Testing for causality in reconstructed state spaces by an optimized mixed prediction method

In this study, a method of causality detection was designed to reveal coupling between dynamical systems represented by time series. The method is based on the predictions in reconstructed state spaces. The results of the proposed method were compared with outcomes of two other methods, the Granger VAR test of causality and the convergent cross-mapping. We used two types of test data. The first test example is a unidirectional connection of chaotic systems of Rössler and Lorenz type. The second one, the fishery model, is an example of two correlated observables without a causal relationship. The results showed that the proposed method of optimized mixed prediction was able to reveal the presence and the direction of coupling and distinguish causality from mere correlation as well.

Figure 1

Rössler system driving Lorenz system (1). 20 000 data points of $x_1$ and $x_2$ variables of the driving Rössler system (left, top); $y_1$ and $y_2$ variables of the Lorenz system itself (right, top); $y_1$ and $y_2$ variables of the driven Lorenz system with coupling strength C = 2 (left, bottom) and C = 4 (right, bottom).

Figure 2

Interaction graph for Rössler system driving Lorenz system (1).

Figure 3

Granger VAR test statistics $S$ estimated in both directions using variables $x_2,y_2$ of the coupled Rössler and Lorenz systems.

Figure 4

Estimates of the CM measure in both directions using variables $x_2,y_2$ of the coupled Rössler and Lorenz systems.

Figure 5

Test for convergence of cross-mappings for Rössler $\to$ Lorenz system, with coupling $C=1$, using variables $x_2,y_2$. For segments of increasing length, the correlation ${\rho }_{X,Y}$ between the two observables is compared to CM measures. Based on this figure, the correct link of $X\to Y$ cannot be unambiguously identified.

Figure 6

Testing whether $X$ (Rössler system) drives $Y$ (Lorenz system) by the OMP method. Variances of residuals of predictions of $y_2$ with or without knowledge of $x_2$ are depicted. Adding information from $X$ improved predictions, indicating that $X\to Y$.

Figure 7

Testing whether $Y$ (Lorenz system) drives $X$ (Rössler system) by the OMP method. Variances of residuals of predictions of $x_2$ with or without knowledge of $y_2$ are depicted. Adding information from $Y$ did not improve predictions, indicating that $Y$ does not drive $X$.

Figure 8

Interaction graph for fishery model (2) of two populations driven by an environmental variable.

Figure 9

Sample of 100 data points from the behavior of populations X and Y of the fishery model.

Figure 10

Convergence of CM investigated in fishery model. The figure indicates that the $X$ and $Y$ are correlated but not causally related.