In many applications, there are sets of synchronously measured quantities at fixed time intervals termed multivariate time series. Examples of such data sets are the electroencephalograms (EEG, measurements of the brain potential at different locations on the scalp or intracranially) and the financial indices, e.g. indices of world financial markets. The main objective is to understand the mechanism that generates the multivariate time series, that is the underlying complex dynamical system. An important step in this direction is the investigation of the inter-dependence among the observed variables, commonly known with the terms connectivity (stemming from physiology) and Granger causality (stemming from econometrics).
I will present some linear and nonlinear measures of Granger causality, such as the Granger causality index and the transfer entropy. These measures have been modified to capture only direct causal effects in the presence of more than two observed variables. The most recent challenge is in estimating direct Granger causality in the presence of many observed variables, as is the case with many world markets. For this purpose, dimension reduction methods have been proposed and I will present two measures we have developed, one based on linear vector autoregressive models termed restricted conditional Granger causality index (RCGCI) and one based on information theory termed partial mutual information from mixed embedding (PMIME). Any Granger causality measure applied to a multivariate time series determines an undirected or directed connection between two nodes being the observed variables, and in this way networks from multivariate time series can be formed. The study of the networks may reveal structural characteristics of the underlying complex system. I will present these topics with examples from finance.