1
|
Kumar A, Pal A. Universal Framework for Record Ages under Restart. PHYSICAL REVIEW LETTERS 2023; 130:157101. [PMID: 37115866 DOI: 10.1103/physrevlett.130.157101] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 09/07/2022] [Accepted: 03/16/2023] [Indexed: 06/19/2023]
Abstract
We propose a universal framework to compute record age statistics of a stochastic time series that undergoes random restarts. The proposed framework makes minimal assumptions on the underlying process and is furthermore suited to treat generic restart protocols going beyond the Markovian setting. After benchmarking the framework for classical random walks on the 1D lattice, we derive a universal criterion underpinning the impact of restart on the age of the nth record for generic time series with nearest-neighbor transitions. Crucially, the criterion contains a penalty of order n that puts strong constraints on restart expediting the creation of records, as compared to the simple first-passage completion. The applicability of our approach is further demonstrated on an aggregation-shattering process where we compute the typical growth rates of aggregate sizes. This unified framework paves the way to explore record statistics of time series under restart in a wide range of complex systems.
Collapse
Affiliation(s)
- Aanjaneya Kumar
- Department of Physics, Indian Institute of Science Education and Research, Dr. Homi Bhabha Road, Pune 411008, India
| | - Arnab Pal
- The Institute of Mathematical Sciences, CIT Campus, Taramani, Chennai 600113, India
- Homi Bhabha National Institute, Training School Complex, Anushakti Nagar, Mumbai 400094, India
| |
Collapse
|
2
|
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.
Collapse
Affiliation(s)
| | | | - Alberto Rosso
- Université Paris-Saclay, CNRS, LPTMS, 91405 Orsay, France
| |
Collapse
|
3
|
Global-Scale Synchronization in the Meteorological Data: A Vectorial Analysis That Includes Higher-Order Differences. CLIMATE 2020. [DOI: 10.3390/cli8110128] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 11/16/2022]
Abstract
To examine the evidence of global warming, in recent years, there has been a growing interest in the statistical analysis of time-dependent meteorological data. In this paper, for 116 observational stations in the world, sequential variations of the monthly distributions of meteorological data are analyzed vectorially. For specific monthly data, temperatures and precipitations are chosen, both of which are averaged over three decades. Climate change can be revealed through the intersecting angle between two 33-dimensional vectors being composed with monthly mean values. Subsequently, the angle data for the entire stations are analyzed statistically and compared between the former (1931–1980) and the latter (1951–2010) periods. Irrespective of the period and the hemisphere, the variation of the angles is found to show the exponential growth as a function of their latitudes. Furthermore, consistent with other studies, this trend is shown to become stronger in the latter period, indicating that the so-called snow/ice-albedo feedback occurs. In contrast to the temperatures, for the precipitations, no significant correlation is found between the angle and the latitude. To examine the albedo effect in more detail, a regional analysis for 75 stations in Japan is carried out as well. Numerical results show that the effect is significant even for the relatively narrow latitudinal range (19%) of the hemisphere. Finally, a synchronization of the monthly patterns of temperatures is given between the northern district of Japan and both North America and Eastern Europe.
Collapse
|
4
|
Mori F, Majumdar SN, Schehr G. Distribution of the time between maximum and minimum of random walks. Phys Rev E 2020; 101:052111. [PMID: 32575204 DOI: 10.1103/physreve.101.052111] [Citation(s) in RCA: 9] [Impact Index Per Article: 2.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 03/02/2020] [Accepted: 04/17/2020] [Indexed: 06/11/2023]
Abstract
We consider a one-dimensional Brownian motion of fixed duration T. Using a path-integral technique, we compute exactly the probability distribution of the difference τ=t_{min}-t_{max} between the time t_{min} of the global minimum and the time t_{max} of the global maximum. We extend this result to a Brownian bridge, i.e., a periodic Brownian motion of period T. In both cases, we compute analytically the first few moments of τ, as well as the covariance of t_{max} and t_{min}, showing that these times are anticorrelated. We demonstrate that the distribution of τ for Brownian motion is valid for discrete-time random walks with n steps and with a finite jump variance, in the limit n→∞. In the case of Lévy flights, which have a divergent jump variance, we numerically verify that the distribution of τ differs from the Brownian case. For random walks with continuous and symmetric jumps we numerically verify that the probability of the event "τ=n" is exactly 1/(2n) for any finite n, independently of the jump distribution. Our results can be also applied to describe the distance between the maximal and minimal height of (1+1)-dimensional stationary-state Kardar-Parisi-Zhang interfaces growing over a substrate of finite size L. Our findings are confirmed by numerical simulations. Some of these results have been announced in a recent Letter [Phys. Rev. Lett. 123, 200201 (2019)PRLTAO0031-900710.1103/PhysRevLett.123.200201].
Collapse
Affiliation(s)
- Francesco Mori
- LPTMS, CNRS, Université Paris-Sud, Université Paris-Saclay, 91405 Orsay, 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
| |
Collapse
|
5
|
Influence of the Long-Term Temperature Trend on the Number of New Records for Annual Maximum Daily Precipitation in Japan. ATMOSPHERE 2020. [DOI: 10.3390/atmos11040371] [Citation(s) in RCA: 4] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 11/16/2022]
Abstract
Record-breaking precipitation events have been frequent in Japan in recent years. To investigate the statistical characteristics of the frequency of record-breaking events, observations can be compared with the values derived from sampling theory with a stationary state. This study counted the number of record-breaking daily and 3-day total precipitation events at 58 rain-gauge stations in Japan between 1901 and 2018. The average number of record-breaking events over the 118-year period was 5.9 for daily total precipitation, which is larger than the theoretical value of 5.4 derived using the assumption that the climate system over the same period was stationary. Sampling theory was used to incorporate the influence of the long-term temperature trend from the Clausius–Clapeyron relation associated with the saturation vapor pressure. In theory, the long-term temperature trend gives a similar number of observed record-breaking events when the long-term temperature trend is approximately 0.5 Kelvin/100 years.
Collapse
|
6
|
Mori F, Majumdar SN, Schehr G. Time Between the Maximum and the Minimum of a Stochastic Process. PHYSICAL REVIEW LETTERS 2019; 123:200201. [PMID: 31809107 DOI: 10.1103/physrevlett.123.200201] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.4] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 09/13/2019] [Indexed: 06/10/2023]
Abstract
We present an exact solution for the probability density function P(τ=t_{min}-t_{max}|T) of the time difference between the minimum and the maximum of a one-dimensional Brownian motion of duration T. We then generalize our results to a Brownian bridge, i.e., a periodic Brownian motion of period T. We demonstrate that these results can be directly applied to study the position difference between the minimal and the maximal heights of a fluctuating (1+1)-dimensional Kardar-Parisi-Zhang interface on a substrate of size L, in its stationary state. We show that the Brownian motion result is universal and, asymptotically, holds for any discrete-time random walk with a finite jump variance. We also compute this distribution numerically for Lévy flights and find that it differs from the Brownian motion result.
Collapse
Affiliation(s)
- Francesco Mori
- LPTMS, CNRS, Université Paris-Sud, Université Paris-Saclay, 91405 Orsay, 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
| |
Collapse
|
7
|
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.
Collapse
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
| |
Collapse
|
8
|
Kundu M, Mukherjee S, Biswas S. Record-breaking statistics near second-order phase transitions. Phys Rev E 2018; 98:022103. [PMID: 30253517 DOI: 10.1103/physreve.98.022103] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 01/04/2018] [Indexed: 06/08/2023]
Abstract
When a quantity reaches a value higher (or lower) than its value at any time before, it is said to have made a record. We numerically study the statistical properties of records in the time series of order parameters in different models near their critical points. Specifically, we choose the transversely driven Edwards-Wilkinson model for interface depinning in (1+1) dimensions and the Ising model in two dimensions, as paradigmatic and simple examples of nonequilibrium and equilibrium critical behaviors, respectively. The total number of record-breaking events in the time series of the order parameters of the models show maxima when the system is near criticality. The number of record-breaking events and associated quantities, such as the distribution of the waiting time between successive record events, show power-law scaling near the critical point. The exponent values are specific to the universality classes of the respective models. Such behaviors near criticality can be used as a precursor to imminent criticality, i.e., abrupt and catastrophic changes in the system. Due to the extreme nature of the records, its measurements are relatively free of detection errors and thus provide a clear signal regarding the state of the system in which they are measured.
Collapse
Affiliation(s)
- Mily Kundu
- Condensed Matter Physics Division, Saha Institute of Nuclear Physics, 1/AF Bidhannagar, Kolkata 700064, India
| | - Sudip Mukherjee
- Condensed Matter Physics Division, Saha Institute of Nuclear Physics, 1/AF Bidhannagar, Kolkata 700064, India
- Barasat Government College, Barasat, Kolkata 700124, India
| | - Soumyajyoti Biswas
- Max Planck Institute for Dynamics and Self-Organization, Am Fassberg 17, 37077 Göttingen, Germany
| |
Collapse
|
9
|
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.
Collapse
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
| |
Collapse
|
10
|
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….
Collapse
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
| |
Collapse
|
11
|
Pál G, Raischel F, Lennartz-Sassinek S, Kun F, Main IG. Record-breaking events during the compressive failure of porous materials. Phys Rev E 2016; 93:033006. [PMID: 27078440 DOI: 10.1103/physreve.93.033006] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 07/02/2015] [Indexed: 06/05/2023]
Abstract
An accurate understanding of the interplay between random and deterministic processes in generating extreme events is of critical importance in many fields, from forecasting extreme meteorological events to the catastrophic failure of materials and in the Earth. Here we investigate the statistics of record-breaking events in the time series of crackling noise generated by local rupture events during the compressive failure of porous materials. The events are generated by computer simulations of the uniaxial compression of cylindrical samples in a discrete element model of sedimentary rocks that closely resemble those of real experiments. The number of records grows initially as a decelerating power law of the number of events, followed by an acceleration immediately prior to failure. The distribution of the size and lifetime of records are power laws with relatively low exponents. We demonstrate the existence of a characteristic record rank k(*), which separates the two regimes of the time evolution. Up to this rank deceleration occurs due to the effect of random disorder. Record breaking then accelerates towards macroscopic failure, when physical interactions leading to spatial and temporal correlations dominate the location and timing of local ruptures. The size distribution of records of different ranks has a universal form independent of the record rank. Subsequences of events that occur between consecutive records are characterized by a power-law size distribution, with an exponent which decreases as failure is approached. High-rank records are preceded by smaller events of increasing size and waiting time between consecutive events and they are followed by a relaxation process. As a reference, surrogate time series are generated by reshuffling the event times. The record statistics of the uncorrelated surrogates agrees very well with the corresponding predictions of independent identically distributed random variables, which confirms that temporal and spatial correlation in the crackling noise is responsible for the observed unique behavior. In principle the results could be used to improve forecasting of catastrophic failure events, if they can be observed reliably in real time.
Collapse
Affiliation(s)
- Gergő Pál
- Department of Theoretical Physics, University of Debrecen, P.O. Box 5, H-4010 Debrecen, Hungary
| | - Frank Raischel
- Department of Theoretical Physics, University of Debrecen, P.O. Box 5, H-4010 Debrecen, Hungary
| | - Sabine Lennartz-Sassinek
- School of Geosciences, University of Edinburgh, EH9 3FE Edinburgh, United Kingdom
- Institute for Geophysics and Meteorology, University of Cologne, Cologne, Germany
| | - Ferenc Kun
- Department of Theoretical Physics, University of Debrecen, P.O. Box 5, H-4010 Debrecen, Hungary
| | - Ian G Main
- School of Geosciences, University of Edinburgh, EH9 3FE Edinburgh, United Kingdom
| |
Collapse
|
12
|
Ben-Naim E, Krapivsky PL, Lemons NW. Scaling exponents for ordered maxima. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2015; 92:062139. [PMID: 26764664 DOI: 10.1103/physreve.92.062139] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 10/23/2015] [Indexed: 06/05/2023]
Abstract
We study extreme value statistics of multiple sequences of random variables. For each sequence with N variables, independently drawn from the same distribution, the running maximum is defined as the largest variable to date. We compare the running maxima of m independent sequences and investigate the probability S(N) that the maxima are perfectly ordered, that is, the running maximum of the first sequence is always larger than that of the second sequence, which is always larger than the running maximum of the third sequence, and so on. The probability S(N) is universal: it does not depend on the distribution from which the random variables are drawn. For two sequences, S(N)∼N(-1/2), and in general, the decay is algebraic, S(N)∼N(-σ(m)), for large N. We analytically obtain the exponent σ(3)≅1.302931 as root of a transcendental equation. Furthermore, the exponents σ(m) grow with m, and we show that σ(m)∼m for large m.
Collapse
Affiliation(s)
- E Ben-Naim
- Theoretical Division and Center for Nonlinear Studies, Los Alamos National Laboratory, Los Alamos, New Mexico 87545, USA
| | - P L Krapivsky
- Department of Physics, Boston University, Boston, Massachusetts 02215, USA
- Institut de Physique Théorique, Université Paris-Saclay, CEA and CNRS, 91191 Gif-sur-Yvette, France
| | - N W Lemons
- Theoretical Division and Center for Nonlinear Studies, Los Alamos National Laboratory, Los Alamos, New Mexico 87545, USA
| |
Collapse
|
13
|
Ben-Naim E, Krapivsky PL. Statistics of superior records. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2013; 88:022145. [PMID: 24032813 DOI: 10.1103/physreve.88.022145] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 05/18/2013] [Revised: 07/15/2013] [Indexed: 06/02/2023]
Abstract
We study statistics of records in a sequence of random variables. These identical and independently distributed variables are drawn from the parent distribution ρ. The running record equals the maximum of all elements in the sequence up to a given point. We define a superior sequence as one where all running records are above the average record expected for the parent distribution ρ. We find that the fraction of superior sequences S(N) decays algebraically with sequence length N, S(N)~N(-β) in the limit N→∞. Interestingly, the decay exponent β is nontrivial, being the root of an integral equation. For example, when ρ is a uniform distribution with compact support, we find β=0.450265. In general, the tail of the parent distribution governs the exponent β. We also consider the dual problem of inferior sequences, where all records are below average, and find that the fraction of inferior sequences I(N) decays algebraically, albeit with a different decay exponent, I(N)~N(-α). We use the above statistical measures to analyze earthquake data.
Collapse
Affiliation(s)
- E Ben-Naim
- Theoretical Division and Center for Nonlinear Studies, Los Alamos National Laboratory, Los Alamos, New Mexico 87545, USA
| | | |
Collapse
|
14
|
Edery Y, Kostinski AB, Majumdar SN, Berkowitz B. Record-breaking statistics for random walks in the presence of measurement error and noise. PHYSICAL REVIEW LETTERS 2013; 110:180602. [PMID: 23683184 DOI: 10.1103/physrevlett.110.180602] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 02/03/2013] [Indexed: 06/02/2023]
Abstract
We examine distance record setting by a random walker in the presence of a measurement error δ and additive noise γ and show that the mean number of (upper) records up to n steps still grows universally as (R(n)) ~ n(1/2) for large n for all jump densities, including Lévy distributions, and for all δ and γ. In contrast, the pace of record setting, measured by the amplitude of the n(1/2) growth, depends on δ and γ. In the absence of noise (γ=0), the amplitude S(δ) is evaluated explicitly for arbitrary jump distributions and it decreases monotonically with increasing δ whereas, in the case of perfect measurement (δ=0), the corresponding amplitude T(γ) increases with γ. The exact results for S(δ) offer a new perspective for characterizing instrumental precision by means of record counting. Our analytical results are supported by extensive numerical simulations.
Collapse
Affiliation(s)
- Yaniv Edery
- Department of Environmental Sciences and Energy Research, Weizmann Institute of Science, 76100 Rehovot, Israel
| | | | | | | |
Collapse
|
15
|
Shcherbakov R, Davidsen J, Tiampo KF. Record-breaking avalanches in driven threshold systems. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2013; 87:052811. [PMID: 23767588 DOI: 10.1103/physreve.87.052811] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 08/09/2012] [Revised: 02/03/2013] [Indexed: 06/02/2023]
Abstract
Record-breaking avalanches generated by the dynamics of several driven nonlinear threshold models are studied. Such systems are characterized by intermittent behavior, where a slow buildup of energy is punctuated by an abrupt release of energy through avalanche events, which usually follow scale-invariant statistics. From the simulations of these systems it is possible to extract sequences of record-breaking avalanches, where each subsequent record-breaking event is larger in magnitude than all previous events. In the present work, several cellular automata are analyzed, among them the sandpile model, the Manna model, the Olami-Feder-Christensen (OFC) model, and the forest-fire model to investigate the record-breaking statistics of model avalanches that exhibit temporal and spatial correlations. Several statistical measures of record-breaking events are derived analytically and confirmed through numerical simulations. The statistics of record-breaking avalanches for the four models are compared to those of record-breaking events extracted from the sequences of independent and identically distributed (i.i.d.) random variables. It is found that the statistics of record-breaking avalanches for the above cellular automata exhibit behavior different from that observed for i.i.d. random variables, which in turn can be used to characterize complex spatiotemporal dynamics. The most pronounced deviations are observed in the case of the OFC model with a strong dependence on the conservation parameter of the model. This indicates that avalanches in the OFC model are not independent and exhibit spatiotemporal correlations.
Collapse
Affiliation(s)
- Robert Shcherbakov
- Department of Earth Sciences, University of Western Ontario, London, Ontario, Canada N6A 5B7.
| | | | | |
Collapse
|
16
|
Wergen G, Volovik D, Redner S, Krug J. Rounding effects in record statistics. PHYSICAL REVIEW LETTERS 2012; 109:164102. [PMID: 23215081 DOI: 10.1103/physrevlett.109.164102] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 06/19/2012] [Indexed: 06/01/2023]
Abstract
We analyze record-breaking events in time series of continuous random variables that are subsequently discretized by rounding to integer multiples of a discretization scale Δ>0. Rounding leads to ties of an existing record, thereby reducing the number of new records. For an infinite number of random variables that are drawn from distributions with a finite upper limit, the number of discrete records is finite, while for distributions with a thinner than exponential upper tail, fewer discrete records arise compared to continuous variables. In the latter case, the record sequence becomes highly regular at long times.
Collapse
Affiliation(s)
- G Wergen
- Institute for Theoretical Physics, University of Cologne, 50937 Köln, Germany
| | | | | | | |
Collapse
|
17
|
Wergen G, Majumdar SN, Schehr G. Record statistics for multiple random walks. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2012; 86:011119. [PMID: 23005380 DOI: 10.1103/physreve.86.011119] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 04/25/2012] [Indexed: 06/01/2023]
Abstract
We study the statistics of the number of records R(n,N) for N identical and independent symmetric discrete-time random walks of n steps in one dimension, all starting at the origin at step 0. At each time step, each walker jumps by a random length drawn independently from a symmetric and continuous distribution. We consider two cases: (I) when the variance σ(2) of the jump distribution is finite and (II) when σ(2) is divergent as in the case of Lévy flights with index 0<μ<2. In both cases we find that the mean record number R(n,N) grows universally as ~α(N) sqrt[n] for large n, but with a very different behavior of the amplitude α(N) for N>1 in the two cases. We find that for large N, α(N) ≈ 2sqrt[lnN] independently of σ(2) in case I. In contrast, in case II, the amplitude approaches to an N-independent constant for large N, α(N) ≈ 4/sqrt[π], independently of 0<μ<2. For finite σ(2) we argue-and this is confirmed by our numerical simulations-that the full distribution of (R(n,N)/sqrt[n]-2sqrt[lnN])sqrt[lnN] converges to a Gumbel law as n → ∞ and N → ∞. In case II, our numerical simulations indicate that the distribution of R(n,N)/sqrt[n] converges, for n → ∞ and N → ∞, to a universal nontrivial distribution independently of μ. We discuss the applications of our results to the study of the record statistics of 366 daily stock prices from the Standard & Poor's 500 index.
Collapse
Affiliation(s)
- Gregor Wergen
- Institut für Theoretische Physik, Universität zu Köln, 50937 Köln, Germany.
| | | | | |
Collapse
|
18
|
Reynolds A. Olfactory search behaviour in the wandering albatross is predicted to give rise to Lévy flight movement patterns. Anim Behav 2012. [DOI: 10.1016/j.anbehav.2012.02.014] [Citation(s) in RCA: 14] [Impact Index Per Article: 1.2] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 10/28/2022]
|
19
|
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.
Collapse
Affiliation(s)
- N R Moloney
- Max Planck Institute for the Physics of Complex Systems, Nöthnitzer Str 38, D-01187 Dresden, Germany.
| | | | | |
Collapse
|
20
|
Abstract
We develop a theoretical approach to quantify the effect of long-term trends on the expected number of extremes in generic time series, using analytical solutions and Monte Carlo simulations. We apply our method to study the effect of warming trends on heat records. We find that the number of record-breaking events increases approximately in proportion to the ratio of warming trend to short-term standard deviation. Short-term variability thus decreases the number of heat extremes, whereas a climatic warming increases it. For extremes exceeding a predefined threshold, the dependence on the warming trend is highly nonlinear. We further find that the sum of warm plus cold extremes increases with any climate change, whether warming or cooling. We estimate that climatic warming has increased the number of new global-mean temperature records expected in the last decade from 0.1 to 2.8. For July temperature in Moscow, we estimate that the local warming trend has increased the number of records expected in the past decade fivefold, which implies an approximate 80% probability that the 2010 July heat record would not have occurred without climate warming.
Collapse
|
21
|
Newman WI, Malamud BD, Turcotte DL. Statistical properties of record-breaking temperatures. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2010; 82:066111. [PMID: 21230709 DOI: 10.1103/physreve.82.066111] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 08/01/2010] [Revised: 10/08/2010] [Indexed: 05/30/2023]
Abstract
A record-breaking temperature is the highest or lowest temperature at a station since the period of time considered began. The temperatures at a station constitute a time series. After the removal of daily and annual periodicities, the primary considerations are trends (i.e., global warming) and long-range correlations. We first carry out Monte Carlo simulations to determine the influence of trends and long-range correlations on record-breaking statistics. We take a time series that is a Gaussian white noise and give the classic record-breaking theory results for an independent and identically distributed process. We then carry out simulations to determine the influence of long-range correlations and linear temperature trends. For the range of fractional Gaussian noises that are observed to be applicable to temperature time series, the influence on the record-breaking statistics is less than 10%. We next superimpose a linear trend on a Gaussian white noise and extend the theory to include the effect of an additive trend. We determine the ratios of the number of maximum to the number of minimum record-breaking temperatures. We find the single governing parameter to be the ratio of the temperature change per year to the standard deviation of the underlying white noise. To test our approach, we consider a 30 yr record of temperatures at the Mauna Loa Observatory for 1977-2006. We determine the temperature trends by direct measurements and use our simulations to infer trends from the number of record-breaking temperatures. The two approaches give values that are in good agreement. We find that the warming trend is primarily due to an increase in the (overnight) minimum temperatures, while the maximum (daytime) temperatures are approximately constant.
Collapse
Affiliation(s)
- William I Newman
- Department of Earth and Space Sciences, University of California, Los Angeles, California 90095, USA.
| | | | | |
Collapse
|
22
|
Eliazar I, Klafter J. Record events in growing populations: universality, correlation, and aging. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2009; 80:061117. [PMID: 20365128 DOI: 10.1103/physreve.80.061117] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 09/02/2009] [Indexed: 05/29/2023]
Abstract
This paper studies the occurrence of record events in score populations which grow stochastically in time. In Rényi's basic record model, a population of independent and identically distributed (i.i.d.) random scores grows deterministically--a single new score being added at each time step. Rényi's record theorem asserts that the resulting record events are independent, and that their occurrence probabilities decrease harmonically in time. Moreover, Rényi's result is universal--being independent of the distribution of the i.i.d. random scores. This paper considers an arbitrary stochastic growth of the score population--allowing the number of the i.i.d. random scores added at each time step to follow arbitrary stochastic dynamics. Exploring the stochastic growth model we: (i) establish a general analog of Rényi's record theorem; (ii) show that universality with respect to the distribution of the i.i.d. random scores is maintained; (iii) compute the distribution of the waiting times for record events; (iv) analyze the dependencies/independencies of the record events; and (v) analyze the aging/stationarity of the record events.
Collapse
Affiliation(s)
- Iddo Eliazar
- Department of Technology Management, Holon Institute of Technology, PO Box 305, Holon 58102, Israel.
| | | |
Collapse
|
23
|
Le Doussal P, Wiese KJ. Driven particle in a random landscape: disorder correlator, avalanche distribution, and extreme value statistics of records. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2009; 79:051105. [PMID: 19518414 DOI: 10.1103/physreve.79.051105] [Citation(s) in RCA: 10] [Impact Index Per Article: 0.7] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 10/30/2008] [Indexed: 05/27/2023]
Abstract
We review how the renormalized force correlator Delta(micro) , the function computed in the functional renormalization-group (RG) field theory, can be measured directly in numerics and experiments on the dynamics of elastic manifolds in the presence of pinning disorder. We show how this function can be computed analytically for a particle dragged through a one-dimensional random-force landscape. The limit of small velocity allows one to access the critical behavior at the depinning transition. For uncorrelated forces one finds three universality classes, corresponding to the three extreme value statistics, Gumbel, Weibull, and Fréchet. For each class we obtain analytically the universal function Delta(micro) , the corrections to the critical force, and the joint probability distribution of avalanche sizes s and waiting times w . We find P(s)=P(w) for all three cases. All results are checked numerically. For a Brownian force landscape, known as the Alessandro, Beatrice, Bertotti, and Montorsi (ABBM) model, avalanche distributions and Delta(micro) can be computed for any velocity. For two-dimensional disorder, we perform large-scale numerical simulations to calculate the renormalized force correlator tensor Delta_{ij}(micro[over ]) , and to extract the anisotropic scaling exponents zeta_{x}>zeta_{y} . We also show how the Middleton theorem is violated. Our results are relevant for the record statistics of random sequences with linear trends, as encountered, e.g., in some models of global warming. We give the joint distribution of the time s between two successive records and their difference in value w .
Collapse
Affiliation(s)
- Pierre Le Doussal
- CNRS-Laboratoire de Physique Théorique de l'Ecole Normale Supérieure, 24 rue Lhomond, 75231 Paris Cedex, France
| | | |
Collapse
|
24
|
Majumdar SN, Ziff RM. Universal record statistics of random walks and Lévy flights. PHYSICAL REVIEW LETTERS 2008; 101:050601. [PMID: 18764382 DOI: 10.1103/physrevlett.101.050601] [Citation(s) in RCA: 13] [Impact Index Per Article: 0.8] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 05/31/2008] [Indexed: 05/26/2023]
Abstract
It is shown that statistics of records for time series generated by random walks are independent of the details of the jump distribution, as long as the latter is continuous and symmetric. In N steps, the mean of the record distribution grows as the sqrt[4N/pi] while the standard deviation grows as sqrt[(2-4/pi)N], so the distribution is non-self-averaging. The mean shortest and longest duration records grow as sqrt[N/pi] and 0.626 508...N, respectively. The case of a discrete random walker is also studied, and similar asymptotic behavior is found.
Collapse
Affiliation(s)
- Satya N Majumdar
- Laboratoire de Physique Théorique et Modèles Statistiques (UMR 8626 du CNRS), Université Paris-Sud, Bâtiment 100, Orsay Cedex, France
| | | |
Collapse
|
25
|
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.
Collapse
Affiliation(s)
- T J Oliveira
- Instituto de Física, Universidade Federal Fluminense, Avenida Litorânea s/n, 24210-340 Niterói RJ, Brazil
| | | |
Collapse
|
26
|
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.
Collapse
Affiliation(s)
- Sanjib Sabhapandit
- Laboratoire de Physique Théorique et Modèles Statistiques, UMR 8626 du CNRS, Université Paris-Sud, Bâtiment 100, 91405 Orsay Cedex, France
| | | |
Collapse
|