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We propose a new approach, called as functional deep neural network (FDNN), for classifying multidimensional functional data. Specifically, a deep neural network is trained based on the principal components of the training data which shall be used to predict the class label of a future data function. Unlike the popular functional discriminant analysis approaches which only work for one‐dimensional functional data, the proposed FDNN approach applies to general non‐Gaussian multidimensional functional data. Moreover, when the log density ratio possesses a locally connected functional modular structure, we show that FDNN achieves minimax optimality. The superiority of our approach is demonstrated through both simulated and real‐world datasets.

Change point detection for high‐dimensional data is an important yet challenging problem for many applications. In this article, we consider multiple change point detection in the context of high‐dimensional generalized linear models, allowing the covariate dimension to grow exponentially with the sample size . The model considered is general and flexible in the sense that it covers various specific models as special cases. It can automatically account for the underlying data generation mechanism without specifying any prior knowledge about the number of change points. Based on dynamic programming and binary segmentation techniques, two algorithms are proposed to detect multiple change points, allowing the number of change points to grow with . To further improve the computational efficiency, a more efficient algorithm designed for the case of a single change point is proposed. We present theoretical properties of our proposed algorithms, including estimation consistency for the number and locations of change points as well as consistency and asymptotic distributions for the underlying regression coefficients. Finally, extensive simulation studies and application to the Alzheimer's Disease Neuroimaging Initiative data further demonstrate the competitive performance of our proposed methods.

In this work, we propose a deep neural networks‐based method to perform non‐parametric regression for functional data. The proposed estimators are based on sparsely connected deep neural networks with rectifier linear unit (ReLU) activation function. We provide the convergence rate of the proposed deep neural networks estimator in terms of the empirical norm. Through Monte Carlo simulation studies, we examine the finite sample performance of the proposed method. Finally, the proposed method is applied to analyse positron emission tomography images of patients with Alzheimer's disease obtained from the Alzheimer Disease Neuroimaging Initiative database.

A multistage variable selection method is introduced for detecting association signals in structured brain‐wide and genome‐wide association studies (brain‐GWAS). Compared to conventional methods that link one voxel to one single nucleotide polymorphism (SNP), our approach is more efficient and powerful in selecting the important signals by integrating anatomic and gene grouping structures in the brain and the genome, respectively. It avoids resorting to a large number of multiple comparisons while effectively controlling the false discoveries. Validity of the proposed approach is demonstrated by both theoretical investigation and numerical simulations. We apply our proposed method to a brain‐GWAS using Alzheimer's Disease Neuroimaging Initiative positron emission tomography (ADNI PET) imaging and genomic data. We confirm previously reported association signals and also uncover several novel SNPs and genes that are either associated with brain glucose metabolism or have their association significantly modified by Alzheimer's disease status.

With the abundance of high‐dimensional data, sparse regularization techniques are very popular in practice because of the built‐in sparsity of their corresponding estimators. However, the success of sparse methods heavily relies on the assumption of sparsity in the underlying model. For models where the sparsity assumption fails, the performance of these sparse methods can be unsatisfactory and misleading. In this article, we consider the perturbed linear model, where the signal is given by the sum of sparse and dense signals. We propose a new penalization‐based method, called Gava, to tackle this kind of signal by making use of a graphical structure among model predictors. The proposed Gava method covers several existing methods as special cases. Our numerical examples and theoretical studies demonstrate the effectiveness of the proposed Gava for estimation and prediction.

With the rapid growth of modern technology, many biomedical studies are being conducted to collect massive datasets with volumes of multi‐modality imaging, genetic, neurocognitive and clinical information from increasingly large cohorts. Simultaneously extracting and integrating rich and diverse heterogeneous information in neuroimaging and/or genomics from these big datasets could transform our understanding of how genetic variants impact brain structure and function, cognitive function and brain‐related disease risk across the lifespan. Such understanding is critical for diagnosis, prevention and treatment of numerous complex brain‐related disorders (e.g., schizophrenia and Alzheimer's disease). However, the development of analytical methods for the joint analysis of both high‐dimensional imaging phenotypes and high‐dimensional genetic data, a big data squared (BD²) problem, presents major computational and theoretical challenges for existing analytical methods. Besides the high‐dimensional nature of BD², various neuroimaging measures often exhibit strong spatial smoothness and dependence and genetic markers may have a natural dependence structure arising from linkage disequilibrium. We review some recent developments of various statistical techniques for imaging genetics, including massive univariate and voxel‐wise approaches, reduced rank regression, mixture models and group sparse multi‐task regression. By doing so, we hope that this review may encourage others in the statistical community to enter into this new and exciting field of research.The Canadian Journal of Statistics47: 108–131; 2019 © 2019 Statistical Society of Canada

Alzheimer's disease (AD) is the most common type of dementia in the elderly with no effective treatment currently. Recent studies of noninvasive neuroimaging, resting‐state functional magnetic resonance imaging (rs‐fMRI) with graph theoretical analysis have shown that patients with AD and mild cognitive impairment (MCI) exhibit disrupted topological organization in large‐scale brain networks. In previous work, it is a common practice to threshold such networks. However, it is not only difficult to make a principled choice of threshold values, but also worse is the discard of potential important information. To address this issue, we propose a threshold‐free feature by integrating a prior persistent homology‐based topological feature (the zeroth Betti number) and a newly defined connected component aggregation cost feature to model brain networks over all possible scales. We show that the induced topological feature (Integrated Persistent Feature) follows a monotonically decreasing convergence function and further propose to use its slope as a concise and persistent brain network topological measure. We apply this measure to study rs‐fMRI data from the Alzheimer's Disease Neuroimaging Initiative and compare our approach with five other widely used graph measures across five parcellation schemes ranging from 90 to 1,024 region‐of‐interests. The experimental results demonstrate that the proposed network measure shows more statistical power and stronger robustness in group difference studies in that the absolute values of the proposed measure of AD are lower than MCI and much lower than normal controls, providing empirical evidence for decreased functional integration in AD dementia and MCI.