A hierarchical Bayesian model for flexible module discovery in three-way time-series data

Abstract

Motivation: Detecting modules of co-ordinated activity is fundamental in the analysis of large biological studies. For two-dimensional data (e.g. genes × patients), this is often done via clustering or biclustering. More recently, studies monitoring patients over time have added another dimension. Analysis is much more challenging in this case, especially when time measurements are not synchronized. New methods that can analyze three-way data are thus needed.

Results: We present a new algorithm for finding coherent and flexible modules in three-way data. Our method can identify both core modules that appear in multiple patients and patient-specific augmentations of these core modules that contain additional genes. Our algorithm is based on a hierarchical Bayesian data model and Gibbs sampling. The algorithm outperforms extant methods on simulated and on real data. The method successfully dissected key components of septic shock response from time series measurements of gene expression. Detected patient-specific module augmentations were informative for disease outcome. In analyzing brain functional magnetic resonance imaging time series of subjects at rest, it detected the pertinent brain regions involved.

Availability and implementation: R code and data are available at http://acgt.cs.tau.ac.il/twigs/.

Fig. 1.

Overview of the model. (A) A toy example of a core module (A) and its private modules (B, C). (B) An overview of the dependencies in the hierarchical model. P is the vector of subject-specific probabilities P_s

Fig. 2.

Simulation results for data with a single module. Each bar represents the average over 10 repeats. (A) Case 1: no subject-specific signal. (B) Case 2: with subject-specific signals. The Bimax-Gibbs variant was later chosen as the default TWIGS algorithm

Fig. 3.

Simulation results for data with five core modules. Each bar represents the average over 10 repeats. (A) Binary data. (B) Normal data. The Bimax-Gibbs-masker variant was later chosen as the default TWIGS algorithm

Fig. 4.

A module summarizing patient response to sepsis. Top: the first core module heatmap. Bottom: the subject-specific enrichments. The red stripes in each patient’s node represent the time points that were covered by its private module. An edge between a subject and a category (blue node) indicates that the subject-specific module was enriched for that category

Fig. 5.

Results of the fMRI analysis. (A) The core module rows of the solution with ${\mu }_1=2$. (B) The core module rows of the solution with ${\mu }_1=1.5$. (C, D) Examples of subject-specific statistics. This example shows the results for core module 4B. (C) The percent of core module parcels covered by the private modules. Asterisks indicate subjects whose private module had a significant overlap (hyper-geometric $P\le 0.001$) with the core module. (D) The number of time points in each private module