Using several methods for detection of causality in time series, we show in a numerical study that coupled chaotic dynamical systems violate the first principle of Granger causality that the cause precedes the effect. While such a violation can be observed in formal applications of time series analysis methods, it cannot occur in nature, due to the relation between entropy production and temporal irreversibility. The obtained knowledge, however, can help to understand the type of causal relations observed in experimental data, namely, it can help to distinguish linear transfer of time-delayed signals from nonlinear interactions. We illustrate these findings in causality detected in experimental time series from the climate system and mammalian cardio-respiratory interactions.
References
1.
N.
Wiener
, “The theory of prediction
,” in Modern Mathematics for the Engineer
, edited by E. F.
Beckenbach
(McGraw-Hill
, New York, NY
, 1956
), pp. 125
–139
.2.
C. W. J.
Granger
, “Time series analysis, cointegration, and applications. Nobel Lecture, December 8, 2003
,” in Les Prix Nobel. The Nobel Prizes 2003
, edited by T.
Frängsmyr
(Nobel Foundation
, Stockholm
, 2004
), pp. 360
–366
.3.
K.
Hlaváčková-Schindler
, M.
Paluš
, M.
Vejmelka
, and J.
Bhattacharya
, “Causality detection based on information-theoretic approaches in time series analysis
,” Phys. Rep.
441
, 1
–46
(2007
).4.
R.
Shaw
, “Strange attractors, chaotic behavior, and information flow
,” Z. Naturforsch. A
36
, 80
–112
(1981
).5.
A. M.
Fraser
, “Information and entropy in strange attractors
,” IEEE Trans. Inf. Theory
35
, 245
–262
(1989
).6.
M.
Paluš
, “Kolmogorov entropy from time series using information-theoretic functionals
,” Neural Netw. World
7
, 269
–292
(1997
). http://www.cs.cas.cz/mp/papers/rd1a.pdf7.
A.
Pikovsky
, M.
Rosenblum
, J.
Kurths
, Synchronization: A Universal Concept in Nonlinear Sciences
(Cambridge university press
, 2003
), Vol. 12
.8.
S.
Boccaletti
, J.
Kurths
, G.
Osipov
, D.
Valladares
, and C.
Zhou
, “The synchronization of chaotic systems
,” Phys. Rep.
366
, 1
–101
(2002
).9.
M.
Paluš
, V.
Komarek
, Z.
Hrncir
, and K.
Sterbova
, “Synchronization as adjustment of information rates: Detection from bivariate time series
,” Phys. Rev. E
63
, 046211
(2001
).10.
J.
Arnhold
, P.
Grassberger
, K.
Lehnertz
, and C. E.
Elger
, “A robust method for detecting interdependences: Application to intracranially recorded EEG
,” Phys. D: Nonlin. Phenom.
134
, 419
–430
(1999
).11.
R. Q.
Quiroga
, A.
Kraskov
, T.
Kreuz
, and P.
Grassberger
, “Performance of different synchronization measures in real data: A case study on electroencephalographic signals
,” Phys. Rev. E
65
, 041903
(2002
).12.
R. G.
Andrzejak
, A.
Kraskov
, H.
Stögbauer
, F.
Mormann
, and T.
Kreuz
, “Bivariate surrogate techniques: Necessity, strengths, and caveats
,” Phys. Rev. E
68
, 066202
(2003
).13.
D. A.
Smirnov
and R. G.
Andrzejak
, “Detection of weak directional coupling: Phase-dynamics approach versus state-space approach
,” Phys. Rev. E
71
, 036207
(2005
).14.
D.
Chicharro
and R. G.
Andrzejak
, “Reliable detection of directional couplings using rank statistics
,” Phys. Rev. E
80
, 026217
(2009
).15.
G.
Sugihara
, R.
May
, H.
Ye
, C.-h.
Hsieh
, E.
Deyle
, M.
Fogarty
, and S.
Munch
, “Detecting causality in complex ecosystems
,” Science
338
, 496
–500
(2012
).16.
J.
Runge
, J.
Heitzig
, V.
Petoukhov
, and J.
Kurths
, “Escaping the curse of dimensionality in estimating multivariate transfer entropy
,” Phys. Rev. Lett.
108
, 258701
(2012
).17.
I.
Vlachos
and D.
Kugiumtzis
, “Nonuniform state-space reconstruction and coupling detection
,” Phys. Rev. E
82
, 016207
(2010
).18.
J.
Sun
, D.
Taylor
, and E. M.
Bollt
, “Causal network inference by optimal causation entropy
,” SIAM J. Appl. Dyn. Syst.
14
, 73
–106
(2015
).19.
C. W. J.
Granger
, “Investigating causal relations by econometric models and cross-spectral methods
,” Econometrica
37
, 424
–438
(1969
).20.
T. M.
Cover
and J. A.
Thomas
, Elements of Information Theory
(Wiley-Interscience
, New York, NY, USA
, 1991
).21.
T.
Schreiber
, “Measuring information transfer
,” Phys. Rev. Lett.
85
, 461
–464
(2000
).22.
M.
Paluš
and M.
Vejmelka
, “Directionality of coupling from bivariate time series: How to avoid false causalities and missed connections
,” Phys. Rev. E
75
, 056211
(2007
).23.
L.
Barnett
, A. B.
Barrett
, and A. K.
Seth
, “Granger causality and transfer entropy are equivalent for gaussian variables
,” Phys. Rev. Lett.
103
, 238701
(2009
).24.
F.
Takens
, “Detecting strange attractors in turbulence
,” in Dynamical Systems and Turbulence, Warwick 1980
(Springer
, 1981
), pp. 366
–381
.25.
A. M.
Fraser
and H. L.
Swinney
, “Independent coordinates for strange attractors from mutual information
,” Phys. Rev. A
33
, 1134
–1140
(1986
).26.
M.
Wibral
, N.
Pampu
, V.
Priesemann
, F.
Siebenhühner
, H.
Seiwert
, M.
Lindner
, and J. T.
Lizier
, “Measuring information-transfer delays
,” PLoS ONE.
2
, e55809
(2013
).27.
A.
Krakovská
and F.
Hanzely
, “Testing for causality in reconstructed state spaces by an optimized mixed prediction method
,” Phys. Rev. E
94
, 052203
(2016
).28.
E. N.
Lorenz
, “Atmospheric predictability as revealed by naturally occurring analogues
,” J. Atmos. Aci.
26
, 636
–646
(1969
).29.
M.
Wiesenfeldt
, U.
Parlitz
, and W.
Lauterborn
, “Mixed state analysis of multivariate time series
,” Int. J. Bif. Chaos
11
, 2217
–2226
(2001
).30.
U.
Feldmann
and J.
Bhattacharya
, “Predictability improvement as an asymmetrical measure of interdependence in bivariate time series
,” Int. J. Bif. Chaos
14
, 505
–514
(2004
).31.
M.
Paluš
, “From nonlinearity to causality: Statistical testing and inference of physical mechanisms underlying complex dynamics
,” Contemp. Phys.
48
, 307
–348
(2007
).32.
A. A.
Tsonis
, E. R.
Deyle
, R. M.
May
, G.
Sugihara
, K.
Swanson
, J. D.
Verbeten
, and G.
Wang
, “Dynamical evidence for causality between galactic cosmic rays and interannual variation in global temperature
,” Proc. Natl. Acad. Sci.
112
, 3253
–3256
(2015
).33.
H. G.
Schuster
and J.
Wolfram
, Deterministic Chaos: An Introduction
(Wiley-VCH
, 2005
).34.
D.
Coufal
, J.
Jakubík
, N.
Jajcay
, J.
Hlinka
, A.
Krakovská
, and M.
Paluš
, “Detection of coupling delay: A problem not yet solved
,” Chaos: Interdiscip. J. Nonlin. Sci.
27
, 083109
(2017
).35.
X. S.
Liang
, “Unraveling the cause-effect relation between time series
,” Phys. Rev. E
90
, 052150
(2014
).36.
X. S.
Liang
, “Information flow and causality as rigorous notions ab initio
,” Phys. Rev. E
94
, 052201
(2016
).37.
K.
Friston
, L.
Harrison
, and W.
Penny
, “Dynamic causal modelling
,” NeuroImage
19
, 1273
–1302
(2003
).38.
M. G.
Rosenblum
and A. S.
Pikovsky
, “Detecting direction of coupling in interacting oscillators
,” Phys. Rev. E
64
, 045202
(2001
).39.
T.
Stankovski
, A.
Duggento
, P. V. E.
McClintock
, and A.
Stefanovska
, “Inference of time-evolving coupled dynamical systems in the presence of noise
,” Phys. Rev. Lett.
109
, 024101
(2012
).40.
I. T.
Tokuda
, S.
Jain
, I. Z.
Kiss
, and J. L.
Hudson
, “Inferring phase equations from multivariate time series
,” Phys. Rev. Lett.
99
, 064101
(2007
).41.
A.
Krakovská
, J.
Jakubík
, M.
Chvosteková
, D.
Coufal
, N.
Jajcay
, and M.
Paluš
, “Comparison of six methods for the detection of causality in a bivariate time series
,” Phys. Rev. E
97
, 042207
(2018
).42.
M.
Paluš
, “Nonlinearity in normal human EEG: Cycles, temporal asymmetry, nonstationarity and randomness, not chaos
,” Biol. Cybern.
75
, 389
–396
(1996
).43.
R.
Kawai
, J. M. R.
Parrondo
, and C. V.
den Broeck
, “Dissipation: The phase-space perspective
,” Phys. Rev. Lett.
98
, 080602
(2007
).44.
J. M. R.
Parrondo
, C. V.
den Broeck
, and R.
Kawai
, “Entropy production and the arrow of time
,” New. J. Phys.
11
, 073008
(2009
).45.
P.
Gaspard
, “Time-reversed dynamical entropy and irreversibility in Markovian random processes
,” J. Stat. Phys.
117
, 599
–615
(2004
).46.
E.
Roldán
and J. M. R.
Parrondo
, “Estimating dissipation from single stationary trajectories
,” Phys. Rev. Lett.
105
, 150607
(2010
).47.
Y. B.
Pesin
, “Characteristic Lyapunov exponents and smooth ergodic theory
,” Russ. Math. Surv.
32
, 55
(1977
).48.
M.
Paluš
, “Linked by dynamics: Wavelet-based mutual information rate as a connectivity measure and scale-specific networks
,” in Advances in Nonlinear Geosciences
, edited by A. A.
Tsonis
(Springer
, 2018
), pp. 427
–463
.49.
J.
Runge
, V.
Petoukhov
, J. F.
Donges
, J.
Hlinka
, N.
Jajcay
, M.
Vejmelka
, D.
Hartman
, N.
Marwan
, M.
Paluš
, and J.
Kurths
, “Identifying causal gateways and mediators in complex spatio-temporal systems
,” Nat. Commun.
6
, 8502
(2015
).50.
M.
Vejmelka
, L.
Pokorná
, J.
Hlinka
, D.
Hartman
, N.
Jajcay
, and M.
Paluš
, “Non-random correlation structures and dimensionality reduction in multivariate climate data
,” Clim. Dyn.
44
, 2663
–2682
(2015
).51.
J.
Hlinka
, N.
Jajcay
, D.
Hartman
, and M.
Paluš
, “Smooth information flow in temperature climate network reflects mass transport
,” Chaos: Interdiscip. J. Nonlin. Sci.
27
, 035811
(2017
).52.
M.
Paluš
, D.
Novotná
, and P.
Tichavský
, “Shifts of seasons at the European mid-latitudes: Natural fluctuations correlated with the North Atlantic Oscillation
,” Geophys. Res. Lett.
32
, L12805
(2005
).53.
M.
Paluš
and D.
Novotná
, “Enhanced Monte Carlo singular system analysis and detection of period 7.8 years oscillatory modes in the monthly NAO index and temperature records
,” Nonlin. Process. Geophys.
11
, 721
–729
(2004
).54.
B.
Musizza
, A.
Stefanovska
, P. V. E.
McClintock
, M.
Paluš
, J.
Petrovčič
, S.
Ribarič
, and F. F.
Bajrović
, “Interactions between cardiac, respiratory and EEG- oscillations in rats during anaesthesia
,” J. Physiol. (Lond.)
580
, 315
–326
(2007
).55.
A.
Stefanovska
and M.
Bračič
, “Physics of the human cardiovascular system
,” Contemp. Phys.
40
, 31
–55
(1999
).56.
A.
Stefanovska
, M.
Bračič-Lotrič
, S.
Strle
, and H.
Haken
, “The cardiovascular system as coupled oscillators?
,” Physiol. Meas.
22
, 535
(2001
).57.
M.
Paluš
, “Testing for nonlinearity using redundancies: Quantitative and qualitative aspects
,” Phys. D: Nonlin. Phenom.
80
, 186
–205
(1995
).58.
M.
Paluš
, “Detecting nonlinearity in multivariate time series
,” Phys. Lett. A
213
, 138
–147
(1996b
).59.
M. G.
Rosenblum
, A. S.
Pikovsky
, and J.
Kurths
, “From phase to lag synchronization in coupled chaotic oscillators
,” Phys. Rev. Lett.
78
, 4193
–4196
(1997
).60.
M.
Paluš
, “Multiscale atmospheric dynamics: Cross-frequency phase-amplitude coupling in the air temperature
,” Phys. Rev. Lett.
112
, 078702
(2014
).61.
W.
Press
, B.
Flannery
, S.
Teukolsky
, and W.
Vetterling
, Numerical Recipes. The Art of Scientific Computations
(Cambridge University Press
, 1989
).© 2018 Author(s).
2018
Author(s)
You do not currently have access to this content.
