Bayesian sparse reduced rank multivariate regression

TENSOR REGRESSION FOR INCOMPLETE OBSERVATIONS WITH APPLICATION TO LONGITUDINAL STUDIES

Multivariate longitudinal data are frequently encountered in practice such as in our motivating longitudinal microbiome study. It is of general interest to associate such high-dimensional, longitudinal measures with some univariate continuous outcome. However, incomplete observations are common in a regular study design, as not all samples are measured at every time point, giving rise to the so-called blockwise missing values. Such missing structure imposes significant challenges for association analysis and defies many existing methods that require complete samples. In this paper we propose to represent multivariate longitudinal data as a three-way tensor array (i.e., sample-by-feature-by-time) and exploit a parsimonious scalar-on-tensor regression model for association analysis. We develop a regularized covariance-based estimation procedure that effectively leverages all available observations without imputation. The method achieves variable selection and smooth estimation of time-varying effects. The application to the motivating microbiome study reveals interesting links between the preterm infant's gut microbiome dynamics and their neurodevelopment. Additional numerical studies on synthetic data and a longitudinal aging study further demonstrate the efficacy of the proposed method.

A fully Bayesian approach to sparse reduced-rank multivariate regression

In the context of high-dimensional multivariate linear regression, sparse reduced-rank regression (SRRR) provides a way to handle both variable selection and low-rank estimation problems. Although there has been extensive research on SRRR, statistical inference procedures that deal with the uncertainty due to variable selection and rank reduction are still limited. To fill this research gap, we develop a fully Bayesian approach to SRRR. A major difficulty that occurs in a fully Bayesian framework is that the dimension of parameter space varies with the selected variables and the reduced-rank. Due to the varying-dimensional problems, traditional Markov chain Monte Carlo (MCMC) methods such as Gibbs sampler and Metropolis-Hastings algorithm are inapplicable in our Bayesian framework. To address this issue, we propose a new posterior computation procedure based on the Laplace approximation within the collapsed Gibbs sampler. A key feature of our fully Bayesian method is that the model uncertainty is automatically integrated out by the proposed MCMC computation. The proposed method is examined via simulation study and real data analysis.

High Dimensional Bayesian Regularization in Regressions Involving Symmetric Tensors

This article develops a regression framework with a symmetric tensor response and vector predictors. The existing literature involving symmetric tensor response and vector predictors proceeds by vectorizing the tensor response to a multivariate vector, thus ignoring the structural information in the tensor. A few recent approaches have proposed novel regression frameworks exploiting the structure of the symmetric tensor and assume symmetric tensor coefficients corresponding to scalar predictors to be low-rank. Although low-rank constraint on coefficient tensors are computationally efficient, they might appear to be restrictive in some real data applications. Motivated by this, we propose a novel class of regularization or shrinkage priors for the symmetric tensor coefficients. Our modeling framework a-priori expresses a symmetric tensor coefficient as sum of low rank and sparse structures, with both these structures being suitably regularized using Bayesian regularization techniques. The proposed framework allows identification of tensor nodes significantly influenced by each scalar predictor. Our framework is implemented using an efficient Markov Chain Monte Carlo algorithm. Empirical results in simulation studies show competitive performance of the proposed approach over its competitors.

Microbiome Multi-Omics Network Analysis: Statistical Considerations, Limitations, and Opportunities

The advent of large-scale microbiome studies affords newfound analytical opportunities to understand how these communities of microbes operate and relate to their environment. However, the analytical methodology needed to model microbiome data and integrate them with other data constructs remains nascent. This emergent analytical toolset frequently ports over techniques developed in other multi-omics investigations, especially the growing array of statistical and computational techniques for integrating and representing data through networks. While network analysis has emerged as a powerful approach to modeling microbiome data, oftentimes by integrating these data with other types of omics data to discern their functional linkages, it is not always evident if the statistical details of the approach being applied are consistent with the assumptions of microbiome data or how they impact data interpretation. In this review, we overview some of the most important network methods for integrative analysis, with an emphasis on methods that have been applied or have great potential to be applied to the analysis of multi-omics integration of microbiome data. We compare advantages and disadvantages of various statistical tools, assess their applicability to microbiome data, and discuss their biological interpretability. We also highlight on-going statistical challenges and opportunities for integrative network analysis of microbiome data.

SOFAR: Large-Scale Association Network Learning

Many modern big data applications feature large scale in both numbers of responses and predictors. Better statistical efficiency and scientific insights can be enabled by understanding the large-scale response-predictor association network structures via layers of sparse latent factors ranked by importance. Yet sparsity and orthogonality have been two largely incompatible goals. To accommodate both features, in this paper we suggest the method of sparse orthogonal factor regression (SOFAR) via the sparse singular value decomposition with orthogonality constrained optimization to learn the underlying association networks, with broad applications to both unsupervised and supervised learning tasks such as biclustering with sparse singular value decomposition, sparse principal component analysis, sparse factor analysis, and spare vector autoregression analysis. Exploiting the framework of convexity-assisted nonconvex optimization, we derive nonasymptotic error bounds for the suggested procedure characterizing the theoretical advantages. The statistical guarantees are powered by an efficient SOFAR algorithm with convergence property. Both computational and theoretical advantages of our procedure are demonstrated with several simulations and real data examples.