如何从 R 中的二维 copula 中找到联合累积分布函数？

Question

我现在正在研究 R 中的 copula，我想知道如何在 R 中找到联合累积分布？

D = c(1,3,2,2,8,2,1,3,1,1,3,3,1,1,2,1,2,1,1,3,4,1,1,3,1,1,2,1,3,7,1,4,6,1,2,1,1,3,1,2,2,3,4,1,1,1,1,2,2,12,1,1,2,1,1,1,3,4)
S = c(1.42,5.15,2.52,2.29,12.36,2.82,1.49,3.53,1.17,1.03,4.03,5.26,1.65,1.41,3.75,1.09,3.44,1.36,1.19,4.76,5.58,1.23,2.29,7.71,1.12,1.26,2.78,1.13,3.87,15.43,1.19,4.95,7.69,1.17,3.27,1.44,1.05,3.94,1.58,2.29,2.73,3.75,6.80,1.16,1.01,1.00,1.02,2.32,2.86,22.90,1.42,1.10,2.78,1.23,1.61,1.33,3.53,10.44)

经过一番探索，我发现 Gamma 分布最能描述上述数据。

library(fitdistrplus)
fg_d <- fitdist(data = Dur, distr = "gamma", method = "mle")
fg_s <- fitdist(data = Sev, distr = "gamma", method = "mle")

然后，我尝试使用 VineCopula packge 选择 copula 系列：

mydata <- cbind(D=D, S=S)
u1 <- pobs(mydata[,1]) 
u2 <- pobs(mydata[,2])
fitCopula <- BiCopSelect(u1, u2, familyset=NA)
summary(fitCopula) 

结果显示为“生存克莱顿”。然后，我尝试构建以下 copula：

library(copula)
cop_model <- surClaytonCopula(param = 5.79)

现在，根据下面的等式（假设 E(L) 是一个常数）：

对于给定的 D 和 S 值，我需要找到 FD(d)、FS(s) 和 C(FD(d),FS(s))。

例如，如果我们取 D=3 和 S=2，那么我们必须找到 F(DandScopula 在 R 中执行此操作？

另外，我们如何找到 C(DorS

Answer 1

这是一个仅使用 base R 和 copula 包的答案：

---

• F_D(d) 是一个伽马 CDF。根据您的代码，它的形状为 2.20，速率为 0.98，因此 F_D(3) 为 pgamma(3, 2.20, 0.98) = 0.7495596

• F_S(s) 是一个伽马 CDF。根据您的代码，它的形状为 1.56，速率为 0.45，因此 F_S(2) 为 pgamma(2, 1.56, 0.45) = 0.3631978

• C(F_D(d), F_S(s)) 是用上述边际。在 R 中这是

library(copula)
D_shape <- 2.20
D_rate  <- 0.98
S_shape <- 1.56
S_rate  <- 0.45
surv_clay <- rotCopula(claytonCopula(5.79))
pCopula(c(pgamma(3, D_shape, D_rate),pgamma(2, S_shape, S_rate)), surv_clay)

• OP cmets 中 Shiau 2006 论文的第 810 页上的公式 (23) 的分母表明 P(D>=3 或 S>=2) = 1- C(F_D (d), F_S(s)) 即：

1 - pCopula(c(pgamma(3, D_shape, D_rate),pgamma(2, S_shape, S_rate)), surv_clay)

• P(D

 pgamma(3, D_shape, D_rate) + 
 pgamma(2, S_shape, S_rate) - 
 pCopula(c(pgamma(3, D_shape, D_rate),pgamma(2, S_shape, S_rate)), surv_clay)

---

来源

• R-bloggers post on Modelling Dependence with Copulas
• 通过阅读 copula 手册实现 VineCopula::surClaytonCopula(5.79) 对应于 copula::rotCopula(copula::claytonCopula(5.79))

---

下面是一些代码，通过模拟来仔细检查一些事情。

library(fitdistrplus)
library(copula)
library(VineCopula)

D = c(1,3,2,2,8,2,1,3,1,1,3,3,1,1,2,1,2,1,1,3,4,1,1,3,1,1,2,1,3,7,1,4,6,1,2,1,1,3,1,2,2,3,4,1,1,1,1,2,2,12,1,1,2,1,1,1,3,4)
S = c(1.42,5.15,2.52,2.29,12.36,2.82,1.49,3.53,1.17,1.03,4.03,5.26,1.65,1.41,3.75,1.09,3.44,1.36,1.19,4.76,5.58,1.23,2.29,7.71,1.12,1.26,2.78,1.13,3.87,15.43,1.19,4.95,7.69,1.17,3.27,1.44,1.05,3.94,1.58,2.29,2.73,3.75,6.80,1.16,1.01,1.00,1.02,2.32,2.86,22.90,1.42,1.10,2.78,1.23,1.61,1.33,3.53,10.44)

(fg_d <- fitdist(data = D, distr = "gamma", method = "mle"))
(fg_s <- fitdist(data = S, distr = "gamma", method = "mle"))

mydata <- cbind(D=D, S=S)
u1 <- pobs(mydata[,1]) 
u2 <- pobs(mydata[,2])
fitCopula <- BiCopSelect(u1, u2, familyset=NA)
summary(fitCopula) 

D_shape <- fg_d$estimate[1]
D_rate <-  fg_d$estimate[2]
S_shape <- fg_s$estimate[1]
S_rate <-  fg_s$estimate[2]

copula_dist <- mvdc(copula=rotCopula(claytonCopula(5.79)), margins=c("gamma","gamma"),
                    paramMargins=list(list(shape=D_shape, rate=D_rate),
                                      list(shape=S_shape, rate=S_rate)))

sim <- rMvdc(n = 1e5,
             copula_dist)

plot(sim, col="red")
points(D,S, col="black")
legend('bottomright',c('Observed','Simulated'),col=c('black','red'),pch=21)

并回答有关具体计算的问题：

## F_D(d) for d=3
mean(sim[,1] <=3)          ## simulated
pgamma(3, D_shape, D_rate) ## theory

## F_S(s) for s=2
mean(sim[,2] <=2)          ## simulated
pgamma(2, S_shape, S_rate) ## theory

## C(F_D(d) for d=3 AND F_S(s) for s=2)
## simulated value:
mean(sim[,1] <=3 & sim[,2] <=2)
## with copula:
surv_clay <- rotCopula(claytonCopula(5.79))
pCopula(c(pgamma(3, D_shape, D_rate),pgamma(2, S_shape, S_rate)), surv_clay)

## P(D>=3 or S>=2)
## simulated
mean(sim[,1] >= 3 | sim[,2] >=2)
## with copula:
1-pCopula(c(pgamma(3, D_shape, D_rate),pgamma(2, S_shape, S_rate)), surv_clay)

## In case you want:
## P(D<=3 or S<=2) = P(D<=3) + P(S<=2) - P(D<=3,S<=2)
## simulated:
mean(sim[,1] <= 3 | sim[,2] <= 2)
## theory with copula:
pgamma(3, D_shape, D_rate) + pgamma(2, S_shape, S_rate) - pCopula(c(pgamma(3, D_shape, D_rate),pgamma(2, S_shape, S_rate)), surv_clay)

运行上面的两段代码会得到以下输出：

> (fg_d <- fitdist(data = D, distr = "gamma", method = "mle"))
Fitting of the distribution ' gamma ' by maximum likelihood 
Parameters:
       estimate Std. Error
shape 2.2082572  0.3831383
rate  0.9775783  0.1903410
> (fg_s <- fitdist(data = S, distr = "gamma", method = "mle"))
Fitting of the distribution ' gamma ' by maximum likelihood 
Parameters:
       estimate Std. Error
shape 1.5628338 0.26500235
rate  0.4494518 0.08964724
> 
> mydata <- cbind(D=D, S=S)
> u1 <- pobs(mydata[,1]) 
> u2 <- pobs(mydata[,2])
> fitCopula <- BiCopSelect(u1, u2, familyset=NA)
Warning message:
In cor(x[(x[, 1] < 0) & (x[, 2] < 0), ]) : the standard deviation is zero
> summary(fitCopula) 
Family
------ 
No:    13
Name:  Survival Clayton

Parameter(s)
------------
par:  5.79

Dependence measures
-------------------
Kendall's tau:    0.74 (empirical = 0.82, p value < 0.01)
Upper TD:         0.89 
Lower TD:         0 

Fit statistics
--------------
logLik:  57.68 
AIC:    -113.37 
BIC:    -111.31 

> 
> D_shape <- fg_d$estimate[1]
> D_rate <-  fg_d$estimate[2]
> S_shape <- fg_s$estimate[1]
> S_rate <-  fg_s$estimate[2]
> 
> copula_dist <- mvdc(copula=rotCopula(claytonCopula(5.79)), margins=c("gamma","gamma"),
+                     paramMargins=list(list(shape=D_shape, rate=D_rate),
+                                       list(shape=S_shape, rate=S_rate)))
> 
> sim <- rMvdc(n = 1e5,
+              copula_dist)
> 
> plot(sim, col="red")
> points(D,S, col="black")
> legend('bottomright',c('Observed','Simulated'),col=c('black','red'),pch=21)

还有——

> ## F_D(d) for d=3
> mean(sim[,1] <=3)          ## simulated
[1] 0.74759
> pgamma(3, D_shape, D_rate) ## theory
[1] 0.746482
> 
> ## F_S(s) for s=2
> mean(sim[,2] <=2)          ## simulated
[1] 0.36233
> pgamma(2, S_shape, S_rate) ## theory
[1] 0.3617122
> 
> ## C(F_D(d) for d=3 AND F_S(s) for s=2)
> ## simulated value:
> mean(sim[,1] <=3 & sim[,2] <=2)
[1] 0.362
> ## with copula:
> surv_clay <- rotCopula(claytonCopula(5.79))
> pCopula(c(pgamma(3, D_shape, D_rate),pgamma(2, S_shape, S_rate)), surv_clay)
[1] 0.3615195
> 
> ## P(D>=3 or S>=2)
> ## simulated
> mean(sim[,1] >= 3 | sim[,2] >=2)
[1] 0.638
> ## with copula:
> 1-pCopula(c(pgamma(3, D_shape, D_rate),pgamma(2, S_shape, S_rate)), surv_clay)
[1] 0.6384805

> ## In case you want:
> ## P(D<=3 or S<=2) = P(D<=3) + P(S<=2) - P(D<=3,S<=2)
> ## simulated:
> mean(sim[,1] <= 3 | sim[,2] <= 2)
[1] 0.74792
> ## theory with copula:
> pgamma(3, D_shape, D_rate) + pgamma(2, S_shape, S_rate) - pCopula(c(pgamma(3, D_shape, D_rate),pgamma(2, S_shape, S_rate)), surv_clay)
[1] 0.7466747