Extreme Value Theory And Its Application In Rainfall

In daily life,people often ignore things that rarely happen,but those things that happen rarely will have a great impact on people's daily lives.And the extreme value theory exactly study the law of these very few things.The extreme value theory originated in Germany.It was presented by statistician Bortkiewicz,and mainly studies the approximate distribution of the maximum and minimum values of random variables.In other words,the extreme value theory studys the distribution of extreme events that rarely occur.It has developed many research results and has been widely used in various fields.For example,finance,engineering,floods,droughts,climate,etc.Extreme weather or rare natural disasters have always plagued people's lives.Abnormal rainfall is an important factor in extreme weather,so it is very meaningful to use reasonable methods to establish models of extreme rainfall.This paper is the basis of the extreme value theory.The extreme value statistical method is applied to the study of daily rainfall in a certain area of the United Kingdom.And it is discussed whether the rainfall model in the region established by various methods is reasonable.At the beginning of the paper,the extreme value distribution and the generalized extreme value distribution are summarized.The various classifications and related results of the extreme value distribution are summarized in detail,and the density function graph of each type is visually drawn.Then the distribution of threshold exceedance is summarized,including block model and the threshold exceedance theoretical model(POT),in which the threshold exceedance theoretical model is based on the generalized Pareto distribution(GPD),so the paper summarizes the generalized Pareto distribution in detail.The next,the steps and methods for establishing a model based on the GPD are summarized.The main steps are three steps.The first step is to select the threshold,including three methods:the mean residual life plot,the Hill graph and estimating the model over a threshold range.The second step is parameter estimation,which mainly estimates shape parameters and scale parameters.The methods include maximum likelihood estimation,probability weight estimation,nonlinear least squares estimation and nonlinear weighted least squares estimation.The first two methods are classical methods.The latter two methods were proposed in recent years.The third step is the model test.There are four methods:The P-P map,the goodness of fit test,the Kolmogorov-Smirnov(K-S)test,the Gramervo statisticsW~2 and the Anderon-Darling statistics~2A test.In the application process,the daily rainfall in a certain area of southwestern England was used as the research object for case analysis.Based on the POT model,the steps and methods of establishing the model were applied to the rainfall data.According to the modeling steps,the threshold is first determined by the average residual life graph,and then the four methods of parameter estimation are respectively used in the data exceeding the threshold.Each method corresponds to a set of estimated parameters,and four generalized Pareto distribution with different parameters are obtained.Finally,the resulting distribution was tested by the K-S test method.