Bayesian Methods for Structured, Heterogeneous and High-Dimensional Neuroimaging Data

We consider the problem of analyzing multivariate time series collected on multiple subjects, with the goal of identifying groups of subjects exhibiting similar trends in their recorded measurements over time as well as time-varying groups of associated measurements. To this end, we propose a Bayesian model for temporal biclustering featuring nested partitions, where a time-invariant partition of subjects induces a time-varying partition of measurements. Our approach allows for data-driven determination of the number of subject and measurement clusters as well as estimation of the number and location of changepoints in measurement partitions. To efficiently perform model fitting and posterior estimation with Markov Chain Monte Carlo, we derive a blocked update of measurements' cluster-assignment sequences. We illustrate the performance of our model in two applications to functional magnetic resonance imaging data and to an electroencephalogram dataset. The results indicate that the proposed model can combine information from potentially many subjects to discover a set of interpretable, dynamic patterns. Experiments on simulated data compare the estimation performance of the proposed model against ground-truth values and other statistical methods, showing that it performs well at identifying ground-truth subject and measurement clusters even when no subject or time dependence is present. 

A common concern in the field of functional data analysis is the challenge of temporal misalignment, which is typically addressed using curve registration methods. Currently, most of these methods assume the data is governed by a single common shape or a finite mixture of population level shapes. We introduce more flexibility using mixed membership models. Individual observations are assumed to partially belong to different clusters, allowing variation across multiple functional features. We propose a Bayesian hierarchical model to estimate the underlying shapes, as well as the individual time-transformation functions and levels of membership. Motivating this work is data from EEG signals in children with autism spectrum disorder (ASD). Our method agrees with the neuroimaging literature, recovering the 1/f pink noise feature distinctly from the peak in the alpha band. Furthermore, the introduction of a regression component in the estimation of time-transformation functions quantifies the effect of age and clinical designation on the location of the peak alpha frequency (PAF). 

There is a rich literature on clustering functional data with applications to time-series modeling, trajectory data, and even spatio-temporal applications. However, existing methods routinely perform global clustering that enforce identical atom values within the same cluster. Such grouping may be inadequate. While there is some limited literature on local clustering approaches to deal with the above problems, these methods are typically not scalable to high-dimensional functions and theoretical properties are not well-investigated. For such high-dimensional clustering problems, units are expected to cluster based on a subset of informative imaging features only, with the remaining imaging resolutions not being instrumental in the clustering process. Focusing on basis expansions for high-dimensional functions, we propose a flexible non-parametric Bayesian approach for multi-resolution clustering. The proposed method imposes independent Dirichlet process (DP) priors on different subsets of basis coefficients that ultimately results in a product of DPM priors inducing local clustering. We generalize the approach to incorporate spatially correlated error terms when modeling random spatial functions that is expected to provide improved model fitting. An efficient Markov chain Monte Carlo (MCMC) algorithm is developed for implementation. We show posterior consistency properties under the local clustering approach that asymptotically recovers the true density of random functions. Extensive simulations illustrate the improved clustering and function estimation under the proposed method compared to classical approaches. We apply the proposed approach to a spatial transcriptomics application where the goal is to infer clusters of genes with distinct spatial patterns of expressions. Our method makes an important contribution by expanding the limited literature on local clustering methods for high-dimensional functions with theoretical guarantees. 

Deep neural networks (DNN) have been adopted in the scalar-on-image regression which predicts the outcome variable using image predictors. However, training DNN often requires a large sample size to achieve a good prediction accuracy and the model fitting results can be difficult to interpret. In this work, we propose a noval Bayesian non-linear scalar-on-image regression framework with a spatially varying neural network (SV-NN) prior. The SV-NN is constructed using a single hidden layer neural network with its weights generated by the soft-thresholded Gaussian process. Our framework is able to select interpretable image regions and to achieve high prediction accuracy with limited training samples. The SV-NN provides large prior support for the imaging effect function, enabling efficient posterior inference on image region selection and automatically determining the network structures. We establish the posterior consistency of model parameters and selection consistency of image regions when the number of voxels/pixels grows much faster than the sample size. We develop an efficient posterior computation algorithm based on stochastic gradient Langevin dynamics (SGLD). We compared our methods with state-of-the-art deep learning methods via analyses of multiple real data sets including the task fMRI data in the Adolescent Brain Cognitive Development (ABCD) study.