  Physicists are starting to work in areas where noisy signal analysis is
required. In these fields, such as Economics, Neuroscience, and Physics, the
notion of causality should be interpreted as a statistical measure. We
introduce to the lay reader the Granger causality between two time series and
illustrate ways of calculating it: a signal $X$ ``Granger-causes'' a signal $Y$
if the observation of the past of $X$ increases the predictability of the
future of $Y$ when compared to the same prediction done with the past of $Y$
alone. In other words, for Granger causality between two quantities it suffices
that information extracted from the past of one of them improves the forecast
of the future of the other, even in the absence of any physical mechanism of
interaction. We present derivations of the Granger causality measure in the
time and frequency domains and give numerical examples using a non-parametric
estimation method in the frequency domain. Parametric methods are addressed in
the Appendix. We discuss the limitations and applications of this method and
other alternatives to measure causality.
