Granger Causality on forward and Reversed Time Series.

In this study, the information flow time arrow is investigated for stochastic data defined by vector autoregressive models. The time series are analyzed forward and backward by different Granger causality detection methods. Besides the normal distribution, which is usually required for the validity of Granger causality analysis, several other distributions of predictive errors are considered. A clear effect of a change in the order of cause and effect on the time-reversed series of unidirectionally connected variables was detected with standard Granger causality test (GC), when the product of the connection strength and the ratio of the predictive errors of the driver and the recipient was below a certain level, otherwise bidirectional causal connection was detected. On the other hand, opposite causal link was detected unconditionally by the methods based on the time reversal testing, but they were not able to detect correct bidirectional connection. The usefulness of the backward analysis is manifested in cases where falsely detected unidirectional connections can be rejected by applying the result obtained after the time reversal, and in cases of uncorrelated causally independent variables, where the absence of a causal link detected by GC on the original series should be confirmed on the time-reversed series.

Causality, dynamical systems and the arrow of time.

Using several methods for detection of causality in time series, we show in a numerical study that coupled chaotic dynamical systems violate the first principle of Granger causality that the cause precedes the effect. While such a violation can be observed in formal applications of time series analysis methods, it cannot occur in nature, due to the relation between entropy production and temporal irreversibility. The obtained knowledge, however, can help to understand the type of causal relations observed in experimental data, namely, it can help to distinguish linear transfer of time-delayed signals from nonlinear interactions. We illustrate these findings in causality detected in experimental time series from the climate system and mammalian cardio-respiratory interactions.

Comparison of six methods for the detection of causality in a bivariate time series.

In this comparative study, six causality detection methods were compared, namely, the Granger vector autoregressive test, the extended Granger test, the kernel version of the Granger test, the conditional mutual information (transfer entropy), the evaluation of cross mappings between state spaces, and an assessment of predictability improvement due to the use of mixed predictions. Seven test data sets were analyzed: linear coupling of autoregressive models, a unidirectional connection of two Hénon systems, a unidirectional connection of chaotic systems of Rössler and Lorenz type and of two different Rössler systems, an example of bidirectionally connected two-species systems, a fishery model as an example of two correlated observables without a causal relationship, and an example of mediated causality. We tested not only 20000 points long clean time series but also noisy and short variants of the data. The standard and the extended Granger tests worked only for the autoregressive models. The remaining methods were more successful with the more complex test examples, although they differed considerably in their capability to reveal the presence and the direction of coupling and to distinguish causality from mere correlation.