New Directions in Functional Data Analysis

功能数据分析是统计学中的一个新兴领域，它处理随机函数的样本。在实践中，这些随机函数的测量通常是在离散的时间点间歇进行的，并且可能受到随机噪声的影响。这导致两种情况，一种是完整或密集的记录，称为“功能数据”，另一种是离散时间点的稀疏测量，称为“纵向数据”，因为此类数据是纵向研究的典型数据。对于这两类数据的统计分析和各自的理论差异很大。研究者主张，没有必要像通常的做法那样，分别为功能数据和纵向数据开发方法。相反，将开发一种在单一平台上处理这两种数据结构的统一方法。该方法植根于主成分分析的降维方法，该方法已扩展到功能/纵向数据，并称为“功能主成分分析（FPCA）”。现有的FPCA方法假设数据来自单一人群，因此不能利用协变量的可用信息，协变量可以是时间无关的，也可以是时间相关的。该提案的第一个主题/目标是通过调整FPCA的协变量信息来填补这一空白，用于功能和纵向数据。PI及其同事最近开发的FPCA方法促进了这种扩展，该方法与非参数和半参数方法相结合。该提案的第二个主题是将目标1中的功能方法学应用于非结构化（非功能）高维数据，方法是重新排序，然后将它们“串”成数据，然后将其解释为功能数据。“串”可以通过多维缩放来完成。如果非结构化数据的变量之间存在足够强的相关性，则可以找到相邻变量高度相关的排序。这种方法将高维数据的诅咒变成了祝福，因为功能数据分析本质上利用了每个主题的高维密集记录数据的邻接性。该项目的总体目标是通过结合FDA和字符串方法，为几种类型的高维数据开发一个有凝聚力的框架。因此，这两个主题紧密相连，并在Aim 3中进一步探讨，以开发和传播这些方法的软件。如今，高维数据在许多学科中都很常见。本文主要关注两类高维数据：功能/纵向数据和非结构化高维数据。Aims 1和Aims 2的组合方法可以处理大量高维数据。拟议的研究是由现实世界的问题驱动的，比如巴尔的摩纵向研究或国家生物技术信息中心数据库中列出的基因表达研究。这些问题的很大一部分源于PI与生物学家和人口统计学家正在进行的合作，其目的是确定：(1)有助于长寿的关键生物和行为因素，(2)疾病的关键风险因素，以及(3)与患者生存有关的基因。通过克服高维数据的挑战，新方法将有助于揭示许多领域的重要问题。

Functional data analysis is an emerging area in statistics that deals with a sample of random functions. In practice, measurements of these random functions are often taken intermittently at discrete time points and may be subject to random noise. This results in two scenarios, one with complete or dense recordings, termed 'functional data', and another with sparse measurements at discrete time points, termed 'longitudinal data' because such data are typical for longitudinal studies. Statistical analysis for these two types of data and the respective theory differ substantially. The investigator advocates that it is not necessary to develop methodology for functional and longitudinal data separately as is common practice. Instead, a unified approach that handles both data structures on a single platform will be developed. The approach is rooted in the dimension reduction approach of principal component analysis, which has been extended to functional/longitudinal data and termed 'functional principal component analysis (FPCA)'. Existing FPCA approaches assume that data are from a single population, thus do not take advantage of available information on covariates, which can be either time-independent or time-dependent. The first theme/aim of the proposal is to fill this gap in the literature by adjusting FPCA for covariate information, for both functional and longitudinal data. A recent FPCA approach developed by the PI and colleagues facilitates this extension, which is coupled with nonparametric and semiparametric approaches. A second theme of the proposal is to apply the functional methodology in Aim 1 to unstructured (non-functional) high dimensional data by reordering and then 'stringing' them into data which can then be interpreted as functional data. 'Stringing' can be accomplished through multidimensional scaling. If sufficiently strong correlations exist among the variables of the unstructured data, an ordering can be found in which neighboring variables are highly correlated. This approach turns the curse of high-dimensional data into a blessing, as functional data analysis inherently takes advantage of the adjacency of high dimensional densely recorded data for each subject. The overarching goal of this project is to develop a cohesive framework for several types of high dimensional data through a combination of FDA and Stringing approaches. The two themes are thus tightly connected and further explored in Aim 3, to develop and disseminate software for these methods. High dimensional data are common nowadays in many disciplines. This proposal focuses on two types of high-dimensional data: functional/longitudinal data and unstructured high-dimensional data. The combined approaches of Aims 1 and 2 can handle a large variety of high-dimensional data. The proposed research is motivated by real world problems, such as those from the Baltimore Longitudinal Study or in gene expression studies listed in the databases of the National Center for Biotechnology Information. A significant portion of the problems originates from ongoing collaborations of the PI with biologists and demographers and aims to identify: (1) key biological and behavioral factors that contribute to longevity, (2) key risk factors to diseases, and (3) genes that are related to patient survivals. The new approaches will help to shed light on important issues in many fields by overcoming the challenges with high dimensional data.