REGULARIZED 3D FUNCTIONAL REGRESSION FOR BRAIN IMAGE DATA VIA HAAR WAVELETS

Matrix-Variate Regression for Sparse, Low-Rank Estimation of Brain Connectivities Associated With a Clinical Outcome

OBJECTIVE

We address the problem of finding brain connectivities that are associated with a clinical outcome or phenotype.

METHODS

The proposed framework regresses a (scalar) clinical outcome on matrix-variate predictors which arise in the form of brain connectivity matrices. For example, in a large cohort of subjects we estimate those regions of functional connectivities that are associated with neurocognitive scores. We approach this high-dimensional yet highly structured estimation problem by formulating a regularized estimation process that results in a low-rank coefficient matrix having a sparse set of nonzero entries which represent regions of biologically relevant connectivities. In contrast to the recent literature on estimating a sparse, low-rank matrix from a single noisy observation, our scalar-on-matrix regression framework produces a data-driven extraction of structures that are associated with a clinical response. The method, called Sparsity Inducing Nuclear-Norm Estimator (SpINNEr), simultaneously constrains the regression coefficient matrix in two ways: a nuclear norm penalty encourages low-rank structure while an l1 norm encourages entry-wise sparsity.

RESULTS

Our simulations show that SpINNEr outperforms other methods in estimation accuracy when the response-related entries (representing the brain's functional connectivity) are arranged in well-connected communities. SpINNEr is applied to investigate associations between HIV-related outcomes and functional connectivity in the human brain.

CONCLUSION AND SIGNIFICANCE

Overall, this work demonstrates the potential of SpINNEr to recover sparse and low-rank estimates under scalar-on-matrix regression framework.

A novel two-way functional linear model with applications in human mortality data analysis

Recently, two-way or longitudinal functional data analysis has attracted much attention in many fields. However, little is known on how to appropriately characterize the association between two-way functional predictor and scalar response. Motivated by a mortality study, in this paper, we propose a novel two-way functional linear model, where the response is a scalar and functional predictor is two-way trajectory. The model is intuitive, interpretable and naturally captures relationship between each way of two-way functional predictor and scalar-type response. Further, we develop a new estimation method to estimate the regression functions in the framework of weak separability. The main technical tools for the construction of the regression functions are product functional principal component analysis and iterative least square procedure. The solid performance of our method is demonstrated in extensive simulation studies. We also analyze the mortality dataset to illustrate the usefulness of the proposed procedure.

Multi-task Learning with High-Dimensional Noisy Images

Recent medical imaging studies have given rise to distinct but inter-related datasets corresponding to multiple experimental tasks or longitudinal visits. Standard scalar-on-image regression models that fit each dataset separately are not equipped to leverage information across inter-related images, and existing multi-task learning approaches are compromised by the inability to account for the noise that is often observed in images. We propose a novel joint scalar-on-image regression framework involving wavelet-based image representations with grouped penalties that are designed to pool information across inter-related images for joint learning, and which explicitly accounts for noise in high-dimensional images via a projection-based approach. In the presence of non-convexity arising due to noisy images, we derive non-asymptotic error bounds under non-convex as well as convex grouped penalties, even when the number of voxels increases exponentially with sample size. A projected gradient descent algorithm is used for computation, which is shown to approximate the optimal solution via well-defined non-asymptotic optimization error bounds under noisy images. Extensive simulations and application to a motivating longitudinal Alzheimer's disease study illustrate significantly improved predictive ability and greater power to detect true signals, that are simply missed by existing methods without noise correction due to the attenuation to null phenomenon.

Classification of EEG signals: An interpretable approach using functional data analysis

Electroencephalography (EEG) is a noninvasive method to record electrical activity of the brain. The EEG data is continuous flow of voltages, in this paper, we consider them as functional data, and propose a three-stage algorithm based on functional data analysis, with the advantage of interpretability. Specifically, the time and frequency information are extracted by wavelet transform in the first stage. Then, functional testing is utilized to select EEG channels and frequencies that show significant differences for different human behaviors. In the third stage, we propose to use penalized multiple functional logistic regression to interpretably classify human behaviors. With simulation and a scalp EEG data as validation set, we show that the proposed three-stage algorithm provides an interpretable classification of the scalp EEG signals.

A Bayesian group lasso classification for ADNI volumetrics data

The primary objective of this paper is to develop a statistically valid classification procedure for analyzing brain image volumetrics data obtained from the Alzheimer's Disease Neuroimaging Initiative (ADNI) in elderly subjects with cognitive impairments. The Bayesian group lasso method thereby proposed for logistic regression efficiently selects an optimal model with the use of a spike and slab type prior. This method selects groups of attributes of a brain subregion encouraged by the group lasso penalty. We conduct simulation studies for high- and low-dimensional scenarios where our method is always able to select the true parameters that are truly predictive among a large number of parameters. The method is then applied on dichotomous response ADNI data which selects predictive atrophied brain regions and classifies Alzheimer's disease patients from healthy controls. Our analysis is able to give an accuracy rate of 80% for classifying Alzheimer's disease. The suggested method selects 29 brain subregions. The medical literature indicates that all these regions are associated with Alzheimer's patients. The Bayesian method of model selection further helps selecting only the subregions that are statistically significant, thus obtaining an optimal model.

Joint Bayesian Estimation of Voxel Activation and Inter-regional Connectivity in fMRI Experiments

Brain activation and connectivity analyses in task-based functional magnetic resonance imaging (fMRI) experiments with multiple subjects are currently at the forefront of data-driven neuroscience. In such experiments, interest often lies in understanding activation of brain voxels due to external stimuli and strong association or connectivity between the measurements on a set of pre-specified groups of brain voxels, also known as regions of interest (ROI). This article proposes a joint Bayesian additive mixed modeling framework that simultaneously assesses brain activation and connectivity patterns from multiple subjects. In particular, fMRI measurements from each individual obtained in the form of a multi-dimensional array/tensor at each time are regressed on functions of the stimuli. We impose a low-rank parallel factorization decomposition on the tensor regression coefficients corresponding to the stimuli to achieve parsimony. Multiway stick-breaking shrinkage priors are employed to infer activation patterns and associated uncertainties in each voxel. Further, the model introduces region-specific random effects which are jointly modeled with a Bayesian Gaussian graphical prior to account for the connectivity among pairs of ROIs. Empirical investigations under various simulation studies demonstrate the effectiveness of the method as a tool to simultaneously assess brain activation and connectivity. The method is then applied to a multi-subject fMRI dataset from a balloon-analog risk-taking experiment, showing the effectiveness of the model in providing interpretable joint inference on voxel-level activations and inter-regional connectivity associated with how the brain processes risk. The proposed method is also validated through simulation studies and comparisons to other methods used within the neuroscience community.

Tucker Tensor Regression and Neuroimaging Analysis

Neuroimaging data often take the form of high dimensional arrays, also known as tensors. Addressing scientific questions arising from such data demands new regression models that take multidimensional arrays as covariates. Simply turning an image array into a vector would both cause extremely high dimensionality and destroy the inherent spatial structure of the array. In a recent work, Zhou et al. (2013) proposed a family of generalized linear tensor regression models based upon the CP (CANDECOMP/PARAFAC) decomposition of regression coefficient array. Low rank approximation brings the ultrahigh dimensionality to a manageable level and leads to efficient estimation. In this article, we propose a tensor regression model based on the more flexible Tucker decomposition. Compared to the CP model, Tucker regression model allows different number of factors along each mode. Such flexibility leads to several advantages that are particularly suited to neuroimaging analysis, including further reduction of the number of free parameters, accommodation of images with skewed dimensions, explicit modeling of interactions, and a principled way of image downsizing. We also compare the Tucker model with CP numerically on both simulated data and a real magnetic resonance imaging data, and demonstrate its effectiveness in finite sample performance.

Methods for scalar-on-function regression

Recent years have seen an explosion of activity in the field of functional data analysis (FDA), in which curves, spectra, images, etc. are considered as basic functional data units. A central problem in FDA is how to fit regression models with scalar responses and functional data points as predictors. We review some of the main approaches to this problem, categorizing the basic model types as linear, nonlinear and nonparametric. We discuss publicly available software packages, and illustrate some of the procedures by application to a functional magnetic resonance imaging dataset.

Generalized Scalar-on-Image Regression Models via Total Variation

The use of imaging markers to predict clinical outcomes can have a great impact in public health. The aim of this paper is to develop a class of generalized scalar-on-image regression models via total variation (GSIRM-TV), in the sense of generalized linear models, for scalar response and imaging predictor with the presence of scalar covariates. A key novelty of GSIRM-TV is that it is assumed that the slope function (or image) of GSIRM-TV belongs to the space of bounded total variation in order to explicitly account for the piecewise smooth nature of most imaging data. We develop an efficient penalized total variation optimization to estimate the unknown slope function and other parameters. We also establish nonasymptotic error bounds on the excess risk. These bounds are explicitly specified in terms of sample size, image size, and image smoothness. Our simulations demonstrate a superior performance of GSIRM-TV against many existing approaches. We apply GSIRM-TV to the analysis of hippocampus data obtained from the Alzheimers Disease Neuroimaging Initiative (ADNI) dataset.

Classification of ADNI PET Images via Regularized 3D Functional Data Analysis

We propose a penalized Haar wavelet approach for the classification of 3D brain images in the framework of functional data analysis, which treats each entire 3D brain image as a single functional input thus automatically takes into account the spatial correlations of voxel level imaging measures. We validate the proposed approach through extensive simulations and compare its classification performance with other commonly used machine learning methods, which show that the proposed method outperforms other methods in both classification accuracy and identification of the relevant voxels. We then apply the proposed method to the practical classification problems for Alzheimer's disease using PET images obtained from the Alzheimer's Disease Neuroimaging Initiative (ADNI) database to highlight the advantages of our approach.

Generalized reduced rank latent factor regression for high dimensional tensor fields, and neuroimaging-genetic applications

We propose a generalized reduced rank latent factor regression model (GRRLF) for the analysis of tensor field responses and high dimensional covariates. The model is motivated by the need from imaging-genetic studies to identify genetic variants that are associated with brain imaging phenotypes, often in the form of high dimensional tensor fields. GRRLF identifies from the structure in the data the effective dimensionality of the data, and then jointly performs dimension reduction of the covariates, dynamic identification of latent factors, and nonparametric estimation of both covariate and latent response fields. After accounting for the latent and covariate effects, GRLLF performs a nonparametric test on the remaining factor of interest. GRRLF provides a better factorization of the signals compared with common solutions, and is less susceptible to overfitting because it exploits the effective dimensionality. The generality and the flexibility of GRRLF also allow various statistical models to be handled in a unified framework and solutions can be efficiently computed. Within the field of neuroimaging, it improves the sensitivity for weak signals and is a promising alternative to existing approaches. The operation of the framework is demonstrated with both synthetic datasets and a real-world neuroimaging example in which the effects of a set of genes on the structure of the brain at the voxel level were measured, and the results compared favorably with those from existing approaches.

WAVELET-DOMAIN REGRESSION AND PREDICTIVE INFERENCE IN PSYCHIATRIC NEUROIMAGING

An increasingly important goal of psychiatry is the use of brain imaging data to develop predictive models. Here we present two contributions to statistical methodology for this purpose. First, we propose and compare a set of wavelet-domain procedures for fitting generalized linear models with scalar responses and image predictors: sparse variants of principal component regression and of partial least squares, and the elastic net. Second, we consider assessing the contribution of image predictors over and above available scalar predictors, in particular via permutation tests and an extension of the idea of confounding to the case of functional or image predictors. Using the proposed methods, we assess whether maps of a spontaneous brain activity measure, derived from functional magnetic resonance imaging, can meaningfully predict presence or absence of attention deficit/hyperactivity disorder (ADHD). Our results shed light on the role of confounding in the surprising outcome of the recent ADHD-200 Global Competition, which challenged researchers to develop algorithms for automated image-based diagnosis of the disorder.

REGULARIZED 3D FUNCTIONAL REGRESSION FOR BRAIN IMAGE DATA VIA HAAR WAVELETS

The primary motivation and application in this article come from brain imaging studies on cognitive impairment in elderly subjects with brain disorders. We propose a regularized Haar wavelet-based approach for the analysis of three-dimensional brain image data in the framework of functional data analysis, which automatically takes into account the spatial information among neighboring voxels. We conduct extensive simulation studies to evaluate the prediction performance of the proposed approach and its ability to identify related regions to the outcome of interest, with the underlying assumption that only few relatively small subregions are truly predictive of the outcome of interest. We then apply the proposed approach to searching for brain subregions that are associated with cognition using PET images of patients with Alzheimer's disease, patients with mild cognitive impairment, and normal controls.