MGL-regression-mixed

Case II - modelling Chinese earthquake loss data with covariates

This example implements the MGL and MGL-EV copula mixed regression models of Zhengxiao Li et al. (2021, “MGL copulas”).

survival MGL copual, survival MGL-EV and Gumbel copula regression

library(rMGLReg)

u <- cbind(earthqCHI$u1, earthqCHI$u2)
u_ <- cbind(earthqCHI$u1, earthqCHI$u2_)
y <- cbind(earthqCHI$y1, earthqCHI$y2)
f <- cbind(earthqCHI$f1, earthqCHI$f2)

obs <- y
U <- u
U_ <- u_
umin <- 20

library(splines)
X <- splines::ns(earthqCHI$year, knots = quantile(earthqCHI$year, c(0.333, 0.667)), intercept = T)

m.MGL180 <- MGL.reg.mixed(obs = y, U = U, U_ = U_, umin = umin, f = f, X = X,
                             copula = "MGL180", 
                             method = "Nelder-Mead",
                             initpar = c(-0.32, 1, 1, 1))

m.MGLEV180 <- MGL.reg.mixed(obs = y, U = U, U_ = U_, umin = umin, f = f, X = X,
                             copula = "MGL-EV180", 
                             method = "Nelder-Mead",
                             initpar = c(-0.32, 1, 1, 1))

m.gumbel <- MGL.reg.mixed(obs = y, U = U, U_ = U_, umin = umin, f = f, X = X,
                             copula = "Gumbel", 
                             method = "Nelder-Mead",
                             initpar = c(-0.32, 1, 1, 1))

m.MGB2 <- MGL.reg.mixed(obs = y, U = U, U_ = U_, umin = umin, f = f, X = X,
                             copula = "MGB2", 
                             method = "Nelder-Mead",
                             initpar = c(-0.1, 0.5, -0.5, -0.5, 0, 0))

Estimates and goodness fit

  estimates.copula <- rbind(c(m.MGL180$estimates, NA, NA), 
                            c(m.MGLEV180$estimates, NA, NA), 
                            c(m.gumbel$estimates, NA, NA),
                            c(m.MGB2$estimates)
                            )
  sd.copula <- rbind(c(m.MGL180$se, NA, NA), 
                     c(m.MGLEV180$se, NA, NA), 
                     c(m.gumbel$se, NA, NA),
                     c(m.MGB2$se)
                     )
  ll.copula <- rbind((m.MGL180)$loglike, (m.MGLEV180)$loglike, (m.gumbel)$loglike, (m.MGB2)$loglike)
  AIC.copula <- rbind((m.MGL180)$AIC, (m.MGLEV180)$AIC, (m.gumbel)$AIC, (m.MGB2)$AIC)
  BIC.copula <- rbind((m.MGL180)$BIC, (m.MGLEV180)$BIC, (m.gumbel)$BIC, (m.MGB2)$BIC)
  table1 <- cbind(estimates.copula, sd.copula, ll.copula, AIC.copula, BIC.copula)
  
  row.names(table1) <- c('MGL180', 'MGLEV180', 'Gumbel', 'MGB2') 
  colnames(table1) <- c("estimates", "estimates","estimates", "estimates","estimates", "estimates",
                        "se", "se","se","se","se","se",
                        'loglike', 
                        'AIC', 'BIC')
  knitr::kable(t(table1), digits = 3)
  

The predicted value of copula parameter with different value of the covariate for the Chinese earthquake loss data set.

library(MASS)
# Survival MGL regression model
par.est <- m.MGL180$estimates
agepred <- seq(from = 1980, to = 1990)
Xcopula <-  splines::ns(agepred, knots = quantile(agepred, c(0.333, 0.667)), intercept = T)
delta.est <- exp(Xcopula%*%par.est)
cov.est <- -solve(m.MGL180$hessian)
beta.sim <- matrix(0, nrow = 100, ncol = 4)
delta.mat <- matrix(0, nrow = length(agepred), ncol = 100)
beta.sim <- mvrnorm(100, mu = par.est, Sigma = cov.est)
for(i in 1:100){
  delta.mat[,i] <- exp(Xcopula%*%beta.sim[i,])
}
matplot(delta.mat, col = 'gray', type = 'l', ylim = c(0, 6), main = 'Survival MGL')
lines(delta.est, col = 'red', lwd = 2)


# Surivial MGL-EV regression
par.est <- m.MGLEV180$estimates
delta.est <- exp(Xcopula%*%par.est)
cov.est <- -solve(m.MGLEV180$hessian)
beta.sim <- matrix(0, nrow = 100, ncol = 4)
delta.mat <- matrix(0, nrow = length(agepred), ncol = 100)
beta.sim <- mvrnorm(100, mu = par.est, Sigma = cov.est)
for(i in 1:100){
  delta.mat[,i] <- exp(Xcopula%*%beta.sim[i,])
}
matplot(delta.mat, col = 'gray', type = 'l', ylim = c(0, 6), main = 'Survival MGL-EV')
lines(delta.est, col = 'red', lwd = 2)

# Gumbel regression
par.est <- m.gumbel$estimates
delta.est <- exp(Xcopula%*%par.est) + 1
cov.est <- -solve(m.gumbel$hessian)
beta.sim <- matrix(0, nrow = 100, ncol = 4)
delta.mat <- matrix(0, nrow = length(agepred), ncol = 100)
beta.sim <- mvrnorm(100, mu = par.est, Sigma = cov.est)
for(i in 1:100){
  delta.mat[,i] <- exp(Xcopula%*%beta.sim[i,]) + 1
}
matplot(delta.mat, col = 'gray', type = 'l', ylim = c(0, 6), main = 'Gumbel')
lines(delta.est, col = 'red', lwd = 2)

# MGB2 regression
par.est <- m.MGB2$estimates
delta.est <- exp(Xcopula%*%par.est[1:ncol(X)])

cov.est <- -solve(m.MGB2$hessian)
beta.sim <- matrix(0, nrow = 100, ncol = 6)
delta.mat <- matrix(0, nrow = length(agepred), ncol = 100)
pars.sim <- mvrnorm(100, mu = par.est, Sigma = cov.est)
beta.sim <- pars.sim[,1:ncol(X)]
for(i in 1:100){
  delta.mat[,i] <- exp(Xcopula%*%beta.sim[i,])
}
matplot(delta.mat, col = 'gray', type = 'l', ylim = c(0, 6), main = 'MGB2')
lines(delta.est, col = 'red', lwd = 2)

library(MASS)
# Survival MGL regression model
par.est <- m.MGL180$estimates
agepred <- seq(from = 1990, to = 2015)
Xcopula <-  splines::ns(agepred, knots = quantile(agepred, c(0.333, 0.667)), intercept = T)
delta.est <- exp(Xcopula%*%par.est)
cov.est <- -solve(m.MGL180$hessian)
beta.sim <- matrix(0, nrow = 100, ncol = 4)
delta.mat <- matrix(0, nrow = length(agepred), ncol = 100)
beta.sim <- mvrnorm(100, mu = par.est, Sigma = cov.est)
for(i in 1:100){
  delta.mat[,i] <- exp(Xcopula%*%beta.sim[i,])
}
data <- as.data.frame(delta.mat)
# data <- data.table(data)
#id variable for position in matrix
data$id <- agepred
#reshape to long format
plot_data <- melt(data, id.var = "id")
p1  <- ggplot() +
  theme_bw() + xlab('Year') + ylab(expression(paste('Predicted', "  ",delta))) +
  geom_line(data = plot_data, mapping = aes(y = value,
                                            x = id,
                                            group = variable),
            size = 0.3, linetype = 1, col = 'gray') +
  geom_path(aes_(x = agepred, y = delta.est),
            size = 1, col = 'red'
  ) +
  scale_x_continuous(limits = c(1990, 2015),
                     breaks = seq(1990, 2015, by = 5)) +
  scale_y_continuous(limits = c(0, 6),
                     breaks = seq(0, 6, by = 1.2)) +
  # ggtitle("Survival MGL copula regression") +
  facet_grid(. ~ "Survival MGL copula regression") +
  theme(plot.title = element_text(hjust = 0.5))

p1

# Surivial MGL-EV regression
par.est <- m.MGLEV180$estimates
delta.est <- exp(Xcopula%*%par.est)
cov.est <- -solve(m.MGLEV180$hessian)
beta.sim <- matrix(0, nrow = 100, ncol = 4)
delta.mat <- matrix(0, nrow = length(agepred), ncol = 100)
beta.sim <- mvrnorm(100, mu = par.est, Sigma = cov.est)
for(i in 1:100){
  delta.mat[,i] <- exp(Xcopula%*%beta.sim[i,])
}
data <- as.data.frame(delta.mat)
#id variable for position in matrix
data$id <- agepred
#reshape to long format
plot_data <- melt(data, id.var = "id")
p2  <- ggplot() +
  theme_bw() + xlab('Year') + ylab(expression(paste('Predicted', "  ",delta))) +
  geom_line(data = plot_data, mapping = aes(y = value,
                                            x = id,
                                            group = variable),
            size = 0.3, linetype = 1, col = 'gray') +
  geom_path(aes_(x = agepred, y = delta.est),
            size = 1, col = 'red'
  ) +
  scale_x_continuous(limits = c(1990, 2015),
                     breaks = seq(1990, 2015, by = 5)) +
  scale_y_continuous(limits = c(0, 6),
                     breaks = seq(0, 6, by = 1.2)) +
  # ggtitle("Survival MGL-EV copula regression") +
  facet_grid(. ~ "Survival MGL-EV copula regression") +
  theme(plot.title = element_text(hjust = 0.5))

p2

# Gumbel regression
par.est <- m.gumbel$estimates
delta.est <- exp(Xcopula%*%par.est) + 1
cov.est <- -solve(m.gumbel$hessian)
beta.sim <- matrix(0, nrow = 100, ncol = 4)
delta.mat <- matrix(0, nrow = length(agepred), ncol = 100)
beta.sim <- mvrnorm(100, mu = par.est, Sigma = cov.est)
for(i in 1:100){
  delta.mat[,i] <- exp(Xcopula%*%beta.sim[i,]) + 1
}
data <- as.data.frame(delta.mat)
#id variable for position in matrix
data$id <- agepred
#reshape to long format
plot_data <- melt(data, id.var = "id")
p3  <- ggplot() +
  theme_bw() + xlab('Year') + ylab(expression(paste('Predicted', "  ",delta))) +
  geom_line(data = plot_data, mapping = aes(y = value,
                                            x = id,
                                            group = variable),
            size = 0.3, linetype = 1, col = 'gray') +
  geom_path(aes_(x = agepred, y = delta.est),
            size = 1, col = 'red'
  ) +
  scale_x_continuous(limits = c(1990, 2015),
                     breaks = seq(1990, 2015, by = 5)) +
  scale_y_continuous(limits = c(0, 6),
                     breaks = seq(0, 6, by = 1.2)) +
  facet_grid(. ~ "Gumbel copula regression") +
  theme(plot.title = element_text(hjust = 0.5))

p3


# MGB2 regression
par.est <- m.MGB2$estimates
delta.est <- exp(Xcopula%*%par.est[1:ncol(X)])

cov.est <- -solve(m.MGB2$hessian)
beta.sim <- matrix(0, nrow = 100, ncol = 6)
delta.mat <- matrix(0, nrow = length(agepred), ncol = 100)
pars.sim <- mvrnorm(100, mu = par.est, Sigma = cov.est)
beta.sim <- pars.sim[,1:ncol(X)]
for(i in 1:100){
  delta.mat[,i] <- exp(Xcopula%*%beta.sim[i,])
}

data <- as.data.frame(delta.mat)
#id variable for position in matrix
data$id <- agepred
#reshape to long format
plot_data <- melt(data, id.var = "id")
p4  <- ggplot() +
  theme_bw() + xlab('Year') + ylab(expression(paste('Predicted', "  ", q))) +
  geom_line(data = plot_data, mapping = aes(y = value,
                                            x = id,
                                            group = variable),
            size = 0.3, linetype = 1, col = 'gray') +
  geom_path(aes_(x = agepred, y = delta.est),
            size = 1, col = 'red'
  ) +
  scale_x_continuous(limits = c(1990, 2015),
                     breaks = seq(1990, 2015, by = 5)) +
  scale_y_continuous(limits = c(0, 6),
                     breaks = seq(0, 6, by = 1.2)) +
  facet_grid(. ~ "MGB2 copula regression") +
  theme(plot.title = element_text(hjust = 0.5))

p4


p <- p1 + p2 + p3 + p4

p


# ggsave(p, file = 'C:/Users/Lee/Desktop/pic2_name.pdf', width = 12, height = 8)