Novel Statistical Methods in Functional and Brain Imaging Data Analysis

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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.