Classical Extreme Value Research Articles

Modern extreme value statistics for Utopian extremes. EVA (2023) Conference Data Challenge: Team Yalla

AbstractCapturing the extremal behaviour of data often requires bespoke marginal and dependence models which are grounded in rigorous asymptotic theory, and hence provide reliable extrapolation into the upper tails of the data-generating distribution. We present a modern toolbox of four methodological frameworks, motivated by classical extreme value theory, that can be used to accurately estimate extreme exceedance probabilities or the corresponding level in either a univariate or multivariate setting. Our frameworks were used to facilitate the winning contribution of Team Yalla to the EVA (2023) Conference Data Challenge, which was organised for the 13$$^\text {th}$$ th International Conference on Extreme Value Analysis. This competition comprised seven teams competing across four separate sub-challenges, with each requiring the modelling of data simulated from known, yet highly complex, statistical distributions, and extrapolation far beyond the range of the available samples in order to predict probabilities of extreme events. Data were constructed to be representative of real environmental data, sampled from the fantasy country of “Utopia”.

Risk Assessment of Landfalling Tropical Cyclones in China Based on Hazard Risk Theory

As a frequent hazard, tropical cyclones have a great impact on the social and economic development of China, which is close to the origin of tropical cyclones in the western Pacific Ocean. The primary objective of this study was to construct a comprehensive risk assessment model for tropical cyclone hazards based on natural influencing factors, informing recommendations for hazard prevention and mitigation in affected regions. This research focused on tropical cyclones that made landfall in mainland China and Hainan from 1949 to 2023, utilizing hazard risk theory and classical extreme value theory. The wind speed and rainfall data during the peak cyclone periods (June to October) from 1997 to 2021 gathered from various meteorological stations, as well as altitude and vegetation cover data, were examined. Hierarchical analysis and ArcGIS spatial analysis methods were employed to study the characteristics of the spatiotemporal distribution of landfalling tropical cyclones and the comprehensive risk of tropical cyclone hazards, and the regions of China were delineated according to these methods. The results showed that, during the period from 1949 to 2023, the overall number of landfalling tropical cyclones decreased in a fluctuating manner, while the intensity of the cyclones increased. Furthermore, severe typhoons tended to occur more frequently in the summer than autumn with time, intensifying the challenge to resist short-term hazards. Moreover, the hazard-causing factors in areas affected by tropical cyclones displayed an increasing trend from north to south and from west to east. In detail, the regions sensitive to natural hazards were primarily located in the central part of Liaoning province, Tianjin, central and eastern Hebei province, Shandong province, eastern Henan province, central and northern Anhui province, Jiangsu province, and Shanghai, which are characterized by flat terrain and relatively low vegetation cover. Overall, the comprehensive risk of tropical cyclone hazards showed a geographical distribution that decreases from south to north and from east to west, with coastal cities in provinces such as Hainan, Guangdong, Guangxi, Fujian, and Zhejiang—including Haikou, Zhanjiang, Xiamen, Beihai, and Taizhou—exhibiting the highest levels of risk.

An extreme value estimation method of wind pressures based on change point theory

Estimating the extreme wind pressure accurately on the roofs of low-rise buildings is crucial for wind-resistant design in wind engineering. This study proposes a new method for extreme value estimation, where change-point theory is incorporated into selecting thresholds to determine the optimal threshold and the wind pressure coefficient is converted into an intermediate exponential variable. The measured wind pressure data collected from the roof of a low-rise building at the Pudong experimental base of Tongji University during Typhoon Ampil is used as the sample of the extreme value estimation. Referring to the position of change point, the interpolation method assists in finding the most appropriate threshold for the wind pressure coefficient. The number of independent peaks in a standard wind pressure sample is reduced from 204 to 30, applying the optimal threshold while considering the correlation. This processing is crucial to reducing the estimation error. Then, the error rates of the proposed method are below 5 % at quantile p = 0.95 and the mean error rate of the proposed extreme value estimation method is within (−0.1, 0) for the whole roof regions. For comparison, the upper limit of the mean error rate for the modified Hermite model based on zero-crossing theory reaches 0.41 and that of Quan's method based on classical extreme value theory is 0.28. These results demonstrate the superiority of the proposed method. Meanwhile, the relevant estimation results show that the proposed method's extreme wind pressure values meet engineering requirements at the specified percentile.

Nonparametric tests of extremes of daily maximum temperatures based on the breaking records: A case study in Seoul and Busan during 1961-2022

An observation in a time series is called an upper record if it is greater than all previous observations in the series. Also it is called a lower record than all previous observations in the series. Therefore, the analysis of record-breaking events is of interest in fields such as climatology, economics and sports. As an alternative to the classical extreme value theory, which are based on the fit of generalized extreme value (GEV) distribution, generalized Pareto (GP) distribution and Poisson processes, this work is based on the study of record-breaking events to analyze non-stationarity in the extremes. The contribution of this work is the use of the statistical tools developed by Castillo-Mateo et al. (2023) to assess and analyze the effect of global warming in the extreme and record breaking events using daily maximum temperature series in Seoul and Busan over 1961-2022. These test statistics consider the information based on the four different types of record, that is, forward upper, forward lower, backward upper and backward lower. Also, change point detection is implemented based on records.

Nonstationarity in High and Low‐Temperature Extremes: Insights From a Global Observational Data Set by Merging Extreme‐Value Methods

AbstractWe merge classical extreme value methods to extract high (high temperatures (HT)) and low (low temperatures (LT)) temperatures and form time series having at least one extreme value per year. Observed daily maximum and minimum temperature records are used from 4,797 quality‐controlled, global, surface stations over 1970–2019. We assess changes in the magnitude and frequency of extreme temperatures by introducing and applying novel methods that exploit the definition of stationarity. Analysis shows significant increasing (40.6% of the stations) and decreasing (41.1%) trends in the frequency of high and LT, respectively, and increasing trends in both high‐ and low‐temperature values (35.6% and 49.7%). Globally, HT and LT frequencies are increasing and decreasing, respectively, by 0.9% and 1.1% per year, relative to the expected frequencies under the assumption of stationarity. The global mean annual HT and LT magnitudes are increasing by 0.004 and 0.016°C/year compared to the expected ones under stationarity. The results indicate that the assumption of stationarity fails to explain the observed changes. The proposed methods are an alternative approach to classical extreme value methods and a useful tool to reveal changes in extremes in the era of earth‐system change.

Extreme precipitation return levels for multiple durations on a global scale

Quantifying the magnitude and frequency of extreme precipitation events is key in translating climate observations to planning and engineering design. Past efforts have mostly focused on the estimation of daily extremes using gauge observations. Recent development of high-resolution global precipitation products, now allow estimation of global extremes. This research aims to quantitatively characterize the spatiotemporal behavior of precipitation extremes, by calculating extreme precipitation return levels for multiple durations on the global domain using the Multi-Source Weighted-Ensemble Precipitation (MSWEP) dataset. Both classical and novel extreme value distributions are used to provide insight into the spatial patterns of precipitation extremes. Our results show that the traditional Generalized Extreme Value (GEV) distribution and Peak-Over-Threshold (POT) methods, which only use the largest events to estimate precipitation extremes, are not spatially coherent. The recently developed Metastatistical Extreme Value (MEV) distribution, that includes all precipitation events, leads to smoother spatial patterns of local extremes. For durations of 5 and 10 days, however, there are less events per year to fit the distribution (37 and 22 on average, respectively), leading to larger inter-annual variability and possible overestimation of the extremes. While the GEV and POT methods predict a consistent shift from heavy to thin tails with increasing duration, the MEV method predicts a relatively constant heaviness of the tail for any precipitation duration, opening up an important research question on what is the ‘correct’ tail behavior of extreme precipitation for different durations. The generated extreme precipitation return levels and corresponding parameters are provided as the Global Precipitation EXtremes (GPEX) dataset. These data can be useful for studying the underlying physical processes causing the spatiotemporal variations of the heaviness of extreme precipitation distributions.

Study on the Peak Factor of the Wind-Induced Response of Super-High-Rise Buildings

The wind-induced responses of tall buildings are stochastic processes, and the peak factor is an important parameter to evaluate the extreme value of the wind-induced response in wind-resistant design. The existing research on the peak factor mainly focuses on the wind pressure on the building surface, but rarely concerns the wind-induced response peak factor of the structures. In view of this, the peak factor of the wind-induced response of super-high-rise buildings was studied in this paper. Firstly, a series of wind tunnel tests of the multi-degree-of-freedom aero-elastic models (MDOF) were carried out, wherein the along-wind and cross-wind responses were measured. Thereafter, the peak factor of wind-induced response was calculated using the peak factor method, classical extreme value theory, and the improved peak factor method. It was found that the peak factor calculated by the improved peak factor method is in good agreement with classical extreme value theory, which indicates that the improved peak factor method is applicable to calculate the peak factor of the wind-induced response of high-rise buildings. The results calculated using the improved peak factor method show that the peak factor of cross-wind response varies significantly with the wind speed, varying from about 2.5 to 5.5. The peak factor of cross-wind response first increases and then decreases with the increase in the wind speed, reaches the minimum near the critical wind speed of vortex-induced vibration (VIV), and increases again when the wind speed is larger than the VIV wind speed. Finally, an empirical formula for the cross-wind response peak factor was proposed as a function of the reduced wind speed, aspect ratio, and damping ratio of the structure.

Anomaly detection of power battery pack using gated recurrent units based variational autoencoder

Rapid and accurate detection of battery pack anomalies and timely fault-tolerant control is of great importance to the safe operation of electric vehicles (EVs). The occurrence of battery system fire accidents shows that if an anomaly cannot be detected early in the potential stage before thermal runaway, the anomaly may develop into a serious fault at any time and lead to irreparable damage. This paper presents such a semi-supervised anomaly detection model, by using a gated recurrent unit (GRU) based variational autoencoder (VAE) (GRU-VAE) framework that detects the early potential anomalies of EV power battery packs. Specifically, employing a GRU-based inference network enables the model to learn the robust latent feature representations of the input multivariate time series (MVTS), which are then used by the generator network to reconstruct the input data. In the training phase, minimizing the reconstruction error helps to learn the data distribution of normal samples. In the testing phase, a large reconstruction error of an input sample indicates that it contains potential abnormal events. In particular, the GRU is used to capture the complex time dependencies in MVTS, and the VAE is used to reconstruct the input sample with probability. The Peaks over Threshold (POT) model in the classical extreme value theory (EVT) is used to properly set the anomaly detection threshold. Experimentation over the real EV operation datasets shows that the model can effectively detect the potential anomalies in MVTS, which is expected to provide a reference for intelligent power battery pack anomaly detection technology.

K-th record estimator of the scale parameter of the α-stable distribution

Abstract Various techniques of scale parameter estimation have been proposed in the case of alpha stable distributions. In the paper, the authors present an estimation technique that involves the k-th record theory. Although this theory is over 40 years old, its implementation in the classical extreme value theory – being the other cornerstone of the presented approach – is quite new, and tempting. Several theoretical properties of the introduced scale parameter estimators are presented. With the use of Monte Carlo methods, a comparative analysis is performed between the approach based on k-th records and approaches based on Hill’s and Pickands’ estimators. Additionally, the paper uses a real-life data set to illustrate how to effectively apply the k-th record estimator of the scale parameter. The research indicates several advantages of the k-th record approach over its other counterparts, especially when dealing with incomplete information about the underlying sample.

The tales that the distribution tails of non-Gaussian autocorrelated processes tell: efficient methods for the estimation of the k-length block-maxima distribution

ABSTRACT The focal point of this work is the estimation of the distribution of maxima without the use of classic extreme value theory and asymptotic properties, which may not be ideal for hydrological processes. The problem is revisited from the perspective of non-asymptotic conditions, and regards the so-called exact distribution of block-maxima of finite-sized k-length blocks. First, we review existing non-asymptotic approaches/models, and introduce an alternative and fast model. Next, through simulations and comparisons (using asymptotic and non-asymptotic models), involving intermittent processes (e.g. rainfall), we highlight the capability of non-asymptotic approaches to model the distribution of maxima with reduced uncertainty and variability. Finally, we discuss an alternative use of such models that concerns the theoretical estimation of the multi-scale probability of obtaining a zero value. This is a useful finding when the scope is the multi-scale modelling of intermittent hydrological processes (e.g. intensity–duration–frequency models). The work also provides step-by-step recipes and an R package.

Variability and extremes: statistical validation of the Alfred Wegener Institute Earth System Model (AWI-ESM)

Abstract. Coupled general circulation models are of paramount importance to quantitatively assessing the magnitude of future climate change. Usual methods for validating climate models include the evaluation of mean values and covariances, but less attention is directed to the evaluation of extremal behaviour. This is a problem because many severe consequences of climate change are due to climate extremes. We present a method for model validation in terms of extreme values based on classical extreme value theory. We further discuss a clustering algorithm to detect spatial dependencies and tendencies for concurrent extremes. To illustrate these methods, we analyse precipitation extremes of the Alfred Wegener Institute Earth System Model (AWI-ESM) global climate model and from other models that take part in the Coupled Model Intercomparison Project CMIP6 and compare them to the reanalysis data set CRU TS4.04. The clustering algorithm presented here can be used to determine regions of the climate system that are then subjected to a further in-depth analysis, and there may also be applications in palaeoclimatology.

Extreme Value Estimation for Heterogeneous Data

We develop a universal econometric formulation of empirical power laws possibly driven by parameter heterogeneity. Our approach extends classical extreme value theory to specifying the tail behavior of the empirical distribution of a general dataset with possibly heterogeneous marginal distributions. We discuss several model examples that satisfy our conditions and demonstrate in simulations how heterogeneity may generate empirical power laws. We observe a cross-sectional power law for the U.S. stock losses and show that this tail behavior is largely driven by the heterogeneous volatilities of the individual assets.

Extreme value innovative trend analysis methodology

The main concern of the article is to determine the trend analyses of upper and lower extreme values separately, apart from the classical central trend. The innovative trend analysis (ITA) modification is adapted for extreme value trend analysis using one standard deviation level from the arithmetic mean value. Extreme value trends emerge along completely different monotonous lines. The application of the proposed methodology is presented for the annual total rainfall time series of seven meteorological stations from each geographical and climatological regions of Turkey. In the future, it is recommended to use only the classical central trend extreme value trend analyses for refined forecasts.

Uniqueness and global optimality of the maximum likelihood estimator for the generalized extreme value distribution

Summary The three-parameter generalized extreme value distribution arises from classical univariate extreme value theory, and is in common use for analysing the far tail of observed phenomena, yet important asymptotic properties of likelihood-based estimation under this standard model have not been established. In this paper we prove that the maximum likelihood estimator is global and unique. An interesting secondary result entails the uniform consistency of a class of limit relations in a tight neighbourhood of the true shape parameter.

A Horse Race between the Block Maxima Method and the Peak–over–Threshold Approach

Classical extreme value statistics consists of two fundamental approaches: the block maxima (BM) method and the peak-over-threshold (POT) approach. It seems to be general consensus among researchers in the field that the POT approach makes use of extreme observations more efficiently than the BM method. We shed light on this discussion from three different perspectives. First, based on recent theoretical results for the BM method, we provide a theoretical comparison in i.i.d. scenarios. We argue that the data generating process may favour either one or the other approach. Second, if the underlying data possesses serial dependence, we argue that the choice of a method should be primarily guided by the ultimate statistical interest: for instance, POT is preferable for quantile estimation, while BM is preferable for return level estimation. Finally, we discuss the two approaches for multivariate observations and identify various open ends for future research.

Spatiotemporal variations and extreme value analysis of significant wave height in the South China Sea based on 71-year long ERA5 wave reanalysis

A thorough understanding of the long-term trends and extreme characteristics in the significant wave height (SWH) contributes greatly to coastal and offshore engineering activities and mitigation of marine disasters. In this study, the annual spatiotemporal variability of the SWH in the South China Sea (SCS) and the return periods for six locations are evaluated based on ERA5 wave reanalysis in a long time period (1950–2020). Long-term trends are estimated by using a popular non-parametric method, the Theil-Sen estimator, and then mapped to show the spatial variability of mean and extreme SWH. Basin-averaged analysis is also performed to investigate the general tendency of the mean SWH in the SCS, with an increasing rate of 0.11 cm/year. The well-known Mann-Kendall test is used to assess the significance of the trends. Significant positive trends in extreme SWH are mainly distributed in the eastern part of the central SCS around the Luzon Strait and the southwestern part of the SCS. In the extreme value analysis, by comparing 2 classical extreme value distribution models combined with 5 sampling methods, the Generalized Pareto Distribution incorporating the Peak Over Threshold (GPD-POT) method has turned out to be suitable for evaluating the return periods in the SCS based on 71-year SWH dataset. Extreme value analysis for different time lengths shows a correlation between the return level and the time span. Small and medium samples may lead to unstable parameter estimation and increased errors. The 100-year return values obtained by GPD-POT using the 71-year wave reanalysis are more credible at P1-P6 with 8.86, 11.79, 11.35, 10.52, 6.50 and 7.71 m, respectively. Both the estimated return periods and the number of extreme events quantified by the Method of Independent Storms (MIS) indicate that extreme events are closely related to the number of tropical cyclones. Seasonally, most extreme events in the SCS occur from June to December, with the summertime maximum SWH situated above the 15th degree of northern latitude and rapidly shifting to south in the fall.

Return Period Evaluation of the Largest Possible Earthquake Magnitudes in Mainland China Based on Extreme Value Theory.

The largest possible earthquake magnitude based on geographical characteristics for a selected return period is required in earthquake engineering, disaster management, and insurance. Ground-based observations combined with statistical analyses may offer new insights into earthquake prediction. In this study, to investigate the seismic characteristics of different geographical regions in detail, clustering was used to provide earthquake zoning for Mainland China based on the geographical features of earthquake events. In combination with geospatial methods, statistical extreme value models and the right-truncated Gutenberg–Richter model were used to analyze the earthquake magnitudes of Mainland China under both clustering and non-clustering. The results demonstrate that the right-truncated peaks-over-threshold model is the relatively optimal statistical model compared with classical extreme value theory models, the estimated return level of which is very close to that of the geographical-based right-truncated Gutenberg–Richter model. Such statistical models can provide a quantitative analysis of the probability of future earthquake risks in China, and geographical information can be integrated to locate the earthquake risk accurately.

Extreme first passage times of piecewise deterministic Markov processes

The time it takes the fastest searcher out of N ≫ 1 searchers to find a target determines the timescale of many physical, chemical, and biological processes. This time is called an extreme first passage time (FPT) and is typically much faster than the FPT of a single searcher. Extreme FPTs of diffusion have been studied for decades, but little is known for other types of stochastic processes. In this paper, we study the distribution of extreme FPTs of piecewise deterministic Markov processes (PDMPs). PDMPs are a broad class of stochastic processes that evolve deterministically between random events. Using classical extreme value theory, we prove general theorems which yield the distribution and moments of extreme FPTs in the limit of many searchers based on the short time distribution of the FPT of a single searcher. We then apply these theorems to some canonical PDMPs, including run and tumble searchers in one, two, and three space dimensions. We discuss our results in the context of some biological systems and show how our approach accounts for an unphysical property of diffusion which can be problematic for extreme statistics.

New extreme value theory for maxima of maxima

Although advanced statistical models have been proposed to fit complex data better, the advances of science and technology have generated more complex data, e.g., Big Data, in which existing probability theory and statistical models find their limitations. This work establishes probability foundations for studying extreme values of data generated from a mixture process with the mixture pattern depending on the sample length and data generating sources. In particular, we show that the limit distribution, termed as the accelerated max-stable distribution, of the maxima of maxima of sequences of random variables with the above mixture pattern is a product of three types of extreme value distributions. As a result, our theoretical results are more general than the classical extreme value theory and can be applicable to research problems related to Big Data. Examples are provided to give intuitions of the new distribution family. We also establish mixing conditions for a sequence of random variables to have the limit distributions. The results for the associated independent sequence and the maxima over arbitrary intervals are also developed. We use simulations to demonstrate the advantages of our newly established maxima of maxima extreme value theory.

Extreme value theory for constrained physical systems.

We investigate extreme value theory for physical systems with a global conservation law which describes renewal processes, mass transport models, and long-range interacting spin models. As shown previously, a special feature is that the distribution of the extreme value exhibits a nonanalytical point in the middle of the support. We expose exact relationships between constrained extreme value theory and well-known quantities of the underlying stochastic dynamics, all valid beyond the midpoint in general, i.e., even far from the thermodynamic limit. For example, for renewal processes the distribution of the maximum time between two renewal events is exactly related to the mean number of these events. In the thermodynamic limit, we show how our theory is suitable to describe typical and rare events which deviate from classical extreme value theory. For example, for the renewal process we unravel dual scaling of the extreme value distribution, pointing out two types of limiting laws: a normalizable scaling function for the typical statistics and a non-normalized state describing the rare events.