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Grela J, Nowak MA. Extreme matrices or how an exponential map links classical and free extreme laws. Phys Rev E 2020; 102:022109. [PMID: 32942363 DOI: 10.1103/physreve.102.022109] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 03/27/2020] [Accepted: 07/03/2020] [Indexed: 11/07/2022]
Abstract
Using our proposed approach to describe extreme matrices, we find an explicit exponentiation formula linking the classical extreme laws of Fréchet, Gumbel, and Weibull given by the Fisher-Tippet-Gnedenko classification and free extreme laws of free Fréchet, free Gumbel, and free Weibull of Ben Arous and Voiculescu. We also develop an extreme random matrix formalism, in which refined questions about extreme matrices can be answered. In particular, we demonstrate explicit calculations for several more or less known random matrix ensembles, providing examples of all three free extreme laws. Finally, we present an exact mapping, showing the equivalence of free extreme laws to the Peak-over-Threshold method in classical probability.
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Affiliation(s)
- Jacek Grela
- Institute of Theoretical Physics and Mark Kac Complex Systems Research Centre, Jagiellonian University, Łojasiewicza 11, 30-348 Kraków, Poland
| | - Maciej A Nowak
- Institute of Theoretical Physics and Mark Kac Complex Systems Research Centre, Jagiellonian University, Łojasiewicza 11, 30-348 Kraków, Poland
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2
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Fan J, Zhou D, Shekhtman LM, Shapira A, Hofstetter R, Marzocchi W, Ashkenazy Y, Havlin S. Possible origin of memory in earthquakes: Real catalogs and an epidemic-type aftershock sequence model. Phys Rev E 2019; 99:042210. [PMID: 31108655 DOI: 10.1103/physreve.99.042210] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.4] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 07/27/2018] [Indexed: 06/09/2023]
Abstract
Earthquakes are one of the most devastating natural disasters that plague society. Skilled, reliable earthquake forecasting remains the ultimate goal for seismologists. Using the detrended fluctuation analysis (DFA) and conditional probability (CP) methods, we find that memory exists not only in interoccurrence seismic records but also in released energy as well as in the series of the number of events per unit time. Analysis of a standard epidemic-type aftershock sequences (ETAS) earthquake model indicates that the empirically observed earthquake memory can be reproduced only for a narrow range of the model's parameters. This finding therefore provides tight constraints on the model's parameters and can serve as a testbed for existing earthquake forecasting models. Furthermore, we show that by implementing DFA and CP results, the ETAS model can significantly improve the short-term forecasting rate for the real (Italian) earthquake catalog.
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Affiliation(s)
- Jingfang Fan
- Blaustein Institutes for Desert Research, Ben-Gurion University of the Negev, Midreshet Ben-Gurion 84990, Israel
- Department of Physics, Bar Ilan University, Ramat Gan 52900, Israel
- Potsdam Institute for Climate Impact Research, 14412 Potsdam, Germany
| | - Dong Zhou
- Blaustein Institutes for Desert Research, Ben-Gurion University of the Negev, Midreshet Ben-Gurion 84990, Israel
- Department of Physics, Bar Ilan University, Ramat Gan 52900, Israel
- School of Reliability and Systems Engineering, Beihang University, Beijing, 100191, China
- Science and Technology on Reliability and Environmental Engineering Laboratory, 100191 Beijing, China
| | | | - Avi Shapira
- National Institute for Regulation of Emergency and Disaster, College of Law and Business, Bnei Brak, 511080, Israel
| | | | - Warner Marzocchi
- Department of Earth, Environmental, and Resources Sciences, University of Naples, Federico II, Complesso di Monte Sant'Angelo, Via Cinthia, 21 80126 Napoli, Italy
| | - Yosef Ashkenazy
- Blaustein Institutes for Desert Research, Ben-Gurion University of the Negev, Midreshet Ben-Gurion 84990, Israel
| | - Shlomo Havlin
- Department of Physics, Bar Ilan University, Ramat Gan 52900, Israel
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3
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Balcerek M, Loch-Olszewska H, Torreno-Pina JA, Garcia-Parajo MF, Weron A, Manzo C, Burnecki K. Inhomogeneous membrane receptor diffusion explained by a fractional heteroscedastic time series model. Phys Chem Chem Phys 2019; 21:3114-3121. [DOI: 10.1039/c8cp06781c] [Citation(s) in RCA: 5] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/21/2022]
Abstract
A fractional heteroscedastic time series model explains inhomogeneous membrane receptor diffusion and interprets ergodicity when the diffusion parameter fluctuates in time.
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Affiliation(s)
- Michał Balcerek
- Faculty of Pure and Applied Mathematics, Hugo Steinhaus Center, Wrocław University of Science and Technology
- 50-370 Wrocław
- Poland
| | - Hanna Loch-Olszewska
- Faculty of Pure and Applied Mathematics, Hugo Steinhaus Center, Wrocław University of Science and Technology
- 50-370 Wrocław
- Poland
| | - Juan A. Torreno-Pina
- ICFO-Institut de Ciències Fotòniques, The Barcelona Institute of Science and Technology
- 08860 Castelldefels (Barcelona)
- Spain
| | - Maria F. Garcia-Parajo
- ICFO-Institut de Ciències Fotòniques, The Barcelona Institute of Science and Technology
- 08860 Castelldefels (Barcelona)
- Spain
- ICREA
- 08010 Barcelona
| | - Aleksander Weron
- Faculty of Pure and Applied Mathematics, Hugo Steinhaus Center, Wrocław University of Science and Technology
- 50-370 Wrocław
- Poland
| | - Carlo Manzo
- ICFO-Institut de Ciències Fotòniques, The Barcelona Institute of Science and Technology
- 08860 Castelldefels (Barcelona)
- Spain
- Facultat de Ciències i Tecnologia, Universitat de Vic – Universitat Central de Catalunya
- 08550 Vic
| | - Krzysztof Burnecki
- Faculty of Pure and Applied Mathematics, Hugo Steinhaus Center, Wrocław University of Science and Technology
- 50-370 Wrocław
- Poland
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4
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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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5
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Aliakbari A, Manshour P, Salehi MJ. Records in fractal stochastic processes. CHAOS (WOODBURY, N.Y.) 2017; 27:033116. [PMID: 28364750 DOI: 10.1063/1.4979348] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.4] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/07/2023]
Abstract
The record statistics in stationary and non-stationary fractal time series is studied extensively. By calculating various concepts in record dynamics, we find some interesting results. In stationary fractional Gaussian noises, we observe a universal behavior for the whole range of Hurst exponents. However, for non-stationary fractional Brownian motions, the record dynamics is crucially dependent on the memory, which plays the role of a non-stationarity index, here. Indeed, the deviation from the results of the stationary case increases by increasing the Hurst exponent in fractional Brownian motions. We demonstrate that the memory governs the dynamics of the records as long as it causes non-stationarity in fractal stochastic processes; otherwise, it has no impact on the record statistics.
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Affiliation(s)
- A Aliakbari
- Department of Physics, Faculty of Sciences, Persian Gulf University, 75169 Bushehr, Iran
| | - P Manshour
- Department of Physics, Faculty of Sciences, Persian Gulf University, 75169 Bushehr, Iran
| | - M J Salehi
- Department of Physics, Faculty of Sciences, Persian Gulf University, 75169 Bushehr, Iran
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Manshour P, Anvari M, Reinke N, Sahimi M, Tabar MRR. Interoccurrence time statistics in fully-developed turbulence. Sci Rep 2016; 6:27452. [PMID: 27282347 PMCID: PMC4901275 DOI: 10.1038/srep27452] [Citation(s) in RCA: 9] [Impact Index Per Article: 1.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 02/14/2016] [Accepted: 05/09/2016] [Indexed: 11/23/2022] Open
Abstract
Emergent extreme events are a key characteristic of complex dynamical systems. The main tool for detailed and deep understanding of their stochastic dynamics is the statistics of time intervals of extreme events. Analyzing extensive experimental data, we demonstrate that for the velocity time series of fully-developed turbulent flows, generated by (i) a regular grid; (ii) a cylinder; (iii) a free jet of helium, and (iv) a free jet of air with the Taylor Reynolds numbers Reλ from 166 to 893, the interoccurrence time distributions P(τ) above a positive threshold Q in the inertial range is described by a universal q- exponential function, P(τ) = β(2 − q)[1 − β(1 − q)τ]1/(1−q), which may be due to the superstatistical nature of the occurrence of extreme events. Our analysis provides a universal description of extreme events in turbulent flows.
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Affiliation(s)
- Pouya Manshour
- Department of Physics, Faculty of Sciences, Persian Gulf University, Bushehr 75169, Iran
| | - Mehrnaz Anvari
- Institute of Physics and for wind, Carl von Ossietzky University of Oldenburg, 26111 Oldenburg, Germany
| | - Nico Reinke
- Institute of Physics and for wind, Carl von Ossietzky University of Oldenburg, 26111 Oldenburg, Germany
| | - Muhammad Sahimi
- Mork Family Department of Chemical Engineering Materials Science, University of Southern California, Los Angeles, California 90089-1211, USA
| | - M Reza Rahimi Tabar
- Institute of Physics and for wind, Carl von Ossietzky University of Oldenburg, 26111 Oldenburg, Germany.,Department of Physics, Sharif University of Technology, Tehran 11155-9161, Iran
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7
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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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Matiu M, Ankerst DP, Menzel A. Asymmetric trends in seasonal temperature variability in instrumental records from ten stations in Switzerland, Germany and the UK from 1864 to 2012. INTERNATIONAL JOURNAL OF CLIMATOLOGY : A JOURNAL OF THE ROYAL METEOROLOGICAL SOCIETY 2016; 36:13-27. [PMID: 27478303 PMCID: PMC4950111 DOI: 10.1002/joc.4326] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Key Words] [Grants] [Track Full Text] [Subscribe] [Scholar Register] [Received: 07/31/2014] [Revised: 02/27/2015] [Accepted: 03/04/2015] [Indexed: 05/11/2023]
Abstract
While the rise in global mean temperature over the past several decades is now widely acknowledged, the issue as to whether and to what extent temperature variability is changing continues to undergo debate. Here, variability refers to the spread of the temperature distribution. Much attention has been given to the effects that changes in mean temperature have on extremes, but these changes are accompanied by changes in variability, and it is actually the two together, in addition to all aspects of a changing climate pattern, that influence extremes. Since extremes have some of the largest impacts on society and ecology, changing temperature variability must be considered in tandem with a gradually increasing temperature mean. Previous studies of trends in temperature variability have produced conflicting results. Here we investigated ten long-term instrumental records in Europe of minimum, mean and maximum temperatures, looking for trends in seasonal, annual and decadal measures of variability (standard deviation and various quantile ranges) as well as asymmetries in the trends of extreme versus mean temperatures via quantile regression. We found consistent and accelerating mean warming during 1864-2012. In the last 40 years (1973-2012) trends for Tmax were higher than for Tmin, reaching up to 0.8 °C per 10a in spring. On the other hand, variability trends were not as uniform: significant changes occurred in opposing directions depending on the season, as well as when comparing 1864-2012 trends to those of 1973-2012. Moreover, if variability changed, then it changed asymmetrically, that is only in the part above or below the median. Consequently, trends in the extreme high and low quantiles differed. Regional differences indicated that in winter, high-alpine stations had increasing variability trends for Tmax especially at the upper tail compared to no changes or decreasing variability at low altitude stations. In contrast, summer variability increased at all stations studied.
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Affiliation(s)
- Michael Matiu
- Fachgebiet Ökoklimatologie Technische Universität München Freising Germany
| | - Donna P Ankerst
- Lehrstuhl für Mathematische Modellierung biologischer Systeme Technische Universität München Garching Germany
| | - Annette Menzel
- Fachgebiet Ökoklimatologie Technische Universität München Freising Germany; Institute for Advanced Study Technische Universität München Garching Germany
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9
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Nicodemi M. Extreme Value Statistics. ENCYCLOPEDIA OF COMPLEXITY AND SYSTEMS SCIENCE 2015:1-10. [DOI: 10.1007/978-3-642-27737-5_197-3] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Received: 03/19/2013] [Accepted: 11/12/2015] [Indexed: 09/02/2023]
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Ludescher J, Bunde A. Universal behavior of the interoccurrence times between losses in financial markets: independence of the time resolution. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2014; 90:062809. [PMID: 25615150 DOI: 10.1103/physreve.90.062809] [Citation(s) in RCA: 8] [Impact Index Per Article: 0.8] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 01/31/2014] [Indexed: 06/04/2023]
Abstract
We consider representative financial records (stocks and indices) on time scales between one minute and one day, as well as historical monthly data sets, and show that the distribution P(Q)(r) of the interoccurrence times r between losses below a negative threshold -Q, for fixed mean interoccurrence times R(Q) in multiples of the corresponding time resolutions, can be described on all time scales by the same q exponentials, P(Q)(r)∝1/{[1+(q-1)βr](1/(q-1))}. We propose that the asset- and time-scale-independent analytic form of P(Q)(r) can be regarded as an additional stylized fact of the financial markets and represents a nontrivial test for market models. We analyze the distribution P(Q)(r) as well as the autocorrelation C(Q)(s) of the interoccurrence times for three market models: (i) multiplicative random cascades, (ii) multifractal random walks, and (iii) the generalized autoregressive conditional heteroskedasticity [GARCH(1,1)] model. We find that only one of the considered models, the multifractal random walk model, approximately reproduces the q-exponential form of P(Q)(r) and the power-law decay of C(Q)(s).
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Affiliation(s)
- Josef Ludescher
- Institut für Theoretische Physik, Justus-Liebig-Universität Giessen, D-35392 Giessen, Germany
| | - Armin Bunde
- Institut für Theoretische Physik, Justus-Liebig-Universität Giessen, D-35392 Giessen, Germany
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Masoliver J. Extreme values and the level-crossing problem: an application to the Feller process. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2014; 89:042106. [PMID: 24827192 DOI: 10.1103/physreve.89.042106] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 01/23/2014] [Indexed: 06/03/2023]
Abstract
We review the question of the extreme values attained by a random process. We relate it to level crossings to one boundary (first-passage problems) as well as to two boundaries (escape problems). The extremes studied are the maximum, the minimum, the maximum absolute value, and the range or span. We specialize in diffusion processes and present detailed results for the Wiener and Feller processes.
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Affiliation(s)
- Jaume Masoliver
- Departament de Física Fonamental, Universitat de Barcelona, Diagonal, 647, E-08028 Barcelona, Spain
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Chicheportiche R, Chakraborti A. Copulas and time series with long-ranged dependencies. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2014; 89:042117. [PMID: 24827203 DOI: 10.1103/physreve.89.042117] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 11/20/2013] [Indexed: 06/03/2023]
Abstract
We review ideas on temporal dependencies and recurrences in discrete time series from several areas of natural and social sciences. We revisit existing studies and redefine the relevant observables in the language of copulas (joint laws of the ranks). We propose that copulas provide an appropriate mathematical framework to study nonlinear time dependencies and related concepts-like aftershocks, Omori law, recurrences, and waiting times. We also critically argue, using this global approach, that previous phenomenological attempts involving only a long-ranged autocorrelation function lacked complexity in that they were essentially monoscale.
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Affiliation(s)
- Rémy Chicheportiche
- Chaire de finance quantitative, École Centrale Paris, 92 295 Châtenay-Malabry, France
| | - Anirban Chakraborti
- Chaire de finance quantitative, École Centrale Paris, 92 295 Châtenay-Malabry, France
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LeBlanc M, Angheluta L, Dahmen K, Goldenfeld N. Universal fluctuations and extreme statistics of avalanches near the depinning transition. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2013; 87:022126. [PMID: 23496478 DOI: 10.1103/physreve.87.022126] [Citation(s) in RCA: 9] [Impact Index Per Article: 0.8] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 10/06/2012] [Indexed: 06/01/2023]
Abstract
We derive exact predictions for universal scaling exponents and scaling functions associated with the statistics of maximum velocities v(m) during avalanches described by the mean-field theory of the interface depinning transition. In particular, we find a robust power-law regime in the statistics of maximum events that can explain the observed distribution of the peak amplitudes in acoustic emission experiments of crystal plasticity. Our results are expected to be broadly applicable to a broad range of systems in the mean-field interface depinning universality class, ranging from magnets to earthquakes.
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Affiliation(s)
- Michael LeBlanc
- Department of Physics, University of Illinois at Urbana-Champaign, Loomis Laboratory of Physics, 1110 West Green Street, Urbana, Illinois 61801-3080, USA
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Moloney NR, Ozogány K, Rácz Z. Order statistics of 1/fα signals. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2011; 84:061101. [PMID: 22304034 DOI: 10.1103/physreve.84.061101] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 09/27/2011] [Indexed: 05/31/2023]
Abstract
Order statistics of periodic, Gaussian noise with 1/f(α) power spectrum is investigated. Using simulations and phenomenological arguments, we find three scaling regimes for the average gap d(k) = (x(k) -x(k) + 1) between the kth and (k+1)st largest values of the signal. The result d(k) k(-1), known for independent, identically distributed variables, remains valid for 0 ≤ α < 1. Nontrivial, α-dependent scaling exponents, d(k) k((α-3)/2), emerge for 1 < α < 5, and, finally, α-independent scaling, d(k) ~ k, is obtained for α > 5. The spectra of average ordered values ε(k) =(x(1) - x(k))~ k(β) is also examined. The exponent β is derived from the gap scaling as well as by relating ε(k) to the density of near-extreme states. Known results for the density of near-extreme states combined with scaling suggest that β(α = 2) = 1/2, β(4) = 3/2, and β(∞) = 2 are exact values. We also show that parallels can be drawn between ε(k) and the quantum mechanical spectra of a particle in power-law potentials.
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Affiliation(s)
- N R Moloney
- Max Planck Institute for the Physics of Complex Systems, Nöthnitzer Str 38, D-01187 Dresden, Germany.
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17
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Györgyi G, Moloney NR, Ozogány K, Rácz Z, Droz M. Renormalization-group theory for finite-size scaling in extreme statistics. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2010; 81:041135. [PMID: 20481705 DOI: 10.1103/physreve.81.041135] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 01/31/2010] [Indexed: 05/29/2023]
Abstract
We present a renormalization-group (RG) approach to explain universal features of extreme statistics applied here to independent identically distributed variables. The outlines of the theory have been described in a previous paper, the main result being that finite-size shape corrections to the limit distribution can be obtained from a linearization of the RG transformation near a fixed point, leading to the computation of stable perturbations as eigenfunctions. Here we show details of the RG theory which exhibit remarkable similarities to the RG known in statistical physics. Besides the fixed points explaining universality, and the least stable eigendirections accounting for convergence rates and shape corrections, the similarities include marginally stable perturbations which turn out to be generic for the Fisher-Tippett-Gumbel class. Distribution functions containing unstable perturbations are also considered. We find that, after a transitory divergence, they return to the universal fixed line at the same or at a different point depending on the type of perturbation.
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Affiliation(s)
- G Györgyi
- Department of Theoretical Physics, University of Geneva, Geneva, Switzerland.
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Nicodemi M. Extreme Value Statistics. ENCYCLOPEDIA OF COMPLEXITY AND SYSTEMS SCIENCE 2009:3317-3323. [DOI: 10.1007/978-0-387-30440-3_197] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 09/02/2023]
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Oliveira TJ, Aarão Reis FDA. Maximal- and minimal-height distributions of fluctuating interfaces. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2008; 77:041605. [PMID: 18517633 DOI: 10.1103/physreve.77.041605] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 11/12/2007] [Indexed: 05/26/2023]
Abstract
Maximal- and minimal-height distributions (MAHD, MIHD) of two-dimensional interfaces grown with the nonlinear equations of Kardar-Parisi-Zhang (KPZ, second order) and of Villain-Lai-Das Sarma (VLDS, fourth order) are shown to be different. Two universal curves may be MAHD or MIHD of each class depending on the sign of the relevant nonlinear term, which is confirmed by results of several lattice models in the KPZ and VLDS classes. The difference between MAHD and MIDH is connected with the asymmetry of the local height distribution. A simple, exactly solvable deposition-erosion model is introduced to illustrate this feature. The average extremal heights scale with the same exponent of the average roughness. In contrast to other correlated systems, generalized Gumbel distributions do not fit those MAHD and MIHD, nor those of Edwards-Wilkinson growth.
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Affiliation(s)
- T J Oliveira
- Instituto de Física, Universidade Federal Fluminense, Avenida Litorânea s/n, 24210-340 Niterói RJ, Brazil
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Burkhardt TW, Györgyi G, Moloney NR, Rácz Z. Extreme statistics for time series: distribution of the maximum relative to the initial value. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2007; 76:041119. [PMID: 17994948 DOI: 10.1103/physreve.76.041119] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 07/19/2007] [Indexed: 05/25/2023]
Abstract
The extreme statistics of time signals is studied when the maximum is measured from the initial value. In the case of independent, identically distributed (iid) variables, we classify the limiting distribution of the maximum according to the properties of the parent distribution from which the variables are drawn. Then we turn to correlated periodic Gaussian signals with a 1/falpha power spectrum and study the distribution of the maximum relative height with respect to the initial height (MRHI). The exact MRHI distribution is derived for alpha=0 (iid variables), alpha=2 (random walk), alpha=4 (random acceleration), and alpha=infinity (single sinusoidal mode). For other, intermediate values of alpha , the distribution is determined from simulations. We find that the MRHI distribution is markedly different from the previously studied distribution of the maximum height relative to the average height for all alpha. The two main distinguishing features of the MRHI distribution are the much larger weight for small relative heights and the divergence at zero height for alpha>3. We also demonstrate that the boundary conditions affect the shape of the distribution by presenting exact results for some nonperiodic boundary conditions. Finally, we show that, for signals arising from time-translationally invariant distributions, the density of near extreme states is the same as the MRHI distribution. This is used in developing a scaling theory for the threshold singularities of the two distributions.
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Affiliation(s)
- T W Burkhardt
- Department of Physics, Temple University, Philadelphia, Pennsylvania 19122, USA.
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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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Guclu H, Korniss G, Toroczkai Z. Extreme fluctuations in noisy task-completion landscapes on scale-free networks. CHAOS (WOODBURY, N.Y.) 2007; 17:026104. [PMID: 17614691 DOI: 10.1063/1.2735446] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/16/2023]
Abstract
We study the statistics and scaling of extreme fluctuations in noisy task-completion landscapes, such as those emerging in synchronized distributed-computing networks, or generic causally constrained queuing networks, with scale-free topology. In these networks the average size of the fluctuations becomes finite (synchronized state) and the extreme fluctuations typically diverge only logarithmically in the large system-size limit ensuring synchronization in a practical sense. Provided that local fluctuations in the network are short tailed, the statistics of the extremes are governed by the Gumbel distribution. We present large-scale simulation results using the exact algorithmic rules, supported by mean-field arguments based on a coarse-grained description.
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Affiliation(s)
- H Guclu
- Center for Nonlinear Studies, Theoretical Division, Los Alamos National Laboratory, MS-B258, Los Alamos, New Mexico 87545, USA
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Sabhapandit S, Majumdar SN. Density of near-extreme events. PHYSICAL REVIEW LETTERS 2007; 98:140201. [PMID: 17501249 DOI: 10.1103/physrevlett.98.140201] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 01/05/2007] [Indexed: 05/15/2023]
Abstract
We provide a quantitative analysis of the phenomenon of crowding of near-extreme events by computing exactly the density of states (DOS) near the maximum of a set of independent and identically distributed random variables. We show that the mean DOS converges to three different limiting forms depending on whether the tail of the distribution of the random variables decays slower than pure exponential, faster than pure exponential, or as a pure exponential function. We argue that some of these results would remain valid even for certain correlated cases and verify it for power-law correlated stationary Gaussian sequences. Satisfactory agreement is found between the near-maximum crowding in the summer temperature reconstruction data of western Siberia and the theoretical prediction.
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Affiliation(s)
- Sanjib Sabhapandit
- Laboratoire de Physique Théorique et Modèles Statistiques, UMR 8626 du CNRS, Université Paris-Sud, Bâtiment 100, 91405 Orsay Cedex, France
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Masoliver J, Perelló J. Extreme times for volatility processes. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2007; 75:046110. [PMID: 17500964 DOI: 10.1103/physreve.75.046110] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 09/14/2006] [Indexed: 05/15/2023]
Abstract
Extreme times techniques, generally applied to nonequilibrium statistical mechanical processes, are also useful for a better understanding of financial markets. We present a detailed study on the mean first-passage time for the volatility of return time series. The empirical results extracted from daily data of major indices seem to follow the same law regardless of the kind of index thus suggesting an universal pattern. The empirical mean first-passage time to a certain level L is fairly different from that of the Wiener process showing a dissimilar behavior depending on whether L is higher or lower than the average volatility. All of this indicates a more complex dynamics in which a reverting force drives volatility toward its mean value. We thus present the mean first-passage time expressions of the most common stochastic volatility models whose approach is comparable to the random diffusion description. We discuss asymptotic approximations of these models and confront them to empirical results with a good agreement with the exponential Ornstein-Uhlenbeck model.
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Affiliation(s)
- Jaume Masoliver
- Departament de Física Fonamental, Universitat de Barcelona, Diagonal 647, E-08028 Barcelona, Spain
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Györgyi G, Moloney NR, Ozogány K, Rácz Z. Maximal height statistics for 1/f(alpha) signals. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2007; 75:021123. [PMID: 17358329 DOI: 10.1103/physreve.75.021123] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 10/18/2006] [Indexed: 05/14/2023]
Abstract
Numerical and analytical results are presented for the maximal relative height distribution of stationary periodic Gaussian signals (one-dimensional interfaces) displaying a 1/f(alpha) power spectrum. For 0<or=alpha<1 (regime of decaying correlations), we observe that the mathematically established limiting distribution (Fisher-Tippett-Gumbel distribution) is approached extremely slowly as the sample size increases. The convergence is rapid for alpha>1 (regime of strong correlations) and a highly accurate picture gallery of distribution functions can be constructed numerically. Analytical results can be obtained in the limit alpha-->infinity and, for large alpha, by perturbation expansion. Furthermore, using path integral techniques we derive a trace formula for the distribution function, valid for alpha=2n even integer. From the latter we extract the small argument asymptote of the distribution function whose analytic continuation to arbitrary alpha>1 is found to be in agreement with simulations. Comparison of the extreme and roughness statistics of the interfaces reveals similarities in both the small and large argument asymptotes of the distribution functions.
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Affiliation(s)
- G Györgyi
- Institute for Theoretical Physics - HAS Research Groups, Eötvös University, Pázmány sétány 1/a, 1117 Budapest, Hungary.
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Eichner JF, Kantelhardt JW, Bunde A, Havlin S. Statistics of return intervals in long-term correlated records. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2007; 75:011128. [PMID: 17358131 DOI: 10.1103/physreve.75.011128] [Citation(s) in RCA: 8] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 06/27/2006] [Indexed: 05/14/2023]
Abstract
We consider long-term correlated data with several distribution densities (Gaussian, exponential, power law, and log normal) and various correlation exponents gamma (0<gamma<1) , and study the statistics of the return intervals r_{j} between events above some threshold q . We show that irrespective of the distribution, the return intervals are long-term correlated in the same way as the original record, but with additional uncorrelated noise. Due to this noise, the correlations are difficult to observe by the detrended fluctuation analysis (which exhibits a crossover behavior) but show up very clearly in the autocorrelation function. The distribution P_{q}(r) of the return intervals is characterized at large scales by a stretched exponential with exponent gamma , and at short scales by a power law with exponent gamma-1 . We discuss in detail the occurrence of finite-size effects for large threshold values for all considered distributions. We show that finite-size effects are most pronounced in exponentially distributed data sets where they can even mask the stretched exponential behavior in records of up to 10;{6} data points. Finally, in order to quantify the clustering of extreme events due to the long-term correlations in the return intervals, we study the conditional distribution function and the related moments. We find that they show pronounced memory effects, irrespective of the distribution of the original data.
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Affiliation(s)
- Jan F Eichner
- Institut für Theoretische Physik III, Justus-Liebig-Universität Giessen, 35392 Giessen, Germany
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Redner S, Petersen MR. Role of global warming on the statistics of record-breaking temperatures. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2006; 74:061114. [PMID: 17280045 DOI: 10.1103/physreve.74.061114] [Citation(s) in RCA: 5] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 01/19/2006] [Revised: 10/03/2006] [Indexed: 05/13/2023]
Abstract
We theoretically study the statistics of record-breaking daily temperatures and validate these predictions using both Monte Carlo simulations and 126 years of available data from the city of Philadelphia. Using extreme statistics, we derive the number and the magnitude of record temperature events, based on the observed Gaussian daily temperature distribution in Philadelphia, as a function of the number of years of observation. We then consider the case of global warming, where the mean temperature systematically increases with time. Over the 126-year time range of observations, we argue that the current warming rate is insufficient to measurably influence the frequency of record temperature events, a conclusion that is supported by numerical simulations and by the Philadelphia data. We also study the role of correlations between temperatures on successive days and find that they do not affect the frequency or magnitude of record temperature events.
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Affiliation(s)
- S Redner
- Center for Polymer Studies and Department of Physics, Boston University, Boston, Massachusetts 02215, USA.
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