Novel Statistical Methods in Functional and Brain Imaging Data Analysis

在过去的几十年里，功能数据分析一直是一个越来越受关注的领域，并成功地应用于许多领域，特别是神经成像，它最近成为快速发展的大数据和复杂数据分析领域的一部分。神经影像数据又称脑成像数据，包括弥散张量成像（DTI）、磁共振成像（MRI）等，既可以作为功能数据处理，又有其自身的特点。因此，它的统计分析继承了功能数据分析的一些方法，也提出了新的挑战，由于其复杂的结构。为了应对这些挑战，我们在功能和脑成像数据分析中开发了新的统计方法，以探索其复杂和相关的结构，并整合来自其他资源的数据，如临床数据，遗传学数据等。 我们考虑了两种数据结构：一种是用函数和单变量预测因子对单变量响应进行建模，另一种是用单变量协变量对函数响应进行建模。传统上，响应的条件均值将被建模。然而，为了获得全貌，处理成像数据的异质性，并考虑复杂和相关的结构，我们对响应的条件分位数进行建模。我们新的统计方法的新奇是多方面的。理论上，通过限制再生核Hilbert空间（RKHS）中的函数效应，我们的估计达到了最优的极小极大收敛速度。在计算上，利用表示定理，我们开发了一个有效的算法的基础上交替方向的乘法器（ADMM）。经典的基于分位数残差结构的原始-对偶算法，考虑到函数数据结构的特殊性，具有很高的效率。此外，为了整合基因数据等大规模数据，我们开发了超大规模场景下的快速筛选方法，并添加了惩罚函数来正则化大规模特征。我们选择无偏的非凸罚函数，如平滑剪切绝对偏差（SCAD）等。筛选和选择的一致性，和高效的算法将得到。 我们提出的新的统计方法是及时的，关键的和重要的。它们可以用于分析许多大规模的真实的数据集，例如阿尔茨海默病神经成像倡议（ADNI），人类连接组计划（HCP）等。这将为了解人类大脑铺平道路，并为更好地治疗各种精神障碍，包括自闭症，阿尔茨海默病等提供希望。拟议的统计方法将为研究生以及本科生和博士后研究人员提供极好的培训机会，以获得宝贵的技能，为他们未来的职业生涯做好准备。衍生算法将在R中实现，以公开提供。

Functional data analysis has been an area of increasing interests in the last decades and successfully used in many fields, particularly Neuroimaging, which recently emerged as part of the rapidly evolving field of big and complex data analysis. Neuroimaging data, also known as brain imaging data, include diffusion tensor imaging (DTI), magnetic resonance imaging (MRI), and so on, which could be treated as functional data while having its own features. Therefore, its statistical analysis inherits some methods from functional data analysis and also poses new challenges due to its complex structures. To address those challenges, we develop novel statistical methods in functional and brain imaging data analysis to explore its complex and correlated structures and integrate data from other resources such as clinical data, genetics data, etc. We consider two data structures: one is to model univariate response with functional and univariate predictors, and the other one is to model functional response with univariate covariates. Traditionally, the conditional mean of the response would be modelled. However, to obtain the full picture, to deal with heterogeneity of imaging data, and to account the complex and correlated structure, we model conditional quantiles of the responses. The novelty of our new statistical methods are multifold. Theoretically, by restricting functional effects in reproducing kernel Hilbert space (RKHS), our estimates achieve the optimal minimax convergence rates. Computationally,by taking advantage of the representation theorem, we develop an efficient algorithm based on the alternating direction method of multipliers (ADMM). The classical primal-dual algorithm based on exploring the quantile residual structure could be highly effective taking account of the special functional data structure. In addition, to integrate large-scale data such as genetic data, we develop fast screening methods in ultra large-scale scenario and add penalty functions to regularize large-scale features. We choose unbiased nonconvex penalty functions such as smoothly clipped absolute deviation (SCAD) and others. Screening and selection consistency, and efficient algorithms will be derived. The novel statistical methods we proposed are timely, critical and important. They can be used in analyzing many large-scale real data sets, for example the Alzheimer's Disease Neuroimaging Initiative (ADNI), the Human Connectome Project (HCP), and others. This will pave ways to understand human brains and offer hopes to better treat various mental disorders including autism, Alzheimer's disease, etc. The proposed statistical methods will provide excellent training opportunities to graduate students as well as undergraduate and postdoctoral researchers to gain valuable skills to prepare them for future careers. The derived algorithms will be implemented in R to be available publicly.