Implementation of two causal methods based on predictions in reconstructed state spaces

If deterministic dynamics is dominant in the data, then methods based on predictions in reconstructed state spaces can serve to detect causal relationships between and within the systems. Here we introduce two algorithms for such causal analysis. They are designed to detect causality from two time series but are potentially also applicable in a multivariate context. The first method is based on cross-predictions, and the second one on the so-called mixed predictions. In terms of performance, the cross-prediction method is considerably faster and less prone to false negatives. The predictability improvement method is slower, but in addition to causal detection, in a multivariate scenario, it also reveals which specific observables can help the most if we want to improve prediction. The study also highlights cases where our methods and state-space approaches generally seem to lose reliability. We propose a new perspective on these situations, namely that the variables under investigation have weak observability due to the complex nonlinear information flow in the system. Thus, in such cases, the failure of causality detection cannot be attributed to the methods themselves but to the use of data that do not allow reliable reconstruction of the underlying dynamics.

Figure 1

Presentation of the test data. One hundred point samples of variables of Rössler systems in Eq. (1) coupled with the strength of $C=0.05$. Followed by three-dimensional plots of 3600 points given by variables from the driver $X$ and variables from the driven $Y$. The plots at the bottom are for the data with added noise (signal-to-noise ratio of 10).

Figure 2

Fluence graph of the coupled Rössler systems given by Eq. (1). The situation for interconnected systems with coupling $C$ below the synchronization threshold. Dashed arrows symbolize nonlinearity in the connections.

Figure 3

Results of the CP method. Normalized mean absolute errors of self-predictions and cross-predictions for four pairs of variables of coupled Rössler systems given by Eq. (1). In each case, the results are given for 19 increasing values of the coupling strength $C$. Pictures on the left are for clean data, while pictures on the right are for data with a signal-to-noise ratio of 10. The figures also show concrete conclusions of the method about the type of causality, detected in individual cases. Red is used for symbols of incorrect detection results. Black is for correct detections.

Figure 4

Results of the PI method. Mean absolute errors of self-predictions and mixed predictions for four pairs of variables of coupled Rössler systems given by Eq. (1). As with the previous figure, red is used for symbols of incorrect detection results and black for correct detections.

Figure 5

Connections detected by the CP method in coupled Rössler systems given by Eq. (1) for coupling value of $C=0.05$. The method found all the links it should have found.

Figure 6

Connections detected by the PI method in coupled Rössler systems given by Eq. (1) for coupling value of $C=0.05$. The dotted links remained undetected.

Figure 7

Cross-map skill $\rho$ for four pairs of variables of coupled Rössler systems given by Eq. (1) for increasing coupling values. The amount of 3700 data points was used for the computations. Embedding dimension within the range of 4 to 11 and time delay 1 was used for the reconstruction. For each coupling, a boxplot provides results for 100 surrogate time series. The line with dots (red) shows the result for the causal effect of the first variable from the title to the second. The line without markers represents the opposite direction. Pictures on the left are for clean data, while pictures on the right are for data with a signal-to-noise ratio of 10. The symbols at the bottom of each figure indicate concrete conclusions of the method about the type of causality. Red is used for symbols of incorrect detection results. Black is for correct detections.

Figure 8

Causal analysis for four pairs of variables of coupled Rössler systems given by Eq. (1). The difference compared to the previous figure is in the use of the $L$-measure [7] for the evaluation of the cross-mapping.