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Angelidis AK, Goulas K, Bratsas C, Makris GC, Hanias MP, Stavrinides SG, Antoniou IE. Distinction of Chaos from Randomness Is Not Possible from the Degree Distribution of the Visibility and Phase Space Reconstruction Graphs. ENTROPY (BASEL, SWITZERLAND) 2024; 26:341. [PMID: 38667895 PMCID: PMC11048845 DOI: 10.3390/e26040341] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Subscribe] [Scholar Register] [Received: 03/21/2024] [Revised: 04/11/2024] [Accepted: 04/16/2024] [Indexed: 04/28/2024]
Abstract
We investigate whether it is possible to distinguish chaotic time series from random time series using network theory. In this perspective, we selected four methods to generate graphs from time series: the natural, the horizontal, the limited penetrable horizontal visibility graph, and the phase space reconstruction method. These methods claim that the distinction of chaos from randomness is possible by studying the degree distribution of the generated graphs. We evaluated these methods by computing the results for chaotic time series from the 2D Torus Automorphisms, the chaotic Lorenz system, and a random sequence derived from the normal distribution. Although the results confirm previous studies, we found that the distinction of chaos from randomness is not generally possible in the context of the above methodologies.
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Affiliation(s)
- Alexandros K. Angelidis
- Department of Mathematics, Aristotle University of Thessaloniki, 54124 Thessaloniki, Greece; (K.G.); (G.C.M.); (I.E.A.)
- Department of Information and Electronic Engineering, International Hellenic University, 57400 Thessaloniki, Greece;
| | - Konstantinos Goulas
- Department of Mathematics, Aristotle University of Thessaloniki, 54124 Thessaloniki, Greece; (K.G.); (G.C.M.); (I.E.A.)
| | - Charalampos Bratsas
- Department of Information and Electronic Engineering, International Hellenic University, 57400 Thessaloniki, Greece;
| | - Georgios C. Makris
- Department of Mathematics, Aristotle University of Thessaloniki, 54124 Thessaloniki, Greece; (K.G.); (G.C.M.); (I.E.A.)
| | - Michael P. Hanias
- Department of Physics, International Hellenic University, 65404 Kavala, Greece; (M.P.H.); (S.G.S.)
| | - Stavros G. Stavrinides
- Department of Physics, International Hellenic University, 65404 Kavala, Greece; (M.P.H.); (S.G.S.)
| | - Ioannis E. Antoniou
- Department of Mathematics, Aristotle University of Thessaloniki, 54124 Thessaloniki, Greece; (K.G.); (G.C.M.); (I.E.A.)
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Telesca L, Czechowski Z. Fisher-Shannon Investigation of the Effect of Nonlinearity of Discrete Langevin Model on Behavior of Extremes in Generated Time Series. ENTROPY (BASEL, SWITZERLAND) 2023; 25:1650. [PMID: 38136530 PMCID: PMC10742732 DOI: 10.3390/e25121650] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Grants] [Track Full Text] [Subscribe] [Scholar Register] [Received: 11/09/2023] [Revised: 12/05/2023] [Accepted: 12/11/2023] [Indexed: 12/24/2023]
Abstract
Diverse forms of nonlinearity within stochastic equations give rise to varying dynamics in processes, which may influence the behavior of extreme values. This study focuses on two nonlinear models of the discrete Langevin equation: one with a fixed diffusion function (M1) and the other with a fixed marginal distribution (M2), both characterized by a nonlinearity parameter. Extremes are defined according to the run theory with thresholds based on percentiles. The behavior of inter-extreme times and run lengths is examined by employing Fisher's Information Measure and the Shannon Entropy. Our findings reveal a clear relationship between the entropic and informational measures and the nonlinearity of model M1-these measures decrease as the nonlinearity parameter increases. Similar relationships are evident for the M2 model, albeit to a lesser extent, even though the background data's marginal distribution remains unaffected by this parameter. As thresholds increase, both the values of Fisher's Information Measure and the Shannon Entropy also increase.
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Affiliation(s)
- Luciano Telesca
- Institute of Methodologies for Environmental Analysis, National Research Council, 85050 Tito, Italy
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Zurita-Valencia T, Muñoz V. Characterizing the Solar Activity Using the Visibility Graph Method. ENTROPY (BASEL, SWITZERLAND) 2023; 25:342. [PMID: 36832708 PMCID: PMC9955573 DOI: 10.3390/e25020342] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Figures] [Subscribe] [Scholar Register] [Received: 12/28/2022] [Revised: 02/03/2023] [Accepted: 02/09/2023] [Indexed: 06/18/2023]
Abstract
In this paper, the Sun and its behavior are studied by means of complex networks. The complex network was built using the Visibility Graph algorithm. This method maps time series into graphs in which every element of the time series is considered as a node and a visibility criterion is defined in order to connect them. Using this method, we construct complex networks for magnetic field and sunspots time series encompassing four solar cycles, and various measures such as degree, clustering coefficient, mean path length, betweenness centrality, eigenvector centrality and decay exponents were calculated. In order to study the system in several time scales, we perform both a global, where the network contains information on the four solar cycles, and a local analysis, involving moving windows. Some metrics correlate with solar activity, while others do not. Interestingly, those metric which seem to respond to varying levels of solar activity in the global analysis, also do in the moving windows analysis. Our results suggest that complex networks can provide a useful way to follow solar activity, and reveal new features on solar cycles.
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Wen T, Chen H, Cheong KH. Visibility graph for time series prediction and image classification: a review. NONLINEAR DYNAMICS 2022; 110:2979-2999. [PMID: 36339319 PMCID: PMC9628348 DOI: 10.1007/s11071-022-08002-4] [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: 05/29/2022] [Accepted: 10/09/2022] [Indexed: 06/16/2023]
Abstract
The analysis of time series and images is significant across different fields due to their widespread applications. In the past few decades, many approaches have been developed, including data-driven artificial intelligence methods, mechanism-driven physical methods, and hybrid mechanism and data-driven models. Complex networks have been used to model numerous complex systems due to its characteristics, including time series prediction and image classification. In order to map time series and images into complex networks, many visibility graph algorithms have been developed, such as horizontal visibility graph, limited penetrable visibility graph, multiplex visibility graph, and image visibility graph. The family of visibility graph algorithms will construct different types of complex networks, including (un-) weighted, (un-) directed, and (single-) multi-layered networks, thereby focusing on different kinds of properties. Different types of visibility graph algorithms will be reviewed in this paper. Through exploring the topological structure and information in the network based on statistical physics, the property of time series and images can be discovered. In order to forecast (multivariate) time series, several variations of local random walk algorithms and different information fusion approaches are applied to measure the similarity between nodes in the network. Different forecasting frameworks are also proposed to consider the information in the time series based on the similarity. In order to classify the image, several machine learning models (such as support vector machine and linear discriminant) are used to classify images based on global features, local features, and multiplex features. Through various simulations on a variety of datasets, researchers have found that the visibility graph algorithm outperformed existing algorithms, both in time series prediction and image classification. Clearly, complex networks are closely connected with time series and images by visibility graph algorithms, rendering complex networks to be an important tool for understanding the characteristics of time series and images. Finally, we conclude in the last section with future outlooks for the visibility graph.
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Affiliation(s)
- Tao Wen
- Science, Mathematics and Technology Cluster, Singapore University of Technology and Design (SUTD), Singapore, 487372 Singapore
| | - Huiling Chen
- Department of Computer Science and Artificial Intelligence, Wenzhou University, Wenzhou, 325035 China
| | - Kang Hao Cheong
- Science, Mathematics and Technology Cluster, Singapore University of Technology and Design (SUTD), Singapore, 487372 Singapore
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5
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Assigning Degrees of Stochasticity to Blazar Light Curves in the Radio Band Using Complex Networks. ENTROPY 2022; 24:e24081063. [PMID: 36010727 PMCID: PMC9407290 DOI: 10.3390/e24081063] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Download PDF] [Figures] [Subscribe] [Scholar Register] [Received: 03/31/2022] [Revised: 04/27/2022] [Accepted: 05/01/2022] [Indexed: 11/16/2022]
Abstract
We focus on characterizing the high-energy emission mechanisms of blazars by analyzing the variability in the radio band of the light curves of more than a thousand sources. We are interested in assigning complexity parameters to these sources, modeling the time series of the light curves with the method of the Horizontal Visibility Graph (HVG), which allows us to obtain properties from degree distributions, such as a characteristic exponent to describe its stochasticity and the Kullback–Leibler Divergence (KLD), presenting a new perspective to the methods commonly used to study Active Galactic Nuclei (AGN). We contrast these parameters with the excess variance, which is an astronomical measurement of variability in light curves; at the same time, we use the spectral classification of the sources. While it is not possible to find significant correlations with the excess variance, the degree distributions extracted from the network are detecting differences related to the spectral classification of blazars. These differences suggest a chaotic behavior in the time series for the BL Lac sources and a correlated stochastic behavior in the time series for the FSRQ sources. Our results show that complex networks may be a valuable alternative tool to study AGNs according to the variability of their energy output.
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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: 8] [Impact Index Per Article: 4.0] [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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Dylewsky D, Kaiser E, Brunton SL, Kutz JN. Principal component trajectories for modeling spectrally continuous dynamics as forced linear systems. Phys Rev E 2022; 105:015312. [PMID: 35193205 DOI: 10.1103/physreve.105.015312] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 08/25/2021] [Accepted: 01/07/2022] [Indexed: 05/08/2023]
Abstract
Delay embeddings of time series data have emerged as a promising coordinate basis for data-driven estimation of the Koopman operator, which seeks a linear representation for observed nonlinear dynamics. Recent work has demonstrated the efficacy of dynamic mode decomposition (DMD) for obtaining finite-dimensional Koopman approximations in delay coordinates. In this paper we demonstrate how nonlinear dynamics with sparse Fourier spectra can be (i) represented by a superposition of principal component trajectories and (ii) modeled by DMD in this coordinate space. For continuous or mixed (discrete and continuous) spectra, DMD can be augmented with an external forcing term. We present a method for learning linear control models in delay coordinates while simultaneously discovering the corresponding exogenous forcing signal in a fully unsupervised manner. This extends the existing DMD with control (DMDc) algorithm to cases where a control signal is not known a priori. We provide examples to validate the learned forcing against a known ground truth and illustrate their statistical similarity. Finally, we offer a demonstration of this method applied to real-world power grid load data to show its utility for diagnostics and interpretation on systems in which somewhat periodic behavior is strongly forced by unknown and unmeasurable environmental variables.
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Affiliation(s)
- Daniel Dylewsky
- Department of Physics, University of Washington, Seattle, Washington 98195, USA
| | - Eurika Kaiser
- Department of Mechanical Engineering, University of Washington, Seattle, Washington 98195, USA
| | | | - J Nathan Kutz
- Department of Applied Mathematics, University of Washington, Seattle, Washington 98195, USA
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9
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Muñoz V, Garcés NE. Analysis of pulsating variable stars using the visibility graph algorithm. PLoS One 2021; 16:e0259735. [PMID: 34788340 PMCID: PMC8598052 DOI: 10.1371/journal.pone.0259735] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [MESH Headings] [Grants] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 04/20/2021] [Accepted: 10/25/2021] [Indexed: 11/19/2022] Open
Abstract
We study the light curves of pulsating variable stars using a complex network approach to build visibility graphs. We consider various types of variables stars (e.g., Cepheids, δ Scuti, RR Lyrae), build two types of graphs (the normal visibility graph (VG) and the horizontal visibility graph (HVG)), and calculate various metrics for the resulting networks. We find that all networks have a power-law degree distribution for the VG and an exponential distribution for the HVG, suggesting that it is a universal feature, regardless of the pulsation features. Metrics such as the average degree, the clustering coefficient and the transitivity coefficient, can distinguish between some star types. We also observe that the results are not strongly affected by the presence of observation gaps in the light curves. These findings suggest that the visibility graph algorithm may be a useful technique to study variability in stars.
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Affiliation(s)
- Víctor Muñoz
- Departamento de Física, Facultad de Ciencias, Universidad de Chile, Santiago, Chile
- * E-mail:
| | - N. Elizabeth Garcés
- Departamento de Física, Facultad de Ciencias, Universidad de Chile, Santiago, Chile
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Boaretto BRR, Budzinski RC, Rossi KL, Prado TL, Lopes SR, Masoller C. Evaluating Temporal Correlations in Time Series Using Permutation Entropy, Ordinal Probabilities and Machine Learning. ENTROPY 2021; 23:e23081025. [PMID: 34441165 PMCID: PMC8391825 DOI: 10.3390/e23081025] [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: 07/08/2021] [Revised: 07/28/2021] [Accepted: 08/03/2021] [Indexed: 11/16/2022]
Abstract
Time series analysis comprises a wide repertoire of methods for extracting information from data sets. Despite great advances in time series analysis, identifying and quantifying the strength of nonlinear temporal correlations remain a challenge. We have recently proposed a new method based on training a machine learning algorithm to predict the temporal correlation parameter, α, of flicker noise (FN) time series. The algorithm is trained using as input features the probabilities of ordinal patterns computed from FN time series, xαFN(t), generated with different values of α. Then, the ordinal probabilities computed from the time series of interest, x(t), are used as input features to the trained algorithm and that returns a value, αe, that contains meaningful information about the temporal correlations present in x(t). We have also shown that the difference, Ω, of the permutation entropy (PE) of the time series of interest, x(t), and the PE of a FN time series generated with α=αe, xαeFN(t), allows the identification of the underlying determinism in x(t). Here, we apply our methodology to different datasets and analyze how αe and Ω correlate with well-known quantifiers of chaos and complexity. We also discuss the limitations for identifying determinism in highly chaotic time series and in periodic time series contaminated by noise. The open source algorithm is available on Github.
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Affiliation(s)
- Bruno R. R. Boaretto
- Department of Physics, Universidade Federal do Paraná, Curitiba 81531-980, Brazil; (B.R.R.B.); (T.L.P.); (S.R.L.)
| | - Roberto C. Budzinski
- Department of Mathematics, Western University, London, ON N6A 3K7, Canada;
- Brain and Mind Institute, Western University, London, ON N6A 3K7, Canada
| | - Kalel L. Rossi
- Theoretical Physics/Complex Systems, ICBM, Carl von Ossietzky University Oldenburg, 26129 Oldenburg, Germany;
| | - Thiago L. Prado
- Department of Physics, Universidade Federal do Paraná, Curitiba 81531-980, Brazil; (B.R.R.B.); (T.L.P.); (S.R.L.)
| | - Sergio R. Lopes
- Department of Physics, Universidade Federal do Paraná, Curitiba 81531-980, Brazil; (B.R.R.B.); (T.L.P.); (S.R.L.)
| | - Cristina Masoller
- Department of Physics, Universitat Politecnica de Catalunya, 08034 Barcelona, Spain
- Correspondence:
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11
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Boaretto BRR, Budzinski RC, Rossi KL, Prado TL, Lopes SR, Masoller C. Discriminating chaotic and stochastic time series using permutation entropy and artificial neural networks. Sci Rep 2021; 11:15789. [PMID: 34349134 PMCID: PMC8338970 DOI: 10.1038/s41598-021-95231-z] [Citation(s) in RCA: 6] [Impact Index Per Article: 2.0] [Reference Citation Analysis] [Abstract] [Key Words] [Grants] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 03/23/2021] [Accepted: 07/15/2021] [Indexed: 02/07/2023] Open
Abstract
Extracting relevant properties of empirical signals generated by nonlinear, stochastic, and high-dimensional systems is a challenge of complex systems research. Open questions are how to differentiate chaotic signals from stochastic ones, and how to quantify nonlinear and/or high-order temporal correlations. Here we propose a new technique to reliably address both problems. Our approach follows two steps: first, we train an artificial neural network (ANN) with flicker (colored) noise to predict the value of the parameter, [Formula: see text], that determines the strength of the correlation of the noise. To predict [Formula: see text] the ANN input features are a set of probabilities that are extracted from the time series by using symbolic ordinal analysis. Then, we input to the trained ANN the probabilities extracted from the time series of interest, and analyze the ANN output. We find that the [Formula: see text] value returned by the ANN is informative of the temporal correlations present in the time series. To distinguish between stochastic and chaotic signals, we exploit the fact that the difference between the permutation entropy (PE) of a given time series and the PE of flicker noise with the same [Formula: see text] parameter is small when the time series is stochastic, but it is large when the time series is chaotic. We validate our technique by analysing synthetic and empirical time series whose nature is well established. We also demonstrate the robustness of our approach with respect to the length of the time series and to the level of noise. We expect that our algorithm, which is freely available, will be very useful to the community.
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Affiliation(s)
- B R R Boaretto
- Department of Physics, Universidade Federal do Paraná, Curitiba, 81531-980, Brazil
| | - R C Budzinski
- Department of Physics, Universidade Federal do Paraná, Curitiba, 81531-980, Brazil
| | - K L Rossi
- Department of Physics, Universidade Federal do Paraná, Curitiba, 81531-980, Brazil
| | - T L Prado
- Department of Physics, Universidade Federal do Paraná, Curitiba, 81531-980, Brazil
| | - S R Lopes
- Department of Physics, Universidade Federal do Paraná, Curitiba, 81531-980, Brazil
| | - C Masoller
- Department of Physics, Universitat Politecnica de Catalunya, 08222, Barcelona, Spain.
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Mansour T, Rastegar R, Roitershtein A. Horizontal visibility graph of a random restricted growth sequence. ADVANCES IN APPLIED MATHEMATICS 2021; 124:102145. [PMID: 33664536 PMCID: PMC7929487 DOI: 10.1016/j.aam.2020.102145] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Grants] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/12/2023]
Abstract
We study the distributional properties of horizontal visibility graphs associated with random restrictive growth sequences and random set partitions of size n. Our main results are formulas expressing the expected degree of graph nodes in terms of simple explicit functions of a finite collection of Stirling and Bernoulli numbers.
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Affiliation(s)
- Toufik Mansour
- Department of Mathematics, University of Haifa, 199 Abba Khoushy Ave, 3498838 Haifa, Israel
| | - Reza Rastegar
- Occidental Petroleum Corporation, Houston, TX 77046 and Departments of Mathematics and Engineering, University of Tulsa, OK 74104, USA
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Analysis of Air Mean Temperature Anomalies by Using Horizontal Visibility Graphs. ENTROPY 2021; 23:e23020207. [PMID: 33567715 PMCID: PMC7915483 DOI: 10.3390/e23020207] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Download PDF] [Figures] [Subscribe] [Scholar Register] [Received: 01/21/2021] [Revised: 02/04/2021] [Accepted: 02/04/2021] [Indexed: 11/29/2022]
Abstract
The last decades have been successively warmer at the Earth’s surface. An increasing interest in climate variability is appearing, and many research works have investigated the main effects on different climate variables. Some of them apply complex networks approaches to explore the spatial relation between distinct grid points or stations. In this work, the authors investigate whether topological properties change over several years. To this aim, we explore the application of the horizontal visibility graph (HVG) approach which maps a time series into a complex network. Data used in this study include a 60-year period of daily mean temperature anomalies in several stations over the Iberian Peninsula (Spain). Average degree, degree distribution exponent, and global clustering coefficient were analyzed. Interestingly, results show that they agree on a lack of significant trends, unlike annual mean values of anomalies, which present a characteristic upward trend. The main conclusions obtained are that complex networks structures and nonlinear features, such as weak correlations, appear not to be affected by rising temperatures derived from global climate conditions. Furthermore, different locations present a similar behavior and the intrinsic nature of these signals seems to be well described by network parameters.
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Mukhopadhyay S, Banerjee S. Learning dynamical systems in noise using convolutional neural networks. CHAOS (WOODBURY, N.Y.) 2020; 30:103125. [PMID: 33138462 DOI: 10.1063/5.0009326] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/09/2023]
Abstract
The problem of distinguishing deterministic chaos from non-chaotic dynamics has been an area of active research in time series analysis. Since noise contamination is unavoidable, it renders deterministic chaotic dynamics corrupted by noise to appear in close resemblance to stochastic dynamics. As a result, the problem of distinguishing noise-corrupted chaotic dynamics from randomness based on observations without access to the measurements of the state variables is difficult. We propose a new angle to tackle this problem by formulating it as a multi-class classification task. The task of classification involves allocating the observations/measurements to the unknown state variables in order to find the nature of these unobserved internal state variables. We employ signal and image processing based methods to characterize the different system dynamics. A deep learning technique using a state-of-the-art image classifier known as the Convolutional Neural Network (CNN) is designed to learn the dynamics. The time series are transformed into textured images of spectrogram and unthresholded recurrence plot (UTRP) for learning stochastic and deterministic chaotic dynamical systems in noise. We have designed a CNN that learns the dynamics of systems from the joint representation of the textured patterns from these images, thereby solving the problem as a pattern recognition task. The robustness and scalability of our approach is evaluated at different noise levels. Our approach demonstrates the advantage of applying the dynamical properties of chaotic systems in the form of joint representation of UTRP images along with spectrogram to improve learning dynamical systems in colored noise.
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Affiliation(s)
- Sumona Mukhopadhyay
- Electrical Engineering and Computer Science, York University, 4700 Keele St, Toronto M3J 1P3, Canada
| | - Santo Banerjee
- Department of Mathematical Sciences, Politecnico di Torino, Corso Duca degli Abruzzi, 24, Torino 10129, Italy
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Colonna JG, Carvalho JRH, Rosso OA. Estimating ecoacoustic activity in the Amazon rainforest through Information Theory quantifiers. PLoS One 2020; 15:e0229425. [PMID: 32716981 PMCID: PMC7384625 DOI: 10.1371/journal.pone.0229425] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 02/01/2020] [Accepted: 06/26/2020] [Indexed: 11/19/2022] Open
Abstract
Automatic monitoring of biodiversity by acoustic sensors has become an indispensable tool to assess environmental stress at an early stage. Due to the difficulty in recognizing the Amazon's high acoustic diversity and the large amounts of raw audio data recorded by the sensors, the labeling and manual inspection of this data is not feasible. Therefore, we propose an ecoacoustic index that allows us to quantify the complexity of an audio segment and correlate this measure with the biodiversity of the soundscape. The approach uses unsupervised methods to avoid the problem of labeling each species individually. The proposed index, named the Ecoacoustic Global Complexity Index (EGCI), makes use of Entropy, Divergence and Statistical Complexity. A distinguishing feature of this index is the mapping of each audio segment, including those of varied lengths, as a single point in a 2D-plane, supporting us in understanding the ecoacoustic dynamics of the rainforest. The main results show a regularity in the ecoacoustic richness of a floodplain, considering different temporal granularities, be it between hours of the day or between consecutive days of the monitoring program. We observed that this regularity does a good job of characterizing the soundscape of the environmental protection area of Mamirauá, in the Amazon, differentiating between species richness and environmental phenomena.
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Affiliation(s)
- Juan G. Colonna
- Instituto de Computação (IComp), Universidade Federal do Amazonas (UFAM), Manaus, Amazonas, Brasil
| | - José R. H. Carvalho
- Instituto de Computação (IComp), Universidade Federal do Amazonas (UFAM), Manaus, Amazonas, Brasil
| | - Osvaldo A. Rosso
- Instituto de Física, Universidade Federal de Alagoas (UFAL), Maceió, Alagoas, Brasil
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16
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Wang Y, Shang P. Analysis of Shannon-Fisher information plane in time series based on information entropy. CHAOS (WOODBURY, N.Y.) 2018; 28:103107. [PMID: 30384635 DOI: 10.1063/1.5023031] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 01/20/2018] [Accepted: 09/26/2018] [Indexed: 06/08/2023]
Abstract
In this paper, we propose a Shannon-Fisher information plane based on the information entropy to analyze financial stock markets. In order to evaluate the effectiveness of this method, we apply this method to two types of artificial time series: Autoregressive Fractionally Integrated Moving Average models and Chebyshev map model. The results show that with the embedding dimension m and the number of possible states of the system M increasing, the normalized Shannon entropy increases, and the Fisher information measure (FIM) decreases. When the parameter M is not so big, the embedding dimension m plays a leading role in determining the FIM. In addition, compared with the classical Shannon-Fisher information through permutation entropy, we conclude that the proposed approach can give us more accurate information on the classification of financial stock markets.
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Affiliation(s)
- Yuanyuan Wang
- Department of Mathematics, School of Science, Beijing Jiaotong University, Beijing 100044, People's Republic of China
| | - Pengjian Shang
- Department of Mathematics, School of Science, Beijing Jiaotong University, Beijing 100044, People's Republic of China
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17
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Lange H, Sippel S, Rosso OA. Nonlinear dynamics of river runoff elucidated by horizontal visibility graphs. CHAOS (WOODBURY, N.Y.) 2018; 28:075520. [PMID: 30070506 DOI: 10.1063/1.5026491] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 02/20/2018] [Accepted: 06/19/2018] [Indexed: 06/08/2023]
Abstract
Horizontal Visibility Graphs (HVGs) are a recently developed method to construct networks from time series. The values of the time series are considered as the nodes of the network and are linked to each other if there is no larger value between them, such as they can "see" each other. The network properties reflect the nonlinear dynamics of the time series. For some classes of stochastic processes and for periodic time series, analytical results can be obtained for network-derived quantities such as the degree distribution, the local clustering coefficient distribution, the mean path length, and others. HVGs have the potential to discern between deterministic-chaotic and correlated-stochastic time series. Here, we investigate the sensitivity of the HVG methodology to properties and pre-processing of real-world data, i.e., time series length, the presence of ties, and deseasonalization, using a set of around 150 runoff time series from managed rivers at daily resolution from Brazil with an average length of 65 years. We show that an application of HVGs on real-world time series requires a careful consideration of data pre-processing steps and analysis methodology before robust results and interpretations can be obtained. For example, one recent analysis of the degree distribution of runoff records reported pronounced sub-exponential "long-tailed" behavior of North American rivers, whereas another study of South American rivers showed hyper-exponential "short-tailed" behavior resembling correlated noise. We demonstrate, using the dataset of Brazilian rivers, that these apparently contradictory results can be reconciled by minor differences in data-preprocessing (here: small differences in subtracting the seasonal cycle). Hence, data-preprocessing that is conventional in hydrology ("deseasonalization") changes long-term correlations and the overall runoff dynamics substantially, and we present empirical consequences and extensive simulations to investigate these issues from a HVG methodological perspective. After carefully accounting for these methodological aspects, the HVG analysis reveals that the river runoff dataset shows indeed complex behavior that appears to stem from a superposition of short-term correlated noise and "long-tailed behaviour," i.e., highly connected nodes. Moreover, the construction of a dam along a river tends to increase short-term correlations in runoff series. In summary, the present study illustrates the (often substantial) effects of methodological and data-preprocessing choices for the interpretation of river runoff dynamics in the HVG framework and its general applicability for real-world time series.
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Affiliation(s)
- Holger Lange
- Norwegian Institute of Bioeconomy Research, Postboks 115, N-1433 Ås, Norway
| | - Sebastian Sippel
- Norwegian Institute of Bioeconomy Research, Postboks 115, N-1433 Ås, Norway
| | - Osvaldo A Rosso
- Departamento de Informática en Salud, Hospital Italiano de Buenos Aires and CONICET, C1199ABB Ciudad Autónoma de Buenos Aires, Argentina
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18
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Fasmer EE, Fasmer OB, Berle JØ, Oedegaard KJ, Hauge ER. Graph theory applied to the analysis of motor activity in patients with schizophrenia and depression. PLoS One 2018; 13:e0194791. [PMID: 29668743 PMCID: PMC5905887 DOI: 10.1371/journal.pone.0194791] [Citation(s) in RCA: 13] [Impact Index Per Article: 2.2] [Reference Citation Analysis] [Abstract] [MESH Headings] [Grants] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 06/23/2017] [Accepted: 03/10/2018] [Indexed: 12/11/2022] Open
Abstract
Depression and schizophrenia are defined only by their clinical features, and diagnostic separation between them can be difficult. Disturbances in motor activity pattern are central features of both types of disorders. We introduce a new method to analyze time series, called the similarity graph algorithm. Time series of motor activity, obtained from actigraph registrations over 12 days in depressed and schizophrenic patients, were mapped into a graph and we then applied techniques from graph theory to characterize these time series, primarily looking for changes in complexity. The most marked finding was that depressed patients were found to be significantly different from both controls and schizophrenic patients, with evidence of less regularity of the time series, when analyzing the recordings with one hour intervals. These findings support the contention that there are important differences in control systems regulating motor behavior in patients with depression and schizophrenia. The similarity graph algorithm we have described can easily be applied to the study of other types of time series.
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Affiliation(s)
| | - Ole Bernt Fasmer
- Division of Psychiatry, Haukeland University Hospital, Bergen, Norway
- Department of Clinical Medicine, Section for Psychiatry, University of Bergen, Bergen, Norway
- K.G. Jebsen Centre for Research on Neuropsychiatric Disorders, Bergen, Norway
- * E-mail:
| | - Jan Øystein Berle
- Division of Psychiatry, Haukeland University Hospital, Bergen, Norway
- Department of Clinical Medicine, Section for Psychiatry, University of Bergen, Bergen, Norway
| | - Ketil J. Oedegaard
- Division of Psychiatry, Haukeland University Hospital, Bergen, Norway
- Department of Clinical Medicine, Section for Psychiatry, University of Bergen, Bergen, Norway
- K.G. Jebsen Centre for Research on Neuropsychiatric Disorders, Bergen, Norway
| | - Erik R. Hauge
- Division of Psychiatry, Haukeland University Hospital, Bergen, Norway
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19
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Hasson U, Iacovacci J, Davis B, Flanagan R, Tagliazucchi E, Laufs H, Lacasa L. A combinatorial framework to quantify peak/pit asymmetries in complex dynamics. Sci Rep 2018; 8:3557. [PMID: 29476077 PMCID: PMC5824940 DOI: 10.1038/s41598-018-21785-0] [Citation(s) in RCA: 12] [Impact Index Per Article: 2.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 10/31/2017] [Accepted: 02/08/2018] [Indexed: 12/05/2022] Open
Abstract
We explore a combinatorial framework which efficiently quantifies the asymmetries between minima and maxima in local fluctuations of time series. We first showcase its performance by applying it to a battery of synthetic cases. We find rigorous results on some canonical dynamical models (stochastic processes with and without correlations, chaotic processes) complemented by extensive numerical simulations for a range of processes which indicate that the methodology correctly distinguishes different complex dynamics and outperforms state of the art metrics in several cases. Subsequently, we apply this methodology to real-world problems emerging across several disciplines including cases in neurobiology, finance and climate science. We conclude that differences between the statistics of local maxima and local minima in time series are highly informative of the complex underlying dynamics and a graph-theoretic extraction procedure allows to use these features for statistical learning purposes.
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Affiliation(s)
- Uri Hasson
- Center for Mind and Brain Sciences, University of Trento, Trento, Italy.
- Center for Practical Wisdom, The University of Chicago, Chicago, USA.
| | - Jacopo Iacovacci
- Department of Surgery and Cancer, Division of Computational and Systems Medicine, Imperial College London, London, SW7 2AZ, United Kingdom
- The Molecular Biology of Metabolism Laboratory, The Francis Crick Institute, London, NW1 1AT, United Kingdom
| | - Ben Davis
- Center for Mind and Brain Sciences, University of Trento, Trento, Italy
| | - Ryan Flanagan
- School of Mathematical Sciences, Queen Mary University of London, E14NS, London, United Kingdom
| | - Enzo Tagliazucchi
- Netherlands Institute for Neuroscience, Meibergdreef 47, 1105 BA, Amsterdam-Zuidoost, The Netherlands
| | - Helmut Laufs
- Department of Neurology and Brain Imaging Center, Goethe University Frankfurt am Main, Frankfurt, Germany
- Department of Neurology, University Hospital Kiel, Kiel, Germany
| | - Lucas Lacasa
- School of Mathematical Sciences, Queen Mary University of London, E14NS, London, United Kingdom.
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20
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Gao ZK, Cai Q, Yang YX, Dong N, Zhang SS. Visibility Graph from Adaptive Optimal Kernel Time-Frequency Representation for Classification of Epileptiform EEG. Int J Neural Syst 2017; 27:1750005. [DOI: 10.1142/s0129065717500058] [Citation(s) in RCA: 133] [Impact Index Per Article: 19.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 02/01/2023]
Abstract
Detecting epileptic seizure from EEG signals constitutes a challenging problem of significant importance. Combining adaptive optimal kernel time-frequency representation and visibility graph, we develop a novel method for detecting epileptic seizure from EEG signals. We construct complex networks from EEG signals recorded from healthy subjects and epilepsy patients. Then we employ clustering coefficient, clustering coefficient entropy and average degree to characterize the topological structure of the networks generated from different brain states. In addition, we combine energy deviation and network measures to recognize healthy subjects and epilepsy patients, and further distinguish brain states during seizure free interval and epileptic seizures. Three different experiments are designed to evaluate the performance of our method. The results suggest that our method allows a high-accurate classification of epileptiform EEG signals.
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Affiliation(s)
- Zhong-Ke Gao
- School of Electrical Engineering and Automation, Tianjin University, No. 92, Weijin Road, Nankai District, Tianjin, China, Tianjin 300072, China
| | - Qing Cai
- School of Electrical Engineering and Automation, Tianjin University, No. 92, Weijin Road, Nankai District, Tianjin, China, Tianjin 300072, China
| | - Yu-Xuan Yang
- School of Electrical Engineering and Automation, Tianjin University, No. 92, Weijin Road, Nankai District, Tianjin, China, Tianjin 300072, China
| | - Na Dong
- School of Electrical Engineering and Automation, Tianjin University, No. 92, Weijin Road, Nankai District, Tianjin, China, Tianjin 300072, China
| | - Shan-Shan Zhang
- School of Electrical Engineering and Automation, Tianjin University, No. 92, Weijin Road, Nankai District, Tianjin, China, Tianjin 300072, China
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21
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Bianchi FM, Livi L, Alippi C, Jenssen R. Multiplex visibility graphs to investigate recurrent neural network dynamics. Sci Rep 2017; 7:44037. [PMID: 28281563 PMCID: PMC5345088 DOI: 10.1038/srep44037] [Citation(s) in RCA: 22] [Impact Index Per Article: 3.1] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 09/27/2016] [Accepted: 02/01/2017] [Indexed: 11/18/2022] Open
Abstract
A recurrent neural network (RNN) is a universal approximator of dynamical systems, whose performance often depends on sensitive hyperparameters. Tuning them properly may be difficult and, typically, based on a trial-and-error approach. In this work, we adopt a graph-based framework to interpret and characterize internal dynamics of a class of RNNs called echo state networks (ESNs). We design principled unsupervised methods to derive hyperparameters configurations yielding maximal ESN performance, expressed in terms of prediction error and memory capacity. In particular, we propose to model time series generated by each neuron activations with a horizontal visibility graph, whose topological properties have been shown to be related to the underlying system dynamics. Successively, horizontal visibility graphs associated with all neurons become layers of a larger structure called a multiplex. We show that topological properties of such a multiplex reflect important features of ESN dynamics that can be used to guide the tuning of its hyperparamers. Results obtained on several benchmarks and a real-world dataset of telephone call data records show the effectiveness of the proposed methods.
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Affiliation(s)
- Filippo Maria Bianchi
- Machine Learning Group, Department of Physics and Technology, University of Tromsø, 9019 Tromsø, Norway
| | - Lorenzo Livi
- Department of Computer Science, College of Engineering, Mathematics and Physical Sciences, University of Exeter, Exeter EX4 4QF, United Kingdom
| | - Cesare Alippi
- Department of Electronics, Information, and Bioengineering, Politecnico di Milano, 20133 Milan, Italy
- Faculty of Informatics, Universitá della Svizzera Italiana, 6900 Lugano, Switzerland
| | - Robert Jenssen
- Machine Learning Group, Department of Physics and Technology, University of Tromsø, 9019 Tromsø, Norway
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22
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Rosso OA, Ospina R, Frery AC. Classification and Verification of Handwritten Signatures with Time Causal Information Theory Quantifiers. PLoS One 2016; 11:e0166868. [PMID: 27907014 PMCID: PMC5131934 DOI: 10.1371/journal.pone.0166868] [Citation(s) in RCA: 24] [Impact Index Per Article: 3.0] [Reference Citation Analysis] [Abstract] [MESH Headings] [Grants] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 04/09/2016] [Accepted: 11/05/2016] [Indexed: 11/19/2022] Open
Abstract
We present a new approach for handwritten signature classification and verification based on descriptors stemming from time causal information theory. The proposal uses the Shannon entropy, the statistical complexity, and the Fisher information evaluated over the Bandt and Pompe symbolization of the horizontal and vertical coordinates of signatures. These six features are easy and fast to compute, and they are the input to an One-Class Support Vector Machine classifier. The results are better than state-of-the-art online techniques that employ higher-dimensional feature spaces which often require specialized software and hardware. We assess the consistency of our proposal with respect to the size of the training sample, and we also use it to classify the signatures into meaningful groups.
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Affiliation(s)
- Osvaldo A. Rosso
- Instituto de Física, Universidade Federal de Alagoas (UFAL), Maceió, AL, Brazil
- Instituto Tecnológico de Buenos Aires (ITBA), and CONICET, Ciudad Autónoma de Buenos Aires, Argentina
- Facultad de Ingeniería y Ciencias Aplicadas, Universidad de los Andes, Santiago, Chile
- * E-mail:
| | - Raydonal Ospina
- Centro de Ciências Exatas e da Natureza, Departamento de Estatística, Universidade Federal de Pernambuco (UFPE), Recife, PE, Brasil
| | - Alejandro C. Frery
- Laboratório de Computação Científica e Análise Numérica, Universidade Federal de Alagoas (UFAL), Maceió, AL, Brazil
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23
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Lecca P, Mura I, Re A, Barker GC, Ihekwaba AEC. Time Series Analysis of the Bacillus subtilis Sporulation Network Reveals Low Dimensional Chaotic Dynamics. Front Microbiol 2016; 7:1760. [PMID: 27872618 PMCID: PMC5097912 DOI: 10.3389/fmicb.2016.01760] [Citation(s) in RCA: 6] [Impact Index Per Article: 0.8] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 07/21/2016] [Accepted: 10/19/2016] [Indexed: 01/12/2023] Open
Abstract
Chaotic behavior refers to a behavior which, albeit irregular, is generated by an underlying deterministic process. Therefore, a chaotic behavior is potentially controllable. This possibility becomes practically amenable especially when chaos is shown to be low-dimensional, i.e., to be attributable to a small fraction of the total systems components. In this case, indeed, including the major drivers of chaos in a system into the modeling approach allows us to improve predictability of the systems dynamics. Here, we analyzed the numerical simulations of an accurate ordinary differential equation model of the gene network regulating sporulation initiation in Bacillus subtilis to explore whether the non-linearity underlying time series data is due to low-dimensional chaos. Low-dimensional chaos is expectedly common in systems with few degrees of freedom, but rare in systems with many degrees of freedom such as the B. subtilis sporulation network. The estimation of a number of indices, which reflect the chaotic nature of a system, indicates that the dynamics of this network is affected by deterministic chaos. The neat separation between the indices obtained from the time series simulated from the model and those obtained from time series generated by Gaussian white and colored noise confirmed that the B. subtilis sporulation network dynamics is affected by low dimensional chaos rather than by noise. Furthermore, our analysis identifies the principal driver of the networks chaotic dynamics to be sporulation initiation phosphotransferase B (Spo0B). We then analyzed the parameters and the phase space of the system to characterize the instability points of the network dynamics, and, in turn, to identify the ranges of values of Spo0B and of the other drivers of the chaotic dynamics, for which the whole system is highly sensitive to minimal perturbation. In summary, we described an unappreciated source of complexity in the B. subtilis sporulation network by gathering evidence for the chaotic behavior of the system, and by suggesting candidate molecules driving chaos in the system. The results of our chaos analysis can increase our understanding of the intricacies of the regulatory network under analysis, and suggest experimental work to refine our behavior of the mechanisms underlying B. subtilis sporulation initiation control.
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Affiliation(s)
- Paola Lecca
- Department of Mathematics, University of Trento Trento, Italy
| | - Ivan Mura
- Department of Industrial Engineering, Universidad de los Andes Bogotá, Colombia
| | - Angela Re
- Laboratory of Translational Genomics, Centre for Integrative Biology, University of Trento Trento, Italy
| | - Gary C Barker
- Gut Health and Food Safety, Institute of Food Research Norwich, UK
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24
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Gao ZK, Cai Q, Yang YX, Dang WD, Zhang SS. Multiscale limited penetrable horizontal visibility graph for analyzing nonlinear time series. Sci Rep 2016; 6:35622. [PMID: 27759088 PMCID: PMC5069474 DOI: 10.1038/srep35622] [Citation(s) in RCA: 122] [Impact Index Per Article: 15.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 07/06/2016] [Accepted: 09/28/2016] [Indexed: 12/20/2022] Open
Abstract
Visibility graph has established itself as a powerful tool for analyzing time series.
We in this paper develop a novel multiscale limited penetrable horizontal visibility
graph (MLPHVG). We use nonlinear time series from two typical complex systems, i.e.,
EEG signals and two-phase flow signals, to demonstrate the effectiveness of our
method. Combining MLPHVG and support vector machine, we detect epileptic seizures
from the EEG signals recorded from healthy subjects and epilepsy patients and the
classification accuracy is 100%. In addition, we derive MLPHVGs from oil-water
two-phase flow signals and find that the average clustering coefficient at different
scales allows faithfully identifying and characterizing three typical oil-water flow
patterns. These findings render our MLPHVG method particularly useful for analyzing
nonlinear time series from the perspective of multiscale network analysis.
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Affiliation(s)
- Zhong-Ke Gao
- School of Electrical Engineering and Automation, Tianjin University, Tianjin 300072, China
| | - Qing Cai
- School of Electrical Engineering and Automation, Tianjin University, Tianjin 300072, China
| | - Yu-Xuan Yang
- School of Electrical Engineering and Automation, Tianjin University, Tianjin 300072, China
| | - Wei-Dong Dang
- School of Electrical Engineering and Automation, Tianjin University, Tianjin 300072, China
| | - Shan-Shan Zhang
- School of Electrical Engineering and Automation, Tianjin University, Tianjin 300072, China
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25
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Trifonov M. The structure function as new integral measure of spatial and temporal properties of multichannel EEG. Brain Inform 2016; 3:211-220. [PMID: 27747814 PMCID: PMC5106404 DOI: 10.1007/s40708-016-0040-8] [Citation(s) in RCA: 7] [Impact Index Per Article: 0.9] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 11/15/2015] [Accepted: 02/08/2016] [Indexed: 11/28/2022] Open
Abstract
The first-order temporal structure functions (SFs), i.e., the first-order statistical moment of absolute increments of scaled multichannel resting state EEG signals in healthy children and teenagers over a wide range of temporal separation (time lags) are computed. Our research shows that the sill level (asymptote) of the SF is mainly defined by a determinant of EEG correlation matrix reflecting the EEG spatial structure. The temporal structure of EEG is found to be characterized by power-law scaling or statistical-scale invariance over time scales less than 0.028 s and at least by two dominant frequencies differing by less than 0.3 Hz. These frequencies define the oscillation behavior of the SF and are mainly distributed within the range of 7.5-12.0 Hz. In this paper, we propose the combined Bessel and exponential model that fits well the empirical SF. It provides a good fit with the mean relative error fitting of 2.8 % over the time lag range of 1 s, using a sampling interval of 4 ms, for all cases under analysis. We also show that the hyper gamma distribution (HGD) fits to the empirical probability density functions (PDFs) of absolute increments of scaled multichannel resting state EEG signals at any given time lag. It means that only two parameters (sample mean of absolute increments and relevant coefficient of variation) may approximately define the empirical PDFs for a given number of channels. A three-dimensional feature vector constructed from the shape and scale parameters of the HGD and the sill level may be used to estimate the closeness of the real EEG to the "random" EEG characterized by the absence of temporal and spatial correlation.
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Affiliation(s)
- Mikhail Trifonov
- IM Sechenov Institute of Evolutionary Physiology and Biochemistry of the Russian Academy of Sciences, Saint-Petersburg, Russia.
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26
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Aragoneses A, Carpi L, Tarasov N, Churkin DV, Torrent MC, Masoller C, Turitsyn SK. Unveiling Temporal Correlations Characteristic of a Phase Transition in the Output Intensity of a Fiber Laser. PHYSICAL REVIEW LETTERS 2016; 116:033902. [PMID: 26849599 DOI: 10.1103/physrevlett.116.033902] [Citation(s) in RCA: 10] [Impact Index Per Article: 1.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 05/20/2015] [Indexed: 06/05/2023]
Abstract
We use advanced statistical tools of time-series analysis to characterize the dynamical complexity of the transition to optical wave turbulence in a fiber laser. Ordinal analysis and the horizontal visibility graph applied to the experimentally measured laser output intensity reveal the presence of temporal correlations during the transition from the laminar to the turbulent lasing regimes. Both methods unveil coherent structures with well-defined time scales and strong correlations both, in the timing of the laser pulses and in their peak intensities. Our approach is generic and may be used in other complex systems that undergo similar transitions involving the generation of extreme fluctuations.
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Affiliation(s)
- A Aragoneses
- Departament de Física, Universitat Politècnica de Catalunya, 08222 Terrassa, Spain
- Duke University, Physics Department, Box 90305, Durham, North Carolina 27708, USA
| | - L Carpi
- Departament de Física, Universitat Politècnica de Catalunya, 08222 Terrassa, Spain
| | - N Tarasov
- Aston Institute of Photonic Technologies, Aston University, Birmingham B4 7ET, United Kingdom
- Institute for Computational Technologies, SB RAS, Novosibirsk 630090, Russia
| | - D V Churkin
- Aston Institute of Photonic Technologies, Aston University, Birmingham B4 7ET, United Kingdom
- Institute of Automation and Electrometry SB RAS, Novosibirsk 630090, Russia
- Novosibirsk State University, Novosibirsk 630090, Russia
| | - M C Torrent
- Departament de Física, Universitat Politècnica de Catalunya, 08222 Terrassa, Spain
| | - C Masoller
- Departament de Física, Universitat Politècnica de Catalunya, 08222 Terrassa, Spain
| | - S K Turitsyn
- Aston Institute of Photonic Technologies, Aston University, Birmingham B4 7ET, United Kingdom
- Novosibirsk State University, Novosibirsk 630090, Russia
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27
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Subramaniyam NP, Donges JF, Hyttinen J. Signatures of chaotic and stochastic dynamics uncovered with
ε
-recurrence networks. Proc Math Phys Eng Sci 2015. [DOI: 10.1098/rspa.2015.0349] [Citation(s) in RCA: 8] [Impact Index Per Article: 0.9] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/12/2022] Open
Abstract
An old and important problem in the field of nonlinear time-series analysis entails the distinction between chaotic and stochastic dynamics. Recently,
ε
-recurrence networks have been proposed as a tool to analyse the structural properties of a time series. In this paper, we propose the applicability of local and global
ε
-recurrence network measures to distinguish between chaotic and stochastic dynamics using paradigmatic model systems such as the Lorenz system, and the chaotic and hyper-chaotic Rössler system. We also demonstrate the effect of increasing levels of noise on these network measures and provide a real-world application of analysing electroencephalographic data comprising epileptic seizures. Our results show that both local and global
ε
-recurrence network measures are sensitive to the presence of unstable periodic orbits and other structural features associated with chaotic dynamics that are otherwise absent in stochastic dynamics. These network measures are still robust at high noise levels and short data lengths. Furthermore,
ε
-recurrence network analysis of the real-world epileptic data revealed the capability of these network measures in capturing dynamical transitions using short window sizes.
ε
-recurrence network analysis is a powerful method in uncovering the signatures of chaotic and stochastic dynamics based on the geometrical properties of time series.
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Affiliation(s)
- N. P. Subramaniyam
- Department of Electronics and Communications Engineering, Tampere University of Technology, Tampere, Finland
- BioMediTech, Tampere, Finland
| | - J. F. Donges
- Earth System Analysis, Potsdam Institute for Climate Impact Research, Potsdam, Germany
- Planetary Boundary Research Lab, Stockholm University, Stockholm, Sweden
| | - J. Hyttinen
- Department of Electronics and Communications Engineering, Tampere University of Technology, Tampere, Finland
- BioMediTech, Tampere, Finland
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28
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Li Q, Fu Z, Yuan N. Beyond Benford's Law: Distinguishing Noise from Chaos. PLoS One 2015; 10:e0129161. [PMID: 26030809 PMCID: PMC4452586 DOI: 10.1371/journal.pone.0129161] [Citation(s) in RCA: 17] [Impact Index Per Article: 1.9] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 01/05/2015] [Accepted: 05/05/2015] [Indexed: 11/24/2022] Open
Abstract
Determinism and randomness are two inherent aspects of all physical processes. Time series from chaotic systems share several features identical with those generated from stochastic processes, which makes them almost undistinguishable. In this paper, a new method based on Benford's law is designed in order to distinguish noise from chaos by only information from the first digit of considered series. By applying this method to discrete data, we confirm that chaotic data indeed can be distinguished from noise data, quantitatively and clearly.
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Affiliation(s)
- Qinglei Li
- Laboratory for Climate and Ocean-Atmosphere Studies, Dept. of Atmospheric and Oceanic Sciences, School of Physics, Peking University, Beijing, China
| | - Zuntao Fu
- Laboratory for Climate and Ocean-Atmosphere Studies, Dept. of Atmospheric and Oceanic Sciences, School of Physics, Peking University, Beijing, China
- * E-mail: (ZTF); (NMY)
| | - Naiming Yuan
- Laboratory for Climate and Ocean-Atmosphere Studies, Dept. of Atmospheric and Oceanic Sciences, School of Physics, Peking University, Beijing, China
- Department of Geography, Climatology, Climate Dynamics, and Climate Change, Justus-Liebig University Giessen, Giessen, Germany
- * E-mail: (ZTF); (NMY)
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