Functional Data Analysis: From Univariate to High-Dimensional Functional Data

这个项目处理函数、图像和形状形式的数据。这类数据的一个共同特征是它们是无限维的，这使它们区别于传统数据。这些数据的方法和理论被称为“功能数据分析”，这是一个正在迅速发展的领域。功能数据的无限维性质要求采样和降维方法，由于数据集的规模很大，这是具有挑战性的。本研究通过开发新的降维方法来模拟单变量功能数据（项目1和2）以及下一代功能数据（项目3和4）来解决这些挑战。指导原则是同时实现降维和柔性建模。这项研究的动机是在衰老、生物医学研究和神经影像学研究方面的潜在应用，包括理论、方法、数据分析和应用。该项目的另一个推动因素是，迫切需要让更多的统计学家参与大脑研究，并培训下一代统计学家应对大数据的挑战。软件开发是该项目的关键组成部分，附带的计算机代码将集成到现有的开源包PACE中，供公众访问。函数数据分析是统计学中一个引人入胜的领域，它处理随机函数的样本。然而，实际上，这些随机函数的测量只能在离散的时间点或网格上进行，而且测量结果经常受到噪声的污染。当前的功能数据分析方法通常是针对测量的特定采样计划量身定制的。在项目1中，提出了一个统一的方法，在理论上和实现上，函数回归。重点是功能响应和功能/矢量协变量的降维模型，其中样条基函数将用于估计未知的非参数成分函数。项目2涉及一类新的功能生存模型，该模型可容纳带有功能/向量协变量的删减单变量响应数据。这些模型从根本上不同于现有的生存模型，与当前的功能（广义）线性模型相比，它们可以处理删减的响应数据。项目3和项目4处理多元功能数据，项目3侧重于功能数据对的差异或同步度量，项目4侧重于高维多元功能数据的重构。在项目3中提出了新的距离或同步的功能度量，使用了一个新的概念，旨在功能数据的导数的一致性。这些新措施有利于项目4中高维功能数据的重新配置，使重新配置的空间附近区域的功能平滑连接，从而克服了高维的诅咒。本文提出的重构方法不仅将多维尺度方法从多元数据扩展到函数数据，而且在温和的假设条件下，将高维的诅咒变成了祝福。

This project deals with data that are in the form of functions, images, and shapes. A common feature of such data is that they are infinite dimensional, which distinguishes them from traditional data. Methodology and theory for these data is termed "functional data analysis," an area that is rapidly evolving. The infinite dimensional nature of functional data calls for sampling and dimension reduction approaches, which are challenging due to the large sizes of the data sets. This research tackles these challenges by developing novel dimension reduction methodology to model univariate functional data (Projects 1 and 2), as well as next generation functional data (Projects 3 and 4). The guiding principle is to accomplish dimension reduction and flexible modeling simultaneously. The proposed research, motivated by potential applications to research on aging, biomedical research, and neuroimaging, includes theory, methodology, data analysis, and applications. The project is also driven by the pressing need to involve more statisticians in brain research and to train the next generation of statisticians to tackle the challenges of big data. Software development is a key component of this project, and the accompanying computer code will be integrated into an existing open-source package, PACE, available for public access. Functional data analysis is a fascinating area in statistics that deals with a sample of random functions. However, measurements of these random functions, realistically, can only be taken at discrete time points or grids, and the measurements often are contaminated by noise. Current approaches in functional data analysis are typically tailored toward specific sampling plans for the measurements. In Project 1, a unifying approach, both in theory and in implementation, for functional regression is proposed. The focus is on dimension reduction models for functional responses and functional/vector covariates, where spline basis functions will be used to estimate the unknown nonparametric components functions. Project 2 deals with a new class of functional survival models that accommodate censored univariate response data with functional/vector covariates. These models differ fundamentally from existing survival models, and can handle censored response data in contrast to current functional (generalized) linear models. Projects 3 and 4 deal with multivariate functional data, with Project 3 focusing on measures of disparity or synchronization for pairs of functional data, and Project 4 focusing on reconfiguration of high-dimensional multivariate functional data. New functional measures of distance or synchronization are proposed in Project 3 using a novel concept that aims at concordance of the derivatives of functional data. These new measures facilitate the reconfiguration of high-dimensional functional data in Project 4, so that functions in nearby regions of the reconfigured space are smoothly connected, thereby overcoming the curse of high dimensionality. The proposed reconfiguration methods not only extend the multidimensional scaling method from multivariate to functional data, but also turn the curse of high dimensionality into a blessing under mild assumptions.