Model Functional Data Through a Local FPCA Framework

这项研究是由许多现实生活中的问题推动的，这些问题的建模和分析涉及函数数据，即每个受试者/重复的测量值对应于函数的值的数据(称为样本轨迹)。特别是，本研究的动机是函数聚类问题和函数数据本质上是动态的。函数主成分分析(FPCA)在函数数据分析中得到了广泛应用。尽管FPCA取得了成功，但如果轨迹空间的几何形状是非欧几里得的，特别是当样本轨迹只在稀疏的时间点观察时，FPCA往往效率低下，就像许多科学研究的情况一样。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，或样本轨迹受非线性动力系统控制的存在。研究人员提出了一种新的策略(称为局部FPCA框架)来分析稀疏和噪声观测的功能数据。它的目的是得到更有效的样本轨迹的局部化表示，其中考虑了轨迹空间的几何结构。该框架将函数主成分分析的基本原理与函数聚类和非线性降维的概念相结合。这项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为同质子组，并在每个聚类内应用FPCA以获得更有效的样本轨迹表示。(B)用基于模型的局部FPCA方法对含有随机参数的常微分方程组模型进行拟合。(C)研究拟议方法的理论方面，并将其应用于各种科学问题。这项研究将为从事植物生物学、生态学和流行病学等不同领域的科学家提供一套新的统计工具，他们必须分析纵向/功能数据。特别是，这项研究是理解复杂动力系统的垫脚石。国际艾滋病研究所正在与科学家合作，在人口层面上研究艾滋病毒疾病的动态，这项研究有助于更好地了解这些系统，这些系统在艾滋病的病理中具有重要意义。这项研究产生的计算和分析工具也可能促进相关领域的进一步研究。此外，这项研究开发了开放源码软件，整个科学界都可以免费获得。面对复杂的数据和具有挑战性的问题，新一代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到涉及功能数据的真正科学问题。另一方面，通过合作，相关领域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.