Simulation from a Conditional Bivariate Copula

Simulates from a conditional bivariate copula, where each copula parameter takes a different value, depending on the calibration function and covariates.

condBiCopSim(family, calib.fnc, X, par2 = 0, return.par = TRUE,
  tau = TRUE)

Arguments

family	family A copula family: 1 Gaussian, 2 Student t, 3 Clayton, 4 Gumbel, 5 Frank, 13 Survival Clayton, 14 Survival Gumbel, 23 Rotated (90 degrees) Clayton, 24 Rotated (90 degrees) Gumbel, 33 Rotated (270 degrees) Clayton and 34 Rotated (270 degrees) Gumbel.
calib.fnc	A calibration function.
X	A vector (if calib.fnc takes a single argument) or matrix (if calib.fnc takes multiple arguments) of covariates values.
par2	The second copula parameter (for the Student t), default par2 = 0.
return.par	Should the parameter (and calibration function) be returned as well (default return.par = TRUE)?
tau	Should the calibration function (and the model) be specified for the copula parameter or Kendall's tau (default tau = TRUE)?

Value

If return.par = TRUE, then the function returns a list with:

• data, a matrix with two columns containing the simulated data,

• par, a vector containing the values of the copula parameter,

• and eta, a vector containing the values of the calibration function.

If return.par = FALSE, then the function simply returns data, a matrix with two columns containing the simulated data.

See also

gamBiCopFit and gamBiCopSimulate.

Examples

require(copula)
#> Loading required package: copula
set.seed(0)

## Simulation parameters (sample size, correlation between covariates,
## Gaussian copula family)
n <- 2e2
rho <- 0.5
fam <- 1


## A calibration surface depending on three variables
eta0 <- 1
calib.surf <- list(
  calib.quad <- function(t, Ti = 0, Tf = 1, b = 8) {
    Tm <- (Tf - Ti) / 2
    a <- -(b / 3) * (Tf^2 - 3 * Tf * Tm + 3 * Tm^2)
    return(a + b * (t - Tm)^2)
  },
  calib.sin <- function(t, Ti = 0, Tf = 1, b = 1, f = 1) {
    a <- b * (1 - 2 * Tf * pi / (f * Tf * pi +
      cos(2 * f * pi * (Tf - Ti))
      - cos(2 * f * pi * Ti)))
    return((a + b) / 2 + (b - a) * sin(2 * f * pi * (t - Ti)) / 2)
  },
  calib.exp <- function(t, Ti = 0, Tf = 1, b = 2, s = Tf / 8) {
    Tm <- (Tf - Ti) / 2
    a <- (b * s * sqrt(2 * pi) / Tf) * (pnorm(0, Tm, s) - pnorm(Tf, Tm, s))
    return(a + b * exp(-(t - Tm)^2 / (2 * s^2)))
  }
)

## Display the calibration surface
par(mfrow = c(1, 3), pty = "s", mar = c(1, 1, 4, 1))
u <- seq(0, 1, length.out = 100)
sel <- matrix(c(1, 1, 2, 2, 3, 3), ncol = 2)
jet.colors <- colorRamp(c(
  "#00007F", "blue", "#007FFF", "cyan", "#7FFF7F",
  "yellow", "#FF7F00", "red", "#7F0000"
))
jet <- function(x) rgb(jet.colors(exp(x / 3) / (1 + exp(x / 3))),
    maxColorValue = 255
  )
for (k in 1:3) {
  tmp <- outer(u, u, function(x, y)
    eta0 + calib.surf[[sel[k, 1]]](x) + calib.surf[[sel[k, 2]]](y))
  persp(u, u, tmp,
    border = NA, theta = 60, phi = 30, zlab = "",
    col = matrix(jet(tmp), nrow = 100),
    xlab = paste("X", sel[k, 1], sep = ""),
    ylab = paste("X", sel[k, 2], sep = ""),
    main = paste("eta0+f", sel[k, 1],
      "(X", sel[k, 1], ") +f", sel[k, 2],
      "(X", sel[k, 2], ")",
      sep = ""
    )
  )
}

## 3-dimensional matrix X of covariates
covariates.distr <- mvdc(normalCopula(rho, dim = 3),
  c("unif"), list(list(min = 0, max = 1)),
  marginsIdentical = TRUE
)
X <- rMvdc(n, covariates.distr)

## U in [0,1]x[0,1] with copula parameter depending on X
U <- condBiCopSim(fam, function(x1, x2, x3) {
  eta0 + sum(mapply(function(f, x)
    f(x), calib.surf, c(x1, x2, x3)))
}, X[, 1:3], par2 = 6, return.par = TRUE)

## Merge U and X
data <- data.frame(U$data, X)
names(data) <- c(paste("u", 1:2, sep = ""), paste("x", 1:3, sep = ""))

## Display the data
dev.off()
#> null device 
#>           1 
plot(data[, "u1"], data[, "u2"], xlab = "U1", ylab = "U2")