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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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Zimatore G, Gallotta MC, Innocenti L, Bonavolontà V, Ciasca G, De Spirito M, Guidetti L, Baldari C. Recurrence quantification analysis of heart rate variability during continuous incremental exercise test in obese subjects. CHAOS (WOODBURY, N.Y.) 2020; 30:033135. [PMID: 32237785 DOI: 10.1063/1.5140455] [Citation(s) in RCA: 7] [Impact Index Per Article: 1.8] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 11/28/2019] [Accepted: 02/25/2020] [Indexed: 05/20/2023]
Abstract
The present paper concerns a new description of changing in metabolism during incremental exercises test that permit an individually tailored program of exercises for obese subjects. We analyzed heart rate variability from RR interval time series (tachogram) with an alternative approach, the recurrence quantification analysis, that allows a description of a time series in terms of its dynamic structure and is able to identify the phase transitions. A transition in cardiac signal dynamics was detected and it perfectly reflects the aerobic threshold, as identified by gas exchange during an incremental exercise test, revealing the coupling from the respiratory system toward the heart. Moreover, our analysis shows that, in the recurrence plot of RR interval, it is possible to identify a specific pattern that allows to identify phase transitions between different dynamic regimes. The perfect match of the occurrence of the phase transitions with changes observed in the VO2 consumption, the gold standard approach to estimate thresholds, strongly supports the possibility of using our analysis of RR interval to detect metabolic threshold. In conclusion, we propose a novel nonlinear data analysis method that allows for an easy and personalized detection of thresholds both from professional and even from low-cost wearable devices, without the need of expensive gas analyzers.
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Affiliation(s)
- G Zimatore
- Department of Theoretical and Applied Sciences, eCampus University, Novedrate (CO) 22060, Italy
| | - M C Gallotta
- Department of Human, Movement and Health Sciences, University of Rome "Foro Italico," Rome 00135, Italy
| | - L Innocenti
- Department of Human, Movement and Health Sciences, University of Rome "Foro Italico," Rome 00135, Italy
| | - V Bonavolontà
- Department of Basic Medical Sciences, Neuroscience and Sense Organs, University of Bari "Aldo Moro," Bari 70121, Italy
| | - G Ciasca
- Institute of Physics, Fondazione Policlinico Universitario A. Gemelli IRCCS, Università Cattolica del Sacro Cuore Rome 00168, Italy
| | - M De Spirito
- Institute of Physics, Fondazione Policlinico Universitario A. Gemelli IRCCS, Università Cattolica del Sacro Cuore Rome 00168, Italy
| | - L Guidetti
- Department of Human, Movement and Health Sciences, University of Rome "Foro Italico," Rome 00135, Italy
| | - C Baldari
- Department of Theoretical and Applied Sciences, eCampus University, Novedrate (CO) 22060, Italy
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Gao ZK, Li YL, Yang YX, Ma C. A recurrence network-based convolutional neural network for fatigue driving detection from EEG. CHAOS (WOODBURY, N.Y.) 2019; 29:113126. [PMID: 31779352 DOI: 10.1063/1.5120538] [Citation(s) in RCA: 22] [Impact Index Per Article: 4.4] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 07/18/2019] [Accepted: 10/30/2019] [Indexed: 06/10/2023]
Abstract
Driver fatigue is an important cause of traffic accidents, which has triggered great concern for detecting drivers' fatigue. Numerous methods have been proposed to fulfill this challenging task, including feature methods and machine learning methods. Recently, with the development of deep learning techniques, many studies achieved better results than traditional feature methods, and the combination of traditional methods and deep learning techniques gradually received attention. In this paper, we propose a recurrence network-based convolutional neural network (RN-CNN) method to detect fatigue driving. To be specific, we first conduct a simulated driving experiment to collect electroencephalogram (EEG) signals of subjects under alert state and fatigue state. Then, we construct the multiplex recurrence network (RN) from EEG signals to fuse information from the original time series. Finally, CNN is employed to extract and learn the features of a multiplex RN for realizing a classification task. The results indicate that the proposed RN-CNN method can achieve an average accuracy of 92.95%. To verify the effectiveness of our method, some existing competitive methods are compared with ours. The results show that our method outperforms the existing methods, which demonstrate the effect of the RN-CNN method.
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Affiliation(s)
- Zhong-Ke Gao
- School of Electrical and Information Engineering, Tianjin University, Tianjin 300072, China
| | - Yan-Li Li
- School of Electrical and Information Engineering, Tianjin University, Tianjin 300072, China
| | - Yu-Xuan Yang
- School of Electrical and Information Engineering, Tianjin University, Tianjin 300072, China
| | - Chao Ma
- School of Electrical and Information Engineering, Tianjin University, Tianjin 300072, China
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González-Gómez GH, Infante O, Martínez-García P, Lerma C. Analysis of diagonals in cross recurrence plots between heart rate and systolic blood pressure during supine position and active standing in healthy adults. CHAOS (WOODBURY, N.Y.) 2018; 28:085704. [PMID: 30180620 DOI: 10.1063/1.5024685] [Citation(s) in RCA: 6] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 02/02/2018] [Accepted: 05/07/2018] [Indexed: 06/08/2023]
Abstract
The inter beat interval (IBI) duration and systolic blood pressure (SBP) are cardiovascular variables related through several feedback mechanisms. We propose the analysis of diagonal lines in cross recurrence plots (CRPs) from IBI and SBP embedded within the same phase space to identify events where trajectories of both variables concur. The aim of the study was to describe the relationship between IBI and SBP of healthy subjects using CRP and diagonal analysis during baseline condition-supine position (SP)-and how the relationship changes during the physiological stress of active standing (AS). IBI and SBP time series were obtained from continuous blood pressure recordings during SP and AS (15 min each) in 19 young healthy subjects. IBI and SBP time series were embedded within a five-dimensional phase space using an embedding delay estimated from cross correlation between IBI and SBP. During SP, mean CRP showed high determinism (≥85%) and also brief but repeated events where both variables stay within a reduced space. Most quantitative recurrences analysis indexes of CRP increased significantly (p < 0.05) during AS. CRP analysis showed short diagonals indicating a very strong deterministic relationship between IBI and SBP with intermittent unlocking periods. The strength of IBI and SBP relationship increased during the physiological stress of AS. The CRP method allowed a rigorous quantitative description of the deterministic association between these two variables. Diagonal lines were intermittent and not always parallel, showing that there is not a defined and unique rhythm. This suggests the activation of different influences at different times and with different precedence between the heart rate and blood pressure in response to AS.
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Affiliation(s)
| | - Oscar Infante
- Departamento de Instrumentación Electromecánica, Instituto Nacional de Cardiología Ignacio Chávez, 14080 Mexico D.F., Mexico
| | - Paola Martínez-García
- Servicio de Radio-Oncología, Instituto Nacional de Ciencias Médicas y Nutrición Salvador Zubirán, 14080 Mexico D.F., Mexico
| | - Claudia Lerma
- Departamento de Instrumentación Electromecánica, Instituto Nacional de Cardiología Ignacio Chávez, 14080 Mexico D.F., Mexico
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Meng J, Zhao L, Shen F, Yan R. Gear fault diagnosis based on recurrence network. JOURNAL OF INTELLIGENT & FUZZY SYSTEMS 2018. [DOI: 10.3233/jifs-169540] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/15/2022]
Affiliation(s)
- Jing Meng
- School of Instrument Science and Engineering, Southeast University, Nanjing, Jiangsu, China
| | - Liye Zhao
- School of Instrument Science and Engineering, Southeast University, Nanjing, Jiangsu, China
| | - Fei Shen
- School of Instrument Science and Engineering, Southeast University, Nanjing, Jiangsu, China
| | - Ruqiang Yan
- School of Instrument Science and Engineering, Southeast University, Nanjing, Jiangsu, China
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Goswami B, Boers N, Rheinwalt A, Marwan N, Heitzig J, Breitenbach SFM, Kurths J. Abrupt transitions in time series with uncertainties. Nat Commun 2018; 9:48. [PMID: 29298987 PMCID: PMC5752700 DOI: 10.1038/s41467-017-02456-6] [Citation(s) in RCA: 13] [Impact Index Per Article: 2.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 05/26/2017] [Accepted: 12/03/2017] [Indexed: 12/05/2022] Open
Abstract
Identifying abrupt transitions is a key question in various disciplines. Existing transition detection methods, however, do not rigorously account for time series uncertainties, often neglecting them altogether or assuming them to be independent and qualitatively similar. Here, we introduce a novel approach suited to handle uncertainties by representing the time series as a time-ordered sequence of probability density functions. We show how to detect abrupt transitions in such a sequence using the community structure of networks representing probabilities of recurrence. Using our approach, we detect transitions in global stock indices related to well-known periods of politico-economic volatility. We further uncover transitions in the El Niño-Southern Oscillation which coincide with periods of phase locking with the Pacific Decadal Oscillation. Finally, we provide for the first time an 'uncertainty-aware' framework which validates the hypothesis that ice-rafting events in the North Atlantic during the Holocene were synchronous with a weakened Asian summer monsoon.
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Affiliation(s)
- Bedartha Goswami
- Potsdam Institute for Climate Impact Research, Transdisciplinary Concepts & Methods, 14412, Potsdam, Germany.
- Institute of Earth and Environmental Science, University of Potsdam, Karl-Liebknecht Str. 24-25, 14476, Potsdam, Germany.
| | - Niklas Boers
- Potsdam Institute for Climate Impact Research, Transdisciplinary Concepts & Methods, 14412, Potsdam, Germany
- Grantham Institute - Climate Change and the Environment, Imperial College London, South Kensington Campus, London, SW7 2AZ, UK
| | - Aljoscha Rheinwalt
- Potsdam Institute for Climate Impact Research, Transdisciplinary Concepts & Methods, 14412, Potsdam, Germany
- Institute of Earth and Environmental Science, University of Potsdam, Karl-Liebknecht Str. 24-25, 14476, Potsdam, Germany
| | - Norbert Marwan
- Potsdam Institute for Climate Impact Research, Transdisciplinary Concepts & Methods, 14412, Potsdam, Germany
| | - Jobst Heitzig
- Potsdam Institute for Climate Impact Research, Transdisciplinary Concepts & Methods, 14412, Potsdam, Germany
| | - Sebastian F M Breitenbach
- Sediment and Isotope Geology, Institute for Geology, Mineralogy & Geophysics, Ruhr-Universität Bochum, Universitätsstr. 150, 44801, Bochum, Germany
| | - Jürgen Kurths
- Potsdam Institute for Climate Impact Research, Transdisciplinary Concepts & Methods, 14412, Potsdam, Germany
- Department of Physics, Humboldt-Universität zu Berlin, Newtonstr. 15, 12489, Berlin, Germany
- Saratov State University, 83 Astrakhanskaya Street, Saratov, 410012, Russia
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Eroglu D, Marwan N, Stebich M, Kurths J. Multiplex recurrence networks. Phys Rev E 2018; 97:012312. [PMID: 29448424 DOI: 10.1103/physreve.97.012312] [Citation(s) in RCA: 15] [Impact Index Per Article: 2.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 07/13/2017] [Indexed: 06/08/2023]
Abstract
We have introduced a multiplex recurrence network approach by combining recurrence networks with the multiplex network approach in order to investigate multivariate time series. The potential use of this approach is demonstrated on coupled map lattices and a typical example from palaeobotany research. In both examples, topological changes in the multiplex recurrence networks allow for the detection of regime changes in their dynamics. The method goes beyond classical interpretation of pollen records by considering the vegetation as a whole and using the intrinsic similarity in the dynamics of the different regional vegetation elements. We find that the different vegetation types behave more similarly when one environmental factor acts as the dominant driving force.
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Affiliation(s)
- Deniz Eroglu
- Potsdam Institute for Climate Impact Research (PIK), Potsdam 14473, Germany
- Department of Physics, Humboldt University, 12489 Berlin, Germany
| | - Norbert Marwan
- Potsdam Institute for Climate Impact Research (PIK), Potsdam 14473, Germany
| | - Martina Stebich
- Senckenberg Research Station of Quaternary Palaeontology Weimar, Am Jakobskirchhof 4, Weimar 99423, Germany
| | - Jürgen Kurths
- Potsdam Institute for Climate Impact Research (PIK), Potsdam 14473, Germany
- Department of Physics, Humboldt University, 12489 Berlin, Germany
- Institute for Complex Systems and Mathematical Biology, University of Aberdeen, Aberdeen AB24 3UE, United Kingdom
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Scarsoglio S, Cazzato F, Ridolfi L. From time-series to complex networks: Application to the cerebrovascular flow patterns in atrial fibrillation. CHAOS (WOODBURY, N.Y.) 2017; 27:093107. [PMID: 28964131 DOI: 10.1063/1.5003791] [Citation(s) in RCA: 7] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/07/2023]
Abstract
A network-based approach is presented to investigate the cerebrovascular flow patterns during atrial fibrillation (AF) with respect to normal sinus rhythm (NSR). AF, the most common cardiac arrhythmia with faster and irregular beating, has been recently and independently associated with the increased risk of dementia. However, the underlying hemodynamic mechanisms relating the two pathologies remain mainly undetermined so far; thus, the contribution of modeling and refined statistical tools is valuable. Pressure and flow rate temporal series in NSR and AF are here evaluated along representative cerebral sites (from carotid arteries to capillary brain circulation), exploiting reliable artificially built signals recently obtained from an in silico approach. The complex network analysis evidences, in a synthetic and original way, a dramatic signal variation towards the distal/capillary cerebral regions during AF, which has no counterpart in NSR conditions. At the large artery level, networks obtained from both AF and NSR hemodynamic signals exhibit elongated and chained features, which are typical of pseudo-periodic series. These aspects are almost completely lost towards the microcirculation during AF, where the networks are topologically more circular and present random-like characteristics. As a consequence, all the physiological phenomena at the microcerebral level ruled by periodicity-such as regular perfusion, mean pressure per beat, and average nutrient supply at the cellular level-can be strongly compromised, since the AF hemodynamic signals assume irregular behaviour and random-like features. Through a powerful approach which is complementary to the classical statistical tools, the present findings further strengthen the potential link between AF hemodynamic and cognitive decline.
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Affiliation(s)
- Stefania Scarsoglio
- Department of Mechanical and Aerospace Engineering, Politecnico di Torino, Torino, Italy
| | - Fabio Cazzato
- Medacta International SA, Castel San Pietro, Switzerland
| | - Luca Ridolfi
- Department of Environmental, Land and Infrastructure Engineering, Politecnico di Torino, Torino, Italy
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Jacob R, Harikrishnan KP, Misra R, Ambika G. Measure for degree heterogeneity in complex networks and its application to recurrence network analysis. ROYAL SOCIETY OPEN SCIENCE 2017; 4:160757. [PMID: 28280579 PMCID: PMC5319345 DOI: 10.1098/rsos.160757] [Citation(s) in RCA: 10] [Impact Index Per Article: 1.4] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Subscribe] [Scholar Register] [Received: 09/28/2016] [Accepted: 12/05/2016] [Indexed: 05/13/2023]
Abstract
We propose a novel measure of degree heterogeneity, for unweighted and undirected complex networks, which requires only the degree distribution of the network for its computation. We show that the proposed measure can be applied to all types of network topology with ease and increases with the diversity of node degrees in the network. The measure is applied to compute the heterogeneity of synthetic (both random and scale free (SF)) and real-world networks with its value normalized in the interval [Formula: see text]. To define the measure, we introduce a limiting network whose heterogeneity can be expressed analytically with the value tending to 1 as the size of the network N tends to infinity. We numerically study the variation of heterogeneity for random graphs (as a function of p and N) and for SF networks with γ and N as variables. Finally, as a specific application, we show that the proposed measure can be used to compare the heterogeneity of recurrence networks constructed from the time series of several low-dimensional chaotic attractors, thereby providing a single index to compare the structural complexity of chaotic attractors.
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Affiliation(s)
- Rinku Jacob
- Department of Physics, The Cochin College, Cochin 682 002, India
| | - K. P. Harikrishnan
- Department of Physics, The Cochin College, Cochin 682 002, India
- Author for correspondence: K. P. Harikrishnan e-mail:
| | - R. Misra
- Inter University Centre for Astronomy and Astrophysics, Pune 411 007, India
| | - G. Ambika
- Indian Institute of Science Education and Research, Pune 411 008, India
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Assaf D, Amar E, Marwan N, Neuman Y, Salai M, Rath E. Dynamic Patterns of Expertise: The Case of Orthopedic Medical Diagnosis. PLoS One 2016; 11:e0158820. [PMID: 27414794 PMCID: PMC4945032 DOI: 10.1371/journal.pone.0158820] [Citation(s) in RCA: 5] [Impact Index Per Article: 0.6] [Reference Citation Analysis] [Abstract] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 01/28/2016] [Accepted: 06/22/2016] [Indexed: 12/02/2022] Open
Abstract
The aim of this study was to analyze dynamic patterns for scanning femoroacetabular impingement (FAI) radiographs in orthopedics, in order to better understand the nature of expertise in radiography. Seven orthopedics residents with at least two years of expertise and seven board-certified orthopedists participated in the study. The participants were asked to diagnose 15 anteroposterior (AP) pelvis radiographs of 15 surgical patients, diagnosed with FAI syndrome. Eye tracking data were recorded using the SMI desk-mounted tracker and were analyzed using advanced measures and methodologies, mainly recurrence quantification analysis. The expert orthopedists presented a less predictable pattern of scanning the radiographs although there was no difference between experts and non-experts in the deterministic nature of their scan path. In addition, the experts presented a higher percentage of correct areas of focus and more quickly made their first comparison between symmetric regions of the pelvis. We contribute to the understanding of experts’ process of diagnosis by showing that experts are qualitatively different from residents in their scanning patterns. The dynamic pattern of scanning that characterizes the experts was found to have a more complex and less predictable signature, meaning that experts’ scanning is simultaneously both structured (i.e. deterministic) and unpredictable.
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Affiliation(s)
- Dan Assaf
- Sackler Faculty of Medicine, Tel Aviv University, Tel Aviv, Israel
| | - Eyal Amar
- Sackler Faculty of Medicine, Tel Aviv University, Tel Aviv, Israel.,Orthopedics Division, Tel Aviv Sourasky Medical Center, 6 Weizmann Street, Tel Aviv, Israel
| | - Norbert Marwan
- Potsdam Institute for Climate Impact Research, Potsdam, Germany
| | - Yair Neuman
- Department of Education, Homeland Security Institute, Center for the Study of Conversion and Inter-Religious Encounters, Ben-Gurion University of the Negev, Beer-Sheva, Israel
| | - Moshe Salai
- Sackler Faculty of Medicine, Tel Aviv University, Tel Aviv, Israel.,Orthopedics Division, Tel Aviv Sourasky Medical Center, 6 Weizmann Street, Tel Aviv, Israel
| | - Ehud Rath
- Sackler Faculty of Medicine, Tel Aviv University, Tel Aviv, Israel.,Orthopedics Division, Tel Aviv Sourasky Medical Center, 6 Weizmann Street, Tel Aviv, Israel
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Measuring Electromechanical Coupling in Patients with Coronary Artery Disease and Healthy Subjects. ENTROPY 2016. [DOI: 10.3390/e18040153] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.4] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 11/17/2022]
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Jacob R, Harikrishnan KP, Misra R, Ambika G. Uniform framework for the recurrence-network analysis of chaotic time series. Phys Rev E 2016; 93:012202. [PMID: 26871068 DOI: 10.1103/physreve.93.012202] [Citation(s) in RCA: 6] [Impact Index Per Article: 0.8] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 07/17/2015] [Indexed: 05/27/2023]
Abstract
We propose a general method for the construction and analysis of unweighted ε-recurrence networks from chaotic time series. The selection of the critical threshold ε_{c} in our scheme is done empirically and we show that its value is closely linked to the embedding dimension M. In fact, we are able to identify a small critical range Δε numerically that is approximately the same for the random and several standard chaotic time series for a fixed M. This provides us a uniform framework for the nonsubjective comparison of the statistical measures of the recurrence networks constructed from various chaotic attractors. We explicitly show that the degree distribution of the recurrence network constructed by our scheme is characteristic to the structure of the attractor and display statistical scale invariance with respect to increase in the number of nodes N. We also present two practical applications of the scheme, detection of transition between two dynamical regimes in a time-delayed system and identification of the dimensionality of the underlying system from real-world data with a limited number of points through recurrence network measures. The merits, limitations, and the potential applications of the proposed method are also highlighted.
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Affiliation(s)
- Rinku Jacob
- Department of Physics, The Cochin College, Cochin-682 002, India
| | - K P Harikrishnan
- Department of Physics, The Cochin College, Cochin-682 002, India
| | - R Misra
- Inter University Centre for Astronomy and Astrophysics, Pune-411 007, India
| | - G Ambika
- Indian Institute of Science Education and Research, Pune-411 008, India
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Porta A, Faes L. Assessing causality in brain dynamics and cardiovascular control. PHILOSOPHICAL TRANSACTIONS. SERIES A, MATHEMATICAL, PHYSICAL, AND ENGINEERING SCIENCES 2013; 371:20120517. [PMID: 23858491 PMCID: PMC5397300 DOI: 10.1098/rsta.2012.0517] [Citation(s) in RCA: 22] [Impact Index Per Article: 2.0] [Reference Citation Analysis] [Key Words] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/22/2023]
Affiliation(s)
- Alberto Porta
- Department of Biomedical Sciences for Health, Galeazzi Orthopaedic Institute, University of Milan, 20161 Milan, Italy.
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