copBasic (version 1.7.1)

qua.regressCOP: Perform Quantile Regression using a Copula by Numerical Derivative Method for V with respect to U

Description

Perform quantile regression (Nelsen, 2006, pp. 217--218) using a copula by numerical derivatives of the copula. If $X$ and $Y$ are random variables having quantile functions $x(F)$ and $y(G)$ and letting $y=\tilde{y}(x)$ denote a solution to $\mathrm{Pr}[Y \le y\mid X = x] = F$, where $F$ is a nonexceedance probability. Then the curve $y=\tilde{y}(x)$ is the quantile regression curve of $V$ or $Y$ with respect to $U$ or $X$, respectively. If $F=1/2$, then median regression is performed (med.regressCOP). Using copulas, the quantile regression is expressed as $$\mathrm{Pr}[Y \le y\mid X = x] = \mathrm{Pr}[V \le G(y) \mid U = F(x)] = \mathrm{Pr}[V \le v\mid U = v] = \frac{\delta \mathbf{C}(u,v)}{\delta u}\mbox{,}$$ where $v = G(y)$ and $u = F(x)$. The general algorithm is

1. Set$\delta \mathbf{C}(u,v)/\delta u = F$,
2. Solve the regression curve$v = \tilde{v}(u)$(accomplished byderCOPinv), and
3. Replace$u$by$x(u)$and$v$by$y(v)$.

The last step is optional as step two produces the regression in probability space, which might be desired, and step 3 actually transforms the probability regressions into the quantiles of the respective random variables.

Usage

qua.regressCOP(f=0.5, u=seq(0.01,0.99, by=0.01), cop=NULL, para=NULL, ...)

Arguments

f

A single value of nonexceedance probability $F$ to perform regression at and defaults to median regression $F=1/2$;

u

Nonexceedance probability $u$ in the $X$ direction;

cop

A copula function;

para

Vector of parameters or other data structure, if needed, to pass to the copula; and

...

Additional arguments to pass.

Value

• An Rdata.frame of the regressed probabilities of $V$ and provided $U=u$ values is returned.

References

Nelsen, R.B., 2006, An introduction to copulas: New York, Springer, 269 p.

See Also

med.regressCOP, derCOPinv, qua.regressCOP.draw

Examples

# Use a positively associated Plackett copula and perform quantile regression
theta <- 10
R <- qua.regressCOP(cop=PLACKETTcop, para=theta) # 50th percentile regression
plot(R$U,R$V, type="l", lwd=6, xlim=c(0,1), ylim=c(0,1), col=8)
lines(R$U,(1+(theta-1)*R$U)/(theta+1), col=4, lwd=1) # theoretical for Plackett, see
#                                                             (Nelsen, 2006, p. 218)
R <- qua.regressCOP(f=0.90, cop=PLACKETTcop,para=theta) # 90th-percentile regression
lines(R$U,R$V, col=2, lwd=2)
R <- qua.regressCOP(f=0.10, cop=PLACKETTcop,para=theta) # 10th-percentile regression
lines(R$U,R$V, col=3, lty=2)
mtext("Quantile Regression V wrt U for Plackett copula")
# Use a composite copula with two Placketts with compositing parameters alpha and beta.
para <- list(cop1=PLACKETTcop, cop2=PLACKETTcop,
             para1=0.04, para2=5, alpha=0.9, beta=0.6)
plot(c(0,1),c(0,1), type="n", lwd=3,
     xlab="U, NONEXCEEDANCE PROBABILITY", ylab="V, NONEXCEEDANCE PROBABILITY")
# Draw the regression of V on U and then U on V (swap=TRUE)
qua.regressCOP.draw(cop=composite2COP, para=para, ploton=FALSE)
qua.regressCOP.draw(cop=composite2COP, para=para, swap=TRUE, lty=2, ploton=FALSE)
mtext("Composition of Two Plackett Copulas and Quantile Regression")
# Use a composite copula with two Placketts with compositing parameters alpha and beta.
para <- list(cop1=PLACKETTcop,  cop2=PLACKETTcop,
             para1=0.34, para2=50, alpha=0.63, beta=0.47)
D <- simCOP(n=3000, cop=composite2COP, para=para, cex=0.5)
qua.regressCOP.draw(cop=composite2COP, para=para, ploton=FALSE)
qua.regressCOP.draw(cop=composite2COP, para=para, swap=TRUE, lty=2, ploton=FALSE)
level.curvesCOP(cop=composite2COP, para=para, ploton=FALSE)
mtext("Composition of Two Plackett Copulas, Level Curves, Quantile Regression")

para <- list(cop1=PLACKETTcop,  cop2=PLACKETTcop, # Note the singularity
             para1=0, para2=500, alpha=0.63, beta=0.47)
D <- simCOP(n=3000, cop=composite2COP, para=para, cex=0.5)
qua.regressCOP.draw(cop=composite2COP, para=para, ploton=FALSE)
qua.regressCOP.draw(cop=composite2COP, para=para, swap=TRUE, lty=2, ploton=FALSE)
level.curvesCOP(cop=composite2COP, para=para, ploton=FALSE)
mtext("Composition of Two Plackett Copulas, Level Curves, Quantile Regression")

pdf("quantile_regression_test.pdf")
for(i in 1:10) {
  para <- list(cop1=PLACKETTcop, cop2=PLACKETTcop, alpha=runif(1), beta=runif(1),
               para1=10^runif(1,min=-4,max=0), para2=10^runif(1,min=0,max=4))
  txts <- c("Alpha=",    round(para$alpha,    digits=4),
            "; Beta=",   round(para$beta,     digits=4),
            "; Theta1=", round(para$para1[1], digits=5),
            "; Theta2=", round(para$para2[1], digits=2))

  D <- simCOP(n=3000, cop=composite2COP, para=para, cex=0.5, col=3)
  mtext(paste(txts, collapse=""))
  qua.regressCOP.draw(f=c(seq(0.05, 0.95, by=0.05)),
                      cop=composite2COP, para=para, ploton=FALSE)
  qua.regressCOP.draw(f=c(seq(0.05, 0.95, by=0.05)),
                      cop=composite2COP, para=para, swap=TRUE, ploton=FALSE)
  level.curvesCOP(cop=composite2COP, para=para, ploton=FALSE)
}
dev.off()