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Almendral JA, Leyva I, Sendiña-Nadal I. Unveiling the Connectivity of Complex Networks Using Ordinal Transition Methods. ENTROPY (BASEL, SWITZERLAND) 2023; 25:1079. [PMID: 37510026 PMCID: PMC10378875 DOI: 10.3390/e25071079] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Grants] [Track Full Text] [Subscribe] [Scholar Register] [Received: 06/14/2023] [Revised: 07/10/2023] [Accepted: 07/14/2023] [Indexed: 07/30/2023]
Abstract
Ordinal measures provide a valuable collection of tools for analyzing correlated data series. However, using these methods to understand information interchange in the networks of dynamical systems, and uncover the interplay between dynamics and structure during the synchronization process, remains relatively unexplored. Here, we compare the ordinal permutation entropy, a standard complexity measure in the literature, and the permutation entropy of the ordinal transition probability matrix that describes the transitions between the ordinal patterns derived from a time series. We find that the permutation entropy based on the ordinal transition matrix outperforms the rest of the tested measures in discriminating the topological role of networked chaotic Rössler systems. Since the method is based on permutation entropy measures, it can be applied to arbitrary real-world time series exhibiting correlations originating from an existing underlying unknown network structure. In particular, we show the effectiveness of our method using experimental datasets of networks of nonlinear oscillators.
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Affiliation(s)
- Juan A Almendral
- Complex Systems Group & Grupo Interdisciplinar de Sistemas Complejos (GISC), Universidad Rey Juan Carlos, 28933 Móstoles, Madrid, Spain
- Center for Biomedical Technology, Universidad Politécnica de Madrid, 28223 Pozuelo de Alarcón, Madrid, Spain
| | - I Leyva
- Complex Systems Group & Grupo Interdisciplinar de Sistemas Complejos (GISC), Universidad Rey Juan Carlos, 28933 Móstoles, Madrid, Spain
- Center for Biomedical Technology, Universidad Politécnica de Madrid, 28223 Pozuelo de Alarcón, Madrid, Spain
| | - Irene Sendiña-Nadal
- Complex Systems Group & Grupo Interdisciplinar de Sistemas Complejos (GISC), Universidad Rey Juan Carlos, 28933 Móstoles, Madrid, Spain
- Center for Biomedical Technology, Universidad Politécnica de Madrid, 28223 Pozuelo de Alarcón, Madrid, Spain
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2
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Chen X, Xu G, He B, Zhang S, Su Z, Jia Y, Zhang X, Zhao Z. Capturing synchronization with complexity measure of ordinal pattern transition network constructed by crossplot. ROYAL SOCIETY OPEN SCIENCE 2023; 10:221067. [PMID: 37388315 PMCID: PMC10300663 DOI: 10.1098/rsos.221067] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Grants] [Track Full Text] [Figures] [Subscribe] [Scholar Register] [Received: 08/29/2022] [Accepted: 06/06/2023] [Indexed: 07/01/2023]
Abstract
To evaluate the synchronization of bivariate time series has been a hot topic, and a number of measures have been proposed. In this work, by introducing the ordinal pattern transition network into the crossplot, a new method for measuring the synchronization of bivariate time series is proposed. After the crossplot been partitioned and coded, the coded partitions are defined as network nodes and a directed weighted network is constructed based on the temporal adjacency of the nodes. The crossplot transition entropy of the network is proposed as an indicator of the synchronization between two time series. To test the characteristics and performance of the method, it is used to analyse the unidirectional coupled Lorentz model and compared it with existing methods. The results showed the new method had the advantages of easy parameter setting, efficiency, robustness, good consistency and suitability for short time series. Finally, electroencephalogram (EEG) data from auditory-evoked potential EEG-biometric dataset are investigated, and some useful and interesting results are obtained.
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Affiliation(s)
- Xiaobi Chen
- School of Mechanical Engineering, Xi'an Jiaotong University, Xi'an 710049, People's Republic of China
| | - Guanghua Xu
- School of Mechanical Engineering, Xi'an Jiaotong University, Xi'an 710049, People's Republic of China
- State Key Laboratory for Manufacturing Systems Engineering, Xi'an Jiaotong University, Xi'an 710049, People's Republic of China
- The First Affiliated Hospital of Xi'an Jiaotong University, Xi'an, People's Republic of China
| | - Bo He
- School of Mechanical Engineering, Xi'an Jiaotong University, Xi'an 710049, People's Republic of China
| | - Sicong Zhang
- School of Mechanical Engineering, Xi'an Jiaotong University, Xi'an 710049, People's Republic of China
| | - Zijvn Su
- School of Materials, Sun Yat-sen University, Shenzhen 518107, People's Republic of China
| | - Yaguang Jia
- School of Mechanical Engineering, Xi'an Jiaotong University, Xi'an 710049, People's Republic of China
| | - Xun Zhang
- School of Mechanical Engineering, Xi'an Jiaotong University, Xi'an 710049, People's Republic of China
| | - Zhe Zhao
- School of Microelectronics, Xi'an Jiaotong University, Xi'an 710049, People's Republic of China
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3
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Du M, Wei J, Li MY, Gao ZK, Kurths J. Interconnected ordinal pattern complex network for characterizing the spatial coupling behavior of gas-liquid two-phase flow. CHAOS (WOODBURY, N.Y.) 2023; 33:2894468. [PMID: 37276554 DOI: 10.1063/5.0146259] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 02/12/2023] [Accepted: 05/11/2023] [Indexed: 06/07/2023]
Abstract
The complex phase interactions of the two-phase flow are a key factor in understanding the flow pattern evolutional mechanisms, yet these complex flow behaviors have not been well understood. In this paper, we employ a series of gas-liquid two-phase flow multivariate fluctuation signals as observations and propose a novel interconnected ordinal pattern network to investigate the spatial coupling behaviors of the gas-liquid two-phase flow patterns. In addition, we use two network indices, which are the global subnetwork mutual information (I) and the global subnetwork clustering coefficient (C), to quantitatively measure the spatial coupling strength of different gas-liquid flow patterns. The gas-liquid two-phase flow pattern evolutionary behaviors are further characterized by calculating the two proposed coupling indices under different flow conditions. The proposed interconnected ordinal pattern network provides a novel tool for a deeper understanding of the evolutional mechanisms of the multi-phase flow system, and it can also be used to investigate the coupling behaviors of other complex systems with multiple observations.
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Affiliation(s)
- Meng Du
- School of Electrical Engineering and Automation, Tianjin University of Science and Technology, Tianjin 300222, China
| | - Jie Wei
- School of Electrical Engineering and Automation, Tianjin University of Science and Technology, Tianjin 300222, China
| | - Meng-Yu Li
- School of Electrical Engineering and Automation, Tianjin University, Tianjin 300072, China
| | - Zhong-Ke Gao
- School of Electrical Engineering and Automation, Tianjin University, Tianjin 300072, China
| | - Jürgen Kurths
- Research Department Complexity Science, Potsdam Institute for Climate Impact Research, 14473 Potsdam, Germany
- Institute of Physics, Humboldt University of Berlin, 12489 Berlin, Germany
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4
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Lehnertz K. Ordinal methods for a characterization of evolving functional brain networks. CHAOS (WOODBURY, N.Y.) 2023; 33:022101. [PMID: 36859225 DOI: 10.1063/5.0136181] [Citation(s) in RCA: 6] [Impact Index Per Article: 6.0] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 11/24/2022] [Accepted: 01/06/2023] [Indexed: 06/18/2023]
Abstract
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 noise-makes ordinal time series analysis well suited to improve characterization of the still poorly understood spatiotemporal 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.
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Affiliation(s)
- Klaus Lehnertz
- Department of Epileptology, University of Bonn Medical Centre, Venusberg Campus 1, 53127 Bonn, Germany; Helmholtz Institute for Radiation and Nuclear Physics, University of Bonn, Nussallee 14-16, 53115 Bonn, Germany; and Interdisciplinary Center for Complex Systems, University of Bonn, Brühler Straße 7, 53175 Bonn, Germany
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5
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Zhang Z, Wu J, Chen Y, Wang J, Xu J. Distinguish between Stochastic and Chaotic Signals by a Local Structure-Based Entropy. ENTROPY (BASEL, SWITZERLAND) 2022; 24:1752. [PMID: 36554157 PMCID: PMC9778404 DOI: 10.3390/e24121752] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Figures] [Subscribe] [Scholar Register] [Received: 11/04/2022] [Revised: 11/26/2022] [Accepted: 11/27/2022] [Indexed: 06/17/2023]
Abstract
As a measure of complexity, information entropy is frequently used to categorize time series, such as machinery failure diagnostics, biological signal identification, etc., and is thought of as a characteristic of dynamic systems. Many entropies, however, are ineffective for multivariate scenarios due to correlations. In this paper, we propose a local structure entropy (LSE) based on the idea of a recurrence network. Given certain tolerance and scales, LSE values can distinguish multivariate chaotic sequences between stochastic signals. Three financial market indices are used to evaluate the proposed LSE. The results show that the LSEFSTE100 and LSES&P500 are higher than LSESZI, which indicates that the European and American stock markets are more sophisticated than the Chinese stock market. Additionally, using decision trees as the classifiers, LSE is employed to detect bearing faults. LSE performs higher on recognition accuracy when compared to permutation entropy.
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Affiliation(s)
- Zelin Zhang
- School of Mathematics, Physics and Optoelectronic Engineering, Hubei University of Automotive Technology, Shiyan 442002, China
- Hubei Key Laboratory of Applied Mathematics, Hubei University, Wuhan 430061, China
| | - Jun Wu
- School of Mathematics, Physics and Optoelectronic Engineering, Hubei University of Automotive Technology, Shiyan 442002, China
- Hubei Key Laboratory of Applied Mathematics, Hubei University, Wuhan 430061, China
| | - Yufeng Chen
- School of Electrical and Information Engineering, Hubei University of Automotive Technology, Shiyan 442002, China
| | - Ji Wang
- School of Liberal Arts and Humanities, Sichuan Vocational College of Finance and Economics, Chengdu 610101, China
| | - Jinyu Xu
- School of Electrical and Information Engineering, Hubei University of Automotive Technology, Shiyan 442002, China
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Lu J, Donner RV, Yin D, Guan S, Zou Y. Partial event coincidence analysis for distinguishing direct and indirect coupling in functional network construction. CHAOS (WOODBURY, N.Y.) 2022; 32:063134. [PMID: 35778157 DOI: 10.1063/5.0087607] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 02/07/2022] [Accepted: 06/07/2022] [Indexed: 06/15/2023]
Abstract
Correctly identifying interaction patterns from multivariate time series presents an important step in functional network construction. In this context, the widespread use of bivariate statistical association measures often results in a false identification of links because strong similarity between two time series can also emerge without the presence of a direct interaction due to intermediate mediators or common drivers. In order to properly distinguish such direct and indirect links for the special case of event-like data, we present here a new generalization of event coincidence analysis to a partial version thereof, which is aimed at excluding possible transitive effects of indirect couplings. Using coupled chaotic systems and stochastic processes on two generic coupling topologies (star and chain configuration), we demonstrate that the proposed methodology allows for the correct identification of indirect interactions. Subsequently, we apply our partial event coincidence analysis to multi-channel EEG recordings to investigate possible differences in coordinated alpha band activity among macroscopic brain regions in resting states with eyes open (EO) and closed (EC) conditions. Specifically, we find that direct connections typically correspond to close spatial neighbors while indirect ones often reflect longer-distance connections mediated via other brain regions. In the EC state, connections in the frontal parts of the brain are enhanced as compared to the EO state, while the opposite applies to the posterior regions. In general, our approach leads to a significant reduction in the number of indirect connections and thereby contributes to a better understanding of the alpha band desynchronization phenomenon in the EO state.
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Affiliation(s)
- Jiamin Lu
- School of Physics and Electronic Science, East China Normal University, Shanghai 200062, China
| | - Reik V Donner
- Department of Water, Environment, Construction and Safety, Magdeburg-Stendal University of Applied Sciences, Breitscheidstraße 2, 39114 Magdeburg, Germany
| | - Dazhi Yin
- Key Laboratory of Brain Functional Genomics (Ministry of Education and Shanghai), School of Psychology and Cognitive Science, East China Normal University, Shanghai 200062, China
| | - Shuguang Guan
- School of Physics and Electronic Science, East China Normal University, Shanghai 200062, China
| | - Yong Zou
- School of Physics and Electronic Science, East China Normal University, Shanghai 200062, China
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7
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Varley TF, Sporns O. Network Analysis of Time Series: Novel Approaches to Network Neuroscience. Front Neurosci 2022; 15:787068. [PMID: 35221887 PMCID: PMC8874015 DOI: 10.3389/fnins.2021.787068] [Citation(s) in RCA: 7] [Impact Index Per Article: 3.5] [Reference Citation Analysis] [Abstract] [Key Words] [Grants] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 09/30/2021] [Accepted: 12/20/2021] [Indexed: 12/12/2022] Open
Abstract
In the last two decades, there has been an explosion of interest in modeling the brain as a network, where nodes correspond variously to brain regions or neurons, and edges correspond to structural or statistical dependencies between them. This kind of network construction, which preserves spatial, or structural, information while collapsing across time, has become broadly known as "network neuroscience." In this work, we provide an alternative application of network science to neural data: network-based analysis of non-linear time series and review applications of these methods to neural data. Instead of preserving spatial information and collapsing across time, network analysis of time series does the reverse: it collapses spatial information, instead preserving temporally extended dynamics, typically corresponding to evolution through some kind of phase/state-space. This allows researchers to infer a, possibly low-dimensional, "intrinsic manifold" from empirical brain data. We will discuss three methods of constructing networks from nonlinear time series, and how to interpret them in the context of neural data: recurrence networks, visibility networks, and ordinal partition networks. By capturing typically continuous, non-linear dynamics in the form of discrete networks, we show how techniques from network science, non-linear dynamics, and information theory can extract meaningful information distinct from what is normally accessible in standard network neuroscience approaches.
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Affiliation(s)
- Thomas F. Varley
- Department of Psychological and Brain Sciences, Indiana University, Bloomington, IN, United States
- School of Informatics, Computing, and Engineering, Indiana University, Bloomington, IN, United States
| | - Olaf Sporns
- Department of Psychological and Brain Sciences, Indiana University, Bloomington, IN, United States
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8
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Deep-layer motif method for estimating information flow between EEG signals. Cogn Neurodyn 2022; 16:819-831. [PMID: 35847539 PMCID: PMC9279550 DOI: 10.1007/s11571-021-09759-x] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 04/25/2021] [Revised: 10/04/2021] [Accepted: 11/22/2021] [Indexed: 11/26/2022] Open
Abstract
Accurate identification for the information flow between epileptic seizure signals is the key to construct the directional epileptic brain network which can be used to localize epileptic focus. In this paper, our concern is on how to improve the direction identification of information flow and also investigate how it can be cut off or weakened. In view of this, we propose the deep-layer motif method. Based on the directional index (DI) estimation using permutation conditional mutual information, the effectiveness of the proposed deep-layer motif method is numerically assessed with the coupled mass neural model. Furthermore, we investigate the robustness of this method in considering the interference of autaptic coupling, time delay and short-term plasticity. Results show that compared to the simple 1-layer motif method, the 2nd- and 3rd-layer motif methods have the dominant enhancement effects for the direction identification. In particular, deep-layer motif method possesses good anti-jamming performance and good robustness in calculating DI. In addition, we investigate the effect of deep brain stimulation (DBS) on the information flow. It is found that this deep-layer motif method is still superior to the single-layer motif method in direction identification and is robust to weak DBS. However, the high-frequency strong DBS can effectively decrease the DI suggesting the weakened information flow. These results may give new insights into the seizure detection and control.
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9
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Lainscsek C, Cash SS, Sejnowski TJ, Kurths J. Dynamical ergodicity DDA reveals causal structure in time series. CHAOS (WOODBURY, N.Y.) 2021; 31:103108. [PMID: 34717330 DOI: 10.1063/5.0063724] [Citation(s) in RCA: 3] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 07/16/2021] [Accepted: 09/10/2021] [Indexed: 06/13/2023]
Abstract
Determining synchronization, causality, and dynamical similarity in highly complex nonlinear systems like brains is challenging. Although distinct, these measures are related by the unknown deterministic structure of the underlying dynamical system. For two systems that are not independent on each other, either because they result from a common process or they are already synchronized, causality measures typically fail. Here, we introduce dynamical ergodicity to assess dynamical similarity between time series and then combine this new measure with cross-dynamical delay differential analysis to estimate causal interactions between time series. We first tested this approach on simulated data from coupled Rössler systems where ground truth was known. We then applied it to intracranial electroencephalographic data from patients with epilepsy and found distinct dynamical states that were highly predictive of epileptic seizures.
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Affiliation(s)
- Claudia Lainscsek
- Computational Neurobiology Laboratory, The Salk Institute for Biological Studies, 10010 North Torrey Pines Road, La Jolla, California 92037, USA
| | - Sydney S Cash
- Department of Neurology, Massachusetts General Hospital and Harvard Medical School, Boston, Massachusetts 02114, USA
| | - Terrence J Sejnowski
- Computational Neurobiology Laboratory, The Salk Institute for Biological Studies, 10010 North Torrey Pines Road, La Jolla, California 92037, USA
| | - Jürgen Kurths
- Potsdam Institute for Climate Impact Research (PIK), Potsdam 14473, Germany
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10
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Pessa AAB, Ribeiro HV. ordpy: A Python package for data analysis with permutation entropy and ordinal network methods. CHAOS (WOODBURY, N.Y.) 2021; 31:063110. [PMID: 34241315 DOI: 10.1063/5.0049901] [Citation(s) in RCA: 9] [Impact Index Per Article: 3.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 03/10/2021] [Accepted: 05/18/2021] [Indexed: 06/13/2023]
Abstract
Since Bandt and Pompe's seminal work, permutation entropy has been used in several applications and is now an essential tool for time series analysis. Beyond becoming a popular and successful technique, permutation entropy inspired a framework for mapping time series into symbolic sequences that triggered the development of many other tools, including an approach for creating networks from time series known as ordinal networks. Despite increasing popularity, the computational development of these methods is fragmented, and there were still no efforts focusing on creating a unified software package. Here, we present ordpy (http://github.com/arthurpessa/ordpy), a simple and open-source Python module that implements permutation entropy and several of the principal methods related to Bandt and Pompe's framework to analyze time series and two-dimensional data. In particular, ordpy implements permutation entropy, Tsallis and Rényi permutation entropies, complexity-entropy plane, complexity-entropy curves, missing ordinal patterns, ordinal networks, and missing ordinal transitions for one-dimensional (time series) and two-dimensional (images) data as well as their multiscale generalizations. We review some theoretical aspects of these tools and illustrate the use of ordpy by replicating several literature results.
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Affiliation(s)
- Arthur A B Pessa
- Departamento de Física, Universidade Estadual de Maringá, Maringá, PR 87020-900, Brazil
| | - Haroldo V Ribeiro
- Departamento de Física, Universidade Estadual de Maringá, Maringá, PR 87020-900, Brazil
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11
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A Novel Measure Inspired by Lyapunov Exponents for the Characterization of Dynamics in State-Transition Networks. ENTROPY 2021; 23:e23010103. [PMID: 33445685 PMCID: PMC7828116 DOI: 10.3390/e23010103] [Citation(s) in RCA: 3] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Download PDF] [Figures] [Subscribe] [Scholar Register] [Received: 12/15/2020] [Revised: 01/06/2021] [Accepted: 01/07/2021] [Indexed: 11/17/2022]
Abstract
The combination of network sciences, nonlinear dynamics and time series analysis provides novel insights and analogies between the different approaches to complex systems. By combining the considerations behind the Lyapunov exponent of dynamical systems and the average entropy of transition probabilities for Markov chains, we introduce a network measure for characterizing the dynamics on state-transition networks with special focus on differentiating between chaotic and cyclic modes. One important property of this Lyapunov measure consists of its non-monotonous dependence on the cylicity of the dynamics. Motivated by providing proper use cases for studying the new measure, we also lay out a method for mapping time series to state transition networks by phase space coarse graining. Using both discrete time and continuous time dynamical systems the Lyapunov measure extracted from the corresponding state-transition networks exhibits similar behavior to that of the Lyapunov exponent. In addition, it demonstrates a strong sensitivity to boundary crisis suggesting applicability in predicting the collapse of chaos.
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12
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Wu H, Zou Y, Alves LM, Macau EEN, Sampaio G, Marengo JA. Uncovering episodic influence of oceans on extreme drought events in Northeast Brazil by ordinal partition network approaches. CHAOS (WOODBURY, N.Y.) 2020; 30:053104. [PMID: 32491908 DOI: 10.1063/5.0004348] [Citation(s) in RCA: 4] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 02/11/2020] [Accepted: 04/13/2020] [Indexed: 06/11/2023]
Abstract
Since 2012, the semiarid region of Northeast Brazil (NEB) has been experiencing a continuous dry condition imposing significant social impacts and economic losses. Characterizing the recent extreme drought events and uncovering the influence from the surrounding oceans remain to be big challenges. The physical mechanisms of extreme drought events in the NEB are due to varying interacting time scales from the surrounding tropical oceans (Pacific and Atlantic). From time series observations, we propose a three-step strategy to establish the episodic coupling directions on intraseasonal time scales from the ocean to the precipitation patterns in the NEB, focusing on the distinctive roles of the oceans during the recent extreme drought events of 2012-2013 and 2015-2016. Our algorithm involves the following: (i) computing drought period length from daily precipitation anomalies to capture extreme drought events; (ii) characterizing the episodic coupling delays from the surrounding oceans to the precipitation by applying the Kullback-Leibler divergence (KLD) of complexity measure, which is based on ordinal partition transition network representation of time series; and (iii) calculating the ratio of high temperature in the ocean during the extreme drought events with proper time lags that are identified by KLD measures. From the viewpoint of climatology, our analysis provides data-based evidence of showing significant influence from the North Atlantic in 2012-2013 to the NEB, but in 2015-2016, the Pacific played a dominant role than that of the Atlantic. The episodic intraseasonal time scale properties are potential for monitoring and forecasting droughts in the NEB in order to propose strategies for drought impacts reduction.
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Affiliation(s)
- Huanyu Wu
- School of Physics and Electronic Science, East China Normal University, Shanghai 200062, China
| | - Yong Zou
- School of Physics and Electronic Science, East China Normal University, Shanghai 200062, China
| | - Lincoln M Alves
- Instituto Nacional de Pesquisas Espaciais, São José dos Campos 12246-021, São Paulo, Brazil
| | - Elbert E N Macau
- Instituto Nacional de Pesquisas Espaciais, São José dos Campos 12246-021, São Paulo, Brazil
| | - Gilvan Sampaio
- Instituto Nacional de Pesquisas Espaciais, Cachoeira Paulista 12.630-000, São Paulo, Brazil
| | - Jose A Marengo
- CEMADEN, Brazilian National Center for Monitoring and Early Warning of Natural Disasters, São José dos Campos 12245-320, São Paulo, Brazil
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13
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Sun J. Complex Network Construction of Univariate Chaotic Time Series Based on Maximum Mean Discrepancy. ENTROPY 2020; 22:e22020142. [PMID: 33285917 PMCID: PMC7516554 DOI: 10.3390/e22020142] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Download PDF] [Figures] [Subscribe] [Scholar Register] [Received: 12/11/2019] [Revised: 01/23/2020] [Accepted: 01/23/2020] [Indexed: 12/03/2022]
Abstract
The analysis of chaotic time series is usually a challenging task due to its complexity. In this communication, a method of complex network construction is proposed for univariate chaotic time series, which provides a novel way to analyze time series. In the process of complex network construction, how to measure the similarity between the time series is a key problem to be solved. Due to the complexity of chaotic systems, the common metrics is hard to measure the similarity. Consequently, the proposed method first transforms univariate time series into high-dimensional phase space to increase its information, then uses Gaussian mixture model (GMM) to represent time series, and finally introduces maximum mean discrepancy (MMD) to measure the similarity between GMMs. The Lorenz system is used to validate the correctness and effectiveness of the proposed method for measuring the similarity.
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Affiliation(s)
- Jiancheng Sun
- School of Software and Internet of Things Engineering, Jiangxi University of Finance and Economics, Nanchang 330013, China
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14
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Lainscsek C, Gonzalez CE, Sampson AL, Cash SS, Sejnowski TJ. Causality detection in cortical seizure dynamics using cross-dynamical delay differential analysis. CHAOS (WOODBURY, N.Y.) 2019; 29:101103. [PMID: 31675829 PMCID: PMC6783296 DOI: 10.1063/1.5126125] [Citation(s) in RCA: 5] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [MESH Headings] [Grants] [Track Full Text] [Subscribe] [Scholar Register] [Received: 08/29/2019] [Accepted: 09/17/2019] [Indexed: 05/14/2023]
Abstract
Most natural systems, including the brain, are highly nonlinear and complex, and determining information flow among the components that make up these dynamic systems is challenging. One such example is identifying abnormal causal interactions among different brain areas that give rise to epileptic activities. Here, we introduce cross-dynamical delay differential analysis, an extension of delay differential analysis, as a tool to establish causal relationships from time series signals. Our method can infer causality from short time series signals as well as in the presence of noise. Furthermore, we can determine the onset of generalized synchronization directly from time series data, without having to consult the underlying equations. We first validate our method on simulated datasets from coupled dynamical systems and apply the method to intracranial electroencephalography data obtained from epilepsy patients to better characterize large-scale information flow during epilepsy.
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Affiliation(s)
- Claudia Lainscsek
- Computational Neurobiology Laboratory, The Salk Institute for Biological Studies, 10010 North Torrey Pines Road, La Jolla, California 92037, USA
| | - Christopher E Gonzalez
- Computational Neurobiology Laboratory, The Salk Institute for Biological Studies, 10010 North Torrey Pines Road, La Jolla, California 92037, USA
| | - Aaron L Sampson
- Computational Neurobiology Laboratory, The Salk Institute for Biological Studies, 10010 North Torrey Pines Road, La Jolla, California 92037, USA
| | - Sydney S Cash
- Department of Neurology, Massachusetts General Hospital and Harvard Medical School, Boston, Massachusetts 02114, USA
| | - Terrence J Sejnowski
- Computational Neurobiology Laboratory, The Salk Institute for Biological Studies, 10010 North Torrey Pines Road, La Jolla, California 92037, USA
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