Abstract
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.
Highlights
The notion of causality has been the concern of thinkers at least since the ancient Greeks [1]
Clive Granger [2], in his paper entitled “Investigating Causal Relations by Econometric Models and Cross-spectral Methods” from 1969, elaborated a mathematical framework to describe a form of causality – called Granger Causality1 (GC) in order to distinguish it from other definitions of causality
This work is organized as follows: in Section 2, we present the concept of an autoregressive process – a model of linear regression in which GC is based
Summary
The notion of causality has been the concern of thinkers at least since the ancient Greeks [1]. Granger was inspired by the definition of causality from Norbert Wiener [3], in which Y causes X if knowing the past of Y increases the efficacy of the prediction of the current state of X(t) when compared to the prediction of X(t) by the past values of X alone. In the multidisciplinary science era, more and more physicists are involved in research in other areas, such as Economics and Neuroscience. These areas usually have big data sets. We close the paper by discussing applications, implications and limitations of the method
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