Entropy

Research

22 pages, 676 KiB

Open AccessArticle

Causality in Reversed Time Series: Reversed or Conserved?

by Jakub Kořenek and Jaroslav Hlinka

Entropy 2021, 23(8), 1067; https://doi.org/10.3390/e23081067 - 17 Aug 2021

Cited by 3 | Viewed by 2023

The inference of causal relations between observable phenomena is paramount across scientific disciplines; however, the means for such enterprise without experimental manipulation are limited. A commonly applied principle is that of the cause preceding and predicting the effect, taking into account other circumstances. [...] Read more.

The inference of causal relations between observable phenomena is paramount across scientific disciplines; however, the means for such enterprise without experimental manipulation are limited. A commonly applied principle is that of the cause preceding and predicting the effect, taking into account other circumstances. Intuitively, when the temporal order of events is reverted, one would expect the cause and effect to apparently switch roles. This was previously demonstrated in bivariate linear systems and used in design of improved causal inference scores, while such behaviour in linear systems has been put in contrast with nonlinear chaotic systems where the inferred causal direction appears unchanged under time reversal. The presented work explores the conditions under which the causal reversal happens—either perfectly, approximately, or not at all—using theoretical analysis, low-dimensional examples, and network simulations, focusing on the simplified yet illustrative linear vector autoregressive process of order one. We start with a theoretical analysis that demonstrates that a perfect coupling reversal under time reversal occurs only under very specific conditions, followed up by constructing low-dimensional examples where indeed the dominant causal direction is even conserved rather than reversed. Finally, simulations of random as well as realistically motivated network coupling patterns from brain and climate show that level of coupling reversal and conservation can be well predicted by asymmetry and anormality indices introduced based on the theoretical analysis of the problem. The consequences for causal inference are discussed. Full article

(This article belongs to the Special Issue Information Transfer, Entropy Production, Irreversibility and Time Series Analysis)

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17 pages, 892 KiB

Open AccessArticle

Cross-Scale Causality and Information Transfer in Simulated Epileptic Seizures

by Kajari Gupta and Milan Paluš

Entropy 2021, 23(5), 526; https://doi.org/10.3390/e23050526 - 25 Apr 2021

Cited by 2 | Viewed by 1993

Abstract

An information-theoretic approach for detecting causality and information transfer was applied to phases and amplitudes of oscillatory components related to different time scales and obtained using the wavelet transform from a time series generated by the Epileptor model. Three main time scales and [...] Read more.

An information-theoretic approach for detecting causality and information transfer was applied to phases and amplitudes of oscillatory components related to different time scales and obtained using the wavelet transform from a time series generated by the Epileptor model. Three main time scales and their causal interactions were identified in the simulated epileptic seizures, in agreement with the interactions of the model variables. An approach consisting of wavelet transform, conditional mutual information estimation, and surrogate data testing applied to a single time series generated by the model was demonstrated to be successful in the identification of all directional (causal) interactions between the three different time scales described in the model. Thus, the methodology was prepared for the identification of causal cross-frequency phase–phase and phase–amplitude interactions in experimental and clinical neural data. Full article

(This article belongs to the Special Issue Information Transfer, Entropy Production, Irreversibility and Time Series Analysis)

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13 pages, 608 KiB

Open AccessArticle

Granger Causality on forward and Reversed Time Series

by Martina Chvosteková, Jozef Jakubík and Anna Krakovská

Entropy 2021, 23(4), 409; https://doi.org/10.3390/e23040409 - 30 Mar 2021

Cited by 8 | Viewed by 2111

Abstract

In this study, the information flow time arrow is investigated for stochastic data defined by vector autoregressive models. The time series are analyzed forward and backward by different Granger causality detection methods. Besides the normal distribution, which is usually required for the validity [...] Read more.

In this study, the information flow time arrow is investigated for stochastic data defined by vector autoregressive models. The time series are analyzed forward and backward by different Granger causality detection methods. Besides the normal distribution, which is usually required for the validity of Granger causality analysis, several other distributions of predictive errors are considered. A clear effect of a change in the order of cause and effect on the time-reversed series of unidirectionally connected variables was detected with standard Granger causality test (GC), when the product of the connection strength and the ratio of the predictive errors of the driver and the recipient was below a certain level, otherwise bidirectional causal connection was detected. On the other hand, opposite causal link was detected unconditionally by the methods based on the time reversal testing, but they were not able to detect correct bidirectional connection. The usefulness of the backward analysis is manifested in cases where falsely detected unidirectional connections can be rejected by applying the result obtained after the time reversal, and in cases of uncorrelated causally independent variables, where the absence of a causal link detected by GC on the original series should be confirmed on the time-reversed series. Full article

(This article belongs to the Special Issue Information Transfer, Entropy Production, Irreversibility and Time Series Analysis)

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17 pages, 1936 KiB

Open AccessEditor’s ChoiceArticle

Causality and Information Transfer Between the Solar Wind and the Magnetosphere–Ionosphere System

by Pouya Manshour, Georgios Balasis, Giuseppe Consolini, Constantinos Papadimitriou and Milan Paluš

Entropy 2021, 23(4), 390; https://doi.org/10.3390/e23040390 - 25 Mar 2021

Cited by 19 | Viewed by 3108

Abstract

An information-theoretic approach for detecting causality and information transfer is used to identify interactions of solar activity and interplanetary medium conditions with the Earth’s magnetosphere–ionosphere systems. A causal information transfer from the solar wind parameters to geomagnetic indices is detected. The vertical component [...] Read more.

An information-theoretic approach for detecting causality and information transfer is used to identify interactions of solar activity and interplanetary medium conditions with the Earth’s magnetosphere–ionosphere systems. A causal information transfer from the solar wind parameters to geomagnetic indices is detected. The vertical component of the interplanetary magnetic field (

B z

) influences the auroral electrojet (AE) index with an information transfer delay of 10 min and the geomagnetic disturbances at mid-latitudes measured by the symmetric field in the H component (SYM-H) index with a delay of about 30 min. Using a properly conditioned causality measure, no causal link between AE and SYM-H, or between magnetospheric substorms and magnetic storms can be detected. The observed causal relations can be described as linear time-delayed information transfer. Full article

(This article belongs to the Special Issue Information Transfer, Entropy Production, Irreversibility and Time Series Analysis)

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21 pages, 533 KiB

Open AccessArticle

Time-Reversibility, Causality and Compression-Complexity

by Aditi Kathpalia and Nithin Nagaraj

Entropy 2021, 23(3), 327; https://doi.org/10.3390/e23030327 - 10 Mar 2021

Cited by 6 | Viewed by 2415

Abstract

Detection of the temporal reversibility of a given process is an interesting time series analysis scheme that enables the useful characterisation of processes and offers an insight into the underlying processes generating the time series. Reversibility detection measures have been widely employed in [...] Read more.

Detection of the temporal reversibility of a given process is an interesting time series analysis scheme that enables the useful characterisation of processes and offers an insight into the underlying processes generating the time series. Reversibility detection measures have been widely employed in the study of ecological, epidemiological and physiological time series. Further, the time reversal of given data provides a promising tool for analysis of causality measures as well as studying the causal properties of processes. In this work, the recently proposed Compression-Complexity Causality (CCC) measure (by the authors) is shown to be free of the assumption that the "cause precedes the effect", making it a promising tool for causal analysis of reversible processes. CCC is a data-driven interventional measure of causality (second rung on the Ladder of Causation) that is based on Effort-to-Compress (ETC), a well-established robust method to characterize the complexity of time series for analysis and classification. For the detection of the temporal reversibility of processes, we propose a novel measure called the Compressive Potential based Asymmetry Measure. This asymmetry measure compares the probability of the occurrence of patterns at different scales between the forward-time and time-reversed process using ETC. We test the performance of the measure on a number of simulated processes and demonstrate its effectiveness in determining the asymmetry of real-world time series of sunspot numbers, digits of the transcedental number

π

and heart interbeat interval variability. Full article

(This article belongs to the Special Issue Information Transfer, Entropy Production, Irreversibility and Time Series Analysis)

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18 pages, 36615 KiB

Open AccessEditor’s ChoiceArticle

Coupling between Blood Pressure and Subarachnoid Space Width Oscillations during Slow Breathing

by Agnieszka Gruszecka, Magdalena K. Nuckowska, Monika Waskow, Jacek Kot, Pawel J. Winklewski, Wojciech Guminski, Andrzej F. Frydrychowski, Jerzy Wtorek, Adam Bujnowski, Piotr Lass, Tomislav Stankovski and Marcin Gruszecki

Entropy 2021, 23(1), 113; https://doi.org/10.3390/e23010113 - 15 Jan 2021

Cited by 6 | Viewed by 2893

Abstract

The precise mechanisms connecting the cardiovascular system and the cerebrospinal fluid (CSF) are not well understood in detail. This paper investigates the couplings between the cardiac and respiratory components, as extracted from blood pressure (BP) signals and oscillations of the subarachnoid space width [...] Read more.

The precise mechanisms connecting the cardiovascular system and the cerebrospinal fluid (CSF) are not well understood in detail. This paper investigates the couplings between the cardiac and respiratory components, as extracted from blood pressure (BP) signals and oscillations of the subarachnoid space width (SAS), collected during slow ventilation and ventilation against inspiration resistance. The experiment was performed on a group of 20 healthy volunteers (12 females and 8 males; BMI

= 22.1 \pm 3.2

kg/m

^{2}

; age

25.3 \pm 7.9

years). We analysed the recorded signals with a wavelet transform. For the first time, a method based on dynamical Bayesian inference was used to detect the effective phase connectivity and the underlying coupling functions between the SAS and BP signals. There are several new findings. Slow breathing with or without resistance increases the strength of the coupling between the respiratory and cardiac components of both measured signals. We also observed increases in the strength of the coupling between the respiratory component of the BP and the cardiac component of the

SAS

and vice versa. Slow breathing synchronises the SAS oscillations, between the brain hemispheres. It also diminishes the similarity of the coupling between all analysed pairs of oscillators, while inspiratory resistance partially reverses this phenomenon. BP–SAS and SAS–BP interactions may reflect changes in the overall biomechanical characteristics of the brain. Full article

(This article belongs to the Special Issue Information Transfer, Entropy Production, Irreversibility and Time Series Analysis)

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13 pages, 895 KiB

Open AccessArticle

Measuring Information Coupling between the Solar Wind and the Magnetosphere–Ionosphere System

by Mirko Stumpo, Giuseppe Consolini, Tommaso Alberti and Virgilio Quattrociocchi

Entropy 2020, 22(3), 276; https://doi.org/10.3390/e22030276 - 28 Feb 2020

Cited by 18 | Viewed by 3868

Abstract

The interaction between the solar wind and the Earth’s magnetosphere–ionosphere system is very complex, being essentially the result of the interplay between an external driver, the solar wind, and internal processes to the magnetosphere–ionosphere system. In this framework, modelling the Earth’s magnetosphere–ionosphere response [...] Read more.

The interaction between the solar wind and the Earth’s magnetosphere–ionosphere system is very complex, being essentially the result of the interplay between an external driver, the solar wind, and internal processes to the magnetosphere–ionosphere system. In this framework, modelling the Earth’s magnetosphere–ionosphere response to the changes of the solar wind conditions requires a correct identification of the causality relations between the different parameters/quantities used to monitor this coupling. Nowadays, in the framework of complex dynamical systems, both linear statistical tools and Granger causality models drastically fail to detect causal relationships between time series. Conversely, information theory-based concepts can provide powerful model-free statistical quantities capable of disentangling the complex nature of the causal relationships. In this work, we discuss how to deal with the problem of measuring causal information in the solar wind–magnetosphere–ionosphere system. We show that a time delay of about 30–60 min is found between solar wind and magnetospheric and ionospheric overall dynamics as monitored by geomagnetic indices, with a great information transfer observed between the z component of the interplanetary magnetic field and geomagnetic indices, while a lower transfer is found when other solar wind parameters are considered. This suggests that the best candidate for modelling the geomagnetic response to solar wind changes is the interplanetary magnetic field component

B_{z}

. A discussion of the relevance of our results in the framework of Space Weather is also provided. Full article

(This article belongs to the Special Issue Information Transfer, Entropy Production, Irreversibility and Time Series Analysis)

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16 pages, 1384 KiB

Open AccessArticle

Detecting Causality in Multivariate Time Series via Non-Uniform Embedding

by Ziyu Jia, Youfang Lin, Zehui Jiao, Yan Ma and Jing Wang

Entropy 2019, 21(12), 1233; https://doi.org/10.3390/e21121233 - 16 Dec 2019

Cited by 23 | Viewed by 4099

Abstract

Causal analysis based on non-uniform embedding schemes is an important way to detect the underlying interactions between dynamic systems. However, there are still some obstacles to estimating high-dimensional conditional mutual information and forming optimal mixed embedding vector in traditional non-uniform embedding schemes. In [...] Read more.

Causal analysis based on non-uniform embedding schemes is an important way to detect the underlying interactions between dynamic systems. However, there are still some obstacles to estimating high-dimensional conditional mutual information and forming optimal mixed embedding vector in traditional non-uniform embedding schemes. In this study, we present a new non-uniform embedding method framed in information theory to detect causality for multivariate time series, named LM-PMIME, which integrates the low-dimensional approximation of conditional mutual information and the mixed search strategy for the construction of the mixed embedding vector. We apply the proposed method to simulations of linear stochastic, nonlinear stochastic, and chaotic systems, demonstrating its superiority over partial conditional mutual information from mixed embedding (PMIME) method. Moreover, the proposed method works well for multivariate time series with weak coupling strengths, especially for chaotic systems. In the actual application, we show its applicability to epilepsy multichannel electrocorticographic recordings. Full article

(This article belongs to the Special Issue Information Transfer, Entropy Production, Irreversibility and Time Series Analysis)

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10 pages, 294 KiB

Open AccessArticle

Residual Predictive Information Flow in the Tight Coupling Limit: Analytic Insights from a Minimalistic Model

by Benjamin Wahl, Ulrike Feudel, Jaroslav Hlinka, Matthias Wächter, Joachim Peinke and Jan A. Freund

Entropy 2019, 21(10), 1010; https://doi.org/10.3390/e21101010 - 17 Oct 2019

Cited by 1 | Viewed by 3454

Abstract

In a coupled system, predictive information flows from the causing to the caused variable. The amount of transferred predictive information can be quantified through the use of transfer entropy or, for Gaussian variables, equivalently via Granger causality. It is natural to expect and [...] Read more.

In a coupled system, predictive information flows from the causing to the caused variable. The amount of transferred predictive information can be quantified through the use of transfer entropy or, for Gaussian variables, equivalently via Granger causality. It is natural to expect and has been repeatedly observed that a tight coupling does not permit to reconstruct a causal connection between causing and caused variables. Here, we show that for a model of interacting social groups, carried from the master equation to the Fokker–Planck level, a residual predictive information flow can remain for a pair of uni-directionally coupled variables even in the limit of infinite coupling strength. We trace this phenomenon back to the question of how the synchronizing force and the noise strength scale with the coupling strength. A simplified model description allows us to derive analytic expressions that fully elucidate the interplay between deterministic and stochastic model parts. Full article

(This article belongs to the Special Issue Information Transfer, Entropy Production, Irreversibility and Time Series Analysis)

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12 pages, 1305 KiB

Open AccessArticle

Correlation Dimension Detects Causal Links in Coupled Dynamical Systems

by Anna Krakovská

Entropy 2019, 21(9), 818; https://doi.org/10.3390/e21090818 - 21 Aug 2019

Cited by 7 | Viewed by 3546

Abstract

It is becoming increasingly clear that causal analysis of dynamical systems requires different approaches than, for example, causal analysis of interconnected autoregressive processes. In this study, a correlation dimension estimated in reconstructed state spaces is used to detect causality. If deterministic dynamics plays [...] Read more.

It is becoming increasingly clear that causal analysis of dynamical systems requires different approaches than, for example, causal analysis of interconnected autoregressive processes. In this study, a correlation dimension estimated in reconstructed state spaces is used to detect causality. If deterministic dynamics plays a dominant role in data then the method based on the correlation dimension can serve as a fast and reliable way to reveal causal relationships between and within the systems. This study demonstrates that the method, unlike most other causal approaches, detects causality well, even for very weak links. It can also identify cases of uncoupled systems that are causally affected by a hidden common driver. Full article

(This article belongs to the Special Issue Information Transfer, Entropy Production, Irreversibility and Time Series Analysis)

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23 pages, 1098 KiB

Open AccessArticle

Information Thermodynamics for Time Series of Signal-Response Models

by Andrea Auconi, Andrea Giansanti and Edda Klipp

Entropy 2019, 21(2), 177; https://doi.org/10.3390/e21020177 - 14 Feb 2019

Cited by 9 | Viewed by 4074

Abstract

The entropy production in stochastic dynamical systems is linked to the structure of their causal representation in terms of Bayesian networks. Such a connection was formalized for bipartite (or multipartite) systems with an integral fluctuation theorem in [Phys. Rev. Lett. 111, 180603 (2013)]. [...] Read more.

The entropy production in stochastic dynamical systems is linked to the structure of their causal representation in terms of Bayesian networks. Such a connection was formalized for bipartite (or multipartite) systems with an integral fluctuation theorem in [Phys. Rev. Lett. 111, 180603 (2013)]. Here we introduce the information thermodynamics for time series, that are non-bipartite in general, and we show that the link between irreversibility and information can only result from an incomplete causal representation. In particular, we consider a backward transfer entropy lower bound to the conditional time series irreversibility that is induced by the absence of feedback in signal-response models. We study such a relation in a linear signal-response model providing analytical solutions, and in a nonlinear biological model of receptor-ligand systems where the time series irreversibility measures the signaling efficiency. Full article

(This article belongs to the Special Issue Information Transfer, Entropy Production, Irreversibility and Time Series Analysis)

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