Machine Learning Method to estimate parameters of the GPD distribution: Applied to Lobo flows and Yzeron water levels

Abstract The Peaks Over Threshold (POT) is an extreme value theory method, used in the management of natural disasters such as floods, economic and financial crisis. These disasters have considerable impacts on the environment and justify the extreme data analysis using the POT. The results showed that, under certain conditions, the excesses distribution can be approximated by a Generalized Pareto Distribution (GPD). One of the major difficulties of POT lies in the GPD distribution parameters estimation, particularly the selection of the optimal threshold. In this paper, we propose a machine learning method to estimate the GPD distribution parameters. In the First step, we identify the candidate thresholds using the mrlplot graphic and determine the admissible thresholds by the cross-validation method. The admissible thresholds are those which minimize the squared error between the estimated and empirical quantilites of the training and test samples. The shape and scale parameters of the GPD distribution are estimated by the Modified Newton-Raphson numerical method by setting each candidate threshold of the training sample. This numerical method uses a penalized log likelihood function. In the second step, we select the optimal threshold among the admissible thresholds using the Kolmogorov- Smirnov goodness-of-fit test. The optimal threshold is the one which gives the best adequation based on the pvalue and the Kolmogorow statistic. The numerical results for sample sizes (100, 200, 500, 1000, 2000, 5000) showed the reduction in estimation bias of our method for large sample sizes. This technique is illustrated in a case study of Lobo flows and Yzeron water levels, a water resource of great importance in Cˆote d’Ivoire et France.