Modeling short‐ranged dependence in block extrema with application to polar temperature data

ABSTRACTThe block maxima approach, which consists of dividing a series of observations into equal‐sized blocks to extract the block maxima, is commonly used for identifying and modeling extreme events using the generalized extreme value (GEV) distribution. In the analysis of coastal wave surge levels, the underlying data that generate the block maxima typically have missing observations. Consequently, the observed block maxima may not correspond to the true block maxima, yielding biased estimates of the GEV distribution parameters. Various parametric modeling procedures are proposed to account for the presence of missing observations under a block maxima framework. The performance of these estimators is compared through an extensive simulation study and illustrated by an analysis of extreme wave surges in Atlantic Canada.

India gets three fourths of its annual rainfall during the south-west monsoon season (June–September). The study of extreme events is significant in the stochastic behaviour of rainfall pattern. The aim of the present work is to compare different methods; and find a suitable method to study extreme rainfall trend analysis. In this study, frequency distribution method, generalized extreme value (GEV) distribution, Mann–Kendall and Sen’s slope estimator are used for rainfall trend analysis over the Nethravathi basin located in the southern part of India. The rainfall data during the monsoon months (June–September) were analysed for a period of 1971–2010. The comparison of all the methods had been carried out and it has been observed that there is an increasing trend of frequency in class-1 and decreasing trend in class-2 and class-3, respectively. The interpretation of the results is carried by using the GEV distribution and non-parametric trend analysis (Mann–Kendall and Sen’s slope estimator test). It turns out the best results to identify the extreme rainfall trend are obtained by the statistical techniques – Block Maxima (GEV) distribution, Mann–Kendall and Sen’s slope estimator test as compared to frequency-based method. The above results which help to study climate change will contribute towards sustainable development of the Nethravathi River basin.

This paper focuses on modeling the extreme maximum temperature in Ghana using the extreme value theory. This is to inform decision-makers to help them plan appropriate risk mitigating measures to reduce the damage caused by drought. The block maxima with Generalized Extreme Value (GEV) and the General Pareto (GP) (with “all excesses” and decluster peaks) were used on 113 years of monthly temperature data in Ghana. Two statistical tests for stationarity, namely Augmented Dickey-Fuller (ADF) and Mann-Kendall tests were performed. In the GEV modeling, the model selection criteria (Akaike information criterion and likelihood-ratio test) and the diagnostic checking indicate that the model with linear trend in location parameter is appropriate. In fitting the GP distribution, the results from the parameter estimation show that GP with “all excesses” better fits the data than the decluster peaks. The diagnostic checking also lead to the same conclusion. The GEV estimates of the return level show that the return temperature which exceeds the maximum temperature of the observation period (36.3) starts to appear in the return period of T= 20 over years. This suggests that in 20 years to come, maximum temperature in Ghana will exceed 36.3, which may indicate a drought period.

Some events are difficult to avoid and gives considerable influence to humans and the environment is extreme weather and climate change. Many of the problems that require knowledge about the behavior of extreme values and one of the methods used are the Extreme Value Theory (EVT). EVT used to draw up reliable systems in a variety of conditions, so as to minimize the risk of a major disaster. There are two methods for identifying extreme value, Block Maxima with Generalized Extreme Value (GEV) distribution approach and Peaks over Threshold (POT) with Generalized Pareto Distribution (GPD) approach. This research in Indramayu with January 1961-December 2003 period, the method used is Block Maxima with GEV distribution approach. The result showed that there is no climate change in Indramayu with January 1961-December 2003 period.

A cross-comparison of different extreme value modeling techniques for traffic conflict-based crash risk estimation

Melting of the land-based ice and warming of the oceans around the world has resulted in acceleration of the pace of global mean sea level rise. Higher mean sea level causes more frequent extreme sea levels. This forces coastal cities urgently to do major preparedness and adaptation measures. In Finland, preparedness for coastal flooding hazards is relevant even though protection given by the post-glacial land uplift is helpful. This is especially true on the Finnish southern coast, where mean sea level rise is foreseen and increase of probability of high sea levels within the 21st century is expected (Pellikka et al., 2023, 2018). In the Finnish coastal area, the extreme sea level estimates are used e.g. to support infrastructure planning, flood maps and safe operation of nuclear power plants. On a short timescale, sea level variations are driven on the Finnish coast by storm surges, wind induced oscillations within the bays, and tides (playing a minor role). On the long-term side, global mean sea level rise, land uplift and the water inflow and outflow in the Danish straits (which change the total amount of Baltic Sea water) are the main factors controlling the sea level behaviour. In this presentation, a study aiming at improving extreme sea level estimates in Finland will be illustrated. Tentative results on how different sampling techniques and extrapolation approaches affect the probability estimates of coastal floods will be presented. Altogether 90 years of observations from the 13 Finnish tide gauges are analysed. We apply two different well-known sampling methods (Block Maxima and Peak Over Threshold) to the high tail of the sea level distribution and investigate which extrapolation function belonging to the family of Generalized Extreme Value (GEV) distribution matches best to the Finnish tide gauge observations. The results will be grouped into four coastal regions in Finland: the Gulf of Finland and Archipelago Sea in the south, and the Bothnian Sea and Bay of Bothnia in the west.   Pellikka, H., Johansson, M. M., Nordman, M., and Ruosteenoja, K., 2023: Probabilistic projections and past trends of sea level rise in Finland, Nat. Hazards Earth Syst. Sci., 23, 1613–1630, https://doi.org/10.5194/nhess-23-1613-2023 Pellikka, H., Leijala, U., Johansson, M. M., Leinonen, K., Kahma, K. K., 2018: Future probabilities of coastal floods in Finland, Continental Shelf Research, 157, 32-42, https://doi.org/10.1016/j.csr.2018.02.

Forecasting the block maxima of a future time window is a challenging task due to the difficulty in inferring the tail distribution of a target variable. As the historical observations alone may not be sufficient to train robust models to predict the block maxima, domain-driven process models are often available in many scientific domains to supplement the observation data and improve the forecast accuracy. Unfortunately, coupling the historical observations with process model outputs is a challenge due to their disparate temporal coverage. This paper presents Self-Recover, a deep learning framework to predict the block maxima of a time window by employing self-supervised learning to address the varying temporal data coverage problem. Specifically Self-Recover uses a combination of contrastive and generative self-supervised learning schemes along with a denoising autoencoder to impute the missing values. The framework also combines representations of the historical observations with process model outputs via a residual learning approach and learns the generalized extreme value (GEV) distribution characterizing the block maxima values. This enables the framework to reliably estimate the block maxima of each time window along with its confidence interval. Extensive experiments on real-world datasets demonstrate the superiority of Self-Recover compared to other state-of-the-art forecasting methods.

In modeling spatial extremes, the dependence structure is classically inferred by assuming that block maxima derive from max-stable processes. Weather stations provide daily records rather than just block maxima. The point process approach for univariate extreme value analysis, which uses more historical data and is preferred by some practitioners, does not adapt easily to the spatial setting. We propose a two-step approach with a composite likelihood that utilizes site-wise daily records in addition to block maxima. The procedure separates the estimation of marginal parameters and dependence parameters into two steps. The first step estimates the marginal parameters with an independence likelihood from the point process approach using daily records. Given the marginal parameter estimates, the second step estimates the dependence parameters with a pairwise likelihood using block maxima. In a simulation study, the two-step approach was found to be more efficient than the pairwise likelihood approach using only block maxima. The method was applied to study the effect of El Niño-Southern Oscillation on extreme precipitation in California with maximum daily winter precipitation from 35 sites over 55 years. Using site-specific generalized extreme value models, the two-step approach led to more sites detected with the El Niño effect, narrower confidence intervals for return levels and tighter confidence regions for risk measures of jointly defined events.

The Generalized Pareto (GP) and Generalized extreme value (GEV) distributions play an important role in extreme value analyses, as models for threshold excesses and block maxima respectively. For each of these distributions we consider Bayesian inference using "reference" prior distributions (in the general sense of priors constructed using formal rules) for the model parameters, specifically a Jeffreys prior, the maximal data information (MDI) prior and independent uniform priors on separate model parameters. We investigate the important issue of whether these improper priors lead to proper posterior distributions. We show that, in the GP and GEV cases, the MDI prior, unless modified, never yields a proper posterior and that in the GEV case this also applies to the Jeffreys prior. We also show that a sample size of three (four) is sufficient for independent uniform priors to yield a proper posterior distribution in the GP (GEV) case.

This paper tests the Generalized Extreme Value (GEV) distribution as an EV method using the minimum time-to-collision with the opposing vehicle during passing maneuvers. Detailed trajectory data of the passing, passed and opposite vehicles from a fixed-based driving simulator experiment were used in this study. One hundred experienced drivers from different demographic strata participated in this experiment on a voluntary base. Raw data were collected at a resolution of 0.1 s and included the longitudinal and lateral position, speed and acceleration of all vehicles in the scenario. From this raw data, the minimum time-to-collision with the opposing vehicle at the end of the passing, maneuver was calculated. GEV distribution based on the Block Maxima approach was tested for the estimation of head-on collision probabilities in passing maneuvers. The estimation results achieved good fit with respect to head-on collisions' prediction indicating that this is a promising approach for safety evaluation.

Safety analysis of passing maneuvers using extreme value theory

Extreme value theory (EVT) is useful for modeling the impact of crashes or situations of extreme stress on investor portfolios. EVT is mostly utilized in financial modeling, risk management, insurance, and hydrology. The price of gold fluctuates considerably over time, and this introduces a risk on its own. The goal of this study is to analyze the risk of gold investment by applying the EVT to historical daily data for extreme daily losses and gains in the price of gold. We used daily gold prices in the Pakistan Bullion Market from August 1, 2011 to July 30, 2021. This paper covers two methods such as Block Maxima (BM) and Peak Over Threshold (POT) modeling. The risk measures which are adopted in this paper are Value at Risk (VaR) and Expected Shortfall (ES). The point and interval estimates of VaR and ES are obtained by fitting the Generalized Pareto (GPA) distribution. Moreover, in this paper, return-level forecasting is also included for the next 5 and 10 years by analyzing the Generalized Extreme Value (GEV) distribution.

In this work, we introduce a method of estimating stochastic freeway capacity using elements of both extreme value theory and survival analysis. First, we define capacity data, or estimates of the capacity of the roadway, as the daily maximum flow values. Then, under a survival analysis premise, we introduce censoring into our definition. That is, on days when flows are sufficiently high and congestion occurs, corresponding flow maxima are considered true estimates of capacity; otherwise, for those days that do not observe high flows or congestion, flow maxima are deemed censored observations and capacities must be higher than the observations. By extreme value theory, the collection of flow maxima (block maxima) can be appropriately approximated with a generalized extreme value (GEV) distribution. Because of small sample sizes and the presence of censoring, a Bayesian framework is pursued for model fitting and parameter estimation. To lend credence to our proposed methodology, the procedure is applied to real-world traffic stream data collected by the New Hampshire Department of Transportation (NHDOT) at a busy location on Interstate I-93 near Salem, New Hampshire. Data were collected over a period of 11 months and raw data were aggregated into 15-minute intervals. To assess our procedure, and to provide proof of concept, several validation procedures are presented. First, using distinct training and validation subsets of our data, the procedure yields accurate predictions of highway capacity. Next, our procedure is applied to a training set to yield random capacities which are then used to predict breakdown in the validation set. The frequency of these predicted breakdowns is found to be statistically similar to observed breakdowns observed in our validation set. Lastly, after comparing our methodology to other methods of stochastic capacity estimation, we find our procedure to be highly successful.

Abstract. Accurate estimates of the probability of extreme sea levels are pivotal for assessing risk and for designing coastal defense structures. This probability is typically estimated by modeling observed sea-level records using one of a few statistical approaches. In this study we comparatively apply the generalized-extreme-value (GEV) distribution, based on block maxima (BM) and peaks-over-threshold (POT) formulations, and the recent metastatistical extreme-value distribution (MEVD) to four long time series of sea-level observations distributed along European coastlines. A cross-validation approach, dividing available data into separate calibration and test sub-samples, is used to compare their performances in high-quantile estimation. To address the limitations posed by the length of the observational time series, we quantify the estimation uncertainty associated with different calibration sample sizes from 5 to 30 years. We study extreme values of the coastal water level – the sum of the water level setup induced by meteorological forcing and of the astronomical tide – and we find that the MEVD framework provides robust quantile estimates, especially when longer sample sizes of 10–30 years are considered. However, differences in performance among the approaches explored are subtle, and a definitive conclusion on an optimal solution independent of the return period of interest remains elusive. Finally, we investigate the influence of end-of-century projected mean sea levels on the probability of occurrence of extreme-total-water-level (the sum of the instantaneous water level and the increasing mean sea level) frequencies. The analyses show that increases in the value of total water levels corresponding to a fixed return period are highly heterogeneous across the locations explored.

This chapter reviews the basics of univariate extreme value analysis. The generalized extreme value (GEV) distribution is introduced as the limit distribution of sample maxima with appropriate standardization. The domains of attraction are reviewed and illustrated numerically with simulation for distributions that have different tail behaviors. Common properties of the GEV distribution are summarized. The connection to the generalized Pareto distribution is established by taking the limit of the conditional distribution over a threshold. Statistical inferences are grouped by the type of available data: the block maxima method, r largest order statistics method, the peaks-over-threshold method, and the point process method. For block maxima, the maximum product spacing method, which has been overlooked in extreme value applications, is worth highlighting for its numerical stability and competitive small sample performance in comparison with the likelihood method and the L-moments method. The efficiency gain of using r largest order statistics relative to using block maxima only is shown in a small simulation study. Compared to existing reviews on univariate extreme value analysis, this chapter provides more statistical flavor through numerical studies.