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MEASUREMENT 2017, Proceedings of the 11th International Conference, Smolenice, Slovakia

39

Predictability Improvement As a Tool to Detect Causality

A. Krakovská

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

Email: krakovska@savba.sk

Abstract. A causality method based on predictions in reconstructed state spaces is presented

and tested in relation to the ability to detect coupling and synchronization of interconnected

dynamical systems. The method is demonstrated on a test example of two uni-directionally

coupled chaotic systems of Rössler type. The results show that the predictability of the driven

system can be improved by incorporating information about past values of the driving system

to the prediction process. This predictability improvement reveals the presence and the

direction of the coupling and may help to detect the onset of full synchronization and distinguish

causality from mere correlation.

Keywords: Causality, Reconstructed State Space, Predictability Improvement, Rössler System

1. Introduction

The study of drive-response relationships between dynamical systems is a topic of active

interest. In this paper, the next type of uni-directional coupling in a bivariate dynamical system

is studied: ܩ(t) = F(X(t)),

ܭ(t) = G(Y(t), X(t)),

where X and Y are the state vectors of the driving and the driven systems. If Y(t) =

<

(X(t)) for

some smooth and invertible function

<

then there is said to be a generalized synchronization

between X and Y. If

<

is an identity, the synchronization is called identical. The direction of

coupling can only be uncovered before the emergence of synchronization. Once the systems

are synchronized, there is a one-to-one relation between their states and the future values of the

driver X can be predicted from the response Y equally well as vice versa.

A first mathematical approach to detect causality has been proposed in 1969 by Clive Granger

[1]. In Granger sense, a time series X is said to cause Y if it can be shown, through statistical

hypothesis tests, that past X values provide significant information about future values of Y. To

be able to consider Granger causality, separability is required. Namely, information about the

causative factor X is expected to be available as an explicit variable. This requirement can be

problematic in a case of dynamically linked variables. Moreover, the initially linear concept

also requires generalizations to enable investigation of complex nonlinear processes. Therefore,

new approaches have been proposed, including extended Granger causality, conditional mutual

information (transfer entropy), cross predictability, measures evaluating distances of

conditioned neighbours in reconstructed state spaces, etc. (see [2] for short review). The

method, used in this paper, reminds the original Granger approach in that it also addresses the

usefulness of past values of the first system for characterizing the second one. However, our

approach belongs to the state-space methods. It determines whether a prediction made in a

reconstructed state space of a time series improves when data from another system are included

in the state space reconstruction. The potential of this idea has been identified in [3], [4].

The paper is organized as follows. In Section 2, the example of uni-directionally coupled

Rössler systems, and the method of causality detection are presented. Then the results of the

causal analysis are given and discussed in Section 3.

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2. Subject and Methods

Data

As a testing example, we use the next uni-directionally coupled non-identical Rössler systems

at different coupling strengths C:

ܪ1 íȦ1x2 í[3

ܪ2 Ȧ1x1 + 0.15x2

ܪ3 = 0.2 + x3(x1 í (1)

ܮ1 íȦ2y2 í\3 + C(x1 í\1)

ܮ2 Ȧ2y1 + 0.72y2

ܮ3 = 0.2 + y3(y1 í

where Ȧ1 = 0.5 and Ȧ2 = 2.515.

The two Rössler systems represent distinct dynamical subsystems coupled through the one-way

driving relationship between variables x1 and y1. Mutual interconnections of the variables may

be represented by the next interaction graph:

20000 data points of x1 and y1 were generated by Matlab solver of ordinary differential

equations ode45 at an integration step 0.3. The starting point was [0, 0, 0.4, 0, 0, 0.4]. First 1000

data points were thrown away. The coupling strength C was chosen from 0 to 1.1 with a step

size of 0.1. The synchronization takes place at the coupling of about 1 [6].

The frequency ratio of about 1:5 used in this example reminds cardio-respiratory interactions,

where the influence of the (slower) respiratory dynamics on the (faster) cardiac dynamics is

larger than in the opposite direction. The system was first used in [5] to show that in this case,

the problem of detecting directionality is much more challenging than in cases of two systems

with more closely related oscillations.

a) b) c)

Fig. 1. a) piece of trajectory corresponding to 100 points of the slow system X (x1, x2, x3 of Eq. 1, C=0)

b) piece of trajectory corresponding to 100 points of the fast system Y (y1, y2, y3 of Eq. 1, C=0)

c) X apparently causes a slowdown of Y: piece of trajectory corresponding to 100 points of “enslaved”

Y (y1, y2, y3 of Eq. 1, C>1)

The Method - Predictability Improvement Using Mixed Prediction

Let the systems X and Y be represented by a time series x and y, respectively, and the task be to

detect causal link from X to Y. Similarly, as in the case of Granger meaning, we say that variable

x causes variable y, if the better prediction of y can be obtained using information from both x

and y rather than using only information from y. However, unlike the Granger's method, ours

is operating in reconstructed multidimensional state spaces. Therefore, as a first step, the dY-

MEASUREMENT 2017, Proceedings of the 11th International Conference, Smolenice, Slovakia

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dimensional manifold MY is reconstructed from lags of observable y so that the state of the

system in time t is [y(t), y(t – ĲY\WíĲY), . . . , y(t – (dY-ĲY)]. From Taken’s theorem [7],

we know that given certain conditions, the reconstructed manifold is diffeomorphic to the

original one. The reconstruction is determined by two parameters, namely the size d of the

reconstructed space and the delay Ĳ. The optimal choice of d and Ĳ requires a task-specific

approach [8]. If the task is related to predicting, as in our case, then it is appropriate to evaluate

the prediction error for various combinations of possible embedding parameters. Consequently,

the lowest errors lead us to proper choices of d and Ĳ.

With regard to the actual way of prediction, the method of analogues was used [9]. This is based

on the finding historical data similar to the current situation and assuming the system will

continue the same way as it did in the past. There are many ways to estimate the follower of the

point y(t), the simplest version being finding the time index i of its nearest neighbour from past

states on the reconstructed trajectory and declaring ǔW= y(i+1). A modification, which was

used in this study, improves the simplest version by averaging followers of several neighbours

while considering exponential weighting based on distances from y(t) for the neighbours.

The predictions in the reconstructed spaces MX and MY were compared to predictions computed

in MX+Y, i.e., in a mixed state space built from delayed observables of both systems. If, for

example, the predictability of Y in MX+Y was better than its predictability in MY, we concluded

that X causally affects Y. A method built on this idea, with special attention attributed to the

optimization of the reconstructed space based on the weighting of each of its coordinates, has

been proposed in [2]. Because of its computational demands, we decided to work with a

simplified variant of the former method in this study.

The specific steps of the causality method used here are as follows:

1. The manifold MY is performed. This provides space to get predictions of y without

using additional information from X. The one point predictions ǔ of large enough

statistical sample of N points over the reconstructed trajectory are computed. The

resulting errors are given by eY(t) = y(t) – ǔW.

2. To get predictions of y using information both from X and Y, the manifold MX+Y

(mixed state space) is reconstructed. The reconstruction contains some of coordinates

from MY and some from MX. If we used the full number of coordinates, the state

corresponding to time t would be [y(t), y(t – ĲY\WíĲY), . . . , y(t – (dY-ĲY), wx(t),

wx(t – ĲXZ[W íĲX), . . . , wx(t – (dX-ĲX)], where the weight w represents the

impact of the system X. Analogously as in step 1, one-point predictions of y are

computed. Let us denote the prediction in time t as ǔX+Y(t) and the corresponding error

as eX+Y(t) = y(t) – ǔX+Y(t).

3. To decide whether the addition of information from system X improves the prediction

of Y (X drives Y), the null hypothesis H0: Var(eX+Y9DUHY) is tested. If H0 is rejected

on a 5% significance level, then we accept that Var(eX+Y) < Var(eY), or equivalently,

that inclusion of knowledge of X significantly improves the prediction of Y .

To get instructions to test causality in the opposite direction, i.e., from Y to X, just replace X by

Y and vice versa in the above three steps.

3. Results and Discussion

Knowing 20000 data-points of variable x1 of the driving Rössler system X and variable y1 of

the responsive Rössler system Y (Eq. 1) we were looking for the causal relationship between

the two systems. First of all, we made reconstructions of the state portraits. The embedding

parameters and the impact w of the system X were individually selected for each coupling

strength. For this, we calculated the prediction errors for several combinations of parameters

and selected the ones that led to the lowest error. As a result, the space MX was reconstructed

MEASUREMENT 2017, Proceedings of the 11th International Conference, Smolenice, Slovakia

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with delay ĲX = 1 and embedding dimension dX = 3, while for MY delay ĲY = 1 or 2 and

embedding dimensions dY = 4, 5, 6, or 7 were optimal (the stronger the coupling, the higher the

so called embedding window ĲYdY [8]). Then the corresponding mixed state spaces MX+Y had

dimensions between 7 and 10. The values of the weights w ranged from 0 to 0.35. In total, 1000

one point predictions were computed, with help of 19000 historical points. Figure 1 shows the

prediction errors for increasing couplings. Obviously, for zero coupling, information from X

does not help in the prediction of Y. It is similar for the couplings above a synchronization level.

The predictability improvement was not significant for the two weakest couplings, but

otherwise the observable from X helps to predict Y (p<0.001). Testing the opposite direction

showed no predictability improvement and rejected the causal link of YĺX.

Fig. 2. Average errors of predictions of y1 (Eq.1)

made in MY (black) and in MX+Y (gray) for

increasing coupling. The results show that

the observable from X helps to predict

observable from Y, referring to causality

;ĺY.

Finally, we conclude that monitoring the predictability improvement in reconstructed mixed

state spaces resulted in a successful detection of the presence and the direction of causal links

even for the complicated case of connection of chaotic systems with slow and fast dynamics.

Acknowledgements

Supported by the Slovak Grant Agency for Science (Grant 2/0011/16) and by the Slovak Research and

Development Agency (Grant APVV–15–0295).

References

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