Testing and Deep Learning for Functional Data

所提出的研究涉及两个独特的领域，功能数据分析（FDA）和深度学习。功能数据是随机函数，由于处理海量数据的技术进步，功能数据变得越来越普遍。例子包括在一段时间内收集的气候或空气污染数据。该领域已成为主流研究领域，但文献主要集中在估计问题上，尚未充分利用深度学习方法的优势。该项目旨在填补这些空白。它包括对功能数据的几个新测试，并采用深度学习而不是传统的非参数平滑方法来处理功能数据。提出的方法将应用于各种功能数据，包括评估污染物对肺癌死亡率的影响以及解释体育活动对健康的影响。一个主要的重点是发展新的理论和算法。与研究相关的计算机代码将作为R或Python包公开传播。研究成果将纳入研究生课程、本科生研究项目和讲习班的短期课程。它们还将在专业会议上发表。学生研究人员将接受研究、计算和沟通技能方面的培训。虽然功能数据本质上是无限维的，但测量只能在离散的位置上获得，这可能因主题而异。每个受试者的测量位置数量可以很小（稀疏功能数据），也可以随着样本量的增加而增加（密集采样功能数据）。拟议的研究涵盖所有类型的抽样计划，并在可行的情况下采用普遍适用的单一平台。这种方法很重要，因为判断特定数据集的采样计划是密集的还是稀疏的并不容易。它还具有理论统一和自动揭示相应估计量收敛速率的相变的优点。项目一（Functional Linear Models Hypothesis Testing for Functional Linear Models）旨在建立一个功能线性模型下假设检验的总体框架。现有的方法侧重于使用量身定制的检验来检验特定的零假设，并不适合于检验功能性协变量影响的时间持续时间，例如PM2.5对肺癌的影响。它们都没有被证明是复合零假设的最佳选择。我们提出了一个单一的平台来检验零假设，即函数协变量的回归系数驻留在所有可能的系数函数的封闭子空间中。所提出的检验类似于经典的f检验，它简单，并作为特殊情况包括对系数函数的整体零性、部分零性和定域的检验。项目2（测试功能数据的同质性和独立性）解决了两个基本任务的挑战，测试功能数据的同质性（相等分布）和独立性。当功能过程只能在几个离散的地点取样时，这种测试是不可行的，这种情况在纵向研究中是普遍存在的。对于每个任务，我们提出了一个定制版本，边际同质性或边际独立性，具有实际意义，在理论和实施上都是可行的。项目3 （Deep learning for Functional Data）旨在将深度学习的成功应用于功能数据。令人惊讶的是，深度神经网络对功能数据的应用很少，仍然是一个开放的问题。PI领导的团队最近开发了一种方法，使用神经网络搜索最优基函数来表示自动适应手头预测任务的功能输入。我们建议扩大这种自适应基础方法的范围和理论理解。另一个目标是设计新的方法来计算部分观察到的功能数据，这些数据使用transformer，这是一种深度神经网络，可以将给定的元素序列（如句子中的单词序列）转换为另一个序列。该项目将为未来一代统计学家的跨学科培训提供广泛的新机会，并将有助于加强统计科学中更具包容性的气氛。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 proposed research involves two distinctive fields, functional data analysis (FDA) and deep learning. Functional data are random functions, which have become increasingly common due to technological advances to handle massive data. Examples include climate or air pollution data collected over a period of time. The field has emerged as a mainstream research area, but the literature is mainly focused on estimation problems and has not yet leveraged the advantages of deep learning methods. This project aims to fill these gaps. It includes several new tests for functional data and employs deep learning, instead of the conventional nonparametric smoothing methods, to handle functional data. The proposed approaches will be applied to various functional data, including evaluating the effect of pollutants on lung cancer mortality and explaining the effects of physical activity on health. A major emphasis is the development of new theory and algorithms. Computer code associated with the research will be publicly disseminated as R- or Python packages. The research findings will be incorporated in graduate curricula, undergraduate research projects, and short courses at workshops. They will also be presented at professional meetings. Student researchers will receive training in research, computing and communication skills.Although functional data are intrinsically infinite dimensional, measurements are only available at discrete locations, which may vary from subject to subject. The number of measurement locations per subject can be small (sparse functional data) or grow with the sample size (intensely sampled functional data). The proposed research covers all types of sampling plans and employs, whenever feasible, a single platform that is universally applicable. Such an approach is important as it is not trivial to judge whether the sampling plan for a particular dataset is intense or sparse. It also has the merit that the theory is unified and automatically reveals the phase transitions of the convergence rates of the corresponding estimators. Project 1 (Hypothesis Testing for Functional Linear Models) aims at developing a general framework for hypothesis testing under the setting of functional linear models. Existing methods focus on testing a specific null hypothesis using a tailored test and are not well suited for testing the temporal duration of the effect of a functional covariate, such as the impact of PM2.5 on lung cancer. None of them has been shown to be optimal for a composite null hypothesis. We propose a single platform to test the null hypothesis that the regression coefficient of a functional covariate resides in a closed subspace of all possible coefficient functions. The proposed test, which resembles the classical F-test, is simple and includes tests for global nullity, partial nullity and domain of the coefficient function as special cases. Project 2 (Testing Homogeneity and Independence for Functional Data) addresses the challenges of two fundamental tasks, testing the homogeneity (equal distributions) and independence of functional data. Such tests are infeasible when the functional process can only be sampled at a few discrete locations, a situation that is ubiquitous in longitudinal studies. For each task, we propose a customized version, marginal homogeneity or marginal independence, that has practical implications and is feasible for theory and implementation. Project 3 (Deep learning for Functional Data) aims at bringing the success of deep learning to bear with functional data. Surprisingly, the application of deep neural networks to functional data has been scarce and remains an open problem. A recent approach, developed by a team led by the PI, uses neural networks to search for the optimal basis functions to represent a functional input that automatically adapts to the prediction task in hand. We propose to expand the reach and theoretical understanding of this adaptive basis approach. Another objective is to design new methodology to impute partially observed functional data that uses Transformers, a deep neural network that transforms a given sequence of elements, such as the sequence of words in a sentence, into another sequence. The project will offer a broad range of new opportunities for interdisciplinary training of a future generation of statisticians and will contribute to enhancing a more inclusive atmosphere in statistical sciences.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.