Model Functional Data Through a Local FPCA Framework

这项研究的动机是许多真实的生活问题，其建模和分析涉及功能数据，即，每个受试者/重复的测量值对应于函数值的数据（称为样本轨迹）。 特别是，这项研究的动机是功能聚类问题和功能数据，这是动态的。函数型主成分分析（FPCA）在函数型数据分析中得到了广泛的应用。尽管它的成功，FPCA往往是低效的，如果轨迹空间的几何形状是非欧几里德，特别是当样本轨迹只在稀疏的时间点集观察，是许多科学研究的情况下。这种非线性的来源包括但不限于样本轨迹的潜在集群的存在，或者样本轨迹由非线性动力系统控制。研究者提出了一种新的策略（称为本地FPCA框架），用于分析稀疏和嘈杂的观察功能数据。 它的目的是获得更有效的本地化表示的样本轨迹，考虑到轨迹空间的几何结构。该框架结合了功能性主成分分析的基本原理，以及功能性聚类和非线性降维的概念。本研究的具体目标包括：（a）开发一个局部FPCA框架，该框架将样本轨迹聚类到同质子组中，并在每个聚类中应用FPCA以获得更有效的样本轨迹表示。(b)用基于模型的局部FPCA方法拟合含随机参数的常微分方程模型。 (c)研究所提出的方法的理论方面，并将其应用到各种科学问题。这项研究将产生一套新的统计工具，为科学家在各个领域，如植物生物学，生态学和流行病学谁必须分析纵向/功能数据。特别是，这项研究是理解复杂动力系统的垫脚石。PI正在与科学家合作研究人群水平的艾滋病毒疾病动力学，这项研究有助于更好地了解这些对艾滋病病理学具有重要影响的系统。从这项研究中产生的计算和分析工具也可能刺激相关领域的进一步研究。此外，这项研究还开发了开放源码软件，可供整个科学界免费使用。面对复杂的数据和具有挑战性的问题，新一代的研究人员需要以跨学科的方式进行培训。更广泛的培训内容包括让统计学/生物统计学学生接触涉及功能数据的真实的科学问题。 另一方面，通过合作，在相关领域工作的科学家能够提高他们的定量分析技能。

This research is motivated by numerous real life problems whose modeling and analysis involve functional data, i.e., data where the measurements per subject/replicate correspond to values of a function (referred to as sample trajectory). In particular, this research is motivated by functional clustering problems and functional data which are dynamical in nature. Functional principal components analysis (FPCA) has been widely used in analyzing functional data. In spite of its success, FPCA tends to be inefficient if the geometry of the trajectory space is non-Euclidean, especially when sample trajectories are only observed at sparse sets of time points, as is the case for many scientific studies. Sources for such nonlinearity include but not limited to the existence of underlying clusters of the sample trajectories, or the sample trajectories being governed by a nonlinear dynamical system. The investigator proposes a new strategy (referred to as the local FPCA framework) for analyzing sparsely and noisily observed functional data. It aims to derive more efficient localized representations for sample trajectories which take into account geometric structures of the trajectory space. This framework combines the principles underlying functional principal components analysis with the notions of functional clustering and nonlinear dimensionality reduction. Specific aims of this research include: (a) Develop a local FPCA framework which clusters the sample trajectories into homogeneous subgroups and applies FPCA within each cluster to derive more efficient representations of the sample trajectories. (b) Fit ordinary differential equation models with random parameters by a model-based local FPCA approach. (c) Study theoretical aspects of the proposed methods and apply them to various scientific problems.This research will produce a new set of statistical tools for scientists working in various fields such as plant biology, ecology and epidemiology who must analyze longitudinal/functional data. In particular, this research is a stepping stone toward understanding complex dynamical systems. The PI is collaborating with scientists on studying HIV disease dynamics at a population level, and this research helps achieving a better understanding of these systems which hold important implications in the pathologies of AIDS. The computational and analytical tools resulted from this research are also likely to stimulate further studies in related fields. Moreover, this research develops open source software that is freely available to the whole scientific community. Facing complex data and challenging questions, a new generation of researchers needs to be trained in an inter-disciplinary manner. The broader training component includes exposing statistics/biostatistics students to real scientific problems involving functional data. On the other hand, through collaborations, scientists working in related fields are able to enhance their quantitative analysis skills.