functional linear regression, graphical models and dependence modelling

我的研究项目将侧重于开发功能数据的新模型和新方法，重点是脑成像数据、空气污染数据和金融数据的应用。我们的第一个研究主题是高维泛函线性回归，我们感兴趣的是选择和估计多个函数值（也称为泛函）协变量对标量响应的影响。大脑信号的重复测量、时间过程基因表达以及空气污染物和股票指数的高频观测都可以被视为函数值数据。我们计划开发方法来拟合一个具有高维函数协变量的函数线性模型。本研究项目不仅将变量选择的思想扩展到函数回归模型，而且对各个学科都有重大影响。例如，我们将研究从不同脑区收集的脑信号对预测注意力缺陷多动障碍（ADHD）指数的影响，ADHD指数是衡量ADHD总体水平的一个综合指标。有了这个模型，我们就能识别出与多动症相关的特定区域。我们的第二个研究主题侧重于功能数据的图形模型。传统的图形模型通常用于描述多变量随机变量的条件依赖结构。图形模型由多个节点和边组成；每个节点代表一个随机变量，连接两个节点的边表示这两个随机变量是条件相关的。通过对精度矩阵施加LASSO惩罚，我们能够获得稀疏的图形结构，这意味着许多随机变量对之间没有边。我们将提出一个功能图形模型，其中每个节点代表一个随机函数。该模型可以应用于基因表达，以了解这些基因在规则中如何相互联系。此外，人们还对执行任务时人类大脑不同区域之间的功能连接感兴趣。功能连接的图形结构可以作为识别疾病状态的有用生物标志物。第三个研究主题是对多元函数数据的依赖结构进行建模。典型的相关功能数据包括特定区域内不同地点空气污染物的高频测量。我们将制定一个总体框架，以适应灵活的依赖结构。这一发展使我们能够全面了解空气污染物的时空关系。本研究通过开发处理高维功能数据的工具，将对功能数据分析领域做出重大贡献。更重要的是，环境科学、神经科学、金融和生物学等其他学科也将从我们的研究中受益。

My research program will focus on developing new models and novel methodologies for functional data, with emphasis on applications to brain imaging data, air pollution data and financial data. Our first research theme concerns high dimensional functional linear regression, where we are interested in selecting and estimating the effect of multiple function-valued (also called functional) covariates on a scalar response. Repeated measurements of brain signals, time-course gene expressions and high frequency observations of air pollutants and stock indices can all be treated as function-valued data. We plan to develop methodologies to fit a functional linear model with high dimensional functional covariates. This research program will not only extend the idea of variable selection to functional regression models, but also have a significant impact on various disciplines. For example, we will investigate the effect of brain signals collected from different brain regions on prediction of the attention deficit hyperactivity disorder (ADHD) index, a summary metric of the overall level of ADHD. With this model, we are able to identify specific regions that are related to ADHD. Our second research theme focuses on graphical models for functional data. Traditional graphical models are commonly used to describe the conditional dependence structure of multivariate random variables. A graphical model consists of multiple nodes and edges; each node represents one random variables and an edge connecting two nodes indicates that these two random variables are conditionally dependent. By imposing a LASSO penalty on the precision matrix, we are able to obtain a sparse graphical structure that means there are no edges between many pairs of random variables. We will propose a functional graphical model where each node represents a random function. This model can be applied to gene expressions to see how those genes are connected to each other in regulations. Additionally, people are also interested in functional connectivity between different regions of a human brain when a task is performed. The graphical structure for functional connectivity may serve as a useful biomarker in identification of disease status. The third research theme is to model the dependence structure of multivariate functional data. Typical dependent functional data include high frequency measurements of air pollutants at different sites within a specific region. We will develop a general framework to accommodate flexible dependence structures. This development enables us to acquire a comprehensive understanding of the spatial and temporal association of air pollutants. The proposed research will make significant contributions to the field of functional data analysis through developing tools to handle high dimensional functional data. More importantly, other disciplines such as environmental science, neuroscience, finance and biology will also benefit from our research.