Big Data bring new opportunities to modern society and challenges to data scientists. On one hand, Big Data hold great promises for discovering subtle population patterns and heterogeneities that are not possible with small-scale data. On the other hand, the massive sample size and high dimensionality of Big Data introduce unique computational and statistical challenges, including scalability and storage bottleneck, noise accumulation, spurious correlation, incidental endogeneity, and measurement errors. These challenges are distinguished and require new computational and statistical paradigm. This article gives overviews on the salient features of Big Data and how these features impact on paradigm change on statistical and computational methods as well as computing architectures. We also provide various new perspectives on the Big Data analysis and computation. In particular, we emphasize on the viability of the sparsest solution in high-confidence set and point out that exogeneous assumptions in most statistical methods for Big Data can not be validated due to incidental endogeneity. They can lead to wrong statistical inferences and consequently wrong scientific conclusions.

We propose a semiparametric approach called the nonparanormal skeptic for efficiently and robustly estimating high dimensional undirected graphical models. To achieve modeling flexibility, we consider the nonparanormal graphical models proposed by Liu, Lafferty and Wasserman (2009). To achieve estimation robustness, we exploit nonparametric rank-based correlation coefficient estimators, including the Spearman's rho and Kendall's tau. We prove that the nonparanormal skeptic achieves the optimal parametric rates of convergence for both graph recovery and parameter estimation. This result suggests that the nonparanormal graphical models can be used as a safe replacement of the popular Gaussian graphical models, even when the data are truly Gaussian. Besides theoretical analysis, we also conduct thorough numerical simulations to compare the graph recovery performance of different estimators under both ideal and noisy settings. The proposed methods are then applied on a largescale genomic dataset to illustrate their empirical usefulness. The R package huge implementing the proposed methods is available on the Comprehensive R Archive Network: http://cran.r-project.org/.

Summary In this manuscript we consider the problem of jointly estimating multiple graphical models in high dimensions. We assume that the data are collected from n subjects, each of which consists of T possibly dependent observations. The graphical models of subjects vary, but are assumed to change smoothly corresponding to a measure of closeness between subjects. We propose a kernel based method for jointly estimating all graphical models. Theoretically, under a double asymptotic framework, where both (T, n) and the dimension d can increase, we provide the explicit rate of convergence in parameter estimation. It characterizes the strength one can borrow across different individuals and the impact of data dependence on parameter estimation. Empirically, experiments on both synthetic and real resting state functional magnetic resonance imaging (rs-fMRI) data illustrate the effectiveness of the proposed method.

Successful automated diagnoses of attention deficit hyperactive disorder (ADHD) using imaging and functional biomarkers would have fundamental consequences on the public health impact of the disease. In this work, we show results on the predictability of ADHD using imaging biomarkers and discuss the scientific and diagnostic impacts of the research. We created a prediction model using the landmark ADHD 200 data set focusing on resting state functional connectivity (rs-fc) and structural brain imaging. We predicted ADHD status and subtype, obtained by behavioral examination, using imaging data, intelligence quotients and other covariates. The novel contributions of this manuscript include a thorough exploration of prediction and image feature extraction methodology on this form of data, including the use of singular value decompositions (SVDs), CUR decompositions, random forest, gradient boosting, bagging, voxel-based morphometry, and support vector machines as well as important insights into the value, and potentially lack thereof, of imaging biomarkers of disease. The key results include the CUR-based decomposition of the rs-fc-fMRI along with gradient boosting and the prediction algorithm based on a motor network parcellation and random forest algorithm. We conjecture that the CUR decomposition is largely diagnosing common population directions of head motion. Of note, a byproduct of this research is a potential automated method for detecting subtle in-scanner motion. The final prediction algorithm, a weighted combination of several algorithms, had an external test set specificity of 94% with sensitivity of 21%. The most promising imaging biomarker was a correlation graph from a motor network parcellation. In summary, we have undertaken a large-scale statistical exploratory prediction exercise on the unique ADHD 200 data set. The exercise produced several potential leads for future scientific exploration of the neurological basis of ADHD.

SummaryWe consider the testing of mutual independence among all entries in a \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} }{}$d$\end{document}-dimensional random vector based on \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} }{}$n$\end{document} independent observations. We study two families of distribution-free test statistics, which include Kendall’s tau and Spearman’s rho as important examples. We show that under the null hypothesis the test statistics of these two families converge weakly to Gumbel distributions, and we propose tests that control the Type I error in the high-dimensional setting where \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} }{}$d >n$\end{document}. We further show that the two tests are rate-optimal in terms of power against sparse alternatives and that they outperform competitors in simulations, especially when \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} }{}$d$\end{document} is large.

We propose a semiparametric method for conducting scale-invariant sparse principal component analysis (PCA) on high dimensional non-Gaussian data. Compared with sparse PCA, our method has weaker modeling assumption and is more robust to possible data contamination. Theoretically, the proposed method achieves a parametric rate of convergence in estimating the parameter of interests under a flexible semiparametric distribution family; Computationally, the proposed method exploits a rank-based procedure and is as efficient as sparse PCA; Empirically, our method outperforms most competing methods on both synthetic and real-world datasets.

Brain organoids are important models for mimicking some three-dimensional (3D) cytoarchitectural and functional aspects of the brain. Multielectrode arrays (MEAs) that enable recording and stimulation of activity from electrogenic cells offer notable potential for interrogating brain organoids. However, conventional MEAs, initially designed for monolayer cultures, offer limited recording contact area restricted to the bottom of the 3D organoids. Inspired by the shape of electroencephalography caps, we developed miniaturized wafer-integrated MEA caps for organoids. The optically transparent shells are composed of self-folding polymer leaflets with conductive polymer–coated metal electrodes. Tunable folding of the minicaps’ polymer leaflets guided by mechanics simulations enables versatile recording from organoids of different sizes, and we validate the feasibility of electrophysiology recording from 400- to 600-μm-sized organoids for up to 4 weeks and in response to glutamate stimulation. Our studies suggest that 3D shell MEAs offer great potential for high signal-to-noise ratio and 3D spatiotemporal brain organoid recording.

Rank correlations have found many innovative applications in the last decade. In particular, suitable versions of rank correlations have been used for consistent tests of independence between pairs of random variables. The use of ranks is especially appealing for continuous data as tests become distribution-free. However, the traditional concept of ranks relies on ordering data and is, thus, tied to univariate observations. As a result it has long remained unclear how one may construct distribution-free yet consistent tests of independence between multivariate random vectors. This is the problem we address in this paper, in which we lay out a general framework for designing dependence measures that give tests of multivariate independence that are not only consistent and distribution-free but which we also prove to be statistically efficient. Our framework leverages the recently introduced concept of center-outward ranks and signs, a multivariate generalization of traditional ranks, and adopts a common standard form for dependence measures that encompasses many popular measures from the literature. In a unified study, we derive a general asymptotic representation of center-outward test statistics under independence, extending to the multivariate setting the classical Hájek asymptotic representation results. This representation permits a direct calculation of limiting null distributions for the proposed test statistics. Moreover, it facilitates a local power analysis that provides strong support for the center-outward approach to multivariate ranks by establishing, for the first time, the rate-optimality of center-outward tests within families of Konijn alternatives.