Complex Problems in Functional Data Analysis

函数数据分析(FDA)以随机函数的形式处理无限维数据。由于记录和存储海量数据的新技术，此类数据已变得越来越常见。该领域获得了很大的吸引力，研究速度加快，但仍有许多悬而未决的问题和新的研究机会。这项研究集中在四个项目上：1)在具有函数协变量和标量响应的回归环境中，关于感兴趣区域的选择的公开问题；2)当函数协变量被稀疏观测时，为传统函数线性模型实现RKHS(再生核-希尔伯特空间)方法；3)多变量函数数据的动态建模；以及4)功能片段数据分析的挑战，对于每个对象，每个对象被观察到的不同区间比函数数据的域短得多。所开发的方法将应用于具有功能成分的各种数据，以评估污染物对肺癌死亡率的影响，并探索这些污染物之间的相互作用。因此，拟议的研究对公共卫生研究具有直接影响。此外，建议的功能片段方法在加速纵向研究中有广泛的应用，这在社会科学和健康科学中很常见。开发的算法的计算机代码将被集成到CRAN上现有的R包fdapace中。研究成果将被纳入研究生课程、本科生和研究生研究项目以及研讨会上的短期课程，并在专业会议上展示。项目1对于解释函数协变量的影响很重要，但到目前为止，还没有算法可以可靠地识别相关领域，理论也是不完整的。我们建议通过一个新的框架来解决这些悬而未决的问题，该框架涉及一个动态的RKHS方法来克服这些挑战。这有可能在RKHS这个久负盛名的领域开辟新天地。RKHS方法的一个缺点是它难以处理稀少观察到的函数协变量。在项目2中，我们提出了引入不完全函数协变量的解决方案，并证明了回归系数函数可以通过输入的函数协变量恢复。将发展一种新的理论来处理泛函数据的Karhuen-Lo前夕展开中的近似误差。这些新的结果将促进未来涉及功能数据推算的研究。项目3旨在使用成分过程作为协变量对多变量函数数据的导数进行建模。我们提出了一种并发方法，它避免了不适定的逆问题，并且具有适应预测器组件过程的时滞的优势。项目4涉及FDA的另一个公开问题。我们提出了两种用于功能片段的非参数方法，并将发展支持理论。这些新的方法为FDA提供了一个新的研究前沿，因为一旦可以准确地估计协方差，现有的FDA方法，如主成分分析、分类或聚类，就可以很容易地适应功能片段。这一奖项反映了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。

Functional data analysis (FDA) deals with infinite-dimensional data in the form of random functions. Such data have become increasingly common due to new technology to record and store massive data. The field has gained much traction and research has accelerated, but there remain many unsolved problems and new opportunities for research. This research focuses on four projects that address: 1) an open problem regarding the choice of the domain of interest in a regression setting with a functional covariate and scalar response, 2) implementing the RKHS (reproducing kernel Hilbert space) approach for conventional functional linear models when the functional covariates are observed sparsely, 3) dynamic modeling for multivariate functional data, and 4) challenges for the analysis of functional snippet data, for which each subject is observed in a different interval much shorter than the domain of the functional data. The developed methods will be applied to various data with functional components to evaluate the effect of pollutants on lung cancer mortality and to explore the interaction of these pollutants. The proposed research thus has direct impacts on public health research. In addition, the proposed approaches for functional snippets have broad applications in accelerated longitudinal studies, which are common in social and health sciences. The computer code of developed algorithms will be integrated into an existing R-package, fdapace, on CRAN. The research findings will be incorporated into graduate curricula, undergraduate and graduate research projects, and short courses at workshops, and be presented at professional meetings. Project 1 is important for interpreting the influence of a functional covariate, yet to date, there is no algorithm that can reliably identify the relevant domain and the theory is incomplete. We propose to resolve these open problems through a new framework that involves a dynamic RKHS approach to overcome the challenges. This has the potential to break new ground in the well-established field of RKHS. A weakness of the RKHS approach is that it has difficulty to handle sparsely observed functional covariates. In Project 2, we propose a solution by imputing incomplete functional covariates and show that the regression coefficient function can be recovered through the imputed functional covariates. A new line of theory will be developed to deal with the approximation errors in the Karhunen-Lo\'eve expansion for functional data. These new results will facilitate future research that involves imputation for functional data. Project 3 aims at modeling the derivatives of multivariate functional data using the component processes as covariates. We propose a concurrent approach that avoids an ill-posed inverse problem and has the advantage to accommodate time-lags of the predictor component processes. Project 4 deals with another open problem in FDA. We propose two nonparametric approaches for functional snippets and will develop supporting theory. These new approaches provide a new frontier of research in FDA, as once the covariance can be estimated accurately, existing FDA approaches, such as principal component analysis, classification or clustering, can be readily adapted for functional snippets.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.