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Influential diagnosis is an integral part of data analysis, of which most existing methodological frameworks presume a deterministic submodel and are designed for low-dimensional data (i.e., the number of predictors p smaller than the sample size n). However, the stochastic selection of a submodel from high-dimensional data where p exceeds n has become ubiquitous. Thus, methods for identifying observations that could exert undue influence on the choice of a submodel can play an important role in this setting. To date, discussion of this topic has been limited, falling short in two domains: (i) constrained ability to detect multiple influential points, and (ii) applicability only in restrictive settings. After describing the problem, we characterize and formalize the concept of influential observations on variable selection. Then, we propose a generalized diagnostic measure, extended from an available metric accommodating different model selectors and multiple influential observations, the asymptotic distribution of which is subsequently establish large p, thus providing guidelines to ascertain influential observations. A high-dimensional clustering procedure is further incorporated into our proposed scheme to detect multiple influential points. Simulation is conducted to assess the performances of various diagnostic approaches. The proposed procedure further demonstrates its value in improving predictive power when analyzing thermal-stimulated pain based on fMRI data., Comment: 21 pages, 6 figures

All neuroimaging modalities have their own strengths and limitations. A current trend is toward interdisciplinary approaches that use multiple imaging methods to overcome limitations of each method in isolation. At the same time neuroimaging data is increasingly being combined with other non-imaging modalities, such as behavioral and genetic data. The data structure of many of these modalities can be expressed as time-varying multidimensional arrays (tensors), collected at different time-points on multiple subjects. Here, we consider a new approach for the study of neural correlates in the presence of tensor-valued brain images and tensor-valued predictors, where both data types are collected over the same set of time points. We propose a time-varying tensor regression model with an inherent structural composition of responses and covariates. Regression coefficients are expressed using the B-spline technique, and the basis function coefficients are estimated using CP-decomposition by minimizing a penalized loss function. We develop a varying-coefficient model for the tensor-valued regression model, where both predictors and responses are modeled as tensors. This development is a non-trivial extension of function-on-function concurrent linear models for complex and large structural data where the inherent structures are preserved. In addition to the methodological and theoretical development, the efficacy of the proposed method based on both simulated and real data analysis (e.g., the combination of eye-tracking data and functional magnetic resonance imaging (fMRI) data) is also discussed.

Functional alignment between subjects is an important assumption of functional magnetic resonance imaging (fMRI) group-level analysis. However, it is often violated in practice, even after alignment to a standard anatomical template. Hyperalignment, based on sequential Procrustes orthogonal transformations, has been proposed as a method of aligning shared functional information into a common high-dimensional space and thereby improving inter-subject analysis. Though successful, current hyperalignment algorithms have a number of shortcomings, including difficulties interpreting the transformations, a lack of uniqueness of the procedure, and difficulties performing whole-brain analysis. To resolve these issues, we propose the ProMises (Procrustes von Mises-Fisher) model. We reformulate functional alignment as a statistical model and impose a prior distribution on the orthogonal parameters (the von Mises-Fisher distribution). This allows for the embedding of anatomical information into the estimation procedure by penalizing the contribution of spatially distant voxels when creating the shared functional high-dimensional space. Importantly, the transformations, aligned images, and related results are all unique. In addition, the proposed method allows for efficient whole-brain functional alignment. In simulations and application to data from four fMRI studies we find that ProMises improves inter-subject classification in terms of between-subject accuracy and interpretability compared to standard hyperalignment algorithms., Comment: 28 pages, 9 figures

In recent years, neuroimaging has undergone a paradigm shift, moving away from the traditional brain mapping approach toward developing integrated, multivariate brain models that can predict categories of mental events. However, large interindividual differences in brain anatomy and functional localization after standard anatomical alignment remain a major limitation in performing this analysis, as it leads to feature misalignment across subjects in subsequent predictive models.

Functional Gaussian graphical models (GGM) used for analyzing multivariate functional data customarily estimate an unknown graphical model representing the conditional relationships between the functional variables. However, in many applications of multivariate functional data, the graph is known and existing functional GGM methods cannot preserve a given graphical constraint. In this manuscript, we demonstrate how to conduct multivariate functional analysis that exactly conforms to a given inter-variable graph. We first show the equivalence between partially separable functional GGM and graphical Gaussian processes (GP), proposed originally for constructing optimal covariance functions for multivariate spatial data that retain the conditional independence relations in a given graphical model. The theoretical connection help design a new algorithm that leverages Dempster's covariance selection to calculate the maximum likelihood estimate of the covariance function for multivariate functional data under graphical constraints. We also show that the finite term truncation of functional GGM basis expansion used in practice is equivalent to a low-rank graphical GP, which is known to oversmooth marginal distributions. To remedy this, we extend our algorithm to better preserve marginal distributions while still respecting the graph and retaining computational scalability. The insights obtained from the new results presented in this manuscript will help practitioners better understand the relationship between these graphical models and in deciding on the appropriate method for their specific multivariate data analysis task. The benefits of the proposed algorithms are illustrated using empirical experiments and an application to functional modeling of neuroimaging data using the connectivity graph among regions of the brain., Comment: 23 pages, 6 figures

Recently there has been significant interest in measuring time-varying functional connectivity (TVC) between different brain regions using resting-state functional magnetic resonance imaging (rs-fMRI) data. One way to assess the relationship between signals from different brain regions is to measure their phase synchronization (PS) across time. However, this requires the \textit{a priori} choice of type and cut-off frequencies for the bandpass filter needed to perform the analysis. Here we explore alternative approaches based on the use of various mode decomposition (MD) techniques that circumvent this issue. These techniques allow for the data driven decomposition of signals jointly into narrow-band components at different frequencies, thus fulfilling the requirements needed to measure PS. We explore several variants of MD, including empirical mode decomposition (EMD), bivariate EMD (BEMD), noise-assisted multivariate EMD (na-MEMD), and introduce the use of multivariate variational mode decomposition (MVMD) in the context of estimating time-varying PS. We contrast the approaches using a series of simulations and application to rs-fMRI data. Our results show that MVMD outperforms other evaluated MD approaches, and further suggests that this approach can be used as a tool to reliably investigate time-varying PS in rs-fMRI data., Comment: 57 pages, 11 figures. Please cite the preprint if any or portion of the material is used

Functional magnetic resonance imaging (fMRI) has provided invaluable insight into our understanding of human behavior. However, large inter-individual differences in both brain anatomy and functional localization after anatomical alignment remain a major limitation in conducting group analyses and performing population-level inference. This paper addresses this problem by developing and validating a new computational technique for reducing misalignment across individuals in functional brain systems by spatially transforming each subject's functional data to a common reference map. Our proposed Bayesian functional registration approach allows us to assess differences in brain function across subjects and individual differences in activation topology. It combines intensity-based and feature-based information into an integrated framework and allows inference to be performed on the transformation via the posterior samples. We evaluate the method in a simulation study and apply it to data from a study of thermal pain. We find that the proposed approach provides increased sensitivity for group-level inference.

In recent years there has been growing interest in measuring time-varying functional connectivity between different brain regions using resting-state functional magnetic resonance imaging (rs-fMRI) data. One way to assess the relationship between signals from different brain regions is to measure their phase synchronization (PS) across time. There are several ways to perform such analyses, and here we compare methods that utilize a PS metric together with a sliding window, referred to here as windowed phase synchronization (WPS), with those that directly measure the instantaneous phase synchronization (IPS). In particular, IPS has recently gained popularity as it offers single time-point resolution of time-resolved fMRI connectivity. In this paper, we discuss the underlying assumptions required for performing PS analyses and emphasize the necessity of band-pass filtering the data to obtain valid results. We review various methods for evaluating PS and introduce a new approach within the IPS framework denoted the cosine of the relative phase (CRP). We contrast methods through a series of simulations and application to rs-fMRI data. Our results indicate that CRP outperforms other tested methods and overcomes issues related to undetected temporal transitions from positive to negative associations common in IPS analysis. Further, in contrast to phase coherence, CRP unfolds the distribution of PS measures, which benefits subsequent clustering of PS matrices into recurring brain states.

A key strength of twin studies arises from the fact that there are two types of twins, monozygotic and dizygotic, that share differing amounts of genetic information. Accurate differentiation of twin types allows efficient inference on genetic influences in a population. However, identification of zygosity is often prone to errors without genotying. In this study, we propose a novel pairwise feature representation to classify the zygosity of twin pairs of resting state functional magnetic resonance images (rs-fMRI). For this, we project an fMRI signal to a set of basis functions and use the projection coefficients as the compact and discriminative feature representation of noisy fMRI. We encode the relationship between twins as the correlation between the new feature representations across brain regions. We employ hill climbing variable selection to identify brain regions that are the most genetically affected. The proposed framework was applied to 208 twin pairs and achieved 94.19% classification accuracy in automatically identifying the zygosity of paired images.

Causal mediation analysis aims to quantify the intermediate effect of a mediator on the causal pathway from treatment to outcome. With multiple mediators, which are potentially causally dependent, the possible decomposition of pathway effects grows exponentially with the number of mediators. Huang and Pan (2016) introduced a principal component analysis (PCA) based approach to address this challenge, in which the transformed mediators are conditionally independent given the orthogonality of the PCs. However, the transformed mediator PCs, which are linear combinations of original mediators, are difficult to interpret. In this study, we propose a sparse high-dimensional mediation analysis approach by adopting the sparse PCA method introduced by Zou and others (2006) to the mediation setting. We apply the approach to a task-based functional magnetic resonance imaging study, and show that our proposed method is able to detect biologically meaningful results related to the identified mediator., Comment: 24 pages, 3 figures, 1 table

Causal mediation analysis is widely utilized to separate the causal effect of treatment into its direct effect on the outcome and its indirect effect through an intermediate variable (the mediator). In this study we introduce a functional mediation analysis framework in which the three key variables, the treatment, mediator, and outcome, are all continuous functions. With functional measures, causal assumptions and interpretations are not immediately well-defined. Motivated by a functional magnetic resonance imaging (fMRI) study, we propose two functional mediation models based on the influence of the mediator: (1) a concurrent mediation model and (2) a historical mediation model. We further discuss causal assumptions, and elucidate causal interpretations. Our proposed models enable the estimation of individual causal effect curves, where both the direct and indirect effects vary across time. Applied to a task-based fMRI study, we illustrate how our functional mediation framework provides a new perspective for studying dynamic brain connectivity. The R package cfma is available on CRAN.

Cortical surface fMRI (cs-fMRI) has recently grown in popularity versus traditional volumetric fMRI, as it allows for more meaningful spatial smoothing and is more compatible with the common assumptions of isotropy and stationarity in Bayesian spatial models. However, as no Bayesian spatial model has been proposed for cs-fMRI data, most analyses continue to employ the classical, voxel-wise general linear model (GLM) (Worsley and Friston 1995). Here, we propose a Bayesian GLM for cs-fMRI, which employs a class of sophisticated spatial processes to flexibly model latent activation fields. We use integrated nested Laplacian approximation (INLA), a highly accurate and efficient Bayesian computation technique (Rue et al. 2009). To identify regions of activation, we propose an excursions set method based on the joint posterior distribution of the latent fields, which eliminates the need for multiple comparisons correction. Finally, we address a gap in the existing literature by proposing a novel Bayesian approach for multi-subject analysis. The methods are validated and compared to the classical GLM through simulation studies and a motor task fMRI study from the Human Connectome Project. The proposed Bayesian approach results in smoother activation estimates, more accurate false positive control, and increased power to detect truly active regions.

We propose a voxel-wise general linear model with autoregressive noise and heteroscedastic noise innovations (GLMH) for analyzing functional magnetic resonance imaging (fMRI) data. The model is analyzed from a Bayesian perspective and has the benefit of automatically down-weighting time points close to motion spikes in a data-driven manner. We develop a highly efficient Markov Chain Monte Carlo (MCMC) algorithm that allows for Bayesian variable selection among the regressors to model both the mean (i.e., the design matrix) and variance. This makes it possible to include a broad range of explanatory variables in both the mean and variance (e.g., time trends, activation stimuli, head motion parameters and their temporal derivatives), and to compute the posterior probability of inclusion from the MCMC output. Variable selection is also applied to the lags in the autoregressive noise process, making it possible to infer the lag order from the data simultaneously with all other model parameters. We use both simulated data and real fMRI data from OpenfMRI to illustrate the importance of proper modeling of heteroscedasticity in fMRI data analysis. Our results show that the GLMH tends to detect more brain activity, compared to its homoscedastic counterpart, by allowing the variance to change over time depending on the degree of head motion.

In this paper, we use data from the Human Connectome Project (N=461) to investigate the effect of scan length on reliability of resting-state functional connectivity (rsFC) estimates produced from resting-state functional magnetic resonance imaging (rsfMRI). Additionally, we study the benefits of empirical Bayes shrinkage, in which subject-level estimates borrow strength from the population average by trading a small increase in bias for a greater reduction in variance. For each subject, we compute raw and shrinkage estimates of rsFC between 300 regions identified through independent components analysis (ICA) based on rsfMRI scans varying from 3 to 30 minutes in length. The time course for each region is determined using dual regression, and rsFC is estimated as the Pearson correlation between each pair of time courses. Shrinkage estimates for each subject are computed as a weighted average between the raw subject-level estimate and the population average estimate, where the weight is determined for each connection by the relationship of within-subject variance to between-subject variance. We find that shrinkage estimates exhibit greater reliability than raw estimates for most connections, with 30-40% improvement using scans less than 10 minutes in length and 10-20% improvement using scans of 20-30 minutes. We also observe significant spatial variability in reliability of both raw and shrinkage estimates, with connections within the default mode and motor networks exhibiting the greatest reliability and between-network connections exhibiting the poorest reliability. We conclude that the scan length required for reliable estimation of rsFC depends on the specific connections of interest, and shrinkage can be used to increase reliability of rsFC, even when produced from long, high-quality rsfMRI scans.

Mediation analysis has become an important tool in the behavioral sciences for investigating the role of intermediate variables that lie in the path between a randomized treatment and an outcome variable. The influence of the intermediate variable on the outcome is often explored using structural equation models (SEMs), with model coefficients interpreted as possible effects. While there has been significant research on the topic in recent years, little work has been done on mediation analysis when the intermediate variable (mediator) is a high-dimensional vector. In this work we present a new method for exploratory mediation analysis in this setting called the directions of mediation (DMs). The first DM is defined as the linear combination of the elements of a high-dimensional vector of potential mediators that maximizes the likelihood of the SEM. The subsequent DMs are defined as linear combinations of the elements of the high-dimensional vector that are orthonormal to the previous DMs and maximize the likelihood of the SEM. We provide an estimation algorithm and establish the asymptotic properties of the obtained estimators. This method is well suited for cases when many potential mediators are measured. Examples of high-dimensional potential mediators are brain images composed of hundreds of thousands of voxels, genetic variation measured at millions of SNPs, or vectors of thousands of variables in large-scale epidemiological studies. We demonstrate the method using a functional magnetic resonance imaging (fMRI) study of thermal pain where we are interested in determining which brain locations mediate the relationship between the application of a thermal stimulus and self-reported pain.

High-dimensional time-series data are becoming increasingly abundant across a wide variety of domains, spanning economics, neuroscience, particle physics, and cosmology. Fitting statistical models to such data, to enable parameter estimation and time-series prediction, is an important computational primitive. Existing methods, however, are unable to cope with the high-dimensional nature of these problems, due to both computational and statistical reasons. We mitigate both kinds of issues via proposing an M-estimator for Reduced-rank System IDentification (MR. SID). A combination of low-rank approximations, L-1 and L-2 penalties, and some numerical linear algebra tricks, yields an estimator that is computationally efficient and numerically stable. Simulations and real data examples demonstrate the utility of this approach in a variety of problems. In particular, we demonstrate that MR. SID can estimate spatial filters, connectivity graphs, and time-courses from native resolution functional magnetic resonance imaging data. Other applications and extensions are immediately available, as our approach is a generalization of the classical Kalman Filter-Smoother Expectation-Maximization algorithm.

Outlier detection for high-dimensional (HD) data is a popular topic in modern statistical research. However, one source of HD data that has received relatively little attention is functional magnetic resonance images (fMRI), which consists of hundreds of thousands of measurements sampled at hundreds of time points. At a time when the availability of fMRI data is rapidly growing---primarily through large, publicly available grassroots datasets---automated quality control and outlier detection methods are greatly needed. We propose PCA leverage and demonstrate how it can be used to identify outlying time points in an fMRI run. Furthermore, PCA leverage is a measure of the influence of each observation on the estimation of principal components, which are often of interest in fMRI data. We also propose an alternative measure, PCA robust distance, which is less sensitive to outliers and has controllable statistical properties. The proposed methods are validated through simulation studies and are shown to be highly accurate. We also conduct a reliability study using resting-state fMRI data from the Autism Brain Imaging Data Exchange and find that removal of outliers using the proposed methods results in more reliable estimation of subject-level resting-state networks using ICA.

A recent interest in resting state functional magnetic resonance imaging (rsfMRI) lies in subdividing the human brain into anatomically and functionally distinct regions of interest. For example, brain parcellation is often used for defining the network nodes in connectivity studies. While inference has traditionally been performed on group-level data, there is a growing interest in parcellating single subject data. However, this is difficult due to the low signal-to-noise ratio of rsfMRI data, combined with typically short scan lengths. A large number of brain parcellation approaches employ clustering, which begins with a measure of similarity or distance between voxels. The goal of this work is to improve the reproducibility of single-subject parcellation using shrinkage estimators of such measures, allowing the noisy subject-specific estimator to "borrow strength" in a principled manner from a larger population of subjects. We present several empirical Bayes shrinkage estimators and outline methods for shrinkage when multiple scans are not available for each subject. We perform shrinkage on raw intervoxel correlation estimates and use both raw and shrinkage estimates to produce parcellations by performing clustering on the voxels. Our proposed method is agnostic to the choice of clustering method and can be used as a pre-processing step for any clustering algorithm. Using two datasets---a simulated dataset where the true parcellation is known and is subject-specific and a test-retest dataset consisting of two 7-minute rsfMRI scans from 20 subjects---we show that parcellations produced from shrinkage correlation estimates have higher reliability and validity than those produced from raw estimates. Application to test-retest data shows that using shrinkage estimators increases the reproducibility of subject-specific parcellations of the motor cortex by up to 30%., Comment: body 21 pages, 11 figures

In this paper we introduce a new hierarchical model for the simultaneous detection of brain activation and estimation of the shape of the hemodynamic response in multi-subject fMRI studies. The proposed approach circumvents a major stumbling block in standard multi-subject fMRI data analysis, in that it both allows the shape of the hemodynamic response function to vary across region and subjects, while still providing a straightforward way to estimate population-level activation. An efficient estimation algorithm is presented, as is an inferential framework that not only allows for tests of activation, but also for tests for deviations from some canonical shape. The model is validated through simulations and application to a multi-subject fMRI study of thermal pain.

In this work we perform a meta-analysis of neuroimaging data, consisting of locations of peak activations identified in 162 separate studies on emotion. Neuroimaging meta-analyses are typically performed using kernel-based methods. However, these methods require the width of the kernel to be set a priori and to be constant across the brain. To address these issues, we propose a fully Bayesian nonparametric binary regression method to perform neuroimaging meta-analyses. In our method, each location (or voxel) has a probability of being a peak activation, and the corresponding probability function is based on a spatially adaptive Gaussian Markov random field (GMRF). We also include parameters in the model to robustify the procedure against miscoding of the voxel response. Posterior inference is implemented using efficient MCMC algorithms extended from those introduced in Holmes and Held [Bayesian Anal. 1 (2006) 145--168]. Our method allows the probability function to be locally adaptive with respect to the covariates, that is, to be smooth in one region of the covariate space and wiggly or even discontinuous in another. Posterior miscoding probabilities for each of the identified voxels can also be obtained, identifying voxels that may have been falsely classified as being activated. Simulation studies and application to the emotion neuroimaging data indicate that our method is superior to standard kernel-based methods., Comment: Published in at http://dx.doi.org/10.1214/11-AOAS523 the Annals of Applied Statistics (http://www.imstat.org/aoas/) by the Institute of Mathematical Statistics (http://www.imstat.org)

In recent years there has been explosive growth in the number of neuroimaging studies performed using functional Magnetic Resonance Imaging (fMRI). The field that has grown around the acquisition and analysis of fMRI data is intrinsically interdisciplinary in nature and involves contributions from researchers in neuroscience, psychology, physics and statistics, among others. A standard fMRI study gives rise to massive amounts of noisy data with a complicated spatio-temporal correlation structure. Statistics plays a crucial role in understanding the nature of the data and obtaining relevant results that can be used and interpreted by neuroscientists. In this paper we discuss the analysis of fMRI data, from the initial acquisition of the raw data to its use in locating brain activity, making inference about brain connectivity and predictions about psychological or disease states. Along the way, we illustrate interesting and important issues where statistics already plays a crucial role. We also seek to illustrate areas where statistics has perhaps been underutilized and will have an increased role in the future., Comment: Published in at http://dx.doi.org/10.1214/09-STS282 the Statistical Science (http://www.imstat.org/sts/) by the Institute of Mathematical Statistics (http://www.imstat.org)