Flexible modelling of functional data

本文的研究重点是功能数据分析的两个重要方面的方法和理论发展：1)功能回归分析；2)广义纵向数据与生存时间的联合建模。第一个项目是开发一种新的模型——功能加性模型的方法，它在某种意义上统一和推广了功能回归分析。在这个新的框架中，标量或函数响应在函数预测器上的回归不局限于线性算子，这使得数据自己“说话”。将进行全面的研究，包括模型估计和选择，理论，推理，应用和潜在的扩展。本建议的第二个重点是联合建模广义类型的纵向数据和生存时间。这是因为在许多临床试验中，纵向观察不一定是连续的，但可能是分类的，如二项或泊松结果，而大多数现有方法处理连续的纵向数据。我将从功能数据的角度提出一个新的观点，其中将利用功能主成分技术来实现降维。将开发考虑蒙特卡罗误差和最优停止规则的自动算法和推理程序。这个提议的动机是我与生物学家和医生正在进行的和未来的研究合作。这些新方法将应用于这些合作产生的数据集，并将为未来的实验提供有用的指导。所开发的程序将适用于来自不同学科的一般功能数据，这反过来又促进了许多科学领域的进步。

The proposed research focuses on the development of methodologies and theory on two important aspects of functional data analysis: 1) functional regression analysis; 2) joint modelling of generalized longitudinal data and survival time. The first project is to develop methodologies for a new class of models, Functional Additive Models, which in certain sense unify and generalize functional regression analysis. In this new framework, the regression of the scalar or functional response on functional predictor is not restricted to linear operator, which allows the data "speak" for themselves. Comprehensive investigation will be carried out, including model estimation and selection, theory, inference, applications and potential extensions. The second emphasis of this proposal is to jointly model the generalized type of longitudinal data and survival time. This is motivated by the fact that in many clinical trials longitudinal observations are not necessarily continuous, but may be categorical, such as binomial or Poisson outcomes, while most existing approaches deal with continuous longitudinal data. I will present a new perspective from functional data viewpoint, where functional principal component techniques will be exploited to achieve dimension reduction. Automatic algorithms and inference procedures will be developed, which take Monte Carlo error and optimal stopping rules into account. This proposal is motivated by my ongoing and future research collaborations with biologists and physicians. The new approaches will be applied to the datasets generated from these collaborations and will provide useful guidance for future experiments. The procedures developed will be applicable for general functional data from various disciplines, which in turn facilitate the advancement of many scientific fields.