场景下将其与最先进的竞争对手进行比较。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。

Process data of interest frequently occur in engineering, manufacturing, commerce, environmental science and other arenas. For example, water or air contamination levels, configuration of a drilled metal part, chemical composition of a pharmaceutical product, and operational characteristics of computer network, all changing over time, are routinely monitored in real time. Upsets or shifts away from a stable, consistent flow of process data are indicative of special cause intrusion(s). These special causes can be significantly detrimental to decision making and process understanding in the context of a particular application. Development of reference-free statistical control charts for monitoring multivariate processes for both gradual and abrupt changes in the mean vector has been significantly hampered by a lack of suitable nonparametric regression methodology. In response to this challenge, this project will address the acute need for nonparametric estimators for multivariate process data and will develop new reference-free methods for statistical process monitoring. The outcomes of this project will benefit society through enhanced statistical quality assurance in industrial manufacturing, business, commerce, healthcare, and other domains of societal importance. The results of this project will be implemented in a form of publicly available software. Furthermore, the project will involve multiple research training and career mentoring initiatives at various educational levels and will offer multiple opportunities for interdisciplinary training, with a particular focus on broadening participation in statistical sciences.The project will advance the frontiers of nonparametric multivariate regression by developing new theory and methodology of statistical process control for individuals multivariate process data. In the context of nonparametric estimation for independent sub-Gaussian processes, the goal is to investigate nonparametric total variation (TV) and taut string (TS) estimators for multivariate process data with piecewise smooth process mean, establish well-posedness for associated optimization problems, prove their equivalence, and investigate asymptotic consistency/convergence rates for TV/TS estimators in various practically relevant topologies. These theoretical results will be applied to develop computationally efficient algorithmic implementations of the TV/TS estimator, investigate convergence and complexity of these algorithms, and showcase their performance based on synthetic and real data. Subsequently, algorithmic implementations of the TV/TS estimator will be used to design a new class of reference-free statistical control charts for nonparametric monitoring of multivariate process mean and compare them to state-of-the-art competitors under a variety of practically relevant scenarios.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.