Nonstationary frequency analysis and uncertainty quantification for extreme low lake levels in a large river-lake-catchment system

Extreme hydrological events have become increasingly frequent on a global scale. The middle Yangtze River also faces a substantial challenge in dealing with extreme flooding and drought. However, the long-term characteristics of the extreme hydrological regime have not yet been adequately recognized. Moreover, there is uncertainty in the extreme value estimation, and this uncertainty needs to be distinguished and quantified. In this study, we investigated the nonstationary frequency characteristics of extreme low lake levels (ELLLs), taking the Poyang Lake as an example. Daily lake levels from 1960 to 2022 were utilized to estimate the return level using the generalized Pareto distribution (GPD). The uncertainty from three sources, i.e., the parameter estimator, threshold selection, and covariate, was quantified via variance decomposition. The results indicate that (1) the parameter estimator is the predominant source of uncertainty, with a contribution rate of approximately 87 %. The total uncertainty of the covariate, threshold, and interaction term is only 13 %. (2) Two indexes, namely the annual minimum water level (WLmin) and the days with peak over the 90 % threshold per year (DPOT90), decreased (0.01-0.03 m/year) and increased (0.17-1.39 days/year), respectively, indicating a progressively severe drought trend for Poyang Lake. (3) The return level with return period of 5 to 100 years significantly decreased after the early 21st century. A large spatial heterogeneity was identified for the variation in the return level, and the change rate of the return level with a 100-year return period ranged from 5 % to 40 % for the whole lake. (4) The ELLLs had a stronger correlation with the catchment discharge than with the Yangtze River discharge and the large-scale atmospheric circulation indices. This study provides a methodology with reduced uncertainty for nonstationary frequency analysis (NFA) of ELLLs exemplified in large river-lake systems.

Statistical Inference of Dynamic Conditional Generalized Pareto Distribution with Weather and Air Quality Factors

Air pollution is a major global problem, closely related to economic and social development and ecological environment construction. Air pollution data for most regions of China have a close correlation with time and seasons and are affected by multidimensional factors such as meteorology and air quality. In contrast with classical peaks-over-threshold modeling approaches, we use a deep learning technique and three new dynamic conditional generalized Pareto distribution (DCP) models with weather and air quality factors for fitting the time-dependence of the air pollutant concentration and make statistical inferences about their application in air quality analysis. Specifically, in the proposed three DCP models, a dynamic autoregressive exponential function mechanism is applied for the time-varying scale parameter and tail index of the conditional generalized Pareto distribution, and a sufficiently high threshold is chosen using two threshold selection procedures. The probabilistic properties of the DCP model and the statistical properties of the maximum likelihood estimation (MLE) are investigated, simulating and showing the stability and sensitivity of the MLE estimations. The three proposed models are applied to fit the PM2.5 time series in Beijing from 2015 to 2021. Real data are used to illustrate the advantages of the DCP, especially compared to the estimation volatility of GARCH and AIC or BIC criteria. The DCP model involving both the mixed weather and air quality factors performs better than the other two models with weather factors or air quality factors alone. Finally, a prediction model based on long short-term memory (LSTM) is used to predict PM2.5 concentration, achieving ideal results.

Best-Arm Identification Using Extreme Value Theory Estimates of the CVaR

We consider a risk-aware multi-armed bandit framework with the goal of avoiding catastrophic risk. Such a framework has multiple applications in financial risk management. We introduce a new conditional value-at-risk (CVaR) estimation procedure combining extreme value theory with automated threshold selection by ordered goodness-of-fit tests, and we apply this procedure to a pure exploration best-arm identification problem under a fixed budget. We empirically compare our results with the commonly used sample average estimator of the CVaR, and we show a significant performance improvement when the underlying arm distributions are heavy-tailed.

Extreme events and emergency scales

An event is extreme if its magnitude exceeds the threshold. A choice of a threshold is subject to uncertainty caused by a method, the size of available data, a hypothesis on statistics, etc. We assess the degree of uncertainty by the Shannon's entropy calculated on the probability that the threshold changes at any given time. If the amount of data is not sufficient, an observer is in the state of Lewis Carroll's Red Queen who said "When you say hill, I could show you hills, in comparison with which you'd call that a valley". If we have enough data, the uncertainty curve peaks at two values clearly separating the magnitudes of events into three emergency scales: subcritical, critical, and extreme. Our approach to defining the emergency scale is validated by 39 years of Standard and Poor's 500 (S&P500) historical data.

Threshold selection and trimming in extremes

We consider removing lower order statistics from the classical Hill estimator in extreme value statistics, and compensating for it by rescaling the remaining terms. Trajectories of these trimmed statistics as a function of the extent of trimming turn out to be quite flat near the optimal threshold value. For the regularly varying case, the classical threshold selection problem in tail estimation is then revisited, both visually via trimmed Hill plots and, for the Hall class, also mathematically via minimizing the expected empirical variance. This leads to a simple threshold selection procedure for the classical Hill estimator which circumvents the estimation of some of the tail characteristics, a problem which is usually the bottleneck in threshold selection. As a by-product, we derive an alternative estimator of the tail index, which assigns more weight to large observations, and works particularly well for relatively lighter tails. A simple ratio statistic routine is suggested to evaluate the goodness of the implied selection of the threshold. We illustrate the favourable performance and the potential of the proposed method with simulation studies and real insurance data.

A New Parameter Estimator for the Generalized Pareto Distribution under the Peaks over Threshold Framework

Techniques used to analyze exceedances over a high threshold are in great demand for research in economics, environmental science, and other fields. The generalized Pareto distribution (GPD) has been widely used to fit observations exceeding the tail threshold in the peaks over threshold (POT) framework. Parameter estimation and threshold selection are two critical issues for threshold-based GPD inference. In this work, we propose a new GPD-based estimation approach by combining the method of moments and likelihood moment techniques based on the least squares concept, in which the shape and scale parameters of the GPD can be simultaneously estimated. To analyze extreme data, the proposed approach estimates the parameters by minimizing the sum of squared deviations between the theoretical GPD function and its expectation. Additionally, we introduce a recently developed stopping rule to choose the suitable threshold above which the GPD asymptotically fits the exceedances. Simulation studies show that the proposed approach performs better or similar to existing approaches, in terms of bias and the mean square error, in estimating the shape parameter. In addition, the performance of three threshold selection procedures is assessed by estimating the value-at-risk (VaR) of the GPD. Finally, we illustrate the utilization of the proposed method by analyzing air pollution data. In this analysis, we also provide a detailed guide regarding threshold selection.