Lasso Granger Causal Models: Some Strategies and Their Efficiency for Gene Expression Regulatory Networks

Abstract

The detection of causality in gene regulatory networks from experimental data, such as gene expression measurements, is a challenging problem. Granger causality, based on a vector autoregressive model, is one of the most popular methods for uncovering the temporal dependencies between time series, and so it can be used for estimating the causal relationships between the genes in the network. The application of multivariate Granger causality to the networks with a big number of variables (genes) requires a variable selection procedure. For fighting with lack of informative data, the so called regularization procedures are applied. Lasso method is a well known example of such a procedure and the multivariate Granger causality method with the Lasso is called Graphical Lasso Granger method. It is widely accepted that the Graphical Lasso Granger method with an inappropriate parameter setting tends to select too many causal relationships, which leads to spurious results. In our previous work, we proposed a thresholding strategy for Graphical Lasso Granger method, called two-level-thresholding and demonstrated how the variable over-selection of the Graphical Lasso Granger method can be overcome. Thus, an appropriate thresholding, i.e. an appropriate choice of the thresholding parameter, is crucial for the accuracy of the Graphical Lasso Granger method. In this paper, we compare the performance of the Graphical Lasso Granger method with an appropriate thresholding to two other Lasso Granger methods (the regular Lasso Granger method and Copula Granger method) as well as to the method combining ordinary differential equations with dynamic Bayesian Networks. The comparison of the methods is done on the gene expression data of the human cancer cell line for a regulatory network of nineteen selected genes. We test the causal detection ability of these methods with respect to the selected benchmark network and compare the performance of the mentioned methods on various statistical measures. The discussed methods apply a dynamic decision making. They are scalable and can be easily extended to networks with a higher number of genes. In our tests, the best method with respect to the precision and computational cost turns out to be the Graphical Lasso Granger method with two-level-thresholding. Although the discussed algorithms were motivated by problems coming from genetics, they can be also applied to other real-world problems dealing with interactions in a multi-agent system.