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Schimmenti VM, Majumdar SN, Rosso A. Statistical properties of avalanches via the c-record process. Phys Rev E 2021; 104:064129. [PMID: 35030910 DOI: 10.1103/physreve.104.064129] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 06/04/2021] [Accepted: 10/26/2021] [Indexed: 06/14/2023]
Abstract
We study the statistics of avalanches, as a response to an applied force, undergone by a particle hopping on a one-dimensional lattice where the pinning forces at each site are independent and identically distributed (i.i.d.), each drawn from a continuous f(x). The avalanches in this model correspond to the interrecord intervals in a modified record process of i.i.d. variables, defined by a single parameter c>0. This parameter characterizes the record formation via the recursive process R_{k}>R_{k-1}-c, where R_{k} denotes the value of the kth record. We show that for c>0, if f(x) decays slower than an exponential for large x, the record process is nonstationary as in the standard c=0 case. In contrast, if f(x) has a faster than exponential tail, the record process becomes stationary and the avalanche size distribution π(n) has a decay faster than 1/n^{2} for large n. The marginal case where f(x) decays exponentially for large x exhibits a phase transition from a nonstationary phase to a stationary phase as c increases through a critical value c_{crit}. Focusing on f(x)=e^{-x} (with x≥0), we show that c_{crit}=1 and for c<1, the record statistics is nonstationary. However, for c>1, the record statistics is stationary with avalanche size distribution π(n)∼n^{-1-λ(c)} for large n. Consequently, for c>1, the mean number of records up to N steps grows algebraically ∼N^{λ(c)} for large N. Remarkably, the exponent λ(c) depends continuously on c for c>1 and is given by the unique positive root of c=-ln(1-λ)/λ. We also unveil the presence of nontrivial correlations between avalanches in the stationary phase that resemble earthquake sequences.
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Affiliation(s)
| | | | - Alberto Rosso
- Université Paris-Saclay, CNRS, LPTMS, 91405 Orsay, France
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2
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Ierley G, Kostinski A. Detection of unknown signals in arbitrary noise. Phys Rev E 2020; 102:032221. [PMID: 33075960 DOI: 10.1103/physreve.102.032221] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 04/14/2020] [Accepted: 09/01/2020] [Indexed: 11/07/2022]
Abstract
We devise a simple method for detecting signals of unknown form buried in any noise, including heavy tailed. The method centers on signal-noise decomposition in rank and time: Only stationary white noise generates data with a jointly uniform rank-time probability distribution, U(1,N)×U(1,N), for N data points in a time series. Signals of any kind distort this uniformity. Such distortions are captured by rank-time cumulative distributions permitting all-purpose efficient detection, even for single time series and noise of infinite variance.
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Affiliation(s)
- Glenn Ierley
- Department of Mathematical Sciences, Michigan Technological University, Houghton, Michigan 49931, USA and Scripps Institution of Oceanography, University of California San Diego, San Diego, California 92093, USA
| | - Alex Kostinski
- Department of Physics, Michigan Technological University, Houghton, Michigan 49934, USA
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Majumdar SN, von Bomhard P, Krug J. Exactly Solvable Record Model for Rainfall. PHYSICAL REVIEW LETTERS 2019; 122:158702. [PMID: 31050498 DOI: 10.1103/physrevlett.122.158702] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 08/27/2018] [Indexed: 06/09/2023]
Abstract
Daily precipitation time series are composed of null entries corresponding to dry days and nonzero entries that describe the rainfall amounts on wet days. Assuming that wet days follow a Bernoulli process with success probability p, we show that the presence of dry days induces negative correlations between record-breaking precipitation events. The resulting nonmonotonic behavior of the Fano factor of the record counting process is recovered in empirical data. We derive the full probability distribution P(R,n) of the number of records R_{n} up to time n, and show that for large n, it converges to a Poisson distribution with parameter ln(pn). We also study in detail the joint limit p→0, n→∞, which yields a random record model in continuous time t=pn.
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Affiliation(s)
- Satya N Majumdar
- Université Paris-Sud, CNRS, LPTMS, UMR 8626, 91405 Orsay, France
| | | | - Joachim Krug
- Institute for Biological Physics, University of Cologne, 50937 Köln, Germany
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Kindermann F, Hohmann M, Lausch T, Mayer D, Schmidt F, Widera A. Extreme event statistics in a drifting Markov chain. Phys Rev E 2018; 96:012130. [PMID: 29347186 DOI: 10.1103/physreve.96.012130] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.7] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 03/09/2017] [Indexed: 11/07/2022]
Abstract
We analyze extreme event statistics of experimentally realized Markov chains with various drifts. Our Markov chains are individual trajectories of a single atom diffusing in a one-dimensional periodic potential. Based on more than 500 individual atomic traces we verify the applicability of the Sparre Andersen theorem to our system despite the presence of a drift. We present detailed analysis of four different rare-event statistics for our system: the distributions of extreme values, of record values, of extreme value occurrence in the chain, and of the number of records in the chain. We observe that, for our data, the shape of the extreme event distributions is dominated by the underlying exponential distance distribution extracted from the atomic traces. Furthermore, we find that even small drifts influence the statistics of extreme events and record values, which is supported by numerical simulations, and we identify cases in which the drift can be determined without information about the underlying random variable distributions. Our results facilitate the use of extreme event statistics as a signal for small drifts in correlated trajectories.
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Affiliation(s)
- Farina Kindermann
- Department of Physics and Research Center OPTIMAS, University of Kaiserslautern, 67663 Kaiserslautern, Germany
| | - Michael Hohmann
- Department of Physics and Research Center OPTIMAS, University of Kaiserslautern, 67663 Kaiserslautern, Germany
| | - Tobias Lausch
- Department of Physics and Research Center OPTIMAS, University of Kaiserslautern, 67663 Kaiserslautern, Germany
| | - Daniel Mayer
- Department of Physics and Research Center OPTIMAS, University of Kaiserslautern, 67663 Kaiserslautern, Germany.,Graduate School Materials Science in Mainz, Gottlieb-Daimler-Strasse 47, 67663 Kaiserslautern, Germany
| | - Felix Schmidt
- Department of Physics and Research Center OPTIMAS, University of Kaiserslautern, 67663 Kaiserslautern, Germany.,Graduate School Materials Science in Mainz, Gottlieb-Daimler-Strasse 47, 67663 Kaiserslautern, Germany
| | - Artur Widera
- Department of Physics and Research Center OPTIMAS, University of Kaiserslautern, 67663 Kaiserslautern, Germany.,Graduate School Materials Science in Mainz, Gottlieb-Daimler-Strasse 47, 67663 Kaiserslautern, Germany
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Godrèche C, Majumdar SN, Schehr G. Exact Statistics of Record Increments of Random Walks and Lévy Flights. PHYSICAL REVIEW LETTERS 2016; 117:010601. [PMID: 27419552 DOI: 10.1103/physrevlett.117.010601] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 04/05/2016] [Indexed: 06/06/2023]
Abstract
We study the statistics of increments in record values in a time series {x_{0}=0,x_{1},x_{2},…,x_{n}} generated by the positions of a random walk (discrete time, continuous space) of duration n steps. For arbitrary jump length distribution, including Lévy flights, we show that the distribution of the record increment becomes stationary, i.e., independent of n for large n, and compute it explicitly for a wide class of jump distributions. In addition, we compute exactly the probability Q(n) that the record increments decrease monotonically up to step n. Remarkably, Q(n) is universal (i.e., independent of the jump distribution) for each n, decaying as Q(n)∼A/sqrt[n] for large n, with a universal amplitude A=e/sqrt[π]=1.53362….
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Affiliation(s)
- Claude Godrèche
- Institut de Physique Théorique, Université Paris-Saclay, CEA and CNRS, 91191 Gif-sur-Yvette, France
| | - Satya N Majumdar
- LPTMS, CNRS, Université Paris-Sud, Université Paris-Saclay, 91405 Orsay, France
| | - Grégory Schehr
- LPTMS, CNRS, Université Paris-Sud, Université Paris-Saclay, 91405 Orsay, France
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Radicchi F. Underestimating extreme events in power-law behavior due to machine-dependent cutoffs. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2014; 90:050801. [PMID: 25493726 DOI: 10.1103/physreve.90.050801] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 04/22/2014] [Indexed: 06/04/2023]
Abstract
Power-law distributions are typical macroscopic features occurring in almost all complex systems observable in nature. As a result, researchers in quantitative analyses must often generate random synthetic variates obeying power-law distributions. The task is usually performed through standard methods that map uniform random variates into the desired probability space. Whereas all these algorithms are theoretically solid, in this paper we show that they are subject to severe machine-dependent limitations. As a result, two dramatic consequences arise: (i) the sampling in the tail of the distribution is not random but deterministic; (ii) the moments of the sample distribution, which are theoretically expected to diverge as functions of the sample sizes, converge instead to finite values. We provide quantitative indications for the range of distribution parameters that can be safely handled by standard libraries used in computational analyses. Whereas our findings indicate possible reinterpretations of numerical results obtained through flawed sampling methodologies, they also pave the way for the search for a concrete solution to this central issue shared by all quantitative sciences dealing with complexity.
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Affiliation(s)
- Filippo Radicchi
- Center for Complex Networks and Systems Research, School of Informatics and Computing, Indiana University, Bloomington, Indiana 47408, USA
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Sabir B, Santhanam MS. Record statistics of financial time series and geometric random walks. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2014; 90:032126. [PMID: 25314414 DOI: 10.1103/physreve.90.032126] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 07/01/2014] [Indexed: 06/04/2023]
Abstract
The study of record statistics of correlated series in physics, such as random walks, is gaining momentum, and several analytical results have been obtained in the past few years. In this work, we study the record statistics of correlated empirical data for which random walk models have relevance. We obtain results for the records statistics of select stock market data and the geometric random walk, primarily through simulations. We show that the distribution of the age of records is a power law with the exponent α lying in the range 1.5≤α≤1.8. Further, the longest record ages follow the Fréchet distribution of extreme value theory. The records statistics of geometric random walk series is in good agreement with that obtained from empirical stock data.
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Affiliation(s)
- Behlool Sabir
- Indian Institute of Science Education and Research, Dr. Homi Bhabha Road, Pune 411 008, India
| | - M S Santhanam
- Indian Institute of Science Education and Research, Dr. Homi Bhabha Road, Pune 411 008, India
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Majumdar SN, Mounaix P, Schehr G. Exact statistics of the gap and time interval between the first two maxima of random walks and Lévy flights. PHYSICAL REVIEW LETTERS 2013; 111:070601. [PMID: 23992054 DOI: 10.1103/physrevlett.111.070601] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 03/21/2013] [Indexed: 06/02/2023]
Abstract
We investigate the statistics of the gap G(n) between the two rightmost positions of a Markovian one-dimensional random walker (RW) after n time steps and of the duration L(n) which separates the occurrence of these two extremal positions. The distribution of the jumps η(i)'s of the RW, f(η), is symmetric and its Fourier transform has the small k behavior 1-f[over ^](k)~|k|(μ), with 0<μ≤2. For μ=2, the RW converges, for large n, to Brownian motion, while for 0<μ<2 it corresponds to a Lévy flight of index μ. We compute the joint probability density function (PDF) P(n)(g,l) of G(n) and L(n) and show that, when n→∞, it approaches a limiting PDF p(g,l). The corresponding marginal PDFs of the gap, p(gap)(g), and of L(n), p(time)(l), are found to behave like p(gap)(g)~g(-1-μ) for g>>1 and 0<μ<2, and p(time)(l)~l(-γ(μ)) for l>>1 with γ(1<μ≤2)=1+1/μ and γ(0<μ<1)=2. For l, g>>1 with fixed lg(-μ), p(g,l) takes the scaling form p(g,l)~g(-1-2μ)p[over ˜](μ)(lg(-μ)), where p[over ˜](μ)(y) is a (μ-dependent) scaling function. We also present numerical simulations which verify our analytic results.
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