scholarly journals Extreme value theory and return time statistics for dispersing billiard maps and flows, Lozi maps and Lorenz-like maps

2011 ◽  
Vol 31 (5) ◽  
pp. 1363-1390 ◽  
Author(s):  
CHINMAYA GUPTA ◽  
MARK HOLLAND ◽  
MATTHEW NICOL
Keyword(s):  
Time Series ◽  
Hyperbolic Systems ◽  
Return Time ◽  
Value Theory ◽  
Extreme Value ◽  
Extreme Value Statistics ◽  
Distributed Processes ◽  
Return Time Statistics ◽  
Generic Points ◽  
Lozi Maps

AbstractIn this paper we establish extreme value statistics for observations on a class of hyperbolic systems: planar dispersing billiard maps and flows, Lozi maps and Lorenz-like maps. In particular, we show that for time series arising from Hölder observations on these systems which are maximized at generic points the successive maxima of the time series are distributed according to the corresponding extreme value distributions for independent identically distributed processes. These results imply an exponential law for the hitting and return time statistics of these dynamical systems.

Statistics of Extreme Wind Speeds and Wave Heights by the Bivariate ACER Method

2015 ◽  
Vol 137 (2) ◽  
Author(s):  
Arvid Naess ◽  
Oleh Karpa
Keyword(s):  
Time Series ◽  
Extreme Value Distribution ◽  
Offshore Structures ◽  
Accurate Estimate ◽  
Value Theory ◽  
Extreme Value ◽  
Value Distribution ◽  
Extreme Value Statistics ◽  
Wind Speeds ◽  
Wave Heights

In the reliability engineering and design of offshore structures, probabilistic approaches are frequently adopted. They require the estimation of extreme quantiles of oceanographic data based on the statistical information. Due to strong correlation between such random variables as, e.g., wave heights and wind speeds (WS), application of the multivariate, or bivariate in the simplest case, extreme value theory is sometimes necessary. The paper focuses on the extension of the average conditional exceedance rate (ACER) method for prediction of extreme value statistics to the case of bivariate time series. Using the ACER method, it is possible to provide an accurate estimate of the extreme value distribution of a univariate time series. This is obtained by introducing a cascade of conditioning approximations to the true extreme value distribution. When it has been ascertained that this cascade has converged, an estimate of the extreme value distribution has been obtained. In this paper, it will be shown how the univariate ACER method can be extended in a natural way to also cover the case of bivariate data. Application of the bivariate ACER method will be demonstrated for measured coupled WS and wave height data.

Statistics of Extreme Wind Speeds and Wave Heights by the Bivariate ACER Method

2013 ◽  
Author(s):  
Arvid Naess ◽  
Oleh Karpa
Keyword(s):  
Time Series ◽  
Extreme Value Distribution ◽  
Offshore Structures ◽  
Value Theory ◽  
Extreme Value ◽  
Value Distribution ◽  
Extreme Value Statistics ◽  
Wind Speeds ◽  
Wave Heights ◽  
Extreme Wind Speeds

In the reliability engineering and design of offshore structures probabilistic approaches are frequently adopted. They require the estimation of extreme quantiles of oceanographic data based on the statistical information. Due to strong correlation between such random variables as e.g. wave heights and wind speeds, application of the multivariate, or bivariate in the simplest case, extreme value theory is sometimes necessary. The paper focuses on the extension of the ACER method for prediction of extreme value statistics to the case of bivariate time series. Using the ACER method it is possible to provide an estimate of the exact extreme value distribution of a univariate time series. This is obtained by introducing a cascade of conditioning approximations to the exact extreme value distribution. When this cascade has converged, an estimate of the exact distribution has been obtained. In this paper it will be shown how the univariate ACER method can be extended in a natural way to also cover the case of bivariate data. Application of the bivariate ACER method will also be demonstrated at the measured coupled wind speed and wave height data.

scholarly journals Extreme Value Statistics of the Total Energy in an Intermediate-Complexity Model of the Midlatitude Atmospheric Jet. Part I: Stationary Case

2007 ◽  
Vol 64 (7) ◽  
pp. 2137-2158 ◽  
Author(s):  
Mara Felici ◽  
Valerio Lucarini ◽  
Antonio Speranza ◽  
Renato Vitolo
Keyword(s):  
Time Series ◽  
Total Energy ◽  
Goodness Of Fit ◽  
Extreme Values ◽  
Value Theory ◽  
Extreme Value ◽  
Stationary Time Series ◽  
Return Level ◽  
Extreme Value Statistics ◽  
Intermediate Complexity

Abstract A baroclinic model of intermediate complexity for the atmospheric jet at middle latitudes is used as a stochastic generator of atmosphere-like time series. In this case, time series of the total energy of the system are considered. Statistical inference of extreme values is applied to sequences of yearly maxima extracted from the time series in the rigorous setting provided by extreme value theory. The generalized extreme value (GEV) family of distributions is used here as a basic model, both for its qualities of simplicity and its generality. Several physically plausible values of the parameter TE, which represents the forced equator-to-pole temperature gradient and is responsible for setting the average baroclinicity in the atmospheric model, are used to generate stationary time series of the total energy. Estimates of the three GEV parameters—location, scale, and shape—are inferred by maximum likelihood methods. Standard statistical diagnostics, such as return level and quantile–quantile plots, are systematically applied to assess goodness-of-fit. The GEV parameters of location and scale are found to have a piecewise smooth, monotonically increasing dependence on TE. The shape parameter also increases with TE but is always negative, as is required a priori by the boundedness of the total energy. The sensitivity of the statistical inferences is studied with respect to the selection procedure of the maxima: the roles occupied by the length of the sequences of maxima and by the length of data blocks over which the maxima are computed are critically analyzed. Issues related to model sensitivity are also explored by varying the resolution of the system. The method used in this paper is put forward as a rigorous framework for the statistical analysis of extremes of observed data, to study the past and present climate and to characterize its variations.

scholarly journals Extreme events in total ozone over Arosa – Part 1: Application of extreme value theory

2010 ◽  
Vol 10 (20) ◽  
pp. 10021-10031 ◽  
Author(s):  
H. E. Rieder ◽  
J. Staehelin ◽  
J. A. Maeder ◽  
T. Peter ◽  
M. Ribatet ◽  
...  
Keyword(s):  
Time Series ◽  
Frequency Distribution ◽  
Extreme Events ◽  
Extreme Value Theory ◽  
Total Ozone ◽  
Volcanic Eruptions ◽  
Value Theory ◽  
Extreme Value ◽  
Data Set ◽  
Ozone Data

Abstract. In this study ideas from extreme value theory are for the first time applied in the field of stratospheric ozone research, because statistical analysis showed that previously used concepts assuming a Gaussian distribution (e.g. fixed deviations from mean values) of total ozone data do not adequately address the structure of the extremes. We show that statistical extreme value methods are appropriate to identify ozone extremes and to describe the tails of the Arosa (Switzerland) total ozone time series. In order to accommodate the seasonal cycle in total ozone, a daily moving threshold was determined and used, with tools from extreme value theory, to analyse the frequency of days with extreme low (termed ELOs) and high (termed EHOs) total ozone at Arosa. The analysis shows that the Generalized Pareto Distribution (GPD) provides an appropriate model for the frequency distribution of total ozone above or below a mathematically well-defined threshold, thus providing a statistical description of ELOs and EHOs. The results show an increase in ELOs and a decrease in EHOs during the last decades. The fitted model represents the tails of the total ozone data set with high accuracy over the entire range (including absolute monthly minima and maxima), and enables a precise computation of the frequency distribution of ozone mini-holes (using constant thresholds). Analyzing the tails instead of a small fraction of days below constant thresholds provides deeper insight into the time series properties. Fingerprints of dynamical (e.g. ENSO, NAO) and chemical features (e.g. strong polar vortex ozone loss), and major volcanic eruptions, can be identified in the observed frequency of extreme events throughout the time series. Overall the new approach to analysis of extremes provides more information on time series properties and variability than previous approaches that use only monthly averages and/or mini-holes and mini-highs.

scholarly journals Extreme value theory in the solar wind: the role of current sheets

2019 ◽  
Vol 490 (2) ◽  
pp. 1879-1893
Author(s):  
Tiago F P Gomes ◽  
Erico L Rempel ◽  
Fernando M Ramos ◽  
Suzana S A Silva ◽  
Pablo R Muñoz
Keyword(s):  
Solar Wind ◽  
Power Spectra ◽  
Value Theory ◽  
Extreme Value ◽  
Extreme Value Statistics ◽  
Intermittent Turbulence ◽  
Current Sheets ◽  
Developed Turbulence ◽  
The Right

ABSTRACT This article provides observational evidence for the direct relation between current sheets, multifractality and fully developed turbulence in the solar wind. In order to study the role of current sheets in extreme-value statistics in the solar wind, the use of magnetic volatility is proposed. The statistical fits of extreme events are based on the peaks-over-threshold (POT) modelling of Cluster 1 magnetic field data. The results reveal that current sheets are the main factor responsible for the behaviour of the tail of the magnetic volatility distributions. In the presence of current sheets, the distributions display a positive shape parameter, which means that the distribution is unbounded in the right tail. Thus the appearance of larger current sheets is to be expected and magnetic reconnection events are more likely to occur. The volatility analysis confirms that current sheets are responsible for the −5/3 Kolmogorov power spectra and the increase in multifractality and non-Gaussianity in solar wind statistics. In the absence of current sheets, the power spectra display a −3/2 Iroshnikov–Kraichnan law. The implications of these findings for the understanding of intermittent turbulence in the solar wind are discussed.

scholarly journals Characteristics of Extreme Value Statistics of Annual Maximum Monthly Precipitation in East Asia Calculated Using an Earth System Model of Intermediate Complexity

Atmosphere ◽  
2020 ◽  
Vol 11 (12) ◽  
pp. 1273
Author(s):  
Tosiyuki Nakaegawa ◽  
Takuro Kobashi ◽  
Hirotaka Kamahori
Keyword(s):  
Time Series ◽  
Extreme Precipitation ◽  
Extreme Value ◽  
Nonstationary Time Series ◽  
Return Level ◽  
Extreme Value Statistics ◽  
Extreme Value Analysis ◽  
Monthly Precipitation ◽  
Annual Maximum ◽  
Value Analysis

Extreme precipitation is no longer stationary under a changing climate due to the increase in greenhouse gas emissions. Nonstationarity must be considered when realistically estimating the amount of extreme precipitation for future prevention and mitigation. Extreme precipitation with a certain return level is usually estimated using extreme value analysis under a stationary climate assumption without evidence. In this study, the characteristics of extreme value statistics of annual maximum monthly precipitation in East Asia were evaluated using a nonstationary historical climate simulation with an Earth system model of intermediate complexity, capable of long-term integration over 12,000 years (i.e., the Holocene). The climatological means of the annual maximum monthly precipitation for each 100-year interval had nonstationary time series, and the ratios of the largest annual maximum monthly precipitation to the climatological mean had nonstationary time series with large spike variations. The extreme value analysis revealed that the annual maximum monthly precipitation with a return level of 100 years estimated for each 100-year interval also presented a nonstationary time series which was normally distributed and not autocorrelated, even with the preceding and following 100-year interval (lag 1). Wavelet analysis of this time series showed that significant periodicity was only detected in confined areas of the time–frequency space.

scholarly journals Volcanic hazard assessment for the Canary Islands (Spain) using extreme value theory

2011 ◽  
Vol 11 (10) ◽  
pp. 2741-2753 ◽  
Author(s):  
R. Sobradelo ◽  
J. Martí ◽  
A. T. Mendoza-Rosas ◽  
G. Gómez
Keyword(s):  
Time Series ◽  
Poisson Process ◽  
Statistical Method ◽  
Canary Islands ◽  
Extreme Value Theory ◽  
Volcanic Hazard ◽  
Value Theory ◽  
Extreme Value ◽  
Homogeneous Poisson Process ◽  
Non Homogeneous Poisson Process

Abstract. The Canary Islands are an active volcanic region densely populated and visited by several millions of tourists every year. Nearly twenty eruptions have been reported through written chronicles in the last 600 yr, suggesting that the probability of a new eruption in the near future is far from zero. This shows the importance of assessing and monitoring the volcanic hazard of the region in order to reduce and manage its potential volcanic risk, and ultimately contribute to the design of appropriate preparedness plans. Hence, the probabilistic analysis of the volcanic eruption time series for the Canary Islands is an essential step for the assessment of volcanic hazard and risk in the area. Such a series describes complex processes involving different types of eruptions over different time scales. Here we propose a statistical method for calculating the probabilities of future eruptions which is most appropriate given the nature of the documented historical eruptive data. We first characterize the eruptions by their magnitudes, and then carry out a preliminary analysis of the data to establish the requirements for the statistical method. Past studies in eruptive time series used conventional statistics and treated the series as an homogeneous process. In this paper, we will use a method that accounts for the time-dependence of the series and includes rare or extreme events, in the form of few data of large eruptions, since these data require special methods of analysis. Hence, we will use a statistical method from extreme value theory. In particular, we will apply a non-homogeneous Poisson process to the historical eruptive data of the Canary Islands to estimate the probability of having at least one volcanic event of a magnitude greater than one in the upcoming years. This is done in three steps: First, we analyze the historical eruptive series to assess independence and homogeneity of the process. Second, we perform a Weibull analysis of the distribution of repose time between successive eruptions. Third, we analyze the non-homogeneous Poisson process with a generalized Pareto distribution as the intensity function.

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