The human microbiome, which plays a key role in health and disease, consists of a dynamic community of microorganisms. There is a keen interest in understanding interactions among these microbes, and how these relations change over time. However, current methods for microbiome network inference exist only for a single time point. We propose a novel method to jointly estimate time-varying network associations for microbiome data, which encourages edge similarity across a neighborhood of time points. To account for the compositional constraint and zero-inflation that typify microbiome data sets, we utilize a modified centered-log ratio transformation, then use a truncated Gaussian copula model to estimate the covariance matrices at each time point. We also propose an extension of this method to analyze multi-site or multi-domain microbiome data. We compare the performance of our method to existing alternative approaches on simulated data and apply the proposed method to learn cross-site dynamic networks based on oral and stool microbiome samples collected from leukemic patients during the course of cancer treatment.

In the first project, we encountered challenges in selecting reasonably sparse models using traditional model selection criteria. While AIC and BIC, two of the most popular model selection information criteria, attempt to balance model fit and sparsity, selected models still tend to be very dense. Other existing approaches were not well suited to handle the selection of multiple hyperparameters to satisfy multiple objectives. We therefore propose multi-objective optimization to allow the user to filter through the model trade-offs to achieve a more desirable model. In this method, we allow for simultaneous hyperparameter tuning, rather than performing a more traditional grid search. In this project, we focus on its use in the selection of both single and joint graphical models, but we note that this method can be generalized to a wide variety of statistical models where competing objectives, such as sparsity or smoothness and fit, need to be optimized. We demonstrate its use for model selection for both the graphical lasso and the joint graphical lasso.