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We discuss some problems encountered in inference of directionality of coupling, or, in the case of two interacting systems, in inference of causality from bivariate time series. We identify factors and in°uences which can lead to either decreased test sensitivity or to false detections and propose ways how to cope with them in order to perform tests with high sensitivity and low rate of false positive results. ### 双变量时间序列中的耦合方向性：避免虚假因果关系与遗漏连接 #### 引言近年来，耦合复杂系统之间的协同行为引起了理论家和实验家们的广泛关注（参考文献[1]）。不仅在物理领域，在许多生物系统中也观察到了同步和其他相关现象，如心血管呼吸交互作用[2-5]以及神经信号的同步[6-10]。在这些系统中，不仅要检测同步状态，还要识别研究系统间的驱动-响应关系。这一问题实际上是更广泛的因果关系或系统、过程或现象间因果联系问题的一个特例。Wiener[11]给出了因果关系在可测量的预测性术语中的数学表述。Granger[12]通过评估双变量自回归模型中的预测性引入了特定的因果关系概念到时间序列分析中。这种线性的因果关系度量和测试框架被广泛应用于经济学和金融学（参考文献[13]），同时也应用于其他科学领域，例如气候学（参见文献[14]及其引用的文献）和神经生理学，在后者中通过将Granger因果关系概念推广到多变量情况来解决多通道脑电图记录的具体问题[15,16]。然而，Granger因果关系概念仅限于线性关系，这限制了其在非线性相互作用系统中的应用。 #### 双变量时间序列中的因果关系推理双变量时间序列是指两个相互作用系统的观测数据集。从这些时间序列中推断出因果关系或耦合的方向性是一项挑战性的任务。在实际应用中，会遇到以下几个关键问题： 1. **降低测试敏感性**：这通常发生在噪声水平较高或信号质量不佳的情况下，导致难以准确区分真实效应和随机波动。 2. **虚假检测**：即错误地认为存在因果关系，但实际上并不存在。这可能是由于模型假设不当、统计检验方法不恰当等原因造成的。 #### 避免虚假因果性和遗漏连接的方法为了提高测试的敏感性和减少虚假阳性结果的发生率，本文提出了几种策略： 1. **增强信号处理技术**：采用先进的信号处理技术可以有效减少噪声的影响，提高信号的质量。例如，使用滤波技术去除无关噪声、采用复杂的模型拟合方法等。 2. **非线性方法的应用**：鉴于大多数实际系统都表现出非线性特性，采用非线性分析方法对于准确推断因果关系至关重要。例如，使用基于信息论的方法（如互信息、条件互信息）或基于相空间重构的技术。 3. **改进的统计检验**：传统的统计检验方法可能无法充分考虑到时间序列数据的特点。因此，开发新的统计检验方法是必要的，以确保测试结果的可靠性和有效性。 4. **交叉验证**：通过将数据分为训练集和测试集，并对模型进行交叉验证，可以有效地评估模型的泛化能力，从而减少过拟合的风险。 5. **多尺度分析**：考虑到系统的复杂性和多尺度特征，采用多尺度分析方法可以帮助揭示不同时间尺度上的因果关系模式。 6. **模型选择与比较**：合理选择模型，并通过比较不同模型的性能来确定最佳模型，这对于准确识别因果关系非常重要。 #### 结论本文讨论了从双变量时间序列推断耦合方向性或因果关系时可能遇到的问题，并提出了一系列应对策略，以提高测试的敏感性和降低虚假阳性结果的发生率。通过综合运用先进的信号处理技术、非线性分析方法、改进的统计检验以及其他策略，可以更准确地推断出两个相互作用系统之间的因果关系。这对于理解和解释自然界中广泛存在的复杂动力学系统具有重要意义。

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Directionality of coupling from bivariate time series:

How to avoid false causalities and missed connections

Milan Paluˇs

∗

and Martin Vejmelka

†

Institute of Computer Science, Academy of Sciences of the Czech Republic,

Pod vod´arenskou vˇeˇz´ı 2, 182 07 Prague 8, Czech Republic

(Dated: October 16, 2007)

We discuss some problems encountered in inference of directionality of coupling, or, in the case

of two interacting systems, in inference of causality from bivariate time series. We identify factors

and inﬂuences which can lead to either decreased test sensitivity or to false detections and propose

ways how to cope with them in order to perform tests with high sensitivity and low rate of false

positive results.

Reference: Phys. Rev. E 75, 056211, 2007

PACS numbers: 05.45.Tp, 05.45.Xt, 02.50.Ey, 87.80.Tq

I. INTRODUCTION

Cooperative behavior of coupled complex systems has

recently attracted considerable interest from theoreti-

cians as well as experimentalists (see e.g. the monograph

[1]), since synchronization and related phenomena have

been observed not only in physical, but also in many

biological systems. Examples include cardio-respiratory

interactions [2–5] and synchronization of neural signals

[6–10]. In such systems it is not only important to detect

synchronized states, but also to identify drive-response

relationships between the systems studied. This problem

is a special case of the general question of causality or

causal relations between systems, processes or phenom-

ena. The mathematical formulation of causality in mea-

surable terms of predictability was given by Wiener [11].

Granger [12] introduced a speciﬁc notion of causality into

time series analysis by evaluation of predictability in bi-

variate autoregressive models. This linear framework for

measuring and testing causality has been widely applied

in economy and ﬁnance (see Geweke [13] for a comprehen-

sive survey of the literature), but also in diﬀerent sciences

such as climatology (see [14] and references therein) or

neurophysiology, where speciﬁc problems of multichan-

nel electroencephalogram recordings were solved by gen-

eralizing the Granger causality concept to multivariate

cases [15, 16]. Nevertheless, the limitation of the Granger

causality concept to linear relations required further gen-

eralizations which emerged especially in the intensively

developing ﬁeld of synchronization of complex systems.

Considering the task of identiﬁcation of drive-response

relationships, a number of asymmetric dependence mea-

sures have been proposed [6, 7, 9, 10, 17–22] and ap-

plied in diverse scientiﬁc areas such as laser physics [23],

∗

Electronic address: [email protected]

†

Also at Department of Cybernetics, Faculty of Electrical En-

gineering, Czech Technical University, Karlovo n´amˇest´ı 13, 121

35 Praha 2 - Nov´e Mˇesto, Czech Republic; Electronic address:

[email protected]

climatology [24, 25], cardiovascular physiology [22, 24],

neurophysiology [6, 7, 9, 10, 26–28], or ﬁnance [29]. In

spite of these wide-spread applications of various cou-

pling asymmetry measures, the task of correct inference

of coupling asymmetry, i.e., the identiﬁcation of the driv-

ing and driven systems from experimental time series is

far from resolved. In this paper we identify some prob-

lems encountered in this task and give some practical

advices how to avoid false detections of coupling asym-

metry or causality. We will consider two interacting sys-

tems, possibly one of them driving the other. Then the

coupling asymmetry, or, as it is called, the directional-

ity of coupling, also identiﬁes causality, or causal rela-

tions between the studied systems. The problem of dis-

tinguishing the true causality from indirect inﬂuences in

interactions of three or more systems is beyond the scope

of this paper and will be addressed elsewhere.

In Section II we introduce three examples of unidirec-

tionally coupled chaotic systems and analyze their cou-

pling using three already published measures. In this

way we demonstrate the importance of choice of an ap-

propriate measure with known properties and a solid

mathematical background. In Section III we review basic

measures deﬁned in information theory and specify ap-

plications of conditional mutual information (CMI) for

detection of causality. Section IV introduces multidi-

mensional conditional mutual information applicable to

amplitudes of dynamical systems or stochastic processes

and a version of CMI for evaluation of coupling asym-

metry using instantaneous phases of coupled oscillatory

systems. Then, in Sec. V we study bias and variance

of CMI estimates and discuss statistical evaluation of

the estimated CMI in order to assure correct inference

of causality/coupling asymmetry from experimental time

series. Further factors inﬂuencing the bias in the CMI

estimates are discussed in Sec. VI, where also an exam-

ple of assessing direction of coupling in cardio-respiratory

interaction is presented. The discussed topics are sum-

marized and conclusions given in Sec. VII. Finally, Ap-

pendix A proves the equivalence of the conditional mu-

tual information and the transfer entropy introduced by

Schreiber [18].

II. ASYMMETRY IN COUPLING: SYSTEMS

AND MEASURES

As the ﬁrst example, let us consider the unidirection-

ally coupled R¨ossler and Lorenz systems, also studied in

Refs. [7, 9, 19], described by the equations

˙x

= −α{x

+ x

}

˙x

= α{x

+ 0.2 x

} (1)

˙x

= α{0.2 + x

− 5.7)}

for the autonomous R¨ossler system, and

˙y

= 10(−y

+ y

)

˙y

= 28 y

− y

+ ² x

(2)

˙y

= y

−

for the driven Lorenz system in which the equation for

˙y

is augmented by a driving term involving x

. We will

analyze the case with α = 6 and β = 2.

For the second example we will use the unidirectionally

coupled Henon maps, also studied in Refs. [6, 9, 19], with

equations

= 1.4 −x

+ b

= x

(3)

for the driving system {X}, and

= 1.4 −(² x

+ (1 − ²) y

) + b

= y

(4)

for the response system {Y }. Here we will study identical

systems with b

= b

= 0.3.

Our third example will be the unidirectionally coupled

R¨ossler systems given by the equations

˙x

= −ω

− x

˙x

= ω

+ a

(5)

˙x

= b

+ x

− c

)

for the autonomous system, and

˙y

= −ω

− y

+ ²(x

− y

)

˙y

= ω

+ a

(6)

˙y

= b

+ y

− c

)

for the response system. In this Section we will use pa-

rameters a

= a

= 0.15, b

= b

= 0.2, c

= c

= 10.0,

and frequencies ω

= 1.015 and ω

= 0.985.

The data from continuous nonlinear dynamical sys-

tems were generated by numerical integration based on

the adaptive Bulirsch-Stoer method [30] using the sam-

pling interval 0.02617 for the systems (1),(2), and 0.1256

for the systems (5),(6). In the latter case this gives 17

– 21 samples per one period. When the R¨ossler systems

with diﬀerent frequencies were used, the sampling was

updated in order to keep 19–20 samples per period of

the faster system (Sec. V, VI).

In all the cases we denote the driving, autonomous

system by {X}, and the driven, response system by {Y }.

For each of the above three examples we deﬁne a set

of coupling strength parameter ² increasing from ² = 0

to an ²-value before the synchronization threshold. As

Paluˇs et al. [9] explain, the direction of coupling can be

inferred from experimental data only when the underly-

ing systems are coupled, but not yet synchronized. In

the numerical examples, the synchronization threshold

can be determined using the plot of Lyapunov exponents

(LE) of the coupled systems as the function of the cou-

pling strength ². With increasing ² the positive Lya-

punov exponent of the response system (also known as

the conditional Lyapunov exponent [31]) decreases and

it becomes negative just with the ²-value giving the syn-

chronization threshold. The plots of the Lyapunov expo-

nents for the R¨ossler-Lorenz systems (1),(2) can be found

in Refs. [9, 19], the LE plots for the coupled Henon sys-

tems (3),(4) in Refs. [6, 9, 19], while further study of the

coupled R¨ossler systems (5),(6), including their LE plot,

can be found in Sec. IV.

For each value of ² from the predeﬁned range we nu-

merically generate time series {x

} and {y

} as outputs

of the systems {X} and {Y }, obtained by recording the

components x

and y

, resp ectively, and analyze them

by using the following three methods.

Two diﬀerent methods exploit the approach suggested

by Rulkov et al. [32] based on the assumption of the

existence of a smooth map between the trajectories of

{X} and {Y }. If such a smooth map exists then closeness

of points in the state space X of the system {X} implies

a closeness of points in the state space Y of the system

{Y }.

One of the methods due to Le Van Quyen et al. [7]

is based on cross-prediction using the well known idea

of mutual neighbors. A known or reconstructed state

space (e.g., using a time-delay embedding [33] X

, x

n−τ

, x

n−2τ

, . . . ]) must be available. However, in-

stead of using k nearest neighbors, a neighborhood size

δ is pre-selected. Considering a map from X to Y , a

prediction is made for the value of y

n+1

one step ahead

using the following formula

ˆy

n+1

j:X

∈V

)

j+1

. (7)

The volume V

) = {X

: |X

− X

| < δ} is δ-

neighborhood of X

and |V

)| denotes the number

of points contained in the neighborhood. Using data

rescaled to the zero mean and the unit variance, the au-

thors deﬁne a crosspredictability index by subtracting the

root-mean-square prediction error from one

P (X → Y ) = 1 −

n=1

(ˆy

n+1

− y

n+1

)

, (8)

which should measure how the system {X} inﬂuences the

evolution of the system {Y }. The crosspredictability in-

dex P (Y → X) in the opposite direction, characterizing

the ability of the system {Y } to inﬂuence the evolution

of the system {X} is deﬁned in full analogy.

The second approach, proposed by Arnhold et al. [17]

and analysed by Quian Quiroga et al. [19], uses mean

square distances instead of the cross-predictions in order

to quantify the closeness of points in both spaces. We use

the implementation according to Ref. [19, 34] in which a

time-delay embedding [33] is ﬁrst constructed in order

to obtain state space vectors X and Y for both time

series {x

} and {y

}, respectively, then the mean squared

distance to k nearest neighbors is deﬁned for each X as

(k)

(X) =

j=1

− X

n,j

, (9)

where r

n,j

denotes the index of the j −th nearest neigh-

bor of X

. The Y-conditioned squared mean distance is

deﬁned by replacing the nearest neighb ors of X

by the

equal time partners of the nearest neighbors of Y

(k)

(X|Y) =

j=1

− X

n,j

, (10)

where s

n,j

denotes the index of the j−th nearest neighbor

of Y

. Then the asymmetric measure

(k)

(X|Y) =

n=1

(k)

(X)

(k)

(X|Y)

(11)

should reﬂect interdependence in the sense that close-

ness of the points in Y implies closeness of their equal

time partners in X and the values of S

(k)

(X|Y) ap-

proach to one, while, in the case of X independent of

Y , S

(k)

(X|Y) ¿ 1. The quantity S

(k)

(Y|X) measuring

the inﬂuence of {X} on {Y } is deﬁned in full analogy.

The third measure, used in this Section, coarse-grained

transinformation rate i(X → Y ) is the average rate of the

net amount of information “transferred” from the process

{X} to the process {Y }, or, in other words, the average

rate of the net information ﬂow by which the process {X}

inﬂuences the process {Y }. The coarse-grained transin-

formation rate (CTIR), introduced by Paluˇs et al. [9], is

based on the conditional mutual information and will be

brieﬂy reviewed in Sec. III.

The above three measures as functions of the coupling

strength ² for the R¨ossler system (1) driving the Lorenz

system (2) are plotted in Fig. 1. With the exemption

of the weakest coupling (² ≤ 0.6) the cross-predictability

of the system {Y } by the system {X} (the solid line

in Fig. 1a) is greater than the cross-predictability of

the system {X} by the system {Y } (the dashed line

in Fig. 1a). Our result in Fig. 1a agrees with that

of Le Van Quyen et al. [7] who interpret the relation

P (X → Y ) > P (Y → X) by the fact, that the au-

tonomous R¨ossler system {X} drives the response Lorenz

0 1 2

0.1

0.2

(c)

COUPLING STRENGTH ε

i(X→Y), i(Y→X)

0 1 2

0.1

0.2

0.4

0.6

(a)

P(X→Y), P(Y→X)

0.2

0.4

0.6

0.03

0.06

0.09

(b)

S(X|Y), S(Y|X)

0.03

0.06

0.09

FIG. 1: (a) Cross-predictability P (X → Y ) (solid line) and

P (Y → X) (dashed line), (b) relative average distance of

the mutual nearest neighbours S

(k)

(Y|X) (solid line) and

(k)

(X|Y) (dashed line), and (c) coarse-grained transinfor-

mation rate i(X → Y ) (solid line) and i(Y → X) (dashed

line) for the R¨ossler system (1) driving the Lorenz system

(2), as functions of the coupling strength ².

system {Y } and therefore the prediction of {Y } from {X}

is better than the prediction in the opposite direction.

In a similar way, with a few exemptions, the rela-

tive average distance of the mutual nearest neighbours

(k)

(Y|X) > S

(k)

(X|Y) (Fig. 1b) agrees with the re-

sults in [19], suggesting that the state of the response

system {Y } depends more on the state of the driver sys-

tem {X} than vice versa, as also claimed by Arnhold et

al. [17]. (Note that the conditioning X|Y reﬂects the in-

ﬂuence Y → X and vice versa.) The same conclusion

about {X} driving {Y } can be drawn from the CTIR

i(X → Y ) > i(Y → X) (Fig. 1c). The latter inequal-

ity holds for all positive values of ² but the ²-values ap-

proaching the synchronization threshold which emerges

for ² slightly above 2 [9, 19].

The same analyses as in Fig. 1, but for the unidi-

rectionally coupled Henon system (3),(4), are presented

in Fig. 2. One can immediately see that in this case

P (X → Y ) < P (Y → X) (Fig. 2a). This result agrees

with that of Ref. [6]. Shiﬀ et al. [6] oﬀer an interpreta-

tion based on the Takens embedding theorem [33]: From

the time series {x

} only the system {X} can be recon-

structed, while from the time series {y

} the whole system

consisting of the coupled systems {X} and {Y } can be

reconstructed and therefore one can predict the driving

system from the response system and not vice versa [6].

Also, the relation of the second interdependence mea-

sure reverses: In this case the inequality S

(k)

(Y|X) <

(k)

(X|Y) holds. (Fig. 2b). Again, our result agrees

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