基于不同R包中的GPD计算返回水平

Question

我正在对气象数据进行极值分析，准确地说是以mm / d为单位的降水数据。我使用阈值过量方法用最大似然法估计广义Pareto分布的参数。

目的是计算每日降水的几个返回水平（即2,5,10,20,50,100年事件）。

虽然R代码工作正常，但我想知道为什么在根据不同包装的拟合GPD的分位数计算返回水平时，我得到明显不同的结果。尽管每个包中GPD的估计参数几乎相同，但分位数差异很大。

我使用的包是： ismev，extRemes，evir和POT。

我猜对GPD参数的不同估计是由于不同的计算程序，但我不明白为什么分位数的计算差异很大，这取决于不同的包。

当lmom，evir和POT返回相同的quanatile值时，从extRemes包派生的返回周期与其他结果不同。

# packages
library(ismev)
library(extRemes)
library(evir)
library(POT)
library(lmom)

th <- 50

# sample data:
potvalues <- c(
  58.5,44.2,49.6,59.3,48.3,60.9,94.5,47.1,45.3,57.6,48.2,46.2,44.2,50.6,42.1,52.7,80.9,
  58.5,51.3,48.4,51.7,71.9,60.1,64.4,43.5,55.5,49.3,58.2,47.5,43.7,45.2,52.8,42.2,46.4,
  96.1,47.5,50.1,42.4,60.9,72.6,51.6,59.4,80.5,63.7,59.9,45.0,66.7,47.6,53.3,43.1,51.0,
  46.2,53.6,59.8,51.7,46.7,42.6,44.5,45.0,50.0,44.0,89.9,44.2,47.8,53.3,43.0,55.7,44.6,
  44.6,54.9,45.1,43.9,78.7,45.5,64.0,42.7,47.4,57.0,105.4,64.3,43.2,50.4,80.2,49.9,71.6,
  47.4,44.1,47.6,55.2,44.4,78.6,50.8,42.4,47.1,43.5,51.4)

#------------------------------------------------------------------------------------------#

# MLE Fitting of GPD - package extRemes

# fit gpd
pot.ext <- fevd(potvalues, method = "MLE", type="GP", threshold=th)

# return levels:
rl.extremes <-  return.level(pot.ext, conf = 0.05,
                             return.period= c(2,5,10,20,50,100))
rl.extremes <- as.numeric(rl.extremes)

#------------------------------------------------------------------------------------------#

# MLE Fitting of GPD - package ismev

pot.gpd <- gpd.fit(potvalues, threshold=th)

s1 <- quagpa(f=.99, para=c(pot.gpd$threshold, pot.gpd$mle[1],-pot.gpd$mle[2])) # 100
s2 <- quagpa(f=.98, para=c(pot.gpd$threshold, pot.gpd$mle[1],-pot.gpd$mle[2])) #  50
s3 <- quagpa(f=.95, para=c(pot.gpd$threshold, pot.gpd$mle[1],-pot.gpd$mle[2])) #  20
s4 <- quagpa(f=.90, para=c(pot.gpd$threshold, pot.gpd$mle[1],-pot.gpd$mle[2])) #  10
s5 <- quagpa(f=.80, para=c(pot.gpd$threshold, pot.gpd$mle[1],-pot.gpd$mle[2])) #   5
s6 <- quagpa(f=.50, para=c(pot.gpd$threshold, pot.gpd$mle[1],-pot.gpd$mle[2])) #   2

rl.ismev <- c(s6, s5, s4, s3, s2, s1)

#------------------------------------------------------------------------------------------#

# MLE Fitting of GPD - package evir

# fit gpd
gpd.evir <- gpd(potvalues, threshold=th)

# plot
evirplot <- plot(gpd.evir)
1 # Excess Distribution
0 # exit

x100 <- gpd.q(pp=.99, x=evirplot) # 100
x050 <- gpd.q(pp=.98, x=evirplot) #  50
x020 <- gpd.q(pp=.95, x=evirplot) #  20
x010 <- gpd.q(pp=.90, x=evirplot) #  10
x005 <- gpd.q(pp=.80, x=evirplot) #   5
x002 <- gpd.q(pp=.50, x=evirplot) #   2

rl.evir <- t(rbind(x002,x005,x010,x020,x050,x100))
rl.evir <- as.numeric(rl.evir[2,])

#------------------------------------------------------------------------------------------#

# MLE Fitting of GPD - package POT

gpd.pot <- fitgpd(potvalues, threshold=th)
quant = c(0.50, 0.80, 0.90, 0.95, 0.98, 0.99)
rtp <- c(2,5,10,20,50,100)

retvec <- vector()
for (i in quant){
  x <- POT::qgpd(i, loc = th, scale = as.numeric(gpd.pot$param[1]),
            shape = as.numeric(gpd.pot$param[2]))
  retvec <- c(retvec,x)
}

rl.pot <- retvec

#------------------------------------------------------------------------------------------#
# comparison of results - return periods
result <- cbind(rl.extremes,rl.ismev, rl.evir, rl.pot)
round(result, 2)

#------------------------------------------------------------------------------------------#
# comparison of estimated parameters
param.extremes <- pot.ext$results$par # extremes
param.ismev <- pot.gpd$mle # ismev
param.evir <- c(gpd.evir$par.ests[2],gpd.evir$par.ests[1])  # evir
param.pot <- gpd.pot$param # POT

parameters <- cbind(param.extremes, param.ismev , param.evir, param.pot)
round(parameters, 4)

#------------------------------------------------------------------------------------------#

Answer 1

例如，描述了该问题的解决方案。在Coles书中（极值统计建模导论，第4.3.3章）。虽然GEV的返回水平可以直接从其分位数得出，但在计算a的返回水平时，必须考虑所谓的超越率（即每年的事件数或事件超过阈值的可能性）。 GP超出阈值范围的范围内。

N年回报水平由

定义

因此，在不考虑超出率的情况下简单计算GP分布的分位数时，不能获得有意义的返回水平结果。 extRemes包考虑了超出率，而POT和evir包中每年事件数的默认值如果未指定则设置为1。

Answer 2

差异也可能来自将分布函数拟合到数据集的不同方法。我在CRAN上有一个包，用于比较几个R包和方法的GPD拟合（或者更确切地说，它们的分位数估计）：

https://cran.r-project.org/web/packages/extremeStat/vignettes/extremeStat.html

您还可以使用该软件包将GPD与其他发行版进行比较。