%0 Journal Article
%T Instantaneous Granger Causality with the Hilbert-Huang Transform
%A Jo£żo Rodrigues
%A Alexandre Andrade
%J ISRN Signal Processing
%D 2013
%R 10.1155/2013/374064
%X Current measures of causality and temporal precedence have limited frequency and time resolution and therefore may not be viable in the detection of short periods of causality in specific frequencies. In addition, the presence of nonstationarities hinders the causality estimation of current techniques as they are based on Fourier transforms or autoregressive model estimation. In this work we present a combination of techniques to measure causality and temporal precedence between stationary and nonstationary time series, that is sensitive to frequency-specific short episodes of causality. This methodology provides a highly informative time-frequency representation of causality with existing causality measures. This is done by decomposing each time series into intrinsic oscillatory modes with an empirical mode decomposition algorithm and, subsequently, calculating their complex Hilbert spectrum. At each time point the cross-spectrum is calculated between time series and used to measure coherency and compute the transfer function and error covariance matrices using the Wilson-Burg method for spectral factorization. The imaginary part of coherency can then be computed as well as several Granger causality measures in the previous matrices. This work covers the most important theoretical background of these techniques and tries to prove the usefulness of this new approach while pointing out some of its qualities and drawbacks. 1. Introduction Identifying interactions of different temporal scales is a recurrent topic in fields such as neuroscience and meteorological or financial research. The concept of functional connectivity, which is defined as the statistical dependence between different variables, is widely used. However, if activity from one variable directly or indirectly exerts influence on other variables, functional connectivity measures will not identify this dependence [1]. This influence is interpreted as the effective connectivity or causal influence, and one solution for this problem in time series inference (TSI) is the concept of causality introduced by Wiener and formulated by Granger [2], the Granger causality (GC) measure. According to the concept of causality, one stochastic process is causal to a second if the autoregressive predictability of the second process at a given time point is improved by including measurements from the past of the first. GC has shown to be suitable for the study of directionality in neuronal interactions by assessment of neurophysiologic data in both the frequency and time domains [3], as well as of simulated
%U http://www.hindawi.com/journals/isrn.signal.processing/2013/374064/