Estimation of multiple networks in Gaussian mixture models

Estimation of multiple networks with common structures in heterogeneous subgroups

Network estimation has been a critical component of high-dimensional data analysis and can provide an understanding of the underlying complex dependence structures. Among the existing studies, Gaussian graphical models have been highly popular. However, they still have limitations due to the homogeneous distribution assumption and the fact that they are only applicable to small-scale data. For example, cancers have various levels of unknown heterogeneity, and biological networks, which include thousands of molecular components, often differ across subgroups while also sharing some commonalities. In this article, we propose a new joint estimation approach for multiple networks with unknown sample heterogeneity, by decomposing the Gaussian graphical model (GGM) into a collection of sparse regression problems. A reparameterization technique and a composite minimax concave penalty are introduced to effectively accommodate the specific and common information across the networks of multiple subgroups, making the proposed estimator significantly advancing from the existing heterogeneity network analysis based on the regularized likelihood of GGM directly and enjoying scale-invariant, tuning-insensitive, and optimization convexity properties. The proposed analysis can be effectively realized using parallel computing. The estimation and selection consistency properties are rigorously established. The proposed approach allows the theoretical studies to focus on independent network estimation only and has the significant advantage of being both theoretically and computationally applicable to large-scale data. Extensive numerical experiments with simulated data and the TCGA breast cancer data demonstrate the prominent performance of the proposed approach in both subgroup and network identifications.

Simultaneous clustering and estimation of networks in multiple graphical models

Gaussian graphical models are widely used to study the dependence structure among variables. When samples are obtained from multiple conditions or populations, joint analysis of multiple graphical models are desired due to their capacity to borrow strength across populations. Nonetheless, existing methods often overlook the varying levels of similarity between populations, leading to unsatisfactory results. Moreover, in many applications, learning the population-level clustering structure itself is of particular interest. In this article, we develop a novel method, called Simultaneous Clustering and Estimation of Networks via Tensor decomposition (SCENT), that simultaneously clusters and estimates graphical models from multiple populations. Precision matrices from different populations are uniquely organized as a three-way tensor array, and a low-rank sparse model is proposed for joint population clustering and network estimation. We develop a penalized likelihood method and an augmented Lagrangian algorithm for model fitting. We also establish the clustering accuracy and norm consistency of the estimated precision matrices. We demonstrate the efficacy of the proposed method with comprehensive simulation studies. The application to the Genotype-Tissue Expression multi-tissue gene expression data provides important insights into tissue clustering and gene coexpression patterns in multiple brain tissues.

Simultaneous cluster structure learning and estimation of heterogeneous graphs for matrix-variate fMRI data

Graphical models play an important role in neuroscience studies, particularly in brain connectivity analysis. Typically, observations/samples are from several heterogenous groups and the group membership of each observation/sample is unavailable, which poses a great challenge for graph structure learning. In this paper, we propose a method which can achieve Simultaneous Clustering and Estimation of Heterogeneous Graphs (briefly denoted as SCEHG) for matrix-variate functional magnetic resonance imaging (fMRI) data. Unlike the conventional clustering methods which rely on the mean differences of various groups, the proposed SCEHG method fully exploits the group differences of conditional dependence relationships among brain regions for learning cluster structure. In essence, by constructing individual-level between-region network measures, we formulate clustering as penalized regression with grouping and sparsity pursuit, which transforms the unsupervised learning into supervised learning. A modified difference of convex programming with the alternating direction method of multipliers (DC-ADMM) algorithm is proposed to solve the corresponding optimization problem. We also propose a generalized criterion to specify the number of clusters. Extensive simulation studies illustrate the superiority of the SCEHG method over some state-of-the-art methods in terms of both clustering and graph recovery accuracy. We also apply the SCEHG procedure to analyze fMRI data associated with attention-deficit hyperactivity disorder (ADHD), which illustrates its empirical usefulness.

Inferring cell diversity in single cell data using consortium-scale epigenetic data as a biological anchor for cell identity

Methods for cell clustering and gene expression from single-cell RNA sequencing (scRNA-seq) data are essential for biological interpretation of cell processes. Here, we present TRIAGE-Cluster which uses genome-wide epigenetic data from diverse bio-samples to identify genes demarcating cell diversity in scRNA-seq data. By integrating patterns of repressive chromatin deposited across diverse cell types with weighted density estimation, TRIAGE-Cluster determines cell type clusters in a 2D UMAP space. We then present TRIAGE-ParseR, a machine learning method which evaluates gene expression rank lists to define gene groups governing the identity and function of cell types. We demonstrate the utility of this two-step approach using atlases of in vivo and in vitro cell diversification and organogenesis. We also provide a web accessible dashboard for analysis and download of data and software. Collectively, genome-wide epigenetic repression provides a versatile strategy to define cell diversity and study gene regulation of scRNA-seq data.

Combining heterogeneous subgroups with graph-structured variable selection priors for Cox regression

Background

Important objectives in cancer research are the prediction of a patient’s risk based on molecular measurements such as gene expression data and the identification of new prognostic biomarkers (e.g. genes). In clinical practice, this is often challenging because patient cohorts are typically small and can be heterogeneous. In classical subgroup analysis, a separate prediction model is fitted using only the data of one specific cohort. However, this can lead to a loss of power when the sample size is small. Simple pooling of all cohorts, on the other hand, can lead to biased results, especially when the cohorts are heterogeneous.

Results

We propose a new Bayesian approach suitable for continuous molecular measurements and survival outcome that identifies the important predictors and provides a separate risk prediction model for each cohort. It allows sharing information between cohorts to increase power by assuming a graph linking predictors within and across different cohorts. The graph helps to identify pathways of functionally related genes and genes that are simultaneously prognostic in different cohorts.

Conclusions

Results demonstrate that our proposed approach is superior to the standard approaches in terms of prediction performance and increased power in variable selection when the sample size is small.

Supplementary Information

The online version contains supplementary material available at 10.1186/s12859-021-04483-z.

HeteroGGM: an R package for Gaussian graphical model-based heterogeneity analysis

SUMMARY

Heterogeneity is a hallmark of many complex human diseases, and unsupervised heterogeneity analysis has been extensively conducted using high-throughput molecular measurements and histopathological imaging features. 'Classic' heterogeneity analysis has been based on simple statistics such as mean, variance and correlation. Network-based analysis takes interconnections as well as individual variable properties into consideration and can be more informative. Several Gaussian graphical model (GGM)-based heterogeneity analysis techniques have been developed, but friendly and portable software is still lacking. To facilitate more extensive usage, we develop the R package HeteroGGM, which conducts GGM-based heterogeneity analysis using the advanced penaliztaion techniques, can provide informative summary and graphical presentation, and is efficient and friendly.

AVAILABILITYAND IMPLEMENTATION

The package is available at https://CRAN.R-project.org/package=HeteroGGM.

SUPPLEMENTARY INFORMATION

Supplementary data are available at Bioinformatics online.

Penalized model-based clustering of fMRI data

Functional magnetic resonance imaging (fMRI) data have become increasingly available and are useful for describing functional connectivity (FC), the relatedness of neuronal activity in regions of the brain. This FC of the brain provides insight into certain neurodegenerative diseases and psychiatric disorders, and thus is of clinical importance. To help inform physicians regarding patient diagnoses, unsupervised clustering of subjects based on FC is desired, allowing the data to inform us of groupings of patients based on shared features of connectivity. Since heterogeneity in FC is present even between patients within the same group, it is important to allow subject-level differences in connectivity, while still pooling information across patients within each group to describe group-level FC. To this end, we propose a random covariance clustering model (RCCM) to concurrently cluster subjects based on their FC networks, estimate the unique FC networks of each subject, and to infer shared network features. Although current methods exist for estimating FC or clustering subjects using fMRI data, our novel contribution is to cluster or group subjects based on similar FC of the brain while simultaneously providing group- and subject-level FC network estimates. The competitive performance of RCCM relative to other methods is demonstrated through simulations in various settings, achieving both improved clustering of subjects and estimation of FC networks. Utility of the proposed method is demonstrated with application to a resting-state fMRI data set collected on 43 healthy controls and 61 participants diagnosed with schizophrenia.

Pursuing sources of heterogeneity in modeling clustered population

Researchers often have to deal with heterogeneous population with mixed regression relationships, increasingly so in the era of data explosion. In such problems, when there are many candidate predictors, it is not only of interest to identify the predictors that are associated with the outcome, but also to distinguish the true sources of heterogeneity, that is, to identify the predictors that have different effects among the clusters and thus are the true contributors to the formation of the clusters. We clarify the concepts of the source of heterogeneity that account for potential scale differences of the clusters and propose a regularized finite mixture effects regression to achieve heterogeneity pursuit and feature selection simultaneously. We develop an efficient algorithm and show that our approach can achieve both estimation and selection consistency. Simulation studies further demonstrate the effectiveness of our method under various practical scenarios. Three applications are presented, namely, an imaging genetics study for linking genetic factors and brain neuroimaging traits in Alzheimer's disease, a public health study for exploring the association between suicide risk among adolescents and their school district characteristics, and a sport analytics study for understanding how the salary levels of baseball players are associated with their performance and contractual status.

Gaussian graphical model-based heterogeneity analysis via penalized fusion

Heterogeneity is a hallmark of cancer, diabetes, cardiovascular diseases, and many other complex diseases. This study has been partly motivated by the unsupervised heterogeneity analysis for complex diseases based on molecular and imaging data, for which, network-based analysis, by accommodating the interconnections among variables, can be more informative than that limited to mean, variance, and other simple distributional properties. In the literature, there has been very limited research on network-based heterogeneity analysis, and a common limitation shared by the existing techniques is that the number of subgroups needs to be specified a priori or in an ad hoc manner. In this article, we develop a penalized fusion approach for heterogeneity analysis based on the Gaussian graphical model. It applies penalization to the mean and precision matrix parameters to generate regularized and interpretable estimates. More importantly, a fusion penalty is imposed to "automatedly" determine the number of subgroups and generate more concise, reliable, and interpretable estimation. Consistency properties are rigorously established, and an effective computational algorithm is developed. The heterogeneity analysis of non-small-cell lung cancer based on single-cell gene expression data of the Wnt pathway and that of lung adenocarcinoma based on histopathological imaging data not only demonstrate the practical applicability of the proposed approach but also lead to interesting new findings.

Simultaneous Clustering and Estimation of Heterogeneous Graphical Models

We consider joint estimation of multiple graphical models arising from heterogeneous and high-dimensional observations. Unlike most previous approaches which assume that the cluster structure is given in advance, an appealing feature of our method is to learn cluster structure while estimating heterogeneous graphical models. This is achieved via a high dimensional version of Expectation Conditional Maximization (ECM) algorithm (Meng and Rubin, 1993). A joint graphical lasso penalty is imposed on the conditional maximization step to extract both homogeneity and heterogeneity components across all clusters. Our algorithm is computationally efficient due to fast sparse learning routines and can be implemented without unsupervised learning knowledge. The superior performance of our method is demonstrated by extensive experiments and its application to a Glioblastoma cancer dataset reveals some new insights in understanding the Glioblastoma cancer. In theory, a non-asymptotic error bound is established for the output directly from our high dimensional ECM algorithm, and it consists of two quantities: statistical error (statistical accuracy) and optimization error (computational complexity). Such a result gives a theoretical guideline in terminating our ECM iterations.