scholarly journals GLOBAL FLUCTUATIONS IN PHYSICAL SYSTEMS: A SUBTLE INTERPLAY BETWEEN SUM AND EXTREME VALUE STATISTICS

2008 ◽  
Vol 22 (20) ◽  
pp. 3311-3368 ◽  
Author(s):  
MAXIME CLUSEL ◽  
ERIC BERTIN
Keyword(s):  
Extreme Values ◽  
Random Variables ◽  
Extreme Value ◽  
Physical Models ◽  
Extreme Value Statistics ◽  
Specific Class ◽  
Qualitative Properties ◽  
Rigorous Results ◽  
General Mapping ◽  
Global Fluctuations

Fluctuations of global additive quantities, like total energy or magnetization for instance, can in principle be described by statistics of sums of (possibly correlated) random variables. Yet, it turns out that extreme values (the largest value among a set of random variables) may also play a role in the statistics of global quantities, in a direct or indirect way. This review discusses different connections that may appear between problems of sums and of extreme values of random variables, and emphasizes physical situations in which such connections are relevant. Along this line of thought, standard convergence theorems for sums and extreme values of independent and identically distributed random variables are recalled, and some rigorous results as well as more heuristic reasonings are presented for correlated or non-identically distributed random variables. More specifically, the role of extreme values within sums of broadly distributed variables is addressed, and a general mapping between extreme values and sums is presented, allowing us to identify a class of correlated random variables whose sum follows (generalized) extreme value distributions. Possible applications of this specific class of random variables are illustrated on the example of two simple physical models. A few extensions to other related classes of random variables sharing similar qualitative properties are also briefly discussed, in connection with the so-called BHP distribution.

scholarly journals Statistics of the largest geomagnetic storms per solar cycle (1844-1993)

1997 ◽  
Vol 15 (6) ◽  
pp. 719-728 ◽  
Author(s):  
D. M. Willis ◽  
P. R. Stevens ◽  
S. R. Crothers
Keyword(s):  
Statistical Analysis ◽  
Solar Cycle ◽  
Geomagnetic Storm ◽  
Extreme Values ◽  
Geomagnetic Storms ◽  
Extreme Value ◽  
Extreme Value Statistics ◽  
Relative Dispersion ◽  
Solar Cycles ◽  
Aa Index

Abstract. A previous application of extreme-value statistics to the first, second and third largest geomagnetic storms per solar cycle for nine solar cycles is extended to fourteen solar cycles (1844–1993). The intensity of a geomagnetic storm is measured by the magnitude of the daily aa index, rather than the half-daily aa index used previously. Values of the conventional aa index (1868–1993), supplemented by the Helsinki Ak index (1844–1880), provide an almost continuous, and largely homogeneous, daily measure of geomagnetic activity over an interval of 150 years. As in the earlier investigation, analytic expressions giving the probabilities of the three greatest storms (extreme values) per solar cycle, as continuous functions of storm magnitude (aa), are obtained by least-squares fitting of the observations to the appropriate theoretical extreme-value probability functions. These expressions are used to obtain the statistical characteristics of the extreme values; namely, the mode, median, mean, standard deviation and relative dispersion. Since the Ak index may not provide an entirely homogeneous extension of the aa index, the statistical analysis is performed separately for twelve solar cycles (1868–1993), as well as nine solar cycles (1868–1967). The results are utilized to determine the expected ranges of the extreme values as a function of the number of solar cycles. For fourteen solar cycles, the expected ranges of the daily aa index for the first, second and third largest geomagnetic storms per solar cycle decrease monotonically in magnitude, contrary to the situation for the half-daily aa index over nine solar cycles. The observed range of the first extreme daily aa index for fourteen solar cycles is 159–352 nT and for twelve solar cycles is 215–352 nT. In a group of 100 solar cycles the expected ranges are expanded to 137–539 and 177–511 nT, which represent increases of 108% and 144% in the respective ranges. Thus there is at least a 99% probability that the daily aa index will satisfy the condition aa < 550 for the largest geomagnetic storm in the next 100 solar cycles. The statistical analysis is used to infer that remarkable conjugate auroral observations on the night of 16 September 1770, which were recorded during the first voyage of Captain Cook to Australia, occurred during an intense geomagnetic storm.

Extreme Value Metamodeling for System Reliability With Time-Dependent Functions

2015 ◽  
Author(s):  
Zhifu Zhu ◽  
Xiaoping Du
Keyword(s):  
Computational Efficiency ◽  
System Reliability ◽  
Extreme Values ◽  
Random Variables ◽  
Probability Of Failure ◽  
High Accuracy ◽  
Extreme Value ◽  
Time Dependent ◽  
System Failure ◽  
Kriging Method

The reliability of a system is usually measured by the probability that the system performs its intended function in a given period of time. Estimating such reliability is a challenging task when the probability of failure is rare and the responses are nonlinear and time variant. The evaluation of the system reliability defined in a period of time requires the extreme values of the responses in the predefined period of time during which the system is supposed to function. This work builds surrogate models for the extreme values of responses with the Kriging method. For the sake of computational efficiency, the method creates Kriging models with high accuracy only in the region that has high contributions to the system failure; training points of random variables and time are sampled simultaneously so that their interactions could be considered automatically. The example of a mechanism system shows the effectiveness of the proposed method.

Extreme values of phase-type and mixed random variables with parallel-processing examples

1999 ◽  
Vol 36 (01) ◽  
pp. 194-210 ◽  
Author(s):  
Sungyeol Kang ◽  
Richard F. Serfozo
Keyword(s):  
Parallel Processing ◽  
Asymptotic Distribution ◽  
Extreme Values ◽  
Random Variables ◽  
Extreme Value Distribution ◽  
Rates Of Convergence ◽  
Value Theory ◽  
Extreme Value ◽  
Independent Random Variables ◽  
Phase Type

A basic issue in extreme value theory is the characterization of the asymptotic distribution of the maximum of a number of random variables as the number tends to infinity. We address this issue in several settings. For independent identically distributed random variables where the distribution is a mixture, we show that the convergence of their maxima is determined by one of the distributions in the mixture that has a dominant tail. We use this result to characterize the asymptotic distribution of maxima associated with mixtures of convolutions of Erlang distributions and of normal distributions. Normalizing constants and bounds on the rates of convergence are also established. The next result is that the distribution of the maxima of independent random variables with phase type distributions converges to the Gumbel extreme-value distribution. These results are applied to describe completion times for jobs consisting of the parallel-processing of tasks represented by Markovian PERT networks or task-graphs. In these contexts, which arise in manufacturing and computer systems, the job completion time is the maximum of the task times and the number of tasks is fairly large. We also consider maxima of dependent random variables for which distributions are selected by an ergodic random environment process that may depend on the variables. We show under certain conditions that their distributions may converge to one of the three classical extreme-value distributions. This applies to parallel-processing where the subtasks are selected by a Markov chain.

scholarly journals Freezing transitions and extreme values: random matrix theory, and disordered landscapes

2014 ◽  
Vol 372 (2007) ◽  
pp. 20120503 ◽  
Author(s):  
Yan V. Fyodorov ◽  
Jonathan P. Keating
Keyword(s):  
Matrix Theory ◽  
Extreme Values ◽  
Extreme Value ◽  
Extreme Value Statistics ◽  
Constant Length ◽  
Energy Models ◽  
The Riemann Zeta Function ◽  
Riemann Zeta ◽  
Transition Scenario ◽  
Unitary Ensemble

We argue that the freezing transition scenario , previously conjectured to occur in the statistical mechanics of 1/ f -noise random energy models, governs, after reinterpretation, the value distribution of the maximum of the modulus of the characteristic polynomials p N ( θ ) of large N × N random unitary (circular unitary ensemble) matrices U N ; i.e. the extreme value statistics of p N ( θ ) when . In addition, we argue that it leads to multi-fractal-like behaviour in the total length μ N ( x ) of the intervals in which | p N ( θ )|> N x , x >0, in the same limit. We speculate that our results extend to the large values taken by the Riemann zeta function ζ ( s ) over stretches of the critical line of given constant length and present the results of numerical computations of the large values of ). Our main purpose is to draw attention to the unexpected connections between these different extreme value problems.

scholarly journals Extreme value distributions for dependent jointly ln,p-symmetrically distributed random variables

2016 ◽  
Vol 4 (1) ◽  
Author(s):  
K. Müller ◽  
W.-D. Richter
Keyword(s):  
Extreme Values ◽  
Random Variables ◽  
Geometric Approach ◽  
Extreme Value ◽  
Symmetric Distribution ◽  
Tail Behavior ◽  
Extreme Value Distributions ◽  
Symmetric Distributions ◽  
Dependent Random Variables ◽  
Elliptically Contoured

AbstractA measure-of-cone representation of skewed continuous ln,p-symmetric distributions, n ∈ N, p > 0, is proved following the geometric approach known for elliptically contoured distributions. On this basis, distributions of extreme values of n dependent random variables are derived if the latter follow a joint continuous ln,p-symmetric distribution. Light, heavy, and extremely far tails as well as tail indices are discussed, and new parameters of multivariate tail behavior are introduced.

Electro-Mechanical Admittance-Based Damage Detection Using Extreme Value Statistics

2008 ◽  
Vol 385-387 ◽  
pp. 561-564 ◽  
Author(s):  
Costas P. Providakis
Keyword(s):  
Health Monitoring ◽  
Extreme Values ◽  
Simulated Data ◽  
Extreme Value ◽  
Extreme Value Statistics ◽  
Extreme Value Analysis ◽  
Engineering Structures ◽  
Value Analysis ◽  
Electromechanical Impedance ◽  
Tail Distribution

This paper presents the use of statistically rigorous algorithms combined with electromechanical (E/M) impedance approach for health monitoring of engineering structures. In particular, a statistical pattern recognition procedure is developed, based on frequency domain data of electromechanical impedance, to establish a decision boundary for damage identification. In order to diagnose damage with statistical confidence, health monitoring is cast in the context of outlier detection framework. Inappropriate modeling of tail distribution of outliers imposes potentially misleading behavior associated with damage. The present paper attempts to address the problem of establishing decision boundaries based on extreme value statistics so that the extreme values of outliers associated with tail distribution can be properly modeled. The validity of the proposed method is demonstrated using finite element method (FEM) simulated data while a comparison is performed for the extreme value analysis results contrasted with the standard approach where it is assumed that the damage-sensitive features are normally distributed.

Extreme values of phase-type and mixed random variables with parallel-processing examples

1999 ◽  
Vol 36 (1) ◽  
pp. 194-210 ◽  
Author(s):  
Sungyeol Kang ◽  
Richard F. Serfozo
Keyword(s):  
Parallel Processing ◽  
Asymptotic Distribution ◽  
Extreme Values ◽  
Random Variables ◽  
Extreme Value Distribution ◽  
Rates Of Convergence ◽  
Value Theory ◽  
Extreme Value ◽  
Independent Random Variables ◽  
Phase Type

A basic issue in extreme value theory is the characterization of the asymptotic distribution of the maximum of a number of random variables as the number tends to infinity. We address this issue in several settings. For independent identically distributed random variables where the distribution is a mixture, we show that the convergence of their maxima is determined by one of the distributions in the mixture that has a dominant tail. We use this result to characterize the asymptotic distribution of maxima associated with mixtures of convolutions of Erlang distributions and of normal distributions. Normalizing constants and bounds on the rates of convergence are also established. The next result is that the distribution of the maxima of independent random variables with phase type distributions converges to the Gumbel extreme-value distribution. These results are applied to describe completion times for jobs consisting of the parallel-processing of tasks represented by Markovian PERT networks or task-graphs. In these contexts, which arise in manufacturing and computer systems, the job completion time is the maximum of the task times and the number of tasks is fairly large. We also consider maxima of dependent random variables for which distributions are selected by an ergodic random environment process that may depend on the variables. We show under certain conditions that their distributions may converge to one of the three classical extreme-value distributions. This applies to parallel-processing where the subtasks are selected by a Markov chain.

scholarly journals Extreme value statistics of correlated random variables: A pedagogical review

2020 ◽  
Vol 840 ◽  
pp. 1-32 ◽  
Author(s):  
Satya N. Majumdar ◽  
Arnab Pal ◽  
Grégory Schehr
Keyword(s):  
Random Variables ◽  
Extreme Value ◽  
Extreme Value Statistics

Extreme value statistics: potential benefits in water quality management

1997 ◽  
Vol 36 (5) ◽  
pp. 133-140 ◽  
Author(s):  
O. Thas ◽  
P. Vanrolleghem ◽  
B. Kops ◽  
L. Van Vooren ◽  
J. P. Ottoy
Keyword(s):  
Water Quality ◽  
Quality Management ◽  
Extreme Values ◽  
Water Quality Management ◽  
Extreme Value ◽  
Extreme Value Statistics ◽  
Design Stage ◽  
Sea Levels ◽  
Wind Speeds ◽  
Potential Benefits

Recently extreme value statistics have proven useful in environmental applications like the assessment of sea-levels, wind speeds and ozone concentrations. In this paper, after a brief overview of the statistical theory of extreme values, modelling issues are discussed with stress on applications in water quality management. Risk analysis procedures are presented that consider the extremal behaviour of water quality in the design stage of environmental constructions.

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