Nonparametric detection of coupling delay in unidirectionally and bidirectionally coupled nonlinear dynamical systems is examined. Both continuous and discrete-time systems are considered. Two methods of detection are assessed—the method based on conditional mutual information—the CMI method (also known as the transfer entropy method) and the method of convergent cross mapping—the CCM method. Computer simulations show that neither method is generally reliable in the detection of coupling delays. For continuous-time chaotic systems, the CMI method appears to be more sensitive and applicable in a broader range of coupling parameters than the CCM method. In the case of tested discrete-time dynamical systems, the CCM method has been found to be more sensitive, while the CMI method required much stronger coupling strength in order to bring correct results. However, when studied systems contain a strong oscillatory component in their dynamics, results of both methods become ambiguous. The presented study suggests that results of the tested algorithms should be interpreted with utmost care and the nonparametric detection of coupling delay, in general, is a problem not yet solved.
References
1.
T.
Schreiber
, “ Measuring information transfer
,” Phys. Rev. Lett.
85
, 461
–464
(2000
).2.
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
).3.
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
).4.
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
).5.
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
).6.
M.
Wibral
, N.
Pampu
, V.
Priesemann
, F.
Siebenhühner
, H.
Seiwert
, M.
Lindner
, and J. T.
Lizier
, “ Measuring information-transfer delays
,” PLoS One
8
, e55809
(2013
).7.
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
).8.
H.
Ye
, E. R.
Deyle
, L. J.
Gilarranz
, and G.
Sugihara
, “ Distinguishing time-delayed causal interactions using convergent cross mapping
,” Sci. Rep.
5
, 14750
(2015
).9.
T. M.
Cover
and J. A.
Thomas
, Elements of Information Theory
(Wiley-Interscience
, New York, NY, USA
, 1991
).10.
C. E.
Shannon
, “ A mathematical theory of communication
,” Bell Syst. Tech. J.
27
, 379
–423
(1948
).11.
M.
Paluš
, “ Testing for nonlinearity using redundancies: Quantitative and qualitative aspects
,” Physica D
80
, 186
–205
(1995
).12.
A.
Kraskov
, H.
Stögbauer
, and P.
Grassberger
, “ Estimating mutual information
,” Phys. Rev. E
69
, 066138
(2004
).13.
S.
Frenzel
and B.
Pompe
, “ Partial mutual information for coupling analysis of multivariate time series
,” Phys. Rev. Lett.
99
, 204101
(2007
).14.
M.
Vejmelka
and M.
Paluš
, “ Inferring the directionality of coupling with conditional mutual information
,” Phys. Rev. E
77
, 026214
(2008
).15.
F.
Takens
, “ Detecting strange attractors in turbulence
,” in Dynamical Systems and Turbulence, Warwick 1980
(Springer
, 1981
), pp. 366
–381
.16.
A. M.
Fraser
and H. L.
Swinney
, “ Independent coordinates for strange attractors from mutual information
,” Phys. Rev. A
33
, 1134
–1140
(1986
).17.
M.
Paluš
, “ Multiscale atmospheric dynamics: Cross-frequency phase-amplitude coupling in the air temperature
,” Phys. Rev. Lett.
112
, 078702
(2014
).18.
J.
Arnhold
, P.
Grassberger
, K.
Lehnertz
, and C. E.
Elger
, “ A robust method for detecting interdependences: Application to intracranially recorded eeg
,” Physica D
134
, 419
–430
(1999
).19.
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
).20.
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
).21.
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
).22.
D.
Chicharro
and R. G.
Andrzejak
, “ Reliable detection of directional couplings using rank statistics
,” Phys. Rev. E
80
, 026217
(2009
).23.
A.
Krakovská
, J.
Jakubík
, H.
Budáčová
, and M.
Holecyová
, “ Causality studied in reconstructed state space. Examples of uni-directionally connected chaotic systems
,” preprint arXiv:1511.00505 (2015
).24.
H. G.
Schuster
and J.
Wolfram
, Deterministic Chaos: An Introduction
(Wiley-VCH
, 2005
).25.
M.
Paluš
, “ Cross-scale interactions and information transfer
,” Entropy
16
, 5263
–5289
(2014
).26.
M. G.
Rosenblum
and A. S.
Pikovsky
, “ Detecting direction of coupling in interacting oscillators
,” Phys. Rev. E
64
, 045202
(2001
).27.
I.
Vlachos
and D.
Kugiumtzis
, “ Nonuniform state-space reconstruction and coupling detection
,” Phys. Rev. E
82
, 016207
(2010
).28.
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
).29.
L.
Cimponeriu
, M.
Rosenblum
, and A.
Pikovsky
, “ Estimation of delay in coupling from time series
,” Phys. Rev. E
70
, 046213
(2004
).30.
K. J.
Friston
, A.
Bastos
, V.
Litvak
, K. E.
Stephan
, P.
Fries
, and R. J.
Moran
, “ DCM for complex-valued data: Cross-spectra, coherence and phase-delays
,” NeuroImage
59
, 439
–455
(2012
).31.
T.
Stankovski
, V.
Ticcinelli
, P. V. E.
McClintock
, and A.
Stefanovska
, “ Coupling functions in networks of oscillators
,” New J. Phys.
17
, 035002
(2015
).32.
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
).33.
J.
Sun
, D.
Taylor
, and J. M.
Bollt
, “ Causal network inference by optimal causation entropy
,” SIAM J. Appl. Dyn. Syst.
14
(1
), 73
–106
(2015
).34.
A.
Krakovská
, K.
Mezeiová
, and H.
Budáčová
, “ Use of false nearest neighbours for selecting variables and embedding parameters for state space reconstruction
,” J. Complex Syst.
2015
, 932750
(2015
), 12 pp.35.
R. G.
James
, N.
Barnett
, and J. P.
Crutchfield
, “ Information flows? A critique of transfer entropies
,” Phys. Rev. Lett.
116
(23
), 238701
(2016
).© 2017 Author(s).
2017
Author(s)
You do not currently have access to this content.
