Peak Over Threshold Method Research Articles

VALUE AT RISK ESTIMATION USING EXTREME VALUE THEORY APPROACH IN INDONESIA STOCK EXCHANGE

Extreme Value Theory (EVT) is a method used to identify extreme cases in heavy tail data such as financial time series data. This research aimed to obtain an estimate of stock risk through the EVT approach and compare the accuracy of the two EVT approaches, Block Maxima (BM) and Peaks Over Threshold (POT). The method used to estimate stock risk is VaR with the BM and POT approaches, and the Z statistic is used to compare the accuracy. The data used, and the limitation in this research is daily closing price data for non-cyclical consumer stocks included in LQ45 for the period February 01, 2017, to January 31, 2023. Other research limitations are using weekly blocks or 5 working days in dividing BM blocks, using the percentage method in determining threshold values in the POT approach, and using Maximum Likelihood Estimation (MLE) to estimate EVT parameter estimates. The results of the VaR analysis show that the risk level generated by the POT method is greater than the risk level from BM. The results of backtesting between the two EVT approaches in estimating VaR values show that the POT approach is more accurate than the BM approach.

Regional event-based flood quantile estimation method for large climate projection ensembles

Emerging large ensemble climate datasets produced by multiple general circulation models and their downscaling products challenge the limits of hydrodynamic models because of the immense data size. To overcome this new challenge and estimate the discharge quantiles corresponding to different return periods at all river sections in an entire region, this study proposes an event-based regional approach that uses a nationwide distributed rainfall–runoff model as well as large climate projection ensembles. This approach addresses the high computational burden associated with continuous simulations and solves the problem of conventional event-based simulations serving only a single outlet of a basin. For our analysis, we extracted 372 annual maximum 48 h rainfall events that cover the entirety of Shikoku Island and its eight major river basins. Peak discharges were estimated using a 150 m resolution rainfall–runoff–inundation model. These discharges were then screened using either the peak-over-threshold (POT) method or block maxima (BM) method, and frequency curves were subsequently constructed and evaluated. The primary reason for the necessity of POT or BM was to avoid interference from extraneous low discharges. The POT-based frequency curves showed good accuracy when using peak discharges in the range of the top 10–50%, and the results remain stable within this threshold range. The BM method, employing block sizes of 2–5 years, can generate relatively accurate frequency curves, but the choice of block size introduces significant variations in results among certain basins. Generally, the accuracy of results based on the POT method surpasses that of the BM method. Considering the accuracy, computational cost, and result stability, the POT method is preferred. The error introduced by the regional approach was acceptable with more than half of the relative root-mean-square errors remaining within 10% and basically all of the results are within 20%. The results of the regional approach exhibited good accuracy across climate scenarios and provided consistent information regarding future flood quantiles. This study serves as the foundation for high-resolution future flood risk assessment.

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.

Estimation of Large River Design Floods Using the Peaks-Over-Threshold (POT) Method

This research analyzes the peaks-over-threshold (POT) method for designed flood estimation needed to plan river levees, spillways and water facilities. In this study, a one-parameter exponential probability distribution has been modified by including the coefficient of λ, which represents an average number of floods and enables return period calculation within the specified period of time. The study also compares results using the Log-Pearson Type III distribution of maximum annual flows and a standard exponential distribution of the selected peaks over the threshold level. The aforementioned approach represents the standard mathematical tools for river flood design, while the proposed modification of the exponential distribution highlights the estimation of flood quantiles with longer return periods (e.g., 100, 1000 and 10,000 years). Moreover, the sensitivity analysis of the threshold selection is proposed to assist in the flood design flow estimation alongside the proposed modification of the exponential probability distribution. The study was carried out at the Danube River, and the Novi Sad hydrological station (Republic of Serbia) was used for the long-term recorded period from 1876 to 2015. The results suggest that the POT method derives more reliable estimates of design floods than the traditional statistical tools for flood estimation. The results suggest the theoretical values of the water level of the 10,000 years return period is equal to 867 cm, while the Log-Pearson Type III distribution of annual maximum flows underestimated this value for 14 cm.

Extreme value analysis for waves in the Persian Gulf: Skill assessment of different methods for a fetch-limited basin

Estimating the design wave height with specific return period using extreme value analysis (EVA) is of significant importance in coastal and marine engineering. The purpose of this study is to compare different methods such as the Maximum Likelihood, the L-Moment and the least square methods to estimate the parameters of Generalized Pareto Distribution (GPD), Gumbel, and Weibull distributions in order to determine 100 years return period wave height at 16 selected points within the Persian Gulf. The Peaks Over Threshold (POT) method was employed for all tests. The hourly wave data were available for the total period of 30 years from 01/01/1990 to 31/12/2019 using reanalyzed ECMWF-ERA5 dataset as well as hindcast results from a calibrated local WAVEWATCH-III model. In order to find the threshold cutoff value, one of the pivotal parameters in EVA, adaptive approaches were employed which determined the threshold value based on the distribution function and the goodness of fit for extreme data at specific site. It was shown that subjective constant values used in most of previous studies in this water body were much less than what should be used to have the best fit. Moreover, the determined threshold value was not constant at different points and varied depending on the type of distribution employed. The results showed superiority of the Gumbel distribution in most of the points in the Persian Gulf. Sensitivity Analysis to the threshold values showed that using a low threshold value would lead to increasing the return wave height, which was significant in the case of Pareto and Gumbel distributions. Sensitivity analysis to input data confirmed that ECMWF-ERA5 underestimated the wave height and consequently, the underestimation of designed wave height at some locations by more than 40%. Also, the results from Annual Maxima Method (AMM) was similar to POT method but with ±20% differences in values.

Assessing the importance of the choice threshold in quantifying market risk under the POT approach (EVT)

From a theoretical point of view, the selection of thresholds is a critical issue in the framework of the Peaks Over Threshold (POT) approach, which is why in the last decade numerous methodologies have been proposed for its selection. In this paper, we address this subject from an empirical point of view by assessing to what extent the selection of the threshold is decisive in quantifying the market risk. For measuring market risk, we use the Value at Risk (VaR) and the Expected Shortfall (ES) measures. The results obtained indicate that there is a large set of thresholds that provide similar Generalized Pareto Distribution (GPD) quantiles estimators and as a consequence similar market risk measures. Just only, for large thresholds, those corresponding to the 98th and 99th percentile of the GPD some differences are found. It means that the choice of threshold in the framework of the POT method may not be relevant in quantifying market risk when we use the VaR and ES measures for this task.

Extrapolation of Load Histories Based on MCMC and POT

Load spectrum is essential for fatigue life analysis, strength and durability tests of vehicles and components, but the whole life load cannot be acquired by measurement. To extrapolate a short measured load history to a longer time period, a method was presented in this paper based on Markov Chain Monte Carlo (MCMC) in conjunction with the Peak Over Threshold (POT) method. Firstly, we extend the load by MCMC to increase the randomness of load and the number of extreme samples; then fit a generalized Pareto distribution (GPD) using POT method to extrapolate the extreme values. Methods of threshold selection in POT were also presented briefly in this paper. Rain flow curves of the original, the MCMC extended and the extended extrapolated signal comparison verified the effectiveness of this method.

Application of Non-stationary Extreme Value Analysis to Satellite-Observed Sea Surface Temperature Data for Past Decades

Extreme value analysis (EVA) has been extensively used to understand and predict long-term return extreme values. This study provides the first approach to EVA using satellite-observed sea surface temperature (SST) data over the past decades. Representative EVA methods were compared to select an appropriate method to derive SST extremes of the East/Japan Sea (EJS). As a result, the peaks-over-threshold (POT) method showed better performance than the other methods. The Optimum Interpolation Sea Surface Temperature (OISST) database was used to calculate the 100-year-return SST values in the EJS. The calculated SST extremes were 1.60–3.44°C higher than the average value of the upper 5th-percentile satellite-observed SSTs over the past decades (1982–2018). The monthly distribution of the SST extremes was similar to the known seasonal variation of SSTs in the EJS, but enhanced extreme SSTs exceeding 2°C appeared in early summer and late autumn. The calculated 100-year-return SSTs were compared with the simulation results of the Coupled Model Intercomparison Project 5 (CMIP5) climate model. As a result, the extreme SSTs were slightly smaller than the maximum SSTs of the model data with a negative bias of –0.36°C. This study suggests that the POT method can improve our understanding of future oceanic warming based on statistical approaches using SSTs observed by satellites over the past decades.

Chinese Sunspot Drawings and Their Digitizations-(VI) Extreme Value Theory Applied to the Sunspot Number Series from the Purple Mountain Observatory

Based on the daily sunspot number (SN) data (1954–2011) from the Purple Mountain Observatory, the extreme value theory (EVT) is employed for the research of the long-term solar activity. It is the first time that the EVT is applied on the Chinese SN. Two methods are used for the research of the extreme events with EVT. One method is the block maxima (BM) approach, which picks the maximum SN value of each block. Another one is the peaks-over-threshold (POT) approach. After a declustering process, a threshold value (here it is 300) is set to pick the extreme values. The negative shape parameters are obtained by the two methods, respectively, indicating that there is an upper bound for the extreme SN value. Only one value of the N-year return level (RL) is estimated: N = 19 years. For N = 19 years, the RL values of SN obtained by two methods are similar with each other. The RL values are found to be 420 for the POT method and the BM method. Here, the trend of 25th solar cycle is predicted to be stronger, indicating that the length of meridional forms of atmospheric circulation will be increased.

Comparison Between Maximum Annual and Peak Over Threshold Methods for the Determination of Extreme Waves in Moroccan Atlantic Coast

In this article, we present a study that focuses on forecasting the one hundred-years return period waves height using two methods; i) Peak Over Threshold (POT) and ii) Annual Maxima (MA). The analysis is carried out at three points at the Atlantic coast of northern Morocco. The results obtained by these two methods indicate a difference of less than 10 % and have shown that to ensure safe dimensioning of maritime structures, the application of the two statistical methods is necessary. This is explained by the drawbacks encountered in both methods. In the MA method, the inconvenience is linked to the choice of non-extreme events when no significant storm is recorded during one year, furthermore, the non-consideration of important events recorded in other year. In the second method (POT), the difficulty consists in the uncertainty of the calculations by the POT method due to the mistrust associated with the choice of the censoring threshold.

Estimation of Extreme Significant Wave Height in the Northwest Pacific Using Satellite Altimeter Data Focused on Typhoons (1992–2016)

The estimation of extreme ocean wave heights is important for understanding the ocean’s response to long-term changes in the ocean environment and for the effective coastal management of potential disasters in coastal areas. In order to estimate extreme wave height values in the Northwest Pacific Ocean, a 100-year return period were calculated by applying a Peak over Threshold (PoT) method to satellite altimeter SWH data from 1992 to 2016. Satellite altimeter SWH data were validated using in situ measurements from the Ieodo Ocean Research Station (IORS) south of Korea and the Donghae buoy of the Korea Meteorological Administration (KMA) off the eastern coast of Korea. The spatial distribution and seasonal variations of the estimated 100-year return period SWHs in the Northwest Pacific Ocean were presented. To quantitatively analyze the suitability of the PoT method in the Northwest Pacific, where typhoons frequently occur, the estimated 100-year return period SWHs were compared by classifying the regions as containing negligible or significant typhoon effects. Seasonal variations of extreme SWHs within the upper limit of 0.1% and the PoT-based extreme SWHs indicated the effect of typhoons on the high SWHs in the East China Sea and the southern part of the Northwest Pacific during summer and fall. In addition, this study discusses the limitations of satellite altimeter SWH data in the estimation of 100-year extreme SWHs.

The problem of determining the threshold for statistical analysis by the POT method: Application to wave data on the Moroccan Atlantic coast

In a study of extreme waves by the Peak Over Threshold (POT) method, the determination of the threshold of data censoring is an essential step. A wrong choice of the threshold can lead to erroneous results of the wave height design and consequently a bad design of maritime structures such as breakwaters for deep sea ports. In this study, we analyzed the influence of the threshold variation on the results of the hundred-year return period waves, generally considered for the design of maritime structures. The sensitivity study allowed us to confirm that the exponential model is the best probability distribution to describe wave data in two points on the Moroccan Atlantic coast for the wave data period from 1958 to 2019. This study also confirmed that a wrong choice of the statistical distribution and a wrong choice of the threshold lead to significant errors in the estimation of design wave height.

Extreme value analysis dilemma for climate change impact assessment on global flood and extreme precipitation

A reliable estimation of hydrological extremes with potentially severe socio-economic impacts is of crucial importance for efficient planning and design of hydraulic structures. Extreme value theory provides a firm theoretical foundation for the statistical modelling of extreme hydrological events. The dilemma in the modelling is on whether to use block maxima (BM) or peak-over-threshold (POT) method, each with its own cons and pros. It remains unexplored to what extent future projected changes in extreme hydrological events are influenced by the method choice, especially when some simplifications are made to lessen the computational burden for large-scale studies. This study addresses this research question by a comparative analysis between the BM and POT methods on future climate change impact on global flood and extreme precipitation. The extreme precipitation analysis is performed using 24 CMIP5 general circulation models (GCMs), while the flood analysis is based on a multi-model ensemble of 20 members including five global impact models forced by four CMIP5 GCMs. The results reveal that the BM and POT methods agree on the sign of changes in flood and extreme precipitation intensities, but disagree on the magnitude. The discrepancy between the BM and POT results increases with event extremity. The difference also varies with season, where the difference in the global land area with increasing signals peaks in winter at the rates of 11% for extreme precipitation and in summer at the rates of 3.5% for flood.

Effect of Various Factors on the Estimation of T-Year Discharges for Water Management

The paper deals with the effect of various factors on the estimation of designed discharges. As input data, the series of daily discharges and annual peak discharges on the Topľa River at Hanušovce nad Topľou for the period of 1931-2015 were used. The first, maximum annual discharges (AM) approach was applied with the most widely used Log-Pearson III. probability distribution. The second, we analysed the effect of the time series length and the effect of seasonality (winter, summer) on the accuracy of T-year maximum discharges estimation. As an alternative to this approach (AM), the Peak Over Threshold (POT) method was used. We analysed the effect of the threshold level value selection and using of maximum daily discharges on the accuracy of T-year maximum discharges estimation. Results showed that not only the selection of the distribution function to estimate T-year discharges but also the type of used data series may affect the results of the estimation (length or selection of the period). Results also showed that the estimation of designed maximum discharges using by the POT method did not show significant differences at the selected various threshold levels, but for a relatively fast and large increase of discharges during floods, it would be necessary to have peak values for all waves included in the analysis. Determining the specific value of a 500- or 1000-year flood for engineering practice is extremely complex and in interpreting the results, it should be kept in mind that estimated values with very high return periods are extrapolated values. Each statistical method includes some uncertainty that may be caused by the method but also the data may be affected by certain measurement error, therefore, it is also necessary to specify confidence intervals in which the flow of a given 100-, 500-, or 1000-year flood may occur with probability, for example, 90%.

Extreme value modeling of coincident lane load effects for multi-lane factors of bridges using peaks-over-threshold method

Multi-lane factor (MLF) is a probability reduction reflecting unfavorable traffic loads over multiple lanes acting simultaneously on the most adverse position of a bridge. It is one of the key components of traffic load models for bridges. The most recent research established a multi-coefficient MLF model that clearly illustrated the lane load disparity and the probability reduction of their simultaneous actions. However, it used the block maxima (BM) method for extreme value modeling, which requires a large amount of traffic data. This study aims to adopt the peaks-over-threshold (POT) method to obtain more information from short-term traffic data and model the extreme coincident lane load effects (LLEs) for multi-coefficient MLF calibration. First, the multi-coefficient MLF model was reviewed. Thereafter, the bivariate POT method for coincident LLEs modeling using generalized Pareto distribution was proposed and formulated. Critical issues such as bivariate threshold selection and parameter estimation were addressed. Numerical examples were demonstrated to verify and validate the approach. Finally, the proposed approach was applied for calibrating the MLF of an experimental site with four traffic lanes. The results indicated that the coincident LLEs modeling using the POT approach was accurate and more effective than using the BM method when applied to limited data. The calibrated MLFs from the experimental site effectively revealed the lane load disparity of traffic loads over multiple lanes, which is not involved in the traffic load models of current bridge design specifications. Furthermore, the influence of other problems such as weight restriction on coincident LLEs modeling and MLF calibration were discussed. The proposed technique provides a sound approach for multi-coefficient MLF calibration of bridge assessment with short-term site-specific traffic data.

Study of Directional Declustering for Estimating Extreme Wave Heights in the Yellow Sea

The study of extreme waves is important for the protection of coastal and ocean structures. In this work, a 22-year (1990–2011) wave hindcast in the Yellow Sea is employed to perform the assessment of extreme significant wave heights in this area. To extract the independent sample from this database, the fixed window method is used, which takes the peak significant wave height within five d. With the selected samples, directional declustering is studied to extract the homogenous sample. The results show that most of the independent samples (especially large samples) are observed in the North. In this direction, the peak over threshold (POT) method is used to extract the extreme sample from the homogenous sample, and then the generalized Pareto distribution model is used to extrapolate the extreme significant wave height. In addition to this combination, the annual maxima method with the Gumbel model is also used for estimating extreme values. The comparisons show that the return significant wave heights of the first combination are reliable, resulting from a flexible sampling window in the POT method. With this conclusion, the extreme significant wave height is extrapolated from the Yellow Sea, which can be used to protect the structure in the main directional bin.

Application of the non-stationary peak-over-threshold methods for deriving rainfall extremes from temperature projections

Concerns about climate change are amplifying interest in future rainfall extremes. However, there are big differences between the statistics of rainfall extremes obtained using future rainfall time series produced from various climate models. Such large uncertainties create a lot of confusion in establishing climate change adaptation measures. Looking at future rainfall extremes at a particular site yields increasing trends in some climate models and decreasing trends in others. The spatial patterns of rate of change in rainfall extremes also vary widely, depending on the climate model. As a result, they often do not gain the public’s trust. We believe that this difficulty in obtaining credibility does not come from a lack of theory or technique, but from an approach that persuades the public of uncertain future rainfall extremes. In this study, we employed a novel approach to integrate a co-variate of the not-stationary Peak-Over-Threshold (POT) – Generalized Pareto distribution (GPD) model identified at each site with its future projection information for obtaining future rainfall extreme ensembles. Rainfall extremes are obtained from the observed rainfall time series using the POT method, and the scale parameter among GPD parameters are applied as a function of surface air temperature (SAT) or dew-point temperature (DPT). At this time, the threshold of the POT series is set to match the results of frequency analysis of the annual maximum series and the POT series for each site as much as possible. The behavior of future rainfall extremes is analyzed by inputting the future SAT or DAT produced from various climate models into the non-stationary frequency model using the co-variate. As a result of comparing the rainfall extremes obtained using the future rainfall time series directly with the future rainfall extremes obtained indirectly using the proposed method, it was found that the proposed approach projected future design rainfall depths with much less variation between climate models. The spatial pattern of rate of change was also consistent regardless of climate model. The proposed method is expected to contribute to the public’s confidence in future rainfall extremes under climate change scenarios and to be of practical help in formulating reasonable climate change adaptation policies.

Quantifying Flood Frequency Modification Caused by Multi-Reservoir Regulation

The construction and the operation of multi-reservoir has severely altered the downstream design flood. This paper quantified the flood frequency modifications caused by multi-reservoir regulation with recorded data and a numerical model composed of peaks-over-threshold (POT) samples and time-trend models. The whole research was carried out in Dongjiang River Basin, Southern China. Results showed that (1) Sampling with the POT method could eliminate fake changes in change detection and prevent the underestimation of the design floods with an exceeding probability over 90% and the overestimation of the design floods with an exceeding probability less than 80%; (2) Multi-reservoir regulation severely reduced design flood when the exceeding probability was greater than 1% and smaller than 95%. The 10-year floods were mitigated the most. However, when the exceeding probability was over 95%, the impact was insignificant; (3) Flood mitigation was positively correlated to the number of reservoirs, but negatively correlated to the distance between the affecting reservoir and the affected station. However, when multiple reservoirs took effect on the design flood in one station, the closest reservoir did not necessarily take the most effect. The results of this research will amplify the theoretical basis for flood protection and the planning of regulation.

Maximum likelihood estimators based on the block maxima method

The extreme value index is a fundamental parameter in univariate Extreme Value Theory (EVT). It captures the tail behavior of a distribution and is central in the extrapolation beyond observations. Among other semi-parametric methods (such as the popular Hill estimator), the Block Maxima (BM) and Peaks-Over-Threshold (POT) methods are widely used for assessing the extreme value index and related normalizing constants. We provide asymptotic theory for the maximum likelihood estimators (MLE) based on the BM method for independent and identically distributed observations in the max-domain of attraction of some extreme value distribution. Our main result is the asymptotic normality of the MLE with a non-trivial bias depending on the extreme value index and on the so-called second-order parameter. Our approach combines asymptotic expansions of the likelihood process and of the empirical quantile process of block maxima. The results permit to complete the comparison of common semi-parametric estimators in EVT (MLE and probability weighted moment estimators based on the POT or BM methods) through their asymptotic variances, biases and optimal mean square errors.

On second order conditions in the multivariate block maxima and peak over threshold method

Second order conditions provide a natural framework for establishing asymptotic results about estimators for tail related quantities. Such conditions are typically tailored to the estimation principle at hand, and may be vastly different for estimators based on the block maxima (BM) method or the peak-over-threshold (POT) approach. In this paper we provide details on the relationship between typical second order conditions for BM and POT methods in the multivariate case. We show that the two conditions typically imply each other, but with a possibly different second order parameter. The latter implies that, depending on the data generating process, one of the two methods can attain faster convergence rates than the other. The class of multivariate Archimax copulas is examined in detail; we find that this class contains models for which the second order parameter is smaller for the BM method and vice versa. The theory is illustrated by a small simulation study.