Suppose we have N time series available where one time-series could be causally dependent on others. For example, such dependence can be found in economic data or weather data. The goal is to recover the directed causality graph that links these time series. As is well known causality and correlation are not the same and thus one of the important questions is how to address this issue. There are several frameworks such as directed information, the notion of Granger causality, etc. However working with directed information requires too much a priori knowledge about the structure of the time series that is unavailable. In this talk I will show how the notion of Granger causality can be tied to Wiener letting that allows us to recover a directed random graph whose edges are represented by the innovations filters. This approach as well as the directed information approach assuming Gaussianity are however quite computationally intensive. To address this issue we show how it is possible to consider a sparse problem based on a mixed L1-H1 norm, a generalized GLASSO approach that takes into account the temporal dependencies that leads to an approach for selecting edges in a directed graph characterized by complex polynomials. This results in a convex optimization problem and modern techniques such as ADMM are well suited to such problems.We then briefly discuss the sparsification problem associated with Granger causality graphs. The basic issue is to find the sub-graph of the causality graph that is closest to the original in terms of a suitable norm.
Bienvenue à tous!