1
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Kalra DS, Santhanam MS. Inferring long memory using extreme events. CHAOS (WOODBURY, N.Y.) 2021; 31:113131. [PMID: 34881581 DOI: 10.1063/5.0064432] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 07/23/2021] [Accepted: 10/19/2021] [Indexed: 06/13/2023]
Abstract
Many natural and physical processes display long memory and extreme events. In these systems, the measured time series is invariably contaminated by noise and/or missing data. As the extreme events display a large deviation from the mean behavior, noise and/or missing data do not affect the extreme events as much as it affects the typical values. Since the extreme events also carry the information about correlations in the full-time series, we can use them to infer the correlation properties of the latter. In this work, we construct three modified time series using only the extreme events from a given time series. We show that the correlations in the original time series and in the modified time series are related, as measured by the exponent obtained from the detrended fluctuation analysis technique. Hence, the correlation exponents for a long memory time series can be inferred from its extreme events alone. We demonstrate this approach for several empirical time series.
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Affiliation(s)
- Dayal Singh Kalra
- Indian Institute of Science Education and Research, Dr. Homi Bhabha Road, Pune 411008, India
| | - M S Santhanam
- Indian Institute of Science Education and Research, Dr. Homi Bhabha Road, Pune 411008, India
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2
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Estimating the Epicenter of an Impending Strong Earthquake by Combining the Seismicity Order Parameter Variability Analysis with Earthquake Networks and Nowcasting: Application in the Eastern Mediterranean. APPLIED SCIENCES-BASEL 2021. [DOI: 10.3390/app112110093] [Citation(s) in RCA: 4] [Impact Index Per Article: 1.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 11/16/2022]
Abstract
The variance κ1 of the natural time analysis of earthquake catalogs was proposed in 2005 as an order parameter for seismicity, whose fluctuations proved, in 2011, to be minimized a few months before the strongest mainshock when studying the earthquakes in a given area. After the introduction of earthquake networks based on similar activity patterns, in 2012, the study of their higher order cores revealed, in 2019, the selection of appropriate areas in which the precursory minima βmin of the fluctuations β of the seismicity order parameter κ1 could be observed up to six months before all strong earthquakes above a certain threshold. The eastern Mediterranean region was studied in 2019, where all earthquakes of magnitude M≥7.1 were found to be preceded by βmin without any false alarm. Combining these results with the method of nowcasting earthquakes, introduced in 2016, for seismic risk estimation, here, we show that the epicenter of an impending strong earthquake can be estimated. This is achieved by employing—at the time of observing the βmin—nowcasting earthquakes in a square lattice grid in the study area and by averaging, self-consistently, the results obtained for the earthquake potential score. This is understood in the following context: The minimum βmin is ascertained to almost coincide with the onset of Seismic Electric Signals activity, which is accompanied by the development of long range correlations between earthquake magnitudes in the area that is a candidate for a mainshock.
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3
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Telesca L, Czechowski Z. Clustering of extreme events in time series generated by the fractional Ornstein-Uhlenbeck equation. CHAOS (WOODBURY, N.Y.) 2020; 30:093140. [PMID: 33003914 DOI: 10.1063/5.0023301] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 07/28/2020] [Accepted: 09/07/2020] [Indexed: 06/11/2023]
Abstract
We analyze the time clustering phenomenon in sequences of extremes of time series generated by the fractional Ornstein-Uhlenbeck (fO-U) equation as the source of long-term correlation. We used the percentile-based definition of extremes based on the crossing theory or run theory, where a run is a sequence of L contiguous values above a given percentile. Thus, a sequence of extremes becomes a point process in time, being the time of occurrence of the extreme the starting time of the run. We investigate the relationship between the Hurst exponent related to the time series generated by the fO-U equation and three measures of time clustering of the corresponding extremes defined on the base of the 95th percentile. Our results suggest that for persistent pure fractional Gaussian noise, the sequence of the extremes is clusterized, while extremes obtained by antipersistent or Markovian pure fractional Gaussian noise seem to behave more regularly or Poissonianly. However, for the fractional Ornstein-Uhlenbeck equation, the clustering of extremes is evident even for antipersistent and Markovian cases. This is a result of short range correlations caused by differential and drift terms. The drift parameter influences the extremes clustering effect-it drops with increasing value of the parameter.
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Affiliation(s)
- Luciano Telesca
- National Research Council, Institute of Methodologies for Environmental Analysis, C.da S. Loja, 85050 Tito, Potenza, Italy
| | - Zbigniew Czechowski
- Institute of Geophysics, Polish Academy of Sciences, Księcia Janusza 64, 01-452 Warsaw, Poland
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4
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Abstract
The scaling properties of turbulent flows are well established in the inertial sub-range. However, those of the synoptic-scale motions are less known, also because of the difficult analysis of data presenting nonstationary and periodic features. Extensive analysis of experimental wind speed data, collected at the Mauna Loa Observatory of Hawaii, is performed using different methods. Empirical Mode Decomposition, interoccurrence times statistics, and arbitrary-order Hilbert spectral analysis allow to eliminate effects of large-scale modulations, and provide scaling properties of the field fluctuations (Hurst exponent, interoccurrence distribution, and intermittency correction). The obtained results suggest that the mesoscale wind dynamics owns features which are typical of the inertial sub-range turbulence, thus extending the validity of the turbulent cascade phenomenology to scales larger than observed before.
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5
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Ogunshemi A, Hopcraft KI, Preston SP. Lacunarity of the zero crossings of Gaussian processes. Phys Rev E 2019; 99:062109. [PMID: 31330702 DOI: 10.1103/physreve.99.062109] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 08/10/2018] [Indexed: 06/10/2023]
Abstract
A lacunarity analysis of the zero crossings derived from Gaussian stochastic processes with oscillatory autocorrelation functions is evaluated and reveals distinct multiscaling signatures depending on the smoothness and degree of anticorrelation of the process. These bear qualitative similarities and quantitative distinctions from an oscillatory deterministic signal and a Poisson random process both possessing the same mean interval size between crossings. At very small and large scales compared with the correlation length of the random processes, the lacunarity is similar to the Poisson but exhibits significant departures from Poisson behavior if there is a zero-frequency component to the process's power spectrum. A comparison of exact results with the gliding box technique that is frequently used to determine lacunarity demonstrates its inherent bias.
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Affiliation(s)
- A Ogunshemi
- School of Mathematical Sciences, Applied Mathematics Division, University of Nottingham, Nottingham NG7 2RD, United Kingdom
| | - K I Hopcraft
- School of Mathematical Sciences, Applied Mathematics Division, University of Nottingham, Nottingham NG7 2RD, United Kingdom
| | - S P Preston
- School of Mathematical Sciences, Applied Mathematics Division, University of Nottingham, Nottingham NG7 2RD, United Kingdom
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6
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Moloney NR, Faranda D, Sato Y. An overview of the extremal index. CHAOS (WOODBURY, N.Y.) 2019; 29:022101. [PMID: 30823712 DOI: 10.1063/1.5079656] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.8] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 11/01/2018] [Accepted: 01/23/2019] [Indexed: 06/09/2023]
Abstract
For a wide class of stationary time series, extreme value theory provides limiting distributions for rare events. The theory describes not only the size of extremes but also how often they occur. In practice, it is often observed that extremes cluster in time. Such short-range clustering is also accommodated by extreme value theory via the so-called extremal index. This review provides an introduction to the extremal index by working through a number of its intuitive interpretations. Thus, depending on the context, the extremal index may represent (i) the loss of independently and identically distributed degrees of freedom, (ii) the multiplicity of a compound Poisson point process, and (iii) the inverse mean duration of extreme clusters. More recently, the extremal index has also been used to quantify (iv) recurrences around unstable fixed points in dynamical systems.
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Affiliation(s)
- Nicholas R Moloney
- Department of Mathematics and Statistics, University of Reading, Reading RG6 6AX, United Kingdom
| | - Davide Faranda
- Laboratoire de Sciences du Climat et de l'Environnement, UMR 8212 CEA-CNRS-UVSQ, IPSL, Universite Paris-Saclay, 91191 Gif-sur-Yvette, France
| | - Yuzuru Sato
- RIES/Department of Mathematics, Hokkaido University, Kita 20 Nishi 10, Kita-ku, Sapporo 001-0020, Japan
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7
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Allende S, Henry C, Bec J. Stretching and Buckling of Small Elastic Fibers in Turbulence. PHYSICAL REVIEW LETTERS 2018; 121:154501. [PMID: 30362808 DOI: 10.1103/physrevlett.121.154501] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 05/22/2018] [Indexed: 06/08/2023]
Abstract
Small flexible fibers in a turbulent flow are found to be as straight as stiff rods most of the time. This is due to the cooperative action of flexural rigidity and fluid stretching. However, fibers might bend and buckle when they tumble and experience a strong enough local compression. Such events are similar to an activation process, where the role of temperature is played by the inverse of Young's modulus. Numerical simulations show that buckling occurs very intermittently in time. This results from unexpected long-range Lagrangian correlations of the turbulent shear.
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Affiliation(s)
- Sofía Allende
- Université Côte d'Azur, CNRS, OCA, Laboratoire J.-L. Lagrange, 06300 Nice, France
| | - Christophe Henry
- Université Côte d'Azur, CNRS, OCA, Laboratoire J.-L. Lagrange, 06300 Nice, France
| | - Jérémie Bec
- Université Côte d'Azur, CNRS, OCA, Laboratoire J.-L. Lagrange, 06300 Nice, France
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8
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Peng J, Xu G, Shao R, Chen L, Stanley HE. Analysis of fluctuations in the first return times of random walks on regular branched networks. J Chem Phys 2018; 149:024903. [PMID: 30007392 DOI: 10.1063/1.5028123] [Citation(s) in RCA: 9] [Impact Index Per Article: 1.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/15/2022] Open
Abstract
The first return time (FRT) is the time it takes a random walker to first return to its original site, and the global first passage time (GFPT) is the first passage time for a random walker to move from a randomly selected site to a given site. We find that in finite networks, the variance of FRT, Var(FRT), can be expressed as Var(FRT) = 2⟨FRT⟩⟨GFPT⟩ - ⟨FRT⟩2 - ⟨FRT⟩, where ⟨·⟩ is the mean of the random variable. Therefore a method of calculating the variance of FRT on general finite networks is presented. We then calculate Var(FRT) and analyze the fluctuation of FRT on regular branched networks (i.e., Cayley tree) by using Var(FRT) and its variant as the metric. We find that the results differ from those in such other networks as Sierpinski gaskets, Vicsek fractals, T-graphs, pseudofractal scale-free webs, (u, v) flowers, and fractal and non-fractal scale-free trees.
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Affiliation(s)
- Junhao Peng
- School of Math and Information Science, Guangzhou University, Guangzhou 510006, China
| | - Guoai Xu
- School of Cyberspace Security, Beijing University of Posts and Telecommunications, Beijing 100876, China
| | - Renxiang Shao
- School of Math and Information Science, Guangzhou University, Guangzhou 510006, China
| | - Lin Chen
- Center for Polymer Studies and Department of Physics, Boston University, Boston, Massachusetts 02215, USA
| | - H Eugene Stanley
- Center for Polymer Studies and Department of Physics, Boston University, Boston, Massachusetts 02215, USA
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9
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Carbone F, Bruno AG, Naccarato A, De Simone F, Gencarelli CN, Sprovieri F, Hedgecock IM, Landis MS, Skov H, Pfaffhuber KA, Read KA, Martin L, Angot H, Dommergue A, Magand O, Pirrone N. The superstatistical nature and interoccurrence time of atmospheric mercury concentration fluctuations. JOURNAL OF GEOPHYSICAL RESEARCH. ATMOSPHERES : JGR 2018; 123:764-774. [PMID: 30505642 PMCID: PMC6260940 DOI: 10.1002/2017jd027384] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [Grants] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/09/2023]
Abstract
The probability density function (PDF) of the time intervals between subsequent extreme events in atmospheric Hg0 concentration data series from different latitudes has been investigated. The Hg0 dynamic possesses a long-term memory autocorrelation function. Above a fixed threshold Q in the data, the PDFs of the interoccurrence time of the Hg0 data are well described by a Tsallis q-Exponential function. This PDF behavior has been explained in the framework of superstatistics, where the competition between multiple mesoscopic processes affects the macroscopic dynamics. An extensive parameter μ, encompassing all possible fluctuations related to mesoscopic phenomena, has been identified. It follows a χ 2-distribution, indicative of the superstatistical nature of the overall process. Shuffling the data series destroys the long-term memory, the distributions become independent of Q, and the PDFs collapse on to the same exponential distribution. The possible central role of atmospheric turbulence on extreme events in the Hg0 data is highlighted.
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Affiliation(s)
- F. Carbone
- CNR-Institute of Atmospheric Pollution Research, Division of Rende, UNICAL-Polifunzionale, 87036 Rende (CS), Italy
| | - A. G. Bruno
- CNR-Institute of Atmospheric Pollution Research, Division of Rende, UNICAL-Polifunzionale, 87036 Rende (CS), Italy
- Dipartimento di Fisica, Università della Calabria, Ponte Pietro Bucci 31C, I-87036 Rende (CS), Italy
| | - A. Naccarato
- CNR-Institute of Atmospheric Pollution Research, Division of Rende, UNICAL-Polifunzionale, 87036 Rende (CS), Italy
| | - F. De Simone
- CNR-Institute of Atmospheric Pollution Research, Division of Rende, UNICAL-Polifunzionale, 87036 Rende (CS), Italy
| | - C. N. Gencarelli
- CNR-Institute of Atmospheric Pollution Research, Division of Rende, UNICAL-Polifunzionale, 87036 Rende (CS), Italy
| | - F. Sprovieri
- CNR-Institute of Atmospheric Pollution Research, Division of Rende, UNICAL-Polifunzionale, 87036 Rende (CS), Italy
| | - I. M. Hedgecock
- CNR-Institute of Atmospheric Pollution Research, Division of Rende, UNICAL-Polifunzionale, 87036 Rende (CS), Italy
| | - M. S. Landis
- U.S. Environmental Protection Agency, Office of Research and Development, Research Triangle Park, North Carolina 27709, United States
| | - H. Skov
- Aarhus University, Department of Environmental Science, Frederiksborgvej 399, 4000 Roskilde, Denmark
| | - K. A. Pfaffhuber
- Norwegian Institute for Air Research (NILU), P.O. Box 100, 2027 Kjeller, Norway
| | - K. A. Read
- NCAS,National Centre for Atmospheric Sciences, University of York, York, UK
| | - L. Martin
- Cape Point GAW Station, Climate and Environment Research & Monitoring, South African Weather Service, Stellenbosch, South Africa
| | - H. Angot
- University Grenoble Alpes, CNRS, IRD, IGE, Grenoble, France
| | - A. Dommergue
- University Grenoble Alpes, CNRS, IRD, IGE, Grenoble, France
| | - O. Magand
- University Grenoble Alpes, CNRS, IRD, IGE, Grenoble, France
| | - N. Pirrone
- CNR-Institute of Atmospheric Pollution Research, Area della Ricerca di Roma 1, Via Salaria km 29,300, Monterotondo, 00015 Rome, Italy
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10
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Wilson LRM, Hopcraft KI. Periodicity in the autocorrelation function as a mechanism for regularly occurring zero crossings or extreme values of a Gaussian process. Phys Rev E 2018; 96:062129. [PMID: 29347422 DOI: 10.1103/physreve.96.062129] [Citation(s) in RCA: 6] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 03/10/2017] [Indexed: 11/07/2022]
Abstract
The problem of zero crossings is of great historical prevalence and promises extensive application. The challenge is to establish precisely how the autocorrelation function or power spectrum of a one-dimensional continuous random process determines the density function of the intervals between the zero crossings of that process. This paper investigates the case where periodicities are incorporated into the autocorrelation function of a smooth process. Numerical simulations, and statistics about the number of crossings in a fixed interval, reveal that in this case the zero crossings segue between a random and deterministic point process depending on the relative time scales of the periodic and nonperiodic components of the autocorrelation function. By considering the Laplace transform of the density function, we show that incorporating correlation between successive intervals is essential to obtaining accurate results for the interval variance. The same method enables prediction of the density function tail in some regions, and we suggest approaches for extending this to cover all regions. In an ever-more complex world, the potential applications for this scale of regularity in a random process are far reaching and powerful.
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Affiliation(s)
- Lorna R M Wilson
- Institute for Mathematical Innovation, University of Bath, Bath, England, United Kingdom
| | - Keith I Hopcraft
- Department of Mathematical Sciences, University of Nottingham, Nottingham, England, United Kingdom
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11
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Witt A, Ehlers F, Luther S. Extremes of fractional noises: A model for the timings of arrhythmic heart beats in post-infarction patients. CHAOS (WOODBURY, N.Y.) 2017; 27:093942. [PMID: 28964134 DOI: 10.1063/1.5003249] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/07/2023]
Abstract
We have analyzed symbol sequences of heart beat annotations obtained from 24-h electrocardiogram recordings of 184 post-infarction patients (from the Cardiac Arrhythmia Suppression Trial database, CAST). In the symbol sequences, each heart beat was coded as an arrhythmic or as a normal beat. The symbol sequences were analyzed with a model-based approach which relies on two-parametric peaks over the threshold (POT) model, interpreting each premature ventricular contraction (PVC) as an extreme event. For the POT model, we explored (i) the Shannon entropy which was estimated in terms of the Lempel-Ziv complexity, (ii) the shape parameter of the Weibull distribution that best fits the PVC return times, and (iii) the strength of long-range correlations quantified by detrended fluctuation analysis (DFA) for the two-dimensional parameter space. We have found that in the frame of our model the Lempel-Ziv complexity is functionally related to the shape parameter of the Weibull distribution. Thus, two complementary measures (entropy and strength of long-range correlations) are sufficient to characterize realizations of the two-parametric model. For the CAST data, we have found evidence for an intermediate strength of long-range correlations in the PVC timings, which are correlated to the age of the patient: younger post-infarction patients have higher strength of long-range correlations than older patients. The normalized Shannon entropy has values in the range 0.5<hLZ<1.0 which indicates a high degree of randomness in the PVC timings. For the CAST and the model data, the ranges of both measures were found to be in good accordance. The correlation between the age and the persistence strength found for the CAST data could be explained as a change of model parameters.
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Affiliation(s)
- Annette Witt
- Biomedical Physics Group, Max-Planck Institute for Dynamics and Self-Organization, Göttingen, Germany
| | - Frithjof Ehlers
- Biomedical Physics Group, Max-Planck Institute for Dynamics and Self-Organization, Göttingen, Germany
| | - Stefan Luther
- German Center for Cardiovascular Research (DZHK), Partner Site Göttingen, Göttingen, Germany
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12
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Peng J, Agliari E. Scaling laws for diffusion on (trans)fractal scale-free networks. CHAOS (WOODBURY, N.Y.) 2017; 27:083108. [PMID: 28863489 DOI: 10.1063/1.4997761] [Citation(s) in RCA: 9] [Impact Index Per Article: 1.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/07/2023]
Abstract
Fractal (or transfractal) features are common in real-life networks and are known to influence the dynamic processes taking place in the network itself. Here, we consider a class of scale-free deterministic networks, called (u, v)-flowers, whose topological properties can be controlled by tuning the parameters u and v; in particular, for u > 1, they are fractals endowed with a fractal dimension df, while for u = 1, they are transfractal endowed with a transfractal dimension d̃f. In this work, we investigate dynamic processes (i.e., random walks) and topological properties (i.e., the Laplacian spectrum) and we show that, under proper conditions, the same scalings (ruled by the related dimensions) emerge for both fractal and transfractal dimensions.
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Affiliation(s)
- Junhao Peng
- School of Mathematics and Information Science, Guangzhou University, Guangzhou 510006, China
| | - Elena Agliari
- Department of Mathematics, Sapienza Università di Roma, 00198 Rome, Italy
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13
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Serinaldi F, Lombardo F. General simulation algorithm for autocorrelated binary processes. Phys Rev E 2017; 95:023312. [PMID: 28297873 DOI: 10.1103/physreve.95.023312] [Citation(s) in RCA: 13] [Impact Index Per Article: 1.9] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 06/25/2016] [Indexed: 11/07/2022]
Abstract
The apparent ubiquity of binary random processes in physics and many other fields has attracted considerable attention from the modeling community. However, generation of binary sequences with prescribed autocorrelation is a challenging task owing to the discrete nature of the marginal distributions, which makes the application of classical spectral techniques problematic. We show that such methods can effectively be used if we focus on the parent continuous process of beta distributed transition probabilities rather than on the target binary process. This change of paradigm results in a simulation procedure effectively embedding a spectrum-based iterative amplitude-adjusted Fourier transform method devised for continuous processes. The proposed algorithm is fully general, requires minimal assumptions, and can easily simulate binary signals with power-law and exponentially decaying autocorrelation functions corresponding, for instance, to Hurst-Kolmogorov and Markov processes. An application to rainfall intermittency shows that the proposed algorithm can also simulate surrogate data preserving the empirical autocorrelation.
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Affiliation(s)
- Francesco Serinaldi
- School of Civil Engineering and Geosciences, Newcastle University, Newcastle Upon Tyne, NE1 7RU, United Kingdom.,Willis Research Network, 51 Lime St., London EC3M 7DQ, United Kingdom
| | - Federico Lombardo
- Dipartimento di Ingegneria, Università degli Studi Roma Tre, Via Vito Volterra 62, 00146 Rome, Italy
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14
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Understanding Persistence to Avoid Underestimation of Collective Flood Risk. WATER 2016. [DOI: 10.3390/w8040152] [Citation(s) in RCA: 25] [Impact Index Per Article: 3.1] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 11/16/2022]
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15
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Kitsak M, Elmokashfi A, Havlin S, Krioukov D. Long-Range Correlations and Memory in the Dynamics of Internet Interdomain Routing. PLoS One 2015; 10:e0141481. [PMID: 26529312 PMCID: PMC4631327 DOI: 10.1371/journal.pone.0141481] [Citation(s) in RCA: 5] [Impact Index Per Article: 0.6] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 06/29/2015] [Accepted: 10/08/2015] [Indexed: 12/03/2022] Open
Abstract
Data transfer is one of the main functions of the Internet. The Internet consists of a large number of interconnected subnetworks or domains, known as Autonomous Systems (ASes). Due to privacy and other reasons the information about what route to use to reach devices within other ASes is not readily available to any given AS. The Border Gateway Protocol (BGP) is responsible for discovering and distributing this reachability information to all ASes. Since the topology of the Internet is highly dynamic, all ASes constantly exchange and update this reachability information in small chunks, known as routing control packets or BGP updates. In the view of the quick growth of the Internet there are significant concerns with the scalability of the BGP updates and the efficiency of the BGP routing in general. Motivated by these issues we conduct a systematic time series analysis of BGP update rates. We find that BGP update time series are extremely volatile, exhibit long-term correlations and memory effects, similar to seismic time series, or temperature and stock market price fluctuations. The presented statistical characterization of BGP update dynamics could serve as a basis for validation of existing and developing better models of Internet interdomain routing.
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Affiliation(s)
- Maksim Kitsak
- Department of Physics, Northeastern University, Boston, MA, United States of America
- * E-mail:
| | | | - Shlomo Havlin
- Department of Physics, Bar-Ilan University, Ramat Gan, Israel
| | - Dmitri Krioukov
- Department of Physics, Northeastern University, Boston, MA, United States of America
- Department of Mathematics, Northeastern University, Boston, MA, United States of America
- Department of Electrical&Computer Engineering, Northeastern University, Boston, MA, United States of America
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16
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Sarlis NV, Christopoulos SRG, Skordas ES. Minima of the fluctuations of the order parameter of global seismicity. CHAOS (WOODBURY, N.Y.) 2015; 25:063110. [PMID: 26117104 DOI: 10.1063/1.4922300] [Citation(s) in RCA: 8] [Impact Index Per Article: 0.9] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/04/2023]
Abstract
It has been recently shown [N. V. Sarlis, Phys. Rev. E 84, 022101 (2011) and N. V. Sarlis and S.-R. G. Christopoulos, Chaos 22, 023123 (2012)] that earthquakes of magnitude M greater or equal to 7 are globally correlated. Such correlations were identified by studying the variance κ1 of natural time which has been proposed as an order parameter for seismicity. Here, we study the fluctuations of this order parameter using the Global Centroid Moment Tensor catalog for a magnitude threshold Mthres = 5.0 and focus on its behavior before major earthquakes. Natural time analysis reveals that distinct minima of the fluctuations of the order parameter of seismicity appear within almost five and a half months on average before all major earthquakes of magnitude larger than 8.4. This phenomenon corroborates the recent finding [N. V. Sarlis et al., Proc. Natl. Acad. Sci. U.S.A. 110, 13734 (2013)] that similar minima of the seismicity order parameter fluctuations had preceded all major shallow earthquakes in Japan. Moreover, on the basis of these minima a statistically significant binary prediction method for earthquakes of magnitude larger than 8.4 with hit rate 100% and false alarm rate 6.67% is suggested.
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Affiliation(s)
- N V Sarlis
- Department of Solid State Physics and Solid Earth Physics Institute, Faculty of Physics, School of Science, National and Kapodistrian University of Athens, Panepistimiopolis, Zografos 157 84, Athens, Greece
| | - S-R G Christopoulos
- Department of Solid State Physics and Solid Earth Physics Institute, Faculty of Physics, School of Science, National and Kapodistrian University of Athens, Panepistimiopolis, Zografos 157 84, Athens, Greece
| | - E S Skordas
- Department of Solid State Physics and Solid Earth Physics Institute, Faculty of Physics, School of Science, National and Kapodistrian University of Athens, Panepistimiopolis, Zografos 157 84, Athens, Greece
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17
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Tamazian A, Ludescher J, Bunde A. Significance of trends in long-term correlated records. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2015; 91:032806. [PMID: 25871156 DOI: 10.1103/physreve.91.032806] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.4] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 11/14/2014] [Indexed: 06/04/2023]
Abstract
We study the distribution P(x;α,L) of the relative trend x in long-term correlated records of length L that are characterized by a Hurst exponent α between 0.5 and 1.5. The relative trend x is the ratio between the strength of the trend Δ in the record measured by linear regression and the standard deviation σ around the regression line. We consider L between 400 and 2200, which is the typical length scale of monthly local and annual reconstructed global climate records. Extending previous work by Lennartz and Bunde [S. Lennartz and A. Bunde, Phys. Rev. E 84, 021129 (2011)] we show explicitly that x follows the Student's t distribution P∝[1+(x/a)2/l]-(l+1)/2, where the scaling parameter a depends on both L and α, while the effective length l depends, for α below 1.15, only on the record length L. From P we can derive an analytical expression for the trend significance S(x;α,L)=∫(-x)xP(x';α,L)dx' and the border lines of the 95% significance interval. We show that the results are nearly independent of the distribution of the data in the record, holding for Gaussian data as well as for highly skewed non-Gaussian data. For an application, we use our methodology to estimate the significance of central west Antarctic warming.
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Affiliation(s)
- Araik Tamazian
- Institut für Theoretische Physik, Justus-Liebig-Universität Giessen, 35392 Giessen, Germany
| | - Josef Ludescher
- Institut für Theoretische Physik, Justus-Liebig-Universität Giessen, 35392 Giessen, Germany
| | - Armin Bunde
- Institut für Theoretische Physik, Justus-Liebig-Universität Giessen, 35392 Giessen, Germany
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18
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Kishore V, Sonawane AR, Santhanam MS. Manipulation of extreme events on scale-free networks. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2013; 88:014801. [PMID: 23944597 DOI: 10.1103/physreve.88.014801] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 03/25/2013] [Indexed: 06/02/2023]
Abstract
Extreme events taking place on networks are not uncommon. We show that it is possible to manipulate the extreme event occurrence probabilities and distribution over the nodes of a scale-free network by tuning the nodal capacity. This can be used to reduce the number of extreme event occurrences. However, monotonic nodal capacity enhancements, beyond a point, do not lead to any substantial reduction in the number of extreme events. We point out the practical implication of this result for network design in the context of reducing extreme event occurrences.
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Affiliation(s)
- Vimal Kishore
- Indian Institute of Science Education and Research, Homi Bhabha Road, Pune 411 008, India
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19
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Rybski D, Buldyrev SV, Havlin S, Liljeros F, Makse HA. Communication activity in a social network: relation between long-term correlations and inter-event clustering. Sci Rep 2012; 2:560. [PMID: 22876339 PMCID: PMC3413962 DOI: 10.1038/srep00560] [Citation(s) in RCA: 60] [Impact Index Per Article: 5.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 04/12/2012] [Accepted: 07/11/2012] [Indexed: 11/17/2022] Open
Abstract
Human communication in social networks is dominated by emergent statistical laws such as non-trivial correlations and temporal clustering. Recently, we found long-term correlations in the user's activity in social communities. Here, we extend this work to study the collective behavior of the whole community with the goal of understanding the origin of clustering and long-term persistence. At the individual level, we find that the correlations in activity are a byproduct of the clustering expressed in the power-law distribution of inter-event times of single users, i.e. short periods of many events are separated by long periods of no events. On the contrary, the activity of the whole community presents long-term correlations that are a true emergent property of the system, i.e. they are not related to the distribution of inter-event times. This result suggests the existence of collective behavior, possibly arising from nontrivial communication patterns through the embedding social network.
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Affiliation(s)
- Diego Rybski
- Levich Institute and Physics Department, City College of New York, NY 10031, USA
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20
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Sarlis NV, Christopoulos SRG. Natural time analysis of the Centennial Earthquake Catalog. CHAOS (WOODBURY, N.Y.) 2012; 22:023123. [PMID: 22757530 DOI: 10.1063/1.4711374] [Citation(s) in RCA: 8] [Impact Index Per Article: 0.7] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/01/2023]
Abstract
By using the most recent version (1900-2007) of the Centennial Earthquake Catalog, we examine the properties of the global seismicity. Natural time analysis reveals that the fluctuations of the order parameter κ(1) of seismicity exhibit for at least three orders of magnitude a characteristic feature similar to that of the order parameter for other equilibrium or non-equilibrium critical systems-including self-organized critical systems. Moreover, we find non-trivial magnitude correlations for earthquakes of magnitude greater than or equal to 7.
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Affiliation(s)
- N V Sarlis
- Physics Department, Solid State Section and Solid Earth Physics Institute, University of Athens, Panepistimiopolis, Zografos 157 84, Athens, Greece.
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21
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Sarlis NV. Magnitude correlations in global seismicity. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2011; 84:022101. [PMID: 21929043 DOI: 10.1103/physreve.84.022101] [Citation(s) in RCA: 6] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 03/22/2011] [Indexed: 05/31/2023]
Abstract
By employing natural time analysis, we analyze the worldwide seismicity and study the existence of correlations between earthquake magnitudes. We find that global seismicity exhibits nontrivial magnitude correlations for earthquake magnitudes greater than M(w) 6.5.
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Affiliation(s)
- N V Sarlis
- Solid State Section and Solid Earth Physics Institute, Physics Department, University of Athens, Panepistimiopolis, Zografos GR-157 84, Athens, Greece.
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22
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Schweigler T, Davidsen J. Clustering of extreme and recurrent events in deterministic chaotic systems. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2011; 84:016202. [PMID: 21867268 DOI: 10.1103/physreve.84.016202] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 10/16/2010] [Indexed: 05/31/2023]
Abstract
We study the nontrivial clustering properties of extreme or recurrent events in the context of deterministic chaotic systems. We find that correlations between return times of such events can depend nonmonotonically on the threshold used to define the events, which leads to counterintuitive behavior. In particular, the distribution of the conditional return intervals can indicate clustering as well as repelling of extreme events for the same condition but different thresholds-in sharp contrast to what has been observed for stochastic processes with long-range correlations as well as for independent and identically distributed random variables. This has important implications for the time-dependent hazard assessment of extreme events, indicating that possible threshold dependencies should always be taken into account.
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Affiliation(s)
- Thomas Schweigler
- Department of Physics & Astronomy, University of Calgary, Calgary, Alberta, Canada
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23
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Kishore V, Santhanam MS, Amritkar RE. Extreme events on complex networks. PHYSICAL REVIEW LETTERS 2011; 106:188701. [PMID: 21635132 DOI: 10.1103/physrevlett.106.188701] [Citation(s) in RCA: 16] [Impact Index Per Article: 1.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 02/12/2011] [Indexed: 05/30/2023]
Abstract
A wide spectrum of extreme events ranging from traffic jams to floods take place on networks. Motivated by these, we employ a random walk model for transport and obtain analytical and numerical results for the extreme events on networks. They reveal an unforeseen, and yet a robust, feature: small degree nodes of a network are more likely to encounter extreme events than the hubs. Further, we also study the recurrence time distribution and scaling of the probabilities for extreme events. These results suggest a revision of design principles and can be used as an input for designing the nodes of a network so as to smoothly handle extreme events.
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Affiliation(s)
- Vimal Kishore
- Physical Research Laboratory, Navrangpura, Ahmedabad, India
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24
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Omi T, Kanter I, Shinomoto S. Optimal observation time window for forecasting the next earthquake. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2011; 83:026101. [PMID: 21405883 DOI: 10.1103/physreve.83.026101] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 06/22/2010] [Revised: 09/24/2010] [Indexed: 05/30/2023]
Abstract
We report that the accuracy of predicting the occurrence time of the next earthquake is significantly enhanced by observing the latest rate of earthquake occurrences. The observation period that minimizes the temporal uncertainty of the next occurrence is on the order of 10 hours. This result is independent of the threshold magnitude and is consistent across different geographic areas. This time scale is much shorter than the months or years that have previously been considered characteristic of seismic activities.
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Affiliation(s)
- Takahiro Omi
- Department of Physics, Kyoto University, Kyoto 606-8502, Japan.
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25
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Sokolova A, Bogachev MI, Bunde A. Clustering of ventricular arrhythmic complexes in heart rhythm. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2011; 83:021918. [PMID: 21405874 DOI: 10.1103/physreve.83.021918] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 09/08/2010] [Revised: 12/06/2010] [Indexed: 05/30/2023]
Abstract
We study the statistics of intervals τ between ventricular premature complexes (VPCs) in 24-h electrocardiogram records obtained from PhysioNet data source. We find that the long-term memory inherent in the heartbeat intervals leads to power laws in the probability density function P(τ) between VPCs for τ>6 s. As a consequence, the probability W(t,Δt) that at least one VPC will occur within the next time interval Δt, if the last VPC occurred t time units intervals ago, decays by a power law of t. Based on these results, we suggest a method to obtain a priori information about the occurrence of the next VPC, and how to predict it. We think that usage of this a priori information could be useful for the improvement of the algorithms in healthcare monitoring devices with alarm facilities.
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Affiliation(s)
- Anastasia Sokolova
- Institut für Theoretische Physik, Justus-Liebig-Universität Giessen, D-35392 Giessen, Germany
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26
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Sarlis NV, Skordas ES, Varotsos PA. Nonextensivity and natural time: The case of seismicity. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2010; 82:021110. [PMID: 20866778 DOI: 10.1103/physreve.82.021110] [Citation(s) in RCA: 29] [Impact Index Per Article: 2.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 04/18/2010] [Revised: 06/14/2010] [Indexed: 05/29/2023]
Abstract
Nonextensive statistical mechanics, pioneered by Tsallis, has recently achieved a generalization of the Gutenberg-Richter law for earthquakes. This remarkable generalization is combined here with natural time analysis, which enables the distinction of two origins of self-similarity, i.e., the process' memory and the process' increments infinite variance. By using also detrended fluctuation analysis for the detection of long-range temporal correlations, we demonstrate the existence of both temporal and magnitude correlations in real seismic data of California and Japan. Natural time analysis reveals that the nonextensivity parameter q , in contrast to some published claims, cannot be considered as a measure of temporal organization, but the Tsallis formulation does achieve a satisfactory description of real seismic data for Japan for q=1.66 when supplemented by long-range temporal correlations.
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Affiliation(s)
- N V Sarlis
- Solid Earth Physics Institute, Physics Department, University of Athens, Panepistimiopolis, Zografos, Greece
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27
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Liu C, Jiang ZQ, Ren F, Zhou WX. Scaling and memory in the return intervals of energy dissipation rate in three-dimensional fully developed turbulence. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2009; 80:046304. [PMID: 19905433 DOI: 10.1103/physreve.80.046304] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 01/09/2009] [Revised: 06/25/2009] [Indexed: 05/28/2023]
Abstract
We study the statistical properties of return intervals r between successive energy dissipation rates above a certain threshold Q in three-dimensional fully developed turbulence. We find that the distribution function P(Q)(r) scales with the mean return interval R(Q) as P(Q)(r)=R(Q)(-1)f(r/R(Q)) for R(Q) is an element of [50,500], where the scaling function f(x) has two power-law regimes. The scaling behavior is statistically validated by the Cramér-von Mises criterion. The return intervals are short-term and long-term correlated and possess multifractal nature. The Hurst index of the return intervals decays exponentially against R(Q), predicting that rare extreme events with R(Q)-->infinity are also long-term correlated with the Hurst index H(infinity)=0.639. These phenomenological findings have potential applications in risk assessment of extreme events at very large R(Q).
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Affiliation(s)
- Chuang Liu
- School of Business, East China University of Science and Technology, Shanghai, China
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28
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Sarlis NV, Skordas ES, Varotsos PA. Multiplicative cascades and seismicity in natural time. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2009; 80:022102. [PMID: 19792180 DOI: 10.1103/physreve.80.022102] [Citation(s) in RCA: 8] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 01/20/2009] [Revised: 05/18/2009] [Indexed: 05/28/2023]
Abstract
Natural time chi enables the distinction of two origins of self-similarity, i.e., the process memory and the process increments infinite variance. Employing multiplicative cascades in natural time, the most probable value of the variance kappa(1)(is identical to chi(2)-chi(2))is explicitly related with the parameter b of the Gutenberg-Richter law of randomly shuffled earthquake data. Moreover, the existence of temporal and magnitude correlations is studied in the original earthquake data. Magnitude correlations are larger for closer in time earthquakes, when the maximum interoccurrence time varies from half a day to 1 min.
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Affiliation(s)
- N V Sarlis
- Department of Physics, Solid State Section and Solid Earth Physics Institute, University of Athens, Panepistimiopolis, Zografos, 157 84 Athens, Greece
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29
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Lennartz S, Bunde A. Eliminating finite-size effects and detecting the amount of white noise in short records with long-term memory. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2009; 79:066101. [PMID: 19658558 DOI: 10.1103/physreve.79.066101] [Citation(s) in RCA: 7] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 10/24/2008] [Revised: 01/23/2009] [Indexed: 05/25/2023]
Abstract
Long-term memory is ubiquitous in nature and has important consequences for the occurrence of natural hazards, but its detection often is complicated by the short length of the considered records and additive white noise in the data. Here we study synthetic Gaussian distributed records x_{i} of length N that consist of a long-term correlated component (1-a)y_{i} characterized by a correlation exponent gamma , 0<gamma<1 , and a white-noise component aeta_{i} , 0< or =a< or =1 . We show that the autocorrelation function C_{N}(s) has the general form C_{N}(s)=[C_{infinity}(s)-E_{a}]/(1-E_{a}) , where C_{infinity}(0)=1 , C_{infinity}(s>0)=B_{a}s;{-gamma} , and E_{a}={2B_{a}/[(2-gamma)(1-gamma)]}N;{-gamma}+O(N;{-1}) . The finite-size parameter E_{a} also occurs in related quantities, for example, in the variance Delta_{N};{2}(s) of the local mean in time windows of length s : Delta_{N};{2}(s)=[Delta_{infinity};{2}(s)-E_{a}]/(1-E_{a}) . For purely long-term correlated data B_{0} congruent with(2-gamma)(1-gamma)/2 yielding E_{0} congruent withN;{-gamma} , and thus C_{N}(s)=[(2-gamma)(1-gamma)/2s;{-gamma}-N;{-gamma}]/[1-N;{-gamma}] and Delta_{N};{2}(s)=[s;{-gamma}-N;{-gamma}]/[1-N;{-gamma}] . We show how to estimate E_{a} and C_{infinity}(s) from a given data set and thus how to obtain accurately the exponent gamma and the amount of white noise a .
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Affiliation(s)
- Sabine Lennartz
- Institut für Theoretische Physik III, Justus-Liebig-Universität Giessen, 35392 Giessen, Germany.
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30
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Moloney NR, Davidsen J. Extreme value statistics and return intervals in long-range correlated uniform deviates. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2009; 79:041131. [PMID: 19518197 DOI: 10.1103/physreve.79.041131] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 12/12/2008] [Indexed: 05/27/2023]
Abstract
We study extremal statistics and return intervals in stationary long-range correlated sequences for which the underlying probability density function is bounded and uniform. The extremal statistics we consider (e.g., maximum relative to minimum) are such that the reference point from which the maximum is measured is itself a random quantity. We analytically calculate the limiting distributions for independent and identically distributed random variables, and use these as a reference point for correlated cases. The distributions are different from that of the maximum itself (i.e., a Weibull distribution), reflecting the fact that the distribution of the reference point either dominates over or convolves with the distribution of the maximum. The functional form of the limiting distributions is unaffected by correlations, although the convergence is slower. We show that our findings can be directly generalized to a wide class of stochastic processes. We also analyze return interval distributions, and compare them to recent conjectures of their functional form.
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Affiliation(s)
- N R Moloney
- Department of Physics and Astronomy, University of Calgary, 2500 University Drive NW, Calgary, Alberta, Canada AB T2N 1N4.
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31
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Wang F, Yamasaki K, Havlin S, Stanley HE. Multifactor analysis of multiscaling in volatility return intervals. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2009; 79:016103. [PMID: 19257103 DOI: 10.1103/physreve.79.016103] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 08/22/2008] [Indexed: 05/27/2023]
Abstract
We study the volatility time series of 1137 most traded stocks in the U.S. stock markets for the two-year period 2001-2002 and analyze their return intervals tau , which are time intervals between volatilities above a given threshold q . We explore the probability density function of tau , P_(q)(tau) , assuming a stretched exponential function, P_(q)(tau) approximately e;(-tau;(gamma)) . We find that the exponent gamma depends on the threshold in the range between q=1 and 6 standard deviations of the volatility. This finding supports the multiscaling nature of the return interval distribution. To better understand the multiscaling origin, we study how gamma depends on four essential factors, capitalization, risk, number of trades, and return. We show that gamma depends on the capitalization, risk, and return but almost does not depend on the number of trades. This suggests that gamma relates to the portfolio selection but not on the market activity. To further characterize the multiscaling of individual stocks, we fit the moments of tau , mu_(m) identical with(tautau);(m);(1m) , in the range of 10<tau< or =100 by a power law, micro_(m) approximately tau;(delta). The exponent delta is found also to depend on the capitalization, risk, and return but not on the number of trades, and its tendency is opposite to that of gamma . Moreover, we show that delta decreases with increasing gamma approximately by a linear relation. The return intervals demonstrate the temporal structure of volatilities and our findings suggest that their multiscaling features may be helpful for portfolio optimization.
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Affiliation(s)
- Fengzhong Wang
- Center for Polymer Studies and Department of Physics, Boston University, Boston, Massachusetts 02215, USA
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32
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Santhanam MS, Kantz H. Return interval distribution of extreme events and long-term memory. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2008; 78:051113. [PMID: 19113101 DOI: 10.1103/physreve.78.051113] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 02/22/2008] [Revised: 07/08/2008] [Indexed: 05/27/2023]
Abstract
The distribution of recurrence times or return intervals between extreme events is important to characterize and understand the behavior of physical systems and phenomena in many disciplines. It is well known that many physical processes in nature and society display long-range correlations. Hence, in the last few years, considerable research effort has been directed towards studying the distribution of return intervals for long-range correlated time series. Based on numerical simulations, it was shown that the return interval distributions are of stretched exponential type. In this paper, we obtain an analytical expression for the distribution of return intervals in long-range correlated time series which holds good when the average return intervals are large. We show that the distribution is actually a product of power law and a stretched exponential form. We also discuss the regimes of validity and perform detailed studies on how the return interval distribution depends on the threshold used to define extreme events.
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Affiliation(s)
- M S Santhanam
- Max Planck Institute for the Physics of Complex Systems, Nöthnitzer Strasse 38, Dresden 01187, Germany
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33
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Bogachev MI, Bunde A. Memory effects in the statistics of interoccurrence times between large returns in financial records. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2008; 78:036114. [PMID: 18851112 DOI: 10.1103/physreve.78.036114] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 02/11/2008] [Revised: 06/18/2008] [Indexed: 05/26/2023]
Abstract
We study the statistics of the interoccurrence times between events above some threshold Q in two kinds of multifractal data sets (multiplicative random cascades and multifractal random walks) with vanishing linear correlations. We show that in both data sets the relevant quantities (probability density functions and the autocorrelation function of the interoccurrence times, as well as the conditional return period) are governed by power laws with exponents that depend explicitly on the considered threshold. By studying a large number of representative financial records (market indices, stock prices, exchange rates, and commodities), we show explicitly that the interoccurrence times between large daily returns follow the same behavior, in a nearly quantitative manner. We conclude that this kind of behavior is a general consequence of the nonlinear memory inherent in the multifractal data sets.
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Affiliation(s)
- Mikhail I Bogachev
- Institut für Theoretische Physik III, Justus-Liebig-Universität Giessen, 35392 Giessen, Germany
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34
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Smith JM, Hopcraft KI, Jakeman E. Fluctuations in the zeros of differentiable Gaussian processes. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2008; 77:031112. [PMID: 18517334 DOI: 10.1103/physreve.77.031112] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 08/28/2007] [Revised: 12/04/2007] [Indexed: 05/26/2023]
Abstract
The stochastic point processes formed by the zero crossings or extremal points of differentiable, stationary Gaussian processes are studied as a function of their autocorrelation function. The properties of these point processes are mapped to the space formed by the parameters appearing in the autocorrelation function, their adopted form being sensitive to the structure of the autocorrelation function principally in the vicinity of the origin. The distribution for the number of zeros occurring in an asymptotically large interval are approximately negative-binomial or binomial depending upon whether the relative variance or Fano factor is greater or less than unity. The correlation properties of the zeros are such that they are repelled from each other or are "antibunched" if the autocorrelation function of the Gaussian process is characterized by a single scale size, but occur in clusters if more than one characteristic scale size is present. The intervals between zeros can be interpreted in terms of the autocorrelation function of the zeros themselves. When bunching occurs the interval density becomes bimodal, indicating the interval sizes within and between the clusters. The interevent periods are statistically dependent on one another with densities whose asymptotic behavior is governed by that of the autocorrelation function of the Gaussian process at large delay times. Poisson distributed fluctuations of the zeros occur only exceptionally but never form a Poisson process.
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Affiliation(s)
- J M Smith
- School of Mathematical Sciences, Applied Mathematics Division, University of Nottingham, NG7 2RD, United Kingdom
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35
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Bogachev MI, Eichner JF, Bunde A. Effect of nonlinear correlations on the statistics of return intervals in multifractal data sets. PHYSICAL REVIEW LETTERS 2007; 99:240601. [PMID: 18233431 DOI: 10.1103/physrevlett.99.240601] [Citation(s) in RCA: 19] [Impact Index Per Article: 1.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 08/08/2007] [Indexed: 05/25/2023]
Abstract
We study the statistics of return intervals between events above a certain threshold in multifractal data sets without linear correlations. We find that nonlinear correlations in the record lead to a power-law (i) decay of the autocorrelation function of the return intervals, (ii) increase in the conditional return period, and (iii) decay in the probability density function of the return intervals. We show explicitly that all the observed quantities depend both on the threshold value and system size, and hence there is no simple scaling observed. We also demonstrate that this type of behavior can be observed in real economic records and can be used to improve considerably risk estimation.
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Affiliation(s)
- Mikhail I Bogachev
- Institut für Theoretische Physik III, Justus-Liebig-Universität Giessen, 35392 Giessen, Germany
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36
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Olla P. Return times for stochastic processes with power-law scaling. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2007; 76:011122. [PMID: 17677425 DOI: 10.1103/physreve.76.011122] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 03/20/2007] [Indexed: 05/16/2023]
Abstract
An analytical study of the return time distribution of extreme events for stochastic processes with power-law correlation has been carried out. The calculation is based on an epsilon expansion in the correlation exponent: C(t)=/t/-1+epsilon. The fixed point of the theory is associated with stretched exponential scaling of the distribution; analytical expressions have been provided in the preasymptotic regime. Also, the permanence time distribution appears to be characterized by stretched exponential scaling. The conditions for application of the theory to non-Gaussian processes have been analyzed and the relations with the issue of return times in the case of multifractal measures have been discussed.
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Affiliation(s)
- Piero Olla
- Istituto di Scienze dell'Atmosfera e del Clima ed Istituto Nazionale di Fisica Nucleare, Sezione di. Cagliari, I-09042 Monserrato, Italy
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