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Some Peculiarities of Causal Analysis of Coupled Chaotic Systems

Anna Krakovská

Institute of Measurement Science, Slovak Academy of Sciences, Bratislava, Slovakia

Email: krakovska@savba.sk

Abstract. On a test example of uni-directionally coupled Rössler systems we demonstrate some

of the pitfalls of causal analysis of chaotic data. The method based on evaluating predictability

in reconstructed state spaces is used here to detect causality. The results show that the pre-

dictability of the driven Rössler system is improved by incorporating information about the

present state of the driver to the prediction process. The predictability improvement correctly

reveals the presence and the direction of the coupling. However, causal analysis of the time-

reversed test signals does not allow to uncover that the cause precedes the effect. In addition,

causal analysis of complex systems may also encounter other complications such as transient

chaos, or irreversibility of dissipative chaos sometimes masked by the dominance of limit cycles.

Keywords: Causality, Rössler System, Chaos, Irreversibility, Arrow of Time

1. Introduction

Detecting causal relations between systems, based on observations in the form of time series,

often referred to the time series predictions. Namely, if the information on the recent past of the

time series xhelps improve the prediction of y, then we say that the xcauses y. The concept was

introduced by Clive Granger in 1969 [1] but the same idea is essential for more recent methods

such as conditional mutual information or transfer entropy [2], [3].

However, to consider Granger causality, separability is required. It means that information

about the driver is expected to be available as an explicit variable. Separability does not apply

to cases of dynamically linked variables with bidirectional links and cyclic information ﬂow. In

such systems, the causes and effects are entangled. This is reﬂected in Takens’ theorem, which

claims that under some conditions the underlying dynamics is reconstructable from any mea-

sured observable [4]. As we have demonstrated in [5], the Granger causality test produces false

positives for time series coming from coupled dynamical systems. Some other techniques, in-

cluding the method of predictability improvement (PI) used in this paper, may be more success-

ful [6]. The PI determines whether a prediction of an observable from Y, made in a reconstructed

state space, improves when observable from system Xis included in the reconstruction. If the

predictability improves, then, analogously to the idea of Granger’s causality of autoregressive

processes, we hypothesize that Xcauses Y. However, Cummins et al. have introduced a com-

prehensive theory showing that using delay reconstruction techniques on deterministic systems,

we have very limited options. The best we can hope for is ﬁnding the strongly connected com-

ponents of the graph (sets of mutually reachable vertices) which represent distinct subsystems

coupled through one-way driving links [7]. We cannot recover self-loops, and, we cannot dis-

tinguish the direct driving, indirect driving, correlate of a direct or indirect driving.

In the next section, the method of causality detection and the example of uni-directionally

coupled Rössler systems are presented. Then the results of the causal analysis are given and the

complications arising from the presence of chaos are discussed.

2. Subject and Methods

In this study, to detect the causality from Xto Ywe used a simpliﬁed version of The Method of

Predictability Improvement introduced in [6]. The method proceeds as follows:

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1. Based on suitable reconstruction parameters dYand τYthe manifold MYis performed

so that the state corresponding to time twould be [y(t),y(t−τY),...,y(t−(dY−1)τY)]. This

provides space to get prediction ˆ

Yof Ywithout using information from X. The one-point pre-

dictions of a large-enough statistical sample of points over the reconstructed trajectory are com-

puted, using the method of analogues [8]. The resulting errors are eY(t)=Y(t)−ˆ

Y(t).

2. To get the predictions of Yusing information from both Xand Y, we reconstruct the

manifold MX+Ycontaining some of the coordinates from MYand some from MX. If we used all

the coordinates, the state in time twould be [y(t),y(t−τY),...,y(t−(dY−1)τY),w.x(t),w.x(t−

τX),...,w.x(t−(dX−1)τX)], where the weight wrepresents the impact of X. Analogously as

in step 1, the predictions and the corresponding errors eX+Y(t)=Y(t)−ˆ

YX+Y(t)are computed.

3. To decide whether the addition of information from Ximproves signiﬁcantly the predic-

tion of Y, i.e. eX+Y<eY, we use the Welch test.

After exchanging the roles of Xand Yabove we get instructions to test causality in the

opposite direction, i.e., from Yto X.

Rössler 1.015 →Rössler 0.985

Our test data comes from the next uni-directionally connected chaotic Rössler oscillators:

˙x1=−ω1x2−x3

˙x2=ω1x1+0.15x2

˙x3=0.2+x3(x1−10)(1)

˙y1=−ω2y2−y3+C(x1−y1)

˙y2=ω2y1+0.15y2

˙y3=0.2+y3(y1−10)

where ω1=1.015,ω2=0.985.

The interconnections of the variables are represented by the interaction graph on Fig. 1.

The variables form a set of nodes connected by directed edges wherever one variable appears

nontrivially on the right-hand side of the equation for the derivative of the second variable. The

two Rössler systems can be seen as distinct dynamical subsystems Xand Ycoupled through

one-way driving of variable y1by x1.

Fig. 1: Interaction graph for the coupling of two Rössler systems (Eq. (1)).

The coupling strength is controlled by the parameter C, with C=0 for uncoupled systems.

Plots of conditional Lyapunov exponents in [2], similarly as correlation dimension estimates in

[9], show that, synchronization takes place between C=0.11 and C=0.13. The direction of

the coupling can only be uncovered before the emergence of synchronization. In the following,

coupling C=0.07 will be used, which is well below the synchronization value. Starting from the

point [0,0,0.4,0,0,0.4], the solutions were generated by Matlab solver ode45 at an integration

step size of 0.1. This gives about 60 samples per one average orbit around the attractor.

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3. Results and Discussion

We performed the causal analysis between Xand Ywith the assumption that we only know time

series x1and y1.

We started with the reconstruction of the trajectories on manifolds MXand MY. Then, 240

one-point predictions of y1were computed, with help of 1200 historical points. Fig. 2 shows

variances of prediction errors for increasing sampling values.

020 40 60

0

1

2

3

4

5

X → Y

sampling

020 40 60

0

1

2

3

4

5

Y → X

sampling

variance of errors

Fig. 2: Causality analysis of the coupled Rössler systems. On the left: Variances of errors of prediction

(blue) and retrodiction (red) of time series x1. Involvement of y1did not signiﬁcantly lower the errors

(dotted lines). On the right: Variances of errors of prediction (blue) and retrodiction (red) of time series

y1. Involvement of x1signiﬁcantly reduced the errors (dotted lines).

For a wide range of sampling, information from x1helped signiﬁcantly with the one-point

prediction of y1(p<0.01). Testing the opposite direction (see the red lines on Fig. 2) showed

no predictability improvement and correctly rejected the causal link of Y→X.

Also, the fractal dimension based method to investigate causal links, introduced in [11],

conﬁrmed that only the relation X→Yis possible here. Correlation dimension of the state por-

trait reconstructed from y1was markedly higher than that of the portrait from x1[9]. Similarly,

models needed for prediction of y1had a higher degree of freedom than the models to predict

x1.

Although the PI method seems to have done its job, causal analysis of chaotic systems can

bring some unexpected pitfalls. Let us begin with the notion of reversibility, meaning that the

process has the same statistical properties as its time reversal. White noise, or linearly correlated

Gaussian processes and conservative (volume preserving) chaotic systems are reversible, while

dissipative chaotic dynamics belongs to irreversible processes. Looking at it from the perspec-

tive of predictions, the predictability of a reversible signal is the same as the predictability of its

time-reversed version, while in case of irreversible signals, differences between the prediction

and the retrodiction suggest that the arrow of time is playing a role here. One should be able to

make predictions of dissipative chaotic trajectories easier if moving forward, if only because of

the divergence of initially nearby trajectories of chaotic systems trajectories and the tendency

to converge from a bigger space to a smaller attracting set (lower uncertainty).

However, although this was not the case with the system being tested, let us mention that,

despite an underlying irreversible evolution, some systems exhibit the properties of time re-

versible (time symmetric) dynamics, because their long term behavior becomes dominated by

limit cycles. And, on the contrary, in some other systems, the attractor itself is simple (a ﬁxed

point or a limit cycle) but its coexistence with an unstable chaotic repeller or chaotic saddle

can cause that the trajectory ﬁrst shows transient chaos, and only after a relatively long time it

settles down on the simple, stable attractor.

Regarding our test example, Fig. 2 clearly demonstrate the irreversibility related to the ex-

ponential divergence of trajectories. Retrodiction errors are signiﬁcantly higher than prediction

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errors. Now, imagine a simple turn of x1and y1over time and repeating the causal analysis.

Considering the ﬁrst principle of Granger causality that the cause precedes the effect one ex-

pects change of the direction of causality from X→Yto Y→X. Indeed, this happens when

causally linked AR models and their time-reversals are analyzed by the Granger test. It might

even look like a good idea to use this "turning test" routinely to conﬁrm the conclusion about

the direction of the causal link. However, Paluš et al. pointed out that the expected change of

the direction of causality did not happen after time reversing of the tested chaotic signals [10].

The authors suggested that the observed paradox is probably related to the dynamic memory of

the systems. If this paradox appears in the numerical analysis of real data, and Paluš et al. gives

such an example in [10], it might indicate dealing with interactions of dynamical systems.

Many scientists in the history including Henri Poincare and in particular the Nobel Prize

winner Ilya Prigogine were thinking about how to make irreversible processes and the arrow of

time a fundamental part of physics. It is becoming increasingly clear that the problem goes to

the core of complex dynamical systems and with the development of this area, problems around

time-reversed processes are becoming more relevant than ever.

Acknowledgements

Supported by the Slovak Grant Agency for Science (Grant 2/0081/19) and by the Slovak Re-

search and Development Agency (Grant APVV-15-0295).

References

[1] Granger, C.W. (1969). Investigating causal relations by econometric models and cross-spectral

methods. Econometrica: Journal of the Econometric Society, 424–438.

[2] Paluš, M., Vejmelka, M. (2007). Directionality of coupling from bivariate time series: How to avoid

false causalities and missed connections. Physical Review E, 75(5), 056211.

[3] Schreiber, T. (2000). Measuring information transfer. Physical Review Letters, 85(2), 461–464.

[4] Takens, F. (2002). Detecting strange attractors in turbulence. In Dynamical Systems and Turbulence.

Rand, D.A. and Young, L.S., Springer-Verlag, Berlin, 366–381.

[5] Krakovská, A., Jakubík, J., Chvosteková, M., Coufal, D., Jajcay, N., Paluš, M. (2018). Comparison

of six methods for the detection of causality in a bivariate time series. Physical Review E, 97(4),

042207.

[6] Krakovská, A., Hanzely, F. (2016). Testing for causality in reconstructed state spaces by an opti-

mized mixed prediction method. Physical Review E, 94(5), 052203.

[7] Cummins, B., Gedeon, T., Spendlove, K. (2015). On the efﬁcacy of state space reconstruction

methods in determining causality. SIAM Journal on Applied Dynamical Systems, 14(1), 335–381.

[8] Lorenz, E.N. (1969). Atmospheric predictability as revealed by naturally occurring analogues. Jour-

nal of the Atmospheric sciences, 26(4), 636–646.

[9] Krakovská, A., Jakubík, J., Budáˇ

cová, H., Holecyová, M. (2015). Causality studied in recon-

structed state space. Examples of uni-directionally connected chaotic systems. arXiv preprint,

arXiv:1511.00505.

[10] Paluš, M., Krakovská, A., Jakubík, J., Chvosteková, M. (2018). Causality, dynamical systems and

the arrow of time. Chaos: An Interdisciplinary Journal of Nonlinear Science, 28(7), 075307.

[11] Krakovská, A., Budáˇ

cová, H. (2013). Interdependence measure based on correlation dimension. In

Proceedings of the 9th International Conference on Measurement. 31–34.

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