Entropy and time asymmetry are two intertwined aspects of a system’s dynamics, with the production of entropy marking a clear direction in the temporal dimension. In the last few years, metrics to quantify both properties in time series have been designed around the same concept, i.e., the use of ordinal patterns. In spite of this, the relationship between these two families of metrics is yet not well understood. In this contribution, we study this problem by constructing an entropy–time asymmetry plane and evaluating it on a large set of synthetic and real-world time series. We show how the two metrics can at times behave independently, the main reason being the presence of patterns with turning points; due to this, they yield complementary information about the underlying systems, and they have different discriminating performance.
REFERENCES
1.
D. M.
Dinitto
, R. R.
Mcdaniel
, Jr., T. W.
Ruefli
, and J. B.
Thomas
, “The use of ordinal time-series analysis in assessing policy inputs and impacts
,” J. Appl. Behav. Sci.
22
, 77
–93
(1986
). 2.
C.
Bandt
and B.
Pompe
, “Permutation entropy: A natural complexity measure for time series
,” Phys. Rev. Lett.
88
, 174102
(2002
). 3.
M.
Zanin
and F.
Olivares
, “Ordinal patterns-based methodologies for distinguishing chaos from noise in discrete time series
,” Commun. Phys.
4
, 190
(2021
). 4.
J.
Amigó
, Permutation Complexity in Dynamical Systems: Ordinal Patterns, Permutation Entropy and All That
(Springer Science & Business Media
, 2010
).5.
M.
Zanin
, L.
Zunino
, O. A.
Rosso
, and D.
Papo
, “Permutation entropy and its main biomedical and econophysics applications: A review
,” Entropy
14
, 1553
–1577
(2012
). 6.
M.
Riedl
, A.
Müller
, and N.
Wessel
, “Practical considerations of permutation entropy
,” Eur. Phys. J.: Spec. Top.
222
, 249
–262
(2013
). 7.
K.
Keller
, T.
Mangold
, I.
Stolz
, and J.
Werner
, “Permutation entropy: New ideas and challenges
,” Entropy
19
, 134
(2017
). 8.
I.
Leyva
, J. H.
Martínez
, C.
Masoller
, O. A.
Rosso
, and M.
Zanin
, “20 years of ordinal patterns: Perspectives and challenges
,” Europhys. Lett.
138
, 31001
(2022
). 9.
É.
Roldán
and J. M.
Parrondo
, “Estimating dissipation from single stationary trajectories
,” Phys. Rev. Lett.
105
, (9
) 150607
(2010
). 10.
A.
Puglisi
and D.
Villamaina
, “Irreversible effects of memory
,” Europhys. Lett.
88
, 30004
(2009
). 11.
M.
Zanin
and D.
Papo
, “Algorithmic approaches for assessing irreversibility in time series: Review and comparison
,” Entropy
23
, 1474
(2021
). 12.
G.
Graff
, B.
Graff
, A.
Kaczkowska
, D.
Makowiec
, J.
Amigó
, J.
Piskorski
, K.
Narkiewicz
, and P.
Guzik
, “Ordinal pattern statistics for the assessment of heart rate variability
,” Eur. Phys. J.: Spec. Top.
222
, 525
–534
(2013
). 13.
M.
Zanin
, A.
Rodríguez-González
, E.
Menasalvas Ruiz
, and D.
Papo
, “Assessing time series reversibility through permutation patterns
,” Entropy
20
, 665
(2018
). 14.
J. H.
Martínez
, J. L.
Herrera-Diestra
, and M.
Chavez
, “Detection of time reversibility in time series by ordinal patterns analysis
,” Chaos
28
, (14
) 123111
(2018
). 15.
J.
Li
, P.
Shang
, and X.
Zhang
, “Time series irreversibility analysis using Jensen–Shannon divergence calculated by permutation pattern
,” Nonlinear Dyn.
96
, 2637
–2652
(2019
). 16.
W.
Yao
, W.
Yao
, J.
Wang
, and J.
Dai
, “Quantifying time irreversibility using probabilistic differences between symmetric permutations
,” Phys. Lett. A
383
, 738
–743
(2019
). 17.
M.
Martin
, A.
Plastino
, and O.
Rosso
, “Generalized statistical complexity measures: Geometrical and analytical properties
,” Phys. A
369
, 439
–462
(2006
). 18.
J. M.
Amigó
, S.
Zambrano
, and M. A.
Sanjuán
, “True and false forbidden patterns in deterministic and random dynamics
,” Europhys. Lett.
79
, (18
) 50001
(2007
). 19.
A.
Popov
, O.
Avilov
, and O.
Kanaykin
, “Permutation entropy of EEG signals for different sampling rate and time lag combinations,” in 2013 Signal Processing Symposium (SPS) (IEEE, 2013), pp. 1–4.20.
F.
Olivares
and L.
Zunino
, “Multiscale dynamics under the lens of permutation entropy
,” Phys. A
559
, 125081
(2020
). 21.
M.
Chavez
and J. H.
Martínez
, “Ordinal spectrum: A frequency domain characterization of complex time series,” arXiv:2009.02547 (2020).22.
J.
Lin
, “Divergence measures based on the Shannon entropy
,” IEEE Trans. Inf. Theory
37
, 145
–151
(1991
). 23.
H.
Tong
, Threshold Models in Non-Linear Time Series Analysis
(Springer Science & Business Media
, 2012
), Vol. 21.24.
H.
Tong
, “Threshold models in time series analysis—30 years on
,” Stat. Interface
4
, 107
–118
(2011
). 25.
Y.
Xiang
and X.
Gong
, “Efficiency of generalized simulated annealing
,” Phys. Rev. E
62
, 4473
(2000
). 26.
A.
Porporato
, J. R.
Rigby
, and E.
Daly
, “Irreversibility and fluctuation theorem in stationary time series
,” Phys. Rev. Lett.
98
, 094101
(2007
). 27.
L.
Zunino
, F.
Olivares
, A. F.
Bariviera
, and O. A.
Rosso
, “A simple and fast representation space for classifying complex time series
,” Phys. Lett. A
381
, 1021
–1028
(2017
). 28.
S.
Berger
, G.
Schneider
, E. F.
Kochs
, and D.
Jordan
, “Permutation entropy: Too complex a measure for EEG time series?
,” Entropy
19
, 692
(2017
). 29.
C.
Bandt
, “Order patterns, their variation and change points in financial time series and Brownian motion
,” Stat. Pap.
61
, 1565
–1588
(2020
). 30.
E.
Olejarczyk
and W.
Jernajczyk
, “Graph-based analysis of brain connectivity in schizophrenia
,” PLoS One
12
, e0188629
(2017
). 31.
M.
Sabeti
, S.
Katebi
, and R.
Boostani
, “Entropy and complexity measures for EEG signal classification of schizophrenic and control participants
,” Artif. Intell. Med.
47
, 263
–274
(2009
). 32.
B.
Raghavendra
, D. N.
Dutt
, H. N.
Halahalli
, and J. P.
John
, “Complexity analysis of EEG in patients with schizophrenia using fractal dimension
,” Physiol. Meas.
30
, 795
(2009
). 33.
M.
Zanin
, B.
Güntekin
, T.
Aktürk
, L.
Hanoğlu
, and D.
Papo
, “Time irreversibility of resting-state activity in the healthy brain and pathology
,” Front. Physiol.
10
, 1619
(2020
). 34.
K. R.
Patel
, J.
Cherian
, K.
Gohil
, and D.
Atkinson
, “Schizophrenia: Overview and treatment options
,” Pharm. Ther.
39
, 638
(2014
).35.
E. F.
Fama
, “Efficient capital markets: A review of theory and empirical work
,” J. Finance
25
, 383
–417
(1970
). 36.
E. F.
Fama
, “Market efficiency, long-term returns, and behavioral finance
,” J. Financ. Econ.
49
, 283
–306
(1998
). © 2023 Author(s). Published under an exclusive license by AIP Publishing.
2023
Author(s)
You do not currently have access to this content.