20 years of ordinal patterns: Perspectives and challenges

Leyva, I.; Martínez, Johann H.; Masoller, Cristina; Rosso, Osvaldo A.; Zanin, Massimiliano

doi:10.1209/0295-5075/ac6a72

Cited by 28 publications

(12 citation statements)

References 105 publications

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“…Ordinal time series analysis is conceptually simple, computationally fast and comparably robust against measurement noise. Compared to other symbolization techniques 2 , the derivation of ordinal patterns does not require a priori knowledge about the data range, which rendered ordinal time series analysis beneficial for investigations of empirical data from various scientific domains [9][10][11][12][13][14][15][16][17][18][19] . A large proportion of studies was concerned with problems such as distinguishing chaos from noise, improving the detection of determinism [20][21][22] or of dynam-ical changes 23 , system identification 24 , or quantifying time reversibility 25 , thereby employing ordinal-patternderived quantifier for entropy 16,17 , complexity 12 , or combinations thereof 26 .…”

Section: Introductionmentioning

confidence: 99%

Ordinal methods for a characterization of evolving functional brain networks

Lehnertz¹

2023

Preprint

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Ordinal time series analysis is based on the idea to map time series to ordinal patterns, i.e., order relations between the values of a time series and not the values themselves, as introduced in 2002 by C. Bandt and B. Pompe. Despite a resulting loss of information, this approach captures meaningful information about the temporal structure of the underlying system dynamics as well as about properties of interactions between coupled systems. This -together with its conceptual simplicity and robustness against measurement noisemakes ordinal time series analysis well suited to improve characterization of the still poorly understood spatial-temporal dynamics of the human brain. This minireview briefly summarizes the state-of-the-art of uni-and bivariate ordinal time-series-analysis techniques together with applications in the neurosciences. It will highlight current limitations to stimulate further developments which would be necessary to advance characterization of evolving functional brain networks.Deriving evolving functional brain networks from observed, long-lasting, multivariate time series to improve characterization of various physiological and pathophysiological brain dynamics requires suitable and robust time-series-analysis techniques, that are capable of deciphering the multifaceted nature of the brain's complex endogenous and exogenous interactions. I will recapitulate concepts of ordinal time series analysis, showcase its applications in the neurosciences, and will discuss limitations and necessary developments to improve characterization of the complex networked dynamics system human brain.

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Section: Introductionmentioning

confidence: 99%

Ordinal methods for a characterization of evolving functional brain networks

Lehnertz¹

2023

Preprint

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“…However, progress is hampered since calculating ρ dynamics, either using many individual trajectories OR by propagating partial differential equations, prove computationally challenging. Ironically, accurate ρ dynamics require very fine grained calculations but for Hamiltonian evolution coarse-graining (smoothing over fine scales) is necessary for a time-dependent entropy.Inspired by permutation entropy (PE) [11,12] used for time-series, we propose 'PI-Entropy' (the PE of an Indexed ensemble) Π(ρ) which quantifies the shuffling of neighboring ensemble elements as a measure that connects thermodynamic entropy with the mixing and folding due to complex trajectories for ensemble members. The use of indexed ensembles ρ and the focus on 'digitised' shuffling proves to be extremely computationally efficient relative to calculating ρ itself.…”

mentioning

confidence: 99%

“…Inspired by permutation entropy (PE) [11,12] used for time-series, we propose 'PI-Entropy' (the PE of an Indexed ensemble) Π(ρ) which quantifies the shuffling of neighboring ensemble elements as a measure that connects thermodynamic entropy with the mixing and folding due to complex trajectories for ensemble members. The use of indexed ensembles ρ and the focus on 'digitised' shuffling proves to be extremely computationally efficient relative to calculating ρ itself.…”

mentioning

confidence: 99%

Permutation entropy of indexed ensembles: quantifying thermalization dynamics

2023

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We introduce ‘PI-Entropy’ Π(ρ ̃) (the Permutation entropy of an Indexed ensemble) to quantify mixing due to complex dynamics for an ensemble ρ of di erent initial states evolving under identical dynamics. We nd that Π(ρ ̃) acts as an excellent proxy for the thermodynamic entropy S(ρ) but is much more computationally e cient. We study 1-D and 2-D iterative maps and nd that Π(ρ ̃) dynamics distinguish a variety of system time scales and track global loss of information as the ensemble relaxes to equilibrium. There is a universal S-shaped relaxation to equilibrium for generally chaotic systems, and this relaxation is characterized by a shu ing timescale that correlates with the system’s Lyapunov exponent. For the Chirikov Standard Map, a system with a mixed phase space where the chaos grows with nonlinear kick strength K, we nd that for high K, Π(ρ ̃) behaves like the uniformly hyperbolic 2-D Cat Map. For low K we see periodic behavior with a relaxation envelope resembling those of the chaotic regime, but with frequencies that depend on the size and location of the initial ensemble in the mixed phase space as well as K. We discuss how Π(ρ ̃) adapts to experimental work and its general utility in quantifying how complex systems change from a low entropy to a high entropy state.

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“…Inspired by Permutation Entropy [11,12] (PE) used for time-series, we propose 'PI-Entropy' (the Permutation entropy of an Indexed ensemble) Π(ρ) which quantifies the shuffling of neighboring ensemble elements as a measure that connects thermodynamic entropy with the mixing and folding due to complex trajectories for ensemble members. The use of indexed ensembles ρ and the focus on 'digitised' shuffling proves to be extremely computationally efficient relative to calculating ρ itself.…”

mentioning

confidence: 99%

Permutation entropy of indexed ensembles: Quantifying thermalization dynamics

Aragoneses¹,

Kapulkin²,

Pattanayak³

2022

Preprint

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We introduce 'PI-Entropy' Π(ρ) (the Permutation entropy of an Indexed ensemble) to quantify mixing due to complex dynamics for an ensemble ρ of different initial states evolving under identical dynamics. We find that Π(ρ) acts as an excellent proxy for the thermodynamic entropy S(ρ) but is much more computationally efficient. We study 1-D and 2-D iterative maps and find that Π(ρ) dynamics distinguish a variety of system time scales and track global loss of information as the ensemble relaxes to equilibrium. There is a universal S-shaped relaxation to equilibrium for generally chaotic systems, and this relaxation is characterized by a shuffling timescale that correlates with the system's Lyapunov exponent. For the Chirikov Standard Map, a system with a mixed phase space where the chaos grows with nonlinear kick strength K, we find that for high K, Π(ρ) behaves like the uniformly hyperbolic 2-D Cat Map. For low K we see periodic behavior with a relaxation envelope resembling those of the chaotic regime, but with frequencies that depend on the size and location of the initial ensemble in the mixed phase space as well as K. We discuss how Π(ρ) adapts to experimental work and its general utility in quantifying how complex systems change from a low entropy to a high entropy state.

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20 years of ordinal patterns: Perspectives and challenges

Cited by 28 publications

References 105 publications

Ordinal methods for a characterization of evolving functional brain networks

Ordinal methods for a characterization of evolving functional brain networks

Permutation entropy of indexed ensembles: quantifying thermalization dynamics

Permutation entropy of indexed ensembles: Quantifying thermalization dynamics

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