Asymptotic Distribution Of Extreme Values Research Articles

Preconditioning wind speeds for standardised structural design

The statistical treatment of historical wind data has a significant impact on the design wind load, standardisation of which requires characterisation of the wind climate across a large region. Improved spatial resolution is afforded by asymptotic extreme value distributions as they require a lower temporal data resolution. The Gumbel distribution is shown to be preferred, given the available South African data, according to the Akaike Information Criterion. However, its relative inflexibility means that estimates of small probability fractiles could be significantly biased, a difficulty illustrated by disagreement on which is more appropriate: fitting the Gumbel distribution to the wind speed versus to the wind-induced pressure (squared wind speed). By generalising this choice to the problem of finding the most appropriate real-valued exponent to raise the wind speeds by (i.e. not just 1 or 2), this paper aims to reduce this modelling-bias. As estimates of this exponent parameter from a single station would have low confidence given the typical quantity of data, a method of estimating the exponent that maximises the likelihood of observing the entire dataset is developed instead. The method is demonstrated using over 3500 annual gust measurements at 131 stations throughout South Africa, the most likely exponent is found to be 1.59 with confidence intervals sufficiently narrow to reject fitting to both the wind speed as well as to the squared wind speed. It is shown that using an exponent of 1.6 in deriving the design wind load for structural design would result in a substantial reduction in modelling-bias, providing a suitable baseline for future South African loading standards.

The bounded inverse Weibull distribution: An extreme value alternative for application to environmental maxima?

Abstract There has long been interest in making inferences about future low-probability natural events that have magnitudes greater than any in the past record. Given a stationary time series, the unbounded Type 1 and Type 2 asymptotic extreme value distributions are often invoked as giving theoretical justification for extrapolating to large magnitudes and long return periods for hydrological variables such as rainfall and river discharge. However, there is a problem in that environmental extremes are bounded above by the bounded nature of their causal variables. Extrapolation using unbounded asymptotic models therefore cannot be justified from extreme value theory and at some point there will be over-prediction of future magnitudes. This creates the apparent contradiction, for example, of annual rainfall maxima being well approximated by Type 2 extreme value distributions despite the bounded nature of rainfall magnitudes. An alternative asymptotic extreme value approach is suggested for further investigation, with the model being the asymptotic distribution of minima (Weibull distribution) applied to block maxima reciprocals. Two examples are presented where data that are well matched by Type 1 or Type 2 extreme value distributions give reciprocals suggestive of lower bounds (upper bound γ to the original data). The asymptotic model here is a 3-parameter Weibull distribution for the reciprocals, with positive location parameter γ−1. When this situation is demonstrated from data, parameter estimation can be carried out with respect to the distribution of reciprocals of 3-parameter Weibull random variables. This distribution is referenced here as the bounded inverse Weibull distribution. A maximum likelihood parameter estimation methodology is presented, together with a parametric bootstrap approach for obtaining one-sided upper confidence limits to γ. When data permits estimation of γ, the bounded inverse Weibull distribution is suggested as an improved alternative to Type 1 or Type 2 extreme value distributions because the upper bound reality is recognised. However, extensive application to many data sets is required to evaluate the practical utility of the bounded approach for extrapolating beyond the largest recorded event.

Rainfall flood hazard at nuclear power plants in India

ABSTRACTFlood hazard evaluation is an important input for Nuclear Power Plants external events safety studies. In the present study, flood hazard at various nuclear sites in India due to rainfall has been evaluated. Hazard estimation is a statistical procedure by which rainfall intensity versus occurrence frequency is estimated from historical records of rainfall data and extrapolated with asymptotic extreme value distribution. Rainfall data needed for flood hazard assessment are daily annual maximum rainfall (24 h data). The observed data points have been fitted using Gumbel, power law and exponential distribution, and return period has been estimated. To study the stationarity of rainfall data, a moving window estimate of the parameters has been performed. The rainfall pattern is stationary in both coastal and inland regions over the period of observation. The coastal regions show intense rainfall and higher variability than inland regions. Based on the plant layout, catchment area and drainage capacity, the prototype fast breeder reactor (PFBR) site is unlikely to be flooded.

Application of Gumbel I and Monte Carlo methods to assess seismic hazard in and around Pakistan

A proper assessment of seismic hazard is of considerable importance in order to achieve suitable building construction criteria. This paper presents probabilistic seismic hazard assessment in and around Pakistan (23° N–39° N; 59° E–80° E) in terms of peak ground acceleration (PGA). Ground motion is calculated in terms of PGA for a return period of 475 years using a seismogenic-free zone method of Gumbel’s first asymptotic distribution of extreme values and Monte Carlo simulation. Appropriate attenuation relations of universal and local types have been used in this study. The results show that for many parts of Pakistan, the expected seismic hazard is relatively comparable with the level specified in the existing PGA maps.

The effect of the generalized extreme value distribution parameter estimation methods in extreme wind speed prediction

The modeling and prediction of extreme values of geophysical variables, such as wind, ocean surface waves, sea level, temperature and river flow, has always been a field of main concern for engineers and scientists. The analysis of extreme wind speed particularly plays an important role in natural disasters’ preparedness, prevention, mitigation and management and in various ocean, environmental and civil engineering applications, such as the design of offshore platforms and coastal marine structures, coastal management, wind climate analysis and structural safety. The block maxima (BM) approach is fundamental for extreme value analysis. BM method is closely related to the generalized extreme value (GEV) distribution, which unifies the three asymptotic extreme value distributions into a single one. The most common methods used for the estimation of the GEV parameters are maximum likelihood (ML) and probability weighted moments methods. In this work, several very common as well some less known estimation methods are firstly assessed through a simulation analysis. The results of the analysis showed that the maximum product of spacings (MPS), the elemental percentile (EP), the ordinary entropy method and, in a lesser degree, the ML methods seem to be, in general, superior to the other examined methods with respect to bias, mean squared error and variance of the estimated parameters. The effects of the estimation methods have been also assessed with respect to the n-year design values of real wind speed measurements. The obtained results suggest that the MPS and EP methods, which are rather unknown to the engineering community, describe adequately well the extreme quantiles of the wind speed data samples.

Analysis of largest earthquakes in Turkey and its vicinity by application of the Gumbel III distribution

A study of the spatial distribution of seismicity parameters is undertaken along Turkey and its vicinity, using the Gumbel’s third asymptotic distribution of extreme values (GIII). The data set used spans of 111 years (1900-2010). The seismicity of the whole region is subdivided into equal area mesh of 1° lat. × 1° long. Various seismicity parameters examined, resulted from the application of the GIII method. The results show a quite good correlation between the seismicity parameters and the tectonic regime of the studied area. For instance high values concentrated around North Anatolian Fault. The x 2 -test is applied throughout the whole process and in every stage of GIII, in order to check the accuracy of the obtained results. The spatial distribution of upper-bound (ω) formed a W-shape pattern, which shows the difference in the mechanical structure of the materials in the examined area.

Spatial mapping of earthquake hazard parameters in the Hindukush–Pamir Himalaya and adjacent regions: Implication for future seismic hazard

The study deals spatial mapping of earthquake hazard parameters like annual and 100-years mode along with their 90% probability of not being exceeded (NBE) in the Hindukush–Pamir Himalaya and adjoining regions. For this purpose, we applied a straightforward and most robust method known as Gumbel’s third asymptotic distribution of extreme values (GIII). A homogeneous and complete earthquake catalogue during the period 1900–2010 with magnitude MW⩾4.0 is utilized to estimate these earthquake hazard parameters. An equal grid point mesh, of 1° longitude X 1° latitude, is chosen to produce detailed earthquake hazard maps. This performance allows analysis of the localized seismicity parameters and representation of their regional variations as contour maps. The estimated result of annual mode with 90% probability of NBE is expected to exceed the values of MW 6.0 in the Sulaiman–Kirthar ranges of Pakistan and northwestern part of the Nepal and surroundings in the examined region. The 100-years mode with 90% probability of NBE is expected to exceed the value of MW 8.0 in the Hindukush–Pamir Himalaya with Caucasus mountain belt, the Sulaiman–Kirthar ranges of Pakistan, northwestern part of the Nepal and surroundings, the Kangra–Himanchal Pradesh and Kashmir of India. The estimated high values of earthquake hazard parameters are mostly correlated with the main tectonic regimes of the examined region. The spatial variations of earthquake hazard parameters reveal that the examined region exhibits more complexity and has high crustal heterogeneity. The spatial maps provide a brief atlas of the earthquake hazard in the region.

Frequency analysis of the 7–8 December 2010 extreme precipitation in the Panama Canal Watershed

► 7–8/12/2010 Rainfall event over Panama Canal is doubled the previous largest event. ► GEV, Gumbel, and GP distributions indicate that this rain event is unforeseen. ► Uncertainty analyses indicate non-negligible recurrence likelihood of such event. ► Gumbel distribution was found inadequate for very extreme events. ► We propose a method to convey uncertainty estimates to decision makers. The 7–8 December 2010 rainfall event in Panama produced record rainfall and streamflow that are about twice as much as for the previously observed large event in record. In this study we ask whether before the occurrence of this rainfall event, a return period estimate using the historical record and the commonly used statistical asymptotic distributions of extreme values could have indicated that such an event is probable. We examined the daily and 24-h mean areal rainfall over the entire Panama Canal Watershed with the Generalized Extreme Value, Gumbel, and Generalized Pareto distributions using the maximum likelihood approach for the parameter and uncertainty bounds estimation. We found that the solutions that maximized the log likelihood for these three distributions yield return period estimates that are larger than 2000 years. These return periods imply that the 2010 rainfall event was practically unforeseen. It is only the careful implementation of these distributions with full uncertainty analysis to define confidence intervals that yields estimates of return periods with substantial probabilities for such an event to occur. The GEV was found to be the most adequate distribution for this analysis, and the commonly-used Gumbel distribution, although indicated a good fit to the annual maxima series, attributed an extremely low probability for the occurrence of this event.

Predicting violations of national ambient air quality standards using extreme value theory for Delhi city

Type I asymptotic distribution of extreme values was used for making predictions of the expected number of violations of the National Ambient Air Quality Standards (NAAQS) for 24–hour average sulfur dioxide (SO2), nitrogen dioxide (NO2) and suspended particulate matter (SPM), monitored at the four National Ambient Air Quality Monitoring (NAAQM) stations run by the Central Pollution Control Board (CPCB) for Delhi city. The parameters of the distribution were estimated using Gumbel's method. A comparison of the predicted violations of NAAQS and the exceedance of the maximum pollution concentration with that of the observed data indicates that Type I asymptotic distribution adequately fits to the observed extreme value data. The application of extreme value theory as a tool for managing urban air quality was highlighted.

Estimation of the Frèchet Distribution Parameters Based on Record Values

The largest, or smallest, observation in a finite sample is termed an extreme value. Its distribution converges with increasing sample size to one of three types. These three limiting distributions are of considerable interest in engineering. These distributions have been applied to a variety of problems in many fields of practical interest. If the initial distribution f (x, θ) has an unbounded upper tail, but not all of its moments are finite, then the asymptotic extreme value distribution is termed a type II maxima [Frechet distribution (FD)]. In the present paper, we derive the best linear unbiased estimators of location and scale parameters of the FD and we obtain the best linear invariant estimations. Moreover, we use the maximum likelihood method to estimate the parameters of this distribution. Finally, we conduct a numerical experiment and draw conclusions.

Response load extrapolation for wind turbines during operation based on average conditional exceedance rates

ABSTRACTThe paper explores a recently developed method for statistical response load (load effect) extrapolation for application to extreme response of wind turbines during operation. The extrapolation method is based on average conditional exceedance rates and is in the present implementation restricted to cases where the Gumbel distribution is the appropriate asymptotic extreme value distribution. However, two extra parameters are introduced by which a more general and flexible class of extreme value distributions is obtained with the Gumbel distribution as a subclass. The general method is implemented within a hierarchical model where the variables that influence the loading are divided into ergodic variables and time‐invariant non‐ergodic variables.The presented method for statistical response load extrapolation was compared with the existing methods based on peak extrapolation for the blade out‐of‐plane bending moment and the tower mudline bending moment of a pitch‐controlled wind turbine. In general, the results show that the method based on average conditional exceedance rates predicts the extrapolated characteristic response loads at the individual mean wind speeds well and results in more consistent estimates than the methods based on peak extrapolation. Copyright © 2011 John Wiley & Sons, Ltd.

Extreme value statistics and thermodynamics of earthquakes: large earthquakes

A compound Poisson process is used to derive a new shape parameter which can be used to discriminate between large earthquakes and aftershock sequences. Sample exceedance distributions of large earthquakes are fitted to the Pareto tail and the actual distribution of the maximum to the Frechet distribution, while the sample distribution of aftershocks are fitted to a Beta distribution and the distribution of the minimum to the Weibull distribution for the smallest value. The transition between initial sample distributions and asymptotic extreme value distributions shows that self-similar power laws are transformed into nonscaling exponential distributions so that neither self-similarity nor the Gutenberg-Richter law can be considered universal. The energy-magnitude transformation converts the Frechet distribution into the Gumbel distribution, originally proposed by Epstein and Lomnitz, and not the Gompertz distribution as in the Lomnitz-Adler and Lomnitz generalization of the Gutenberg-Richter law. Numerical comparison is made with the Lomnitz-Adler and Lomnitz analysis using the same Catalogue of Chinese Earthquakes. An analogy is drawn between large earthquakes and high energy particle physics. A generalized equation of state is used to transform the Gamma density into the order-statistic Frechet distribution. Earthquaketemperature and volume are determined as functions of the energy. Large insurance claims based on the Pareto distribution, which does not have a right endpoint, show why there cannot be a maximum earthquake energy.

Estimation of Probabilistic Extreme Wind Load Effects: Combination of Aerodynamic and Wind Climate Data

A refined full-order method is presented for estimating the extreme wind load effects of rigid structures with given mean recurrence intervals (MRIs) by combining the distributions of annual maximum wind speed and extreme load coefficients. This refined method is capable of dealing with any type of asymptotic extreme value distribution. With this full-order method, the predictions of wind load effects by using distributions of annual maximum wind velocity pressure and wind speed are compared that provide information on the sensitivity of predictions to the upper tail of wind speed distribution. The efficacy of the first-order method is examined. The influences of the type of distributions and the variations of annual maximum wind speed and extreme load coefficient on the predictions are quantified. Finally, the first- and full-order methods are extended to wind load effects of dynamically sensitive structures which facilitate a comprehensive probabilistic analysis as compared to the Monte Carlo simulation...

The statistical characters of PM 10 in Belgrade area

The concentrations of air pollutants depend on meteorological conditions and pollutant emission level. From the statistical properties of air pollutants the number of times the daily average concentrations exceed the assigned air quality standard (AQS) can be estimated, as well as the level of reduction of particle matter emission sources required to meet the AQS. In this paper three statistical distributions (lognormal, Weibull and type V Pearson distribution) were used to fit the complete set of PM 10 data for the Belgrade urban area during a three-year period (2003–2005). The method of moments and the method of least squares were both used to estimate the parameters of the three theoretical distributions. The type V Pearson distribution represented the PM 10 daily average concentration most closely. However, the parent distributions sometimes diverged in predicting a high PM 10 concentration and therefore asymptotic distributions of extreme values were used to fit the high PM 10 concentration distribution more correctly. This method can successfully predict the return period and exceedances over a critical concentration in succeeding years. The estimated emission source reduction of PM 10 to meet the assigned standard varied from 53% to 63% in the Belgrade urban area. The results provide useful information for air quality management and could be used to examine the similarities and differences among air pollution types in diverse areas.

Monte Carlo Methods for Estimating the Extreme Response of Dynamical Systems

The development of simple accurate, and efficient methods for estimation of the extreme response of dynamical systems subjected to random excitations is discussed in the present paper. The key quantity for calculating the statistical distribution of extreme response is the mean level upcrossing rate function. By exploiting the regularity of the tail behavior of this function, an efficient simulation based methodology for estimating the extreme response distribution function is developed. This makes it possible to avoid the commonly adopted assumption that the extreme value data follow an appropriate asymptotic extreme value distribution, which would be a Gumbel distribution for the models considered in this paper. It is demonstrated that the commonly quoted obstacle against using the standard Monte Carlo method for estimating extreme responses, i.e., excessive CPU time, can be circumvented, bringing the computational efforts down to quite acceptable levels.

Regional Variation of the ω - Upper Bound Magnitude of GIII Distribution in and Around Turkey: Tectonic Implications for Earthquake Hazards

Complete data set of earthquakes in Turkey and the adjacent areas has been used in order to compute the ω values in 24 seismic regions of Turkey. The parameter is obtained through Gumbel’s third asymptotic distribution of extreme values and is well known as upper bound magnitude. This is an interpretation that no earthquake magnitude greater than ω can occur in a region. The results also estimate the most probable magnitude for a time period of 100 years. The estimates of ω exceed the value of 7.00 in 20 of the 24 seismic regions. An effort is also made to find a relation between the magnitude and the length of a fault in the complicated tectonics of Turkey and the surrounding area. Earthquake hazard revealed as tables and maps are also considered for Turkey and the surrounding area.

Time Variant Reliability Analysis of Nonlinear Structural Dynamical Systems using combined Monte Carlo Simulations and Asymptotic Extreme Value Theory

Reliability of nonlinear vibrating systems under stochastic excitations is investigated using a two-stage Monte Carlo simulation strategy. For systems with white noise excitation, the governing equations of motion are interpreted as a set of Ito stochastic differential equations. It is assumed that the probability distribution of the maximum in the steady state response belongs to the basin of attraction of one of the classical asymptotic extreme value distributions. The first stage of the solution strategy consists of selection of the form of the extreme value distribution based on hypothesis tests, and the next stage involves the estimation of parameters of the relevant extreme value distribution. Both these stages are implemented using data from limited Monte Carlo simulations of the system response. The proposed procedure is illustrated with examples of linear/nonlinear systems with single/multiple degrees of freedom, driven by random excitations. The predictions from the proposed method are compared with the results from large-scale Monte Carlo simulations, and also with the classical analytical results, when available, from the theory of out-crossing statistics. Applications of the proposed method for large-scale problems and for vibration data obtained from field/laboratory conditions, are also discussed.

Reliability of Shallow Foundations Subjected to Multidirectional Seismic Loading

A method of assessing the probability of failure of shallow foundations in saturated fine-grained soil under multidirectional seismic loading is presented. The method uses the distributions of two variables, the spectral acceleration at the fundamental period of the structure and the shear strength of the foundation soil, to form the joint probability density function. The performance function, which defines the required soil strength for the foundation to equilibrate the applied loading, is mapped on this domain. Numerical integration is used to ascertain the probability of failure. A bounded probability density function is used, namely a Pearson Type 1 (Beta) distribution, for soil strength. This distribution provides an upper and lower limit to in situ shear strength. The spectral acceleration is represented by the complementary cumulative distribution function of a Type 1 asymptotic extreme value (Gumble) distribution. Such a distribution is shown to be an accurate representation of real earthquake loading and may be found by a probabilistic seismic hazard analysis for a region. The method is based on pseudostatic seismic loading of the foundation and accounts for spectral acceleration acting along the two horizontal axes. The influence of the orientation of the foundation to the earthquake source is incorporated using the concept of principal directions of ground motion. The performance function is formulated for shallow foundations under eccentric and inclined loading using the recommendations in Eurocode 7. The function is shown to be nonlinear and compound, one part pertaining to the bearing mode of failure (for lower values of acceleration) and the remaining part pertaining to the sliding mode of failure (for higher values of acceleration). An equation is presented for the transition acceleration that separates the performance function into the two parts. A case study is presented and conclusions are drawn about the role of the bearing and sliding modes of failure.

Strong ground acceleration seismic hazard in Greece and neighboring regions

In an early paper [Tectonophysics 117 (1985) 259] seismic hazard in Greece was analyzed using a relatively homogeneous earthquake catalogue spanning 1900–1978 and a strong motion attenuation relationship adapted to use in Greece. Improved seismic hazard analyses are obtained here using Gumbel's asymptotic extreme value distribution applied to peak horizontal ground acceleration occurrence, but now taking into account the increased length and quality of earthquake catalogue data spanning 1900–1999 and the burgeoning information on earthquake strong motion data and attenuation relationships appropriate for Europe and, explicitly, Greece. Seismic acceleration hazard results tabulated for six cities reveal (e.g. using arbitrarily the 50-year p.g.a. with 90% probability of not being exceeded) changes of about 10% in the new calculated values: two cities show an increase and four a decrease. These are relatively small and reassuring adjustments. Inspection of the available attenuation relationships leads to a preference for the models of Theodulidis and Papazachos, particularly with the model modification to produce a ‘stiff soil’ site relationship, as these relationships explicitly exploit the Greek strong motion database. Isoacceleration maps are produced for Greece as a whole from each attenuation relationship inspected. The final set of maps based on the Theodulidis and Papazachos models provide a foundation for comparison with the Seismic Hazard Zones adopted in the New Greek Seismic Code where scope can be found to modify zone shape and the level at which p.g.a.s are set. It should be noted that the generation of the present isoacceleration maps is based on a seismogenic zone-free methodology, independent of any Euclidean zoning assumptions.

Troughs under threshold modeling of minimum flows in perennial streams

Troughs under threshold analysis has so far found little application in the modeling of minimum streamflows. In this study, all the troughs under a certain threshold level are considered in deriving the probability distribution of annual minima through the total probability theorem. For the occurrence of minima under the threshold, Poissonian, binomial or negative binomial processes are assumed. The magnitude of minima follows the generalized Pareto, exponential or power distribution. It is shown that asymptotic extreme value distributions for minima or the two-parameter Weibull distribution is obtained for the annual minima, depending on which models are assumed for the occurrence and magnitude of troughs under the threshold. Derived distributions can be used for modeling the minimum flows in streams which do not have zero flows. Expressions for the T-year annual minimum flow are obtained. An example illustrates the application of the troughs under threshold model to the minimum flows observed in a stream.