Peaks-over-threshold Approach 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.

Modeling the risk of single-vehicle run-off-road crashes on horizontal curves using connected vehicle data

Crash risk measures (CRMs) are widely used in safety analysis to complement crash reports. However, none of the existing CRMs are specifically developed for modeling the risk of single-vehicle run-off-road (SVROR) crashes, especially those on horizontal curves. This paper proposes a novel crash risk measure for modeling SVROR crash risk using connected vehicle data. The proposed SVROR crash risk measure (SVROR-CRM) is based on the concept of tetraquark in particle physics. It utilizes the adjusted position deviation risk force (Fposirisk) and adjusted attitude deviation risk moment (Γattirisk) to quantify SVROR crash risk. The SVROR crash risk is then estimated by the joint probability of Fposirisk and Γattirisk using a peak-over threshold approach. The risk threshold is automatically determined via a mean absolute error function. The SVROR-CRM is validated using connected vehicle and crash data from sixteen curves on Interstate 80 in Wyoming. The results suggest that the estimated SVROR crash risks well match historical crash records. Also, it is found that attitude deviation poses a higher risk of SVROR crash than position deviation on horizontal curves. Therefore, it is critical for drivers to steer properly on curves to minimize SVROR crash risks. The proposed approach bridges an important gap in crash risk measure research and can be used to estimate SVROR crash risk and identify unsafe trajectories and high-crash locations and/or periods on highway horizontal curves.

A probabilistic model based on the peak-over-threshold approach for risk assessment of airport controllers' performance

Airport tower control plays an instrumental role in ensuring airport safety. However, obtaining objective, quantitative safety evaluations is challenging due to the unavailability of pertinent human operation data. This study introduces a probabilistic model that combines aircraft dynamics and the peak-over-threshold (POT) approach to assess the safety performance of airport controllers. We applied the POT approach to model reaction times extracted from a radiotelephony dataset via a voice event detection algorithm. The model couples the risks of tower control and aircraft operation to analyze the influence of human factors. Using data from radiotelephony communications and the Base of Aircraft Data (BADA) database, we compared risk levels across scenarios. Our findings revealed heightened airport control risks under low demand (0.374) compared to typical conditions (0.197). Furthermore, the risks associated with coupling under low demand exceeded those under typical demand, with the final approach stage presenting the highest risk (4.929×10−7). Our model underscores the significance of human factors and the implications of mental disconnects between pilots and controllers for safety risks. Collectively, these consistent findings affirm the reliability of our probabilistic model as an evaluative tool for evaluating the safety performance of airport tower controllers. The results also illuminate the path toward quantitative real-time safety evaluations for airport controllers within the industry. We recommend that airport regulators focus on the performance of airport controllers, particularly during the final approach stage.

Extreme Value Theory Modelling of the Behaviour of Johannesburg Stock Exchange Financial Market Data

Financial market data are abundant with outliers, and the search for an appropriate extreme value theory (EVT) approach to apply is an endless debate in the statistics of extremes research. This paper uses EVT methods to model the five-year daily all-share total return index (ALSTRI) and the daily United States dollar (USD) against the South African rand (ZAR) exchange rate of the Johannesburg stock exchange (JSE). The study compares the block maxima approach and the peaks-over-threshold (POT) approach in terms of their ability to model financial market data. The 100-year return levels for the block maxima approach were found to be almost equal to the maximum observations of the financial markets of 10,860 and R18.99 for the ALSTRI and the USD–ZAR, respectively. For the peaks-over-threshold (POT) approach, the results show that the ALSTRI and the USD–ZAR exchange rate will surpass 17,501.63 and R23.72, respectively, at least once in 100 years. The findings in this study reveal a clear distinction between block maxima and POT return level estimates. The POT approach return level estimates were comparably higher than the block maxima estimates. The study further revealed that the blended generalised extreme value (bGEVD) is more suitable for relatively short-term forecasting, since it cuts off at the 50-year return level. Therefore, this study will add value to the literature and knowledge of statistics and econometrics. In the future, more studies on bGEVD, vine copulas, and the r-largest-order bGEVD can be conducted in the financial markets.

Comparison of Extreme Wind and Waves Using Different Statistical Methods in 40 Offshore Wind Energy Lease Areas Worldwide

With the ongoing global drive towards renewable energy, several potential offshore wind energy lease areas worldwide have come into focus. This study aims to estimate the extreme wind and wave conditions across several newly designated offshore wind lease sites spanning six continents that are crucial for risk assessment and the design of offshore wind turbines. Firstly, the raw data of wind speeds and wave heights prevailing in these different lease areas were obtained. Following this, an in-depth extreme value analysis was performed over different return periods. Two principal methodologies were applied for this comparative study: the block-maxima and the peaks-over-threshold (POT) approaches. Various statistical techniques, including the Gumbel method of moments, Gumbel maximum likelihood, Gumbel least-squares, and the three-parameter GEV, were employed under the block-maxima approach to obtain the distribution parameters. The threshold for the POT approach was defined using the mean residual life method, and the distribution parameters were obtained using the maximum likelihood method. The Gumbel least-squares method emerged as the most conservative estimator of extreme values in the majority of cases, while the POT approach generally yielded lower extreme values compared to the block-maxima approach. However, the results from the POT approach showed large variations based on the selected threshold. This comprehensive study’s findings will provide valuable input for the efficient planning, design, and construction of future offshore wind farms.

Mixture Probability Models with Covariates: Applications in Estimating Risk of Hydroclimatic Extremes

Modeling of extreme events is important in many scientific fields, including environmental and civil engineering, and impacts and risk assessments. Among available methods, statistical models that allow estimating extremes’ frequency and intensity are regularly used in procedures to anticipate potential changes in extreme events. Extreme value theory provides a theoretical basis for statistical estimation of extreme events using frequency analysis. The challenge in modeling is knowing when to use the block maxima method or the peaks-over-threshold (POT) method. Each has its drawbacks. POT describes the main characteristics of the observed extreme series; the threshold selection is always challenging and might affect the accuracy of the simulated results and the credibility of changes in extreme values. To encompass this challenge, mixture models offer more flexibility to represent samples with nonhomogeneous data. This study presents the gamma generalized Pareto (GGP) mixture model for estimating risk occurrence of hydroclimatic extremes. The model was developed in its general form, whereas the observed hydrometeorological extreme events depend on multidimensional covariates. A maximum likelihood algorithm is proposed to estimate the parameters with a constraint on the shape parameter of the generalized Pareto (GP) distribution. A Monte Carlo (MC) simulation compared the proposed model with the classical POT approach, with a fixed threshold, and the annual maximum series of streamflow. The approach was applied using a daily hydrological data set from an observed station located in the Saint John River at Fort Kent (01AD002), New Brunswick, Canada. The results show a flexibility to model extremes for dependent or nonstationary time series and adequately describes the central part of the observed frequencies, as well as the tails.

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.

Leading pedestrian intervals – Yay or Nay? A Before-After evaluation of multiple conflict types using an enhanced Non-Stationary framework integrating quantile regression into Bayesian hierarchical extreme value analysis

A pedestrian was estimated to be killed every 85 min and injured every 7 min on US roads in 2019. Targeted safety treatments are particularly required at urban intersections where pedestrians regularly conflict with turning vehicles. Leading Pedestrian Intervals (LPIs) are an innovative, low-cost treatment where the pedestrian and vehicle usage of the potential conflict area (a crosswalk) is staggered in time to give the pedestrians a head start of a few seconds and reduce the “element of surprise” for right-turning vehicles. The effectiveness of LPI treatment on pedestrian safety is mixed, and most importantly, its effect on vehicle-vehicle conflicts is unknown. This study investigates the before-after effects of LPI treatments on vehicle–pedestrian and vehicle-vehicle crash risk by applying traffic conflict techniques. In particular, this study has developed a quantile regression technique within the extreme value model to estimate and compare crash risks before and after the installation of the LPI treatment. The before-after traffic movement video data (504 h in total) were collected from three signalized intersections in the City of Bellevue, Washington. The recorded movements were analyzed using Microsoft’s proprietary computer vision platform, Edge Video Service, and Advanced Mobility Analytics Group’s cloud-based SMART SafetyTM platform to automatedly extract traffic conflicts by analyzing road user trajectories. The treatment effect was measured using a Bayesian hierarchical extreme value model with the peak-over threshold approach. For the extreme value model, a Bayesian quantile regression analysis was conducted to estimate the conflict thresholds corresponding to a high (95th) quantile. Odds ratios were estimated for both conflict types using untreated crossing as a control group. Results indicate that the LPI treatment reduces the crash risk of pedestrians as measured by the reduction in extreme vehicle–pedestrian conflicts by about 42%. The LPI treatment has also been found not to negatively affect rear-end conflicts along the approaches leading to the LPI-treated pedestrian crossing at the signalized intersections. The findings of this study further emphasize the effectiveness of video analytics in proactive safety evaluations of engineering treatments.

Fitting generalized Pareto distribution based on the expected order statistics and its application for ice accretion hazard mapping

The extreme value analysis of the ice accretion thickness can be carried out based on the peaks-over-threshold (POT) approach to take advantage of the available rare samples in assessing the ice accretion thickness hazard. For the POT approach, the generalized Pareto distribution (GPD) with location, scale, and shape parameters has been frequently employed, although the estimators of parameters of GPD by using existing fitting methods could be associated with significant bias or variability. In the present study, we propose a new nonlinear least-squares fitting method by considering the expected order statistics. We compare its performance with some existing methods through simulation analysis, showing that the proposed method outperforms some well-known existing methods for a range of values of the shape parameter.We then apply the POT approach with GPD as well as the block maximum approach with Gumbel distribution to estimate T-year return period value of the annual maximum ice accretion thickness, xT, for Canadian sites. The analysis results indicate that xT is sensitive to whether the POT or the annual maximum approach with Gumbel distribution is employed. For the considered cases, the use of GPD fitted to POT values is preferred. Also, based on the wind-on-ice analysis results, it is suggested a companion wind load factor of about 0.5 to 0.55 could be adopted.

Data-driven predictive corrosion failure model for maintenance planning of process systems

Extreme value analysis (EVA) is occasionally used to predict corrosion progress. This paper adopts EVA to predict the depth of extreme pits to prioritize inspection and maintenance. It considers the peaks over threshold (POT) method to illustrate the predictive capacity of this method in assessing degradation progress based on consecutive inspection reports. The proposed approach uses distribution parameters to establish stochastic corrosion models. Four consecutive inline inspections of a pipeline are used to validate the model. As the block maxima (BM) method is often used in extreme value analysis of corrosion damage depths, the POT approach is compared to the BM's predictive results. The POT approach is considerably more capable (33%) of assessing failures in individual sections than the same workflow implemented with BM. With the downside of increased falsely categorized failures (10.6%). The method's performance in assessing failures makes it most useful for data-driven maintenance of process systems.

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.

Modelling temperature extremes in the Limpopo province: bivariate time-varying threshold excess approach.

A common problem that arises in extreme value theory when dealing with several variables (such as weather or meteorological) is to find an appropriate method to assess their joint or conditional multivariate extremal dependence behaviour. The method for choosing an appropriate threshold in peaks-over threshold approach is also another problem of endless debate. In this era of climate change and global warming, extreme temperatures accompanied by heat waves and cold waves pose serious economic and health challenges particularly in small economies or developing countries like South Africa. The present study attempts to address these problems, in particular, to deal with and capture dependencies in extreme values of two variables, by applying bivariate conditional extremes modelling with a time-varying threshold to Limpopo province’s monthly maximum temperature series. Limpopo and North West provinces are the two hottest provinces in South Africa characterised by heat waves and the present study is carried out in the Limpopo province at Mara, Messina, Polokwane and Thabazimbi meteorological stations for the period 1994–2009. With the aim to model extremal dependence of maximum temperature at these four meteorological stations, two modelling approaches are applied: bivariate conditional extremes model and time-varying threshold. The latter approach was used to capture the climate change effects in the data. The main contribution of this paper is in combining these two approaches in bivariate extremal dependence modelling of maximum temperature extremes in the Limpopo province of South Africa. The findings of the study revealed both significant positive and negative extremal dependence in some pairs of meteorological stations. Among the major findings were the significant strong positive extremal dependence of Thabazimbi on high-temperature values at Mara and the strong negative extremal dependence of Polokwane on high-temperature values at Messina. The findings of this study play an important role in revealing information useful to meteorologists, climatologists, agriculturalists, and planners in the energy sector among others.

Joint Extremes in Temperature and Mortality: A Bivariate POT Approach

This article contributes to insurance risk management by modeling extreme climate risk and extreme mortality risk in an integrated manner via extreme value theory (EVT). We conduct an empirical study using monthly temperature and death data in the United States and find that the joint extremes in cold weather and old-age death counts exhibit the strongest level of dependence. Based on the estimated bivariate generalized Pareto distribution, we quantify the extremal dependence between death counts and temperature indexes. Methodologically, we employ the cutting-edge multivariate peaks over threshold (POT) approach, which is readily applicable to a wide range of topics in extreme risk management.

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.

A hierarchical bayesian peak over threshold approach for conflict-based before-after safety evaluation of leading pedestrian intervals

A hierarchical Bayesian peak over threshold (POT) approach is proposed for conflict-based before-after safety evaluation of Leading Pedestrian Intervals (LPI). The approach combines traffic conflicts of different sites and periods to develop a uniform generalized Pareto distribution (GPD) model for the treatment effect estimation. The hierarchical structure has three levels, a data level that consists of modeling the traffic conflict extremes through the POT approach, a latent process level that relates GPD parameters of the data level to certain covariates, and a prior level with prior distributions to characterize the latent process. The approach was applied to a before-after (BA) safety evaluation of leading pedestrian interval (LPI) in Vancouver, BC. Pedestrian-vehicle traffic conflicts were collected from treatment and control sites during the before and after periods using an automated computer vision analysis technique. The treatment effect was measured by the best fitted GPD model with the calculation of the odds ratio (OR). The overall treatment effect varies from 18.1%–20.9% in terms of reduction in the estimated extreme-serious conflicts. The treatment effect indicates a considerable improvement in pedestrian safety after the implementation of the LPI, and the consistent results demonstrate a reliable BA safety evaluation. As such, the proposed approach is recommended as a promising tool for BA safety studies, particularly in cases where collision data is limited.

Analysis of air/river maximum daily temperature characteristics using the peaks over threshold approach

AbstractThe understanding of river temperature is very important when assessing fish habitat conditions in terms of fish growth as well as fish distribution within river systems. The quantification of extreme water temperatures is also important, especially when studying the suitability of thermal habitats during high air/water temperature events. The present study deals with the analysis of extreme air and water temperatures using the partial duration series approach, also known as the peak over threshold (POT) approach. The advantage of the POT approach is that extreme air and water temperatures can be described not only in magnitude but also in duration and degree day accumulation for each event. In the present study, 74 years of air temperature data (Miramichi meteorological station) were analysed as well as 21 years of river temperature data from the Little Southwest Miramichi River. For example, results showed that the 0.01 exceedance probability (F(x) = .99) for maximum daily air temperature event was calculated at 38.4°C whereas for the same exceedance probability, the maximum daily water temperature event was slightly lower at 32.7°C. The duration of events varied between 1 and 5 days for air temperature and between 1 and 7 days for water temperature. Maximum daily water temperature exceeding 30°C generally occurred over a period of 3 days or more, and the degree day accumulation was higher than 6°C·day. This study showed that the POT approach was effective in quantifying both air and water temperature extremes within the Miramichi River study area.

Joint Extremes in Temperature and Mortality: A Bivariate POT Approach

This paper contributes to insurance risk management by modeling extreme climate risk and extreme mortality risk in an integrated manner via extreme value theory (EVT). We conduct an empirical study using monthly temperature and death data in the U.S., and nd that the joint extremes in cold weather and old-age death counts exhibit the strongest level of dependence. Based on the estimated bivariate generalized Pareto distribution, we quantify the extremal dependence between death counts and temperature indexes. Methodologically, we employ the cutting edge multivariate peaks over threshold (POT) approach, which is readily applicable to a wide range of topics in extreme risk management.

Flood estimation at Hathnikund Barrage, River Yamuna, India using the Peak-Over-Threshold method

ABSTRACT The paper presents a methodology for the estimation of flood magnitude using the peaks-over-threshold (POT) data. A semi-automatic approach for the determination of POT values has been presented. Two frequency analysis methods, namely Gumbel’s method and Log-Pearson type III distribution, have been used. Two major issues in implementing a POT approach are: (i) determining an appropriate threshold and (ii) ensuring the independence of events. The present paper addresses these issues simultaneously. Using the daily discharge data, various return-level flood magnitudes have been estimated. A comparison of flood magnitudes obtained using the annual maxima (AM) approach and POT approach clearly indicates that the flood magnitudes in general are higher when Log-Pearson method is used as compared to when Gumbel’s method is used. The result of the research conducted clearly indicates that the flood magnitudes estimated on the basis of POT sample are more reliable, particularly when the data length is short.

On the Reliability and Tightness of GP and Exponential Models for Probabilistic WCET Estimation

As computer architectures evolve, guaranteeing that Real-Time Systems’ (RTSs’) timing requirements are met through Worst Case Execution Time (WCET) upper bounds becomes increasingly difficult. Techniques such as Measurement-Based Probabilistic Timing Analysis (MBPTA) have emerged that estimate WCET bounds exceeded only with arbitrarily low probabilities (i.e., pWCETs) through Extreme Value Theory (EVT). The Peaks Over Threshold (POT) approach for applying EVT involves adjusting a tail-shaped distribution, e.g., Generalized Pareto (GP) or Exponential, to the values that exceed a carefully selected high threshold. Several works suggest that GP should be used within POT for best representing different tail shapes, while others consider the Exponential model more adequate for providing upper bounds with increased reliability. This work presents empirical reliability and tightness evaluations of the pWCET estimates yielded by the GP and Exponential models while applying MBPTA through the POT approach. It mainly provides counter-evidence to the GP model reliability and evidence of the Exponential model adequacy in this context.

Extreme Value Theory and the New Sunspot Number Series

Abstract Extreme value theory was employed to study solar activity using the new sunspot number index. The block maxima approach was used at yearly (1700–2015), monthly (1749–2016), and daily (1818–2016) scales, selecting the maximum sunspot number value for each solar cycle, and the peaks-over-threshold (POT) technique was used after a declustering process only for the daily data. Both techniques led to negative values for the shape parameters. This implies that the extreme sunspot number value distribution has an upper bound. The return level (RL) values obtained from the POT approach were greater than when using the block maxima technique. Regarding the POT approach, the 110 year (550 and 1100 year) RLs were lower (higher) than the daily maximum observed sunspot number value of 528. Furthermore, according to the block maxima approach, the 10-cycle RL lay within the block maxima daily sunspot number range, as expected, but it was striking that the 50- and 100-cycle RLs were also within that range. Thus, it would seem that the RL is reaching a plateau, and, although one must be cautious, it would be difficult to attain sunspot number values greater than 550. The extreme value trends from the four series (yearly, monthly, and daily maxima per solar cycle, and POT after declustering the daily data) were analyzed with the Mann–Kendall test and Sen’s method. Only the negative trend of the daily data with the POT technique was statistically significant.