Unraveling the cause-effect relation between time series

Abstract

Given two time series, can one faithfully tell, in a rigorous and quantitative way, the cause and effect between them? Based on a recently rigorized physical notion, namely, information flow, we solve an inverse problem and give this important and challenging question, which is of interest in a wide variety of disciplines, a positive answer. Here causality is measured by the time rate of information flowing from one series to the other. The resulting formula is tight in form, involving only commonly used statistics, namely, sample covariances; an immediate corollary is that causation implies correlation, but correlation does not imply causation. It has been validated with touchstone linear and nonlinear series, purportedly generated with one-way causality that evades the traditional approaches. It has also been applied successfully to the investigation of real-world problems; an example presented here is the cause-and-effect relation between the two climate modes, El Niño and the Indian Ocean Dipole (IOD), which have been linked to hazards in far-flung regions of the globe. In general, the two modes are mutually causal, but the causality is asymmetric: El Niño tends to stabilize IOD, while IOD functions to make El Niño more uncertain. To El Niño, the information flowing from IOD manifests itself as a propagation of uncertainty from the Indian Ocean.

The seasonal cycle of the Atlantic Jet dynamics in the Alboran Sea: direct atmospheric forcing versus Mediterranean thermohaline circulation

“…Even if the AJ monthly characteristics (flowing speed and direction) seems to be correlated with the monthly values of local wind and SLP (see Fig. 5), it does not necessarily mean that there is a causality link between the different variables (Liang 2014). SLP and wind did have a seasonal cycle (Candela et al 1989;Dorman et al 1995), but from the experiments above, it seems quite clear that those cycles are not responsible for the seasonality in the AJ.…”

<strong>New insights on land surface-atmosphere feedbacks over tropical South America at interannual timescales</strong>

“…We overcame such limitation by using causality analyses to quantify the connectivity between variables. Such causality or information transfer has been defined within the framework of information theory [69,[84][85][86]. Three basic tenets of the information transfer are the following: (1) Causality implies correlation but correlation does not imply causality, (2) Causality implies directionality, which means that the transfer of information detects the direction of information transfer between two systems, and (3) Asymmetry is a basic property of information transfer.…”

“…As previously discussed in the methodology section, the negative sign of  does not affect the amount of causality but the interpretation of the measure. According to [69] Table 4). The main conclusion of these results is that the T2m is essential in capturing the non-linear interactions at interannual timescale within the LAFs of TropSA.…”

“…To estimate the information flow, [69] proposes a system of two stochastic differential equations: T  implies that y acts upon x by augmenting its uncertainty, while a negative sign indicates that y stabilizes x by diminishing its entropy. This type of interpretation is rather confusing because it goes against the ordinary algebraic interpretation.…”

“…14), as: Estimation of simultaneous correlation does not provide information about directionality, and thus the resulting linking graph is called Undirected Graph. Otherwise, by applying bidirectional metrics, such as Causality described in Appendix A [69], it is possible to establish a Directed Graph, denoting directions with arrows ( Figure 3).…”