# Granger Causality Graph Construction Using Group Lasso

Sergiy Pereverzyev Jr.^{*} and Katerina Schindler

A causality graph consists of quantities of interest (nodes) and the causal (influence) relationships (edges) between them. Detection of such a graph is an important problem that arises in various scientific contexts. Multivariate Granger causality is one of the possibilities to define causal links among quantities using the time dependent observations of the considered quantities. In the case of a large number of quantities and a few causality links among them, the realization of the Granger causality requires the application of some regularization techniques. Recently, as a regularization technique, the so called group Lasso has been shown to be advantageous to detect causal substructures within a graph. In the application of the group Lasso regularization to the construction of the causal graph the following problems arise. First of all, an appropriate choice of the regularization parameter in the group Lasso is needed. Then, the number of the past values of the quantities under investigation, the so called maximal lag of the Granger causality, needs to be chosen. The selection of the maximal lag may have an influence on the detected causalities, so its choice needs to be carefully done. Finally, the application of group Lasso with one regularization parameter suggests that certain properties of the causality graph, such as the number of the so called connected graph components and their sizes, are not well reconstructed. This motivates us to consider an extension of the group Lasso with multiple regularization parameters, where an appropriate choice of the involved regularization parameters becomes especially important. In this talk, the solutions for the mentioned problems will be proposed, and the realization of these will be demonstrated for the problem of the Granger causality graph construction for the so called regulatory interaction networks in biology.

Mathematics Subject Classification: 65S05

Keywords: Granger causality graph; group Lasso; choice of the regularization parameters; multiple regularization parameters; choice of the maximal lag; genes causality graph;

Minisymposion: New Trends in Regularization Theory and Methods for Geomathematical Problems