Collaborative Research: Halfspace Depth for Object and Functional Data

协作研究：对象和功能数据的半空间深度

• 批准号: 2113713
• 负责人: Xiongtao Dai
• 金额: $ 17.5万
• 依托单位: Iowa State University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2021
• 资助国家: 美国
• 起止时间: 2021-07-01 至 2023-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2113713&HistoricalAwards=false
• 关键词: Collaborative; Research; Halfspace; Depth; Object

Complex data objects are increasingly being generated across science and engineering. Non-Euclidean data such as wind directions, neural connectivity networks, and phylogenetic trees draw practical interest, but are challenging to analyze due to their intrinsic constraints. Functional data such as trajectories and images also provide examples of another type of data of high complexity, which are observed on a continuous domain in time or space. In general, practitioners are interested in first exploring the data distributions before any modeling analysis. For instance, given a sample of growth trajectories of children, a first step is to identify typical versus extreme growth patterns, where the latter can be non-trivial to uncover. Also, when analyzing brain connectivity matrices, it is important to find unusual brain networks and differences between healthy and diseased populations. Data-driven methods robust to anomalies are essential in these settings since little is known about the data generating process, and outliers can affect the analysis. Due to the lack of a natural ordering in data objects, exploratory tools such as boxplot and quantile are unavailable for these types of data. The project will address the lack of techniques for exploring non-Euclidean and functional data. Principled statistics and visualization methods will be developed based on a novel way of ranking the observations. The project will also provide training for graduate and undergraduate students. The central research theme is to develop exploratory data analysis tools for non-Euclidean and functional data objects. To overcome the absence of a canonical ordering for object data, the PIs will develop suitable data depth notions to quantify the centrality of data points with respect to the distribution. This will provide a center-outward ranking of the data that will be used as a building block for outlier detection methods, rank tests, and robust classifiers. Analogous to Tukey's halfspace depth for the multivariate Euclidean case, the new depth notions for object data are expected to be intuitive and robust, and have desirable properties well-grounded in theory. Specifically, the research project will investigate a depth notion for non-Euclidean objects; a data visualization and an outlier detection procedure for non-Euclidean data; halfspace depth notions for functional data, one based on theory and another one from an algorithmic perspective; and a depth notion for sparsely observed longitudinal data. Key challenges that will be addressed include a lack of vector space structure when dealing with non-Euclidean objects; the infinite dimensionality and degeneracy when defining depth notions for functional data; detecting outlying trajectories and images in shape and not just at any time point; and the sparsity and irregularity of observations in longitudinal data. Method and theory development will draw from metric geometry, functional data analysis, empirical process, and M-estimation. Software implementing a suite of depth-based methods will be made available to the public as an outcome of the project.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.

复杂的数据对象越来越多地在科学和工程领域产生。非欧几里得数据，如风向、神经连接网络和系统发育树，引起了人们的实际兴趣，但由于其内在的限制，分析起来很有挑战性。轨迹和图像等功能数据也提供了另一种类型的高复杂性数据的例子，这些数据是在时间或空间的连续域上观察到的。一般来说，在进行任何建模分析之前，从业者都对首先探索数据分布感兴趣。例如，给定一个儿童生长轨迹样本，第一步是识别典型和极端的生长模式，后者可能是不容易发现的。此外，在分析大脑连接矩阵时，发现不寻常的大脑网络和健康人群与患病人群之间的差异是很重要的。在这些情况下，数据驱动的方法对异常的鲁棒性至关重要，因为对数据生成过程知之甚少，而异常值可能会影响分析。由于数据对象缺乏自然的排序，诸如箱线图和分位数之类的探索性工具无法用于这些类型的数据。该项目将解决探索非欧几里得和功能数据的技术缺乏问题。原则统计和可视化方法将基于一种对观察结果进行排序的新方法而开发。该项目还将为研究生和本科生提供培训。中心研究主题是为非欧几里德和功能数据对象开发探索性数据分析工具。为了克服对象数据缺乏规范排序的问题，pi将开发合适的数据深度概念，以量化数据点相对于分布的中心性。这将提供数据的中心向外排序，该数据将用作离群检测方法、秩测试和鲁棒分类器的构建块。类似于Tukey的多元欧几里得情况下的半空间深度，目标数据的新深度概念有望直观和健壮，并具有良好的理论基础。具体来说，该研究项目将研究非欧几里得物体的深度概念；非欧几里得数据的数据可视化和离群值检测程序；函数数据的半空间深度概念，一个基于理论，另一个从算法的角度；对于稀疏观测的纵向数据的深度概念。将解决的主要挑战包括：在处理非欧几里得对象时缺乏向量空间结构；定义函数数据深度概念时的无限维数和简并性检测形状上的外围轨迹和图像，而不仅仅是在任何时间点；纵向数据的稀疏性和不规则性。方法和理论的发展将借鉴度量几何、功能数据分析、经验过程和m估计。作为该项目的成果，将向公众提供实现一套基于深度的方法的软件。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。

项目成果

Tukey’s Depth for Object Data

• DOI: 10.1080/01621459.2021.2011298
• 发表时间: 2021-09
• 期刊: Journal of the American Statistical Association
• 影响因子: 3.7
• 作者: Xiongtao Dai;S. López-Pintado