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，它最近作为快速发展的大数据和复杂数据分析领域的一部分而出现。神经成像数据，也称为脑成像数据，包括扩散张量成像(DTI)、磁共振成像(MRI)等，这些数据既可以作为功能数据来处理，又有自己的特点。因此，其统计分析既继承了函数数据分析的一些方法，又因其结构复杂而提出了新的挑战。为了应对这些挑战，我们在功能和脑成像数据分析方面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对单变量反应进行建模，另一种是用单变量协变量对功能反应进行建模。传统上，反应的条件平均值将被模型化。然而，为了获得完整的图像，为了处理成像数据的异质性，以及考虑到复杂和相关的结构，我们对响应的条件分位数进行建模。我们新统计方法的新颖性是多方面的。理论上，通过限制再生核Hilbert空间(RKHS)中的泛函效应，我们的估计获得了最优的极小极大收敛速度。在计算上，利用表示定理，提出了一种基于乘子交替方向法(ADMM)的高效算法。基于分位数残差结构的经典原始-对偶算法，考虑到特殊的函数数据结构，具有很高的效率。此外，为了整合遗传数据等大规模数据，我们开发了超大规模场景下的快速筛选方法，并增加了惩罚函数来规则化大规模特征。我们选择无偏非凸罚函数，如光滑截断绝对偏差(SCAD)等。筛选和选择的一致性，以及高效的算法将被推导出来。 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数据集，例如阿尔茨海默病神经成像倡议(ADNI)、人类连接组项目(HCP)等。这将为了解人类大脑铺平道路，并为更好地治疗包括自闭症、阿尔茨海默病等在内的各种精神疾病带来希望。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，为未来的职业生涯做好准备。派生的算法将在R中实现，并公开提供。

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.