copBasic (version 2.1.5)

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 (derCOPinv). 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 by derCOPinv), 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 R data.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

# NOT RUN {
# 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")#
# }
# NOT RUN {
# }
# NOT RUN {
# 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 (wrtV=TRUE)
qua.regressCOP.draw(cop=composite2COP, para=para, ploton=FALSE)
qua.regressCOP.draw(cop=composite2COP, para=para, wrtV=TRUE, lty=2, ploton=FALSE)
mtext("Composition of Two Plackett Copulas and Quantile Regression")#
# }
# NOT RUN {
# }
# NOT RUN {
# 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, wrtV=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, wrtV=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, wrtV=TRUE, ploton=FALSE)
  level.curvesCOP(cop=composite2COP, para=para, ploton=FALSE)
}
dev.off() # done
# }