Discovering graphical Granger causality using the truncating lasso penalty

Abstract

Motivation: 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.

Results: In this article, 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.

Availability: The proposed truncating lasso method is implemented in the R-package 'grangerTlasso' and is freely available at http://www.stat.lsa.umich.edu/~shojaie/.

Fig. 1.

Mean and standard deviation of performance criteria for lasso, Alasso, Tlasso and TAlasso in estimation of graphical Granger model, with p = 100, n = 50 and d = 2. Top: T = 10, Bottom: T = 20.

Fig. 2.

Images of the adjacency matrix of the true graph, and estimates from lasso, Alasso, Tlasso and TAlasso. Images on the left correspond to the adjacency matrices of graphical Granger models (true and estimates) over time, while images on the right represent the cumulative graphical model (the network structure). In the left panel of the true adjacency matrix, a dark pixel in the (i, j)th entry at time t represents an edge from X^T−t_j to X^T_i. The gray-scale images for the estimates represent percentage of times where an edge is present in 50 simulations. Significant false positives and negatives are marked with rectangles and ovals, respectively.

Fig. 3.

Known transcription regulatory network of E.coli along with estimates based on Alasso, TAlasso and grpLasso. True edges (true positives in estimated networks) are marked with solid black arrows, while false positives are indicated by dashed red arrows.

Fig. 4.

Known BioGRID network of human HeLa cell genes along with the estimates based on TAlasso, grpLasso and CNET. True edges (true positives in estimated networks) are marked with solid black arrows, while false positives are indicated by dashed red arrows.