Learning Graphical Models of Multivariate Functional Data with Applications to Neuroimaging

This dissertation investigates the functional graphical models that infer the functional connectivity based on neuroimaging data, which is noisy, high dimensional and has limited samples. The dissertation provides two recipes to infer the functional graphical model: 1) a fully Bayesian framework 2) an end-to-end deep model. We first propose a fully Bayesian regularization scheme to estimate functional graphical models. We consider a direct Bayesian analog of the functional graphical lasso proposed by Qiao et al. (2019).. We then propose a regularization strategy via the graphical horseshoe. We compare both Bayesian approaches to the frequentist functional graphical lasso, and compare the Bayesian functional graphical lasso to the functional graphical horseshoe. We applied the proposed methods with electroencephalography (EEG) data and diffusion tensor imaging (DTI) data. We find that the Bayesian methods tend to outperform the standard functional graphical lasso, and that the functional graphical horseshoe performs best overall, a procedure for which there is no direct frequentist analog. Then we consider a deep neural network architecture to estimate functional graphical models, by combining two simple off-the-shelf algorithms: adaptive functional principal components analysis (FPCA) Yao et al., 2021a) and convolutional graph estimator (Belilovsky et al., 2016). We train our proposed model with synthetic data which emulate the real world observations and prior knowledge. Based on synthetic data generation process, our model convert an inference problem as a supervised learning problem. Compared with other framework, our proposed deep model which offers a general recipe to infer the functional graphical model based on data-driven approach, take the raw functional dataset as input and avoid deriving sophisticated closed-form. Through simulation studies, we find that our deep functional graph model trained on synthetic data generalizes well and outperform other popular baselines marginally. In addition, we apply deep functional graphical model in the real world EEG data, and our proposed model discover meaningful brain connectivity. Finally, we are interested in estimating casual graph with functional input. In order to process functional covariates in causal estimation, we leverage the similar strategy as our deep functional graphical model. We extend popular deep causal models to infer causal effects with functional confoundings within the potential outcomes framework. Our method is simple yet effective, where we validate our proposed architecture in variety of simulation settings. Our work offers an alternative way to do causal inference with functional data

Discovering Graphical Granger Causality Using the Truncating Lasso Penalty

Components of biological systems interact with each other in order to carry out vital cell functions. Such information can be used to improve estimation and inference, and to obtain better insights into the underlying cellular mechanisms. Discovering regulatory interactions among genes is therefore an important problem in systems biology. Whole-genome expression data over time provides an opportunity to determine how the expression levels of genes are affected by changes in transcription levels of other genes, and can therefore be used to discover regulatory interactions among genes. In this paper, we propose a novel penalization method, called truncating lasso, for estimation of causal relationships from time-course gene expression data. The proposed penalty can correctly determine the order of the underlying time series, and improves the performance of the lasso-type estimators. Moreover, the resulting estimate provides information on the time lag between activation of transcription factors and their effects on regulated genes. We provide an efficient algorithm for estimation of model parameters, and show that the proposed method can consistently discover causal relationships in the large $p$, small $n$ setting. The performance of the proposed model is evaluated favorably in simulated, as well as real, data examples. The proposed truncating lasso method is implemented in the R-package grangerTlasso and is available at http://www.stat.lsa.umich.edu/~shojaie.Comment: 12 pages, 4 figures, 1 tabl

Patterns of Scalable Bayesian Inference

Datasets are growing not just in size but in complexity, creating a demand for rich models and quantification of uncertainty. Bayesian methods are an excellent fit for this demand, but scaling Bayesian inference is a challenge. In response to this challenge, there has been considerable recent work based on varying assumptions about model structure, underlying computational resources, and the importance of asymptotic correctness. As a result, there is a zoo of ideas with few clear overarching principles. In this paper, we seek to identify unifying principles, patterns, and intuitions for scaling Bayesian inference. We review existing work on utilizing modern computing resources with both MCMC and variational approximation techniques. From this taxonomy of ideas, we characterize the general principles that have proven successful for designing scalable inference procedures and comment on the path forward