Quantification of directed influences in multivariate systems by time-series modeling

Identification and analysis of directed dynamic influences in multivariate systems is of interest in many scientific areas. Data-driven methods for identification of dynamic influences in multivariate systems are generally based on time series modelling. Recently introduced such methods, namely, partial directed coherence (PDC) and directed transfer function (DTF) provide qualitative measures for direct and total influences, respectively. These quantities are, however, based on different normalizations. Consequently, they cannot be used to quantify the indirect influence of a signal on another (through hidden paths), which is essential to provide a complete structural description of a process. The prime intent of this paper is to provide a quantitative measure for direct and indirect influences by treating the multivariate process as a jointly stationary process driven by white-noise innovations. The concepts are illustrated through suitable examples.