基于统计推断的GEV分布水文频率计算充分样本长度研究  

Research on sufficient sample length for calculation of hydrological frequency of GEV distribution based on statistical inference

作　　者：戈立婷 宋松柏[1,2] GE Liting;SONG Songbai(College of Water Resources and Architectural Engineering,Northwest A&F University,Yangling 712100,China;Key Laboratory of Agricultural Soil and Water Engineering in Arid Area of Ministry of Education,Northwest A&F University,Yangling 712100,China)

机构地区：[1]西北农林科技大学水利与建筑工程学院,陕西杨凌712100 [2]西北农林科技大学旱区农业水土工程教育部重点实验室,陕西杨凌712100

出　　处：《水利学报》2022年第8期1004-1016,共13页Journal of Hydraulic Engineering

基　　金：国家自然科学基金项目(52079110)。

摘　　要：针对水文频率计算样本长度的充分性问题,本文根据最大似然估计量的渐近正态性质,界定了充分样本长度的概念,将bootstrap与Shapiro-Wilk正态性检验相结合,确定给定水文序列不同设计频率计算所需的充分样本长度。该法不依赖于模型及经验法则,可避免主观因素的影响,能够应用于水文实际计算。为验证方法的可行性,选取广义极值(GEV)分布进行数值模拟分析,计算不同频率下设计值计算所需的充分样本长度n,在此基础上,选用相对置信区间(RCI)、平均相对误差(RMAE)为评定误差指标,应用曲线拟合法,定量分析不同参数、频率、参数估计方法与样本长度间的潜在关系。数值模拟结果表明:(1)通过正态性检验,验证了文中方法的合理性,采用误差指标曲线对推导结果n进行分析,在不同参数、频率下均表现出充分样本长度n规律性,论证了文中方法的正确性。(2)误差指标与样本长度表现为幂率函数关系,当样本长度达到一定样本量时,两种参数估计方法的RCI、RMAE拟合曲线出现交叉点,交叉点后最大似然估计法表现出良好的估计性能。(3)在10%~75%的频率范围内,样本长度30基本满足不同参数、不同参数估计方法下参数估计的充分条件,而在该频率范围外,充分样本长度远大于30,且重现期越大,所需样本长度越大。最后,将文中方法应用于国内8个典型径流、降水和洪峰序列,对实例计算结果进行方差分析(ANOVA)检验,进一步论证了方法的可行性。Aiming at the sufficiency of sample length for hydrological frequency calculation, this paper defines the concept of sufficient sample length according to the asymptotic normality of maximum likelihood estimator, and combines bootstrap with Shapiro-Wilk normality test to determine the sufficient sample length required for different design frequencies of a given hydrological sequence.This method does not depend on the model and empirical rules, and can avoid the influence of subjective factors, which can be applied to hydrological calculation.In order to verify the feasibility of the method, the generalized extreme value(GEV) distribution is selected for numerical simulation analysis, and the sufficient sample length nrequired for the calculation of design values at different frequencies is calculated.To this end, the relative confidence interval(RCI) and the average relative error(RMAE) are selected as the evaluation error indexes, and the curve fitting method is used to analyze the potential relationship between different parameters, frequencies, parameter estimation methods and sample length.Numerical simulation results show that:(ⅰ) Through the normality test, the rationality of the method in this paper is verified.The error index curve is used to analyze the derived result n,showing the regularity of sufficient sample length nunder different parameters and frequencies, which demonstrates the correctness of the method in this paper.(ⅱ) The relationship between the error index and the sample length is a power-law function.When the sample length reaches a certain sample size, the RCI and RMAE fitting curves of the two parameter estimation methods have intersections.After the intersection, the maximum likelihood estimation method shows good estimation performance.(ⅲ) In the frequency range of 10%-75%,under different parameters, the sample length of 30 basically satisfies the sufficient conditions for parameter estimation.Outside the frequency range, the sufficient sample length required by the moment method

关 键 词：GEV分布 充分样本长度 最大似然估计 矩法 渐近正态性质 曲线拟合 正态性检验 

分 类 号：P333[天文地球—水文科学]