Collaborative Research: Halfspace Depth for Object and Functional Data

科学和工程领域越来越多地产生复杂的数据对象。非欧几里德数据，如风向、神经连通性网络和系统发育树，吸引了人们的实际兴趣，但由于其内在的限制，分析起来具有挑战性。诸如轨迹和图像的功能数据还提供了在时间或空间的连续域上观察到的另一种类型的高复杂性数据的例子。一般来说，实践者感兴趣的是在进行任何建模分析之前首先探索数据分布。例如，给出一个儿童成长轨迹的样本，第一步是确定典型和极端的成长模式，后者可能不是很容易发现的。此外，在分析大脑连接矩阵时，发现异常的大脑网络以及健康人群和患病人群之间的差异也很重要。由于对数据生成过程知之甚少，而且离群值可能会影响分析，因此在这些环境中，稳健的数据驱动方法是必不可少的。由于数据对象中缺乏自然的排序，因此不能使用盒图和分位数等探索性工具来处理这些类型的数据。该项目将解决缺乏探索非欧几里得和功能数据的技术的问题。将根据一种新的观测排名方法开发原则性统计和可视化方法。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得和函数式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将制定适当的数据深度概念，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，用作离群值检测方法、排序测试和稳健分类器的构建块。类似于Tukey的多变量欧几里德情形的半空间深度，对象数据的新深度概念被期望是直观和稳健的，并具有良好的理论基础的理想性质。具体地说，该研究项目将研究非欧几里德对象的深度概念；非欧几里德数据的数据可视化和离群点检测程序；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；以及稀疏观测的纵向数据的深度概念。将解决的关键挑战包括：在处理非欧几里得物体时缺乏向量空间结构；在为功能数据定义深度概念时的无限维度和简并性；检测形状而不是仅在任何时间点的离岸轨迹和图像；以及纵向数据中观测的稀疏性和不规则性。方法和理论的发展将借鉴度量几何、函数数据分析、经验过程和M估计。实施一套基于深度的方法的软件将作为项目的成果向公众提供。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。

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.