Unraveling the cause-effect relation between time series

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.