# Quantile Regression of the Grid of the Bivariate Empirical Copula for V with respect to U Perform **quantile regression** from the gridded inversion of the **bivariate empirical copula** of $V$ with respect to $U$. utf8 ```r EMPIRqua.regress(f=0.5, u=seq(0.01,0.99, by=0.01), empinv=NULL, lowess=FALSE, f.lowess=1/5, ...) ``` ## Arguments - `f`: The nonexceedance probability $F$ to perform regression at and defaults to median regression $F=1/2$; - `u`: A vector of $u$ nonexceedance probabilities; - `empinv`: The grid from `EMPIRgridderinv`; - `lowess`: Perform `lowess` smooth on the quantile regression using the smooth factor of `f.lowess`; - `f.lowess`: Smooth factor of almost the same argument name fed to the `lowess()` function in ; and - `...`: Additional arguments to pass. ## Returns The gridded values of the quantile regression of $V$ with respect to $U$. ## Author(s) W.H. Asquith ## See Also `EMPIRgridderinv`, `EMPIRqua.regress2`, `EMPIRmed.regress`, `EMPIRmed.regress2` ## Examples ```r ## Not run: # EXAMPLE 1 theta <- 25 uv <- simCOP(n=1000, cop=PLACKETTcop, para=theta, ploton=FALSE, points=FALSE) fakeU <- lmomco::pp(uv[,1], sort=FALSE) fakeV <- lmomco::pp(uv[,2], sort=FALSE) uv <- data.frame(U=fakeU, V=fakeV) uv.grid <- EMPIRgrid(para=uv, deluv=0.05) # CPU hungry uv.inv1 <- EMPIRgridderinv(empgrid=uv.grid) plot(uv, pch=16, col=rgb(0,0,0,.1), xlim=c(0,1), ylim=c(0,1), xlab="U, NONEXCEEDANCE PROBABILITY", ylab="V, NONEXCEEDANCE PROBABILITY") lines(qua.regressCOP(f=0.5, cop=PLACKETTcop, para=theta), lwd=2) lines(qua.regressCOP(f=0.2, cop=PLACKETTcop, para=theta), lwd=2) lines(qua.regressCOP(f=0.7, cop=PLACKETTcop, para=theta), lwd=2) lines(qua.regressCOP(f=0.1, cop=PLACKETTcop, para=theta), lwd=2) lines(qua.regressCOP(f=0.9, cop=PLACKETTcop, para=theta), lwd=2) med.wrtu <- EMPIRqua.regress(f=0.5, empinv=uv.inv1, lowess=TRUE, f.lowess=0.1) lines(med.wrtu, col=2, lwd=4) qua.wrtu <- EMPIRqua.regress(f=0.2, empinv=uv.inv1, lowess=TRUE, f.lowess=0.1) lines(qua.wrtu, col=2, lwd=2, lty=2) qua.wrtu <- EMPIRqua.regress(f=0.7, empinv=uv.inv1, lowess=TRUE, f.lowess=0.1) lines(qua.wrtu, col=2, lwd=2, lty=2) qua.wrtu <- EMPIRqua.regress(f=0.1, empinv=uv.inv1, lowess=TRUE, f.lowess=0.1) lines(qua.wrtu, col=2, lwd=2, lty=4) qua.wrtu <- EMPIRqua.regress(f=0.9, empinv=uv.inv1, lowess=TRUE, f.lowess=0.1) lines(qua.wrtu, col=2, lwd=2, lty=4) library(quantreg) # Quantile Regression by quantreg U <- seq(0.01, 0.99, by=0.01) rqlm <- rq(V~U, data=uv, tau=0.1) rq.1 <- rqlm$coefficients[1] + rqlm$coefficients[2]*U rqlm <- rq(V~U, data=uv, tau=0.2) rq.2 <- rqlm$coefficients[1] + rqlm$coefficients[2]*U rqlm <- rq(V~U, data=uv, tau=0.5) rq.5 <- rqlm$coefficients[1] + rqlm$coefficients[2]*U rqlm <- rq(V~U, data=uv, tau=0.7) rq.7 <- rqlm$coefficients[1] + rqlm$coefficients[2]*U rqlm <- rq(V~U, data=uv, tau=0.9) rq.9 <- rqlm$coefficients[1] + rqlm$coefficients[2]*U lines(U, rq.1, col=4, lwd=2, lty=4) lines(U, rq.2, col=4, lwd=2, lty=2) lines(U, rq.5, col=4, lwd=4) lines(U, rq.7, col=4, lwd=2, lty=2) lines(U, rq.9, col=4, lwd=2, lty=4)# ## End(Not run) ## Not run: # EXAMPLE 2 # Start again with the PSP copula uv <- simCOP(n=10000, cop=PSP, ploton=FALSE, points=FALSE) fakeU <- lmomco::pp(uv[,1], sort=FALSE) fakeV <- lmomco::pp(uv[,2], sort=FALSE) uv <- data.frame(U=fakeU, V=fakeV) uv.grid <- EMPIRgrid(para=uv, deluv=0.05) # CPU hungry uv.inv1 <- EMPIRgridderinv(empgrid=uv.grid) plot(uv, pch=16, col=rgb(0,0,0,0.1), xlim=c(0,1), ylim=c(0,1), xlab="U, NONEXCEEDANCE PROBABILITY", ylab="V, NONEXCEEDANCE PROBABILITY") lines(qua.regressCOP(f=0.5, cop=PSP), lwd=2) lines(qua.regressCOP(f=0.2, cop=PSP), lwd=2) lines(qua.regressCOP(f=0.7, cop=PSP), lwd=2) lines(qua.regressCOP(f=0.1, cop=PSP), lwd=2) lines(qua.regressCOP(f=0.9, cop=PSP), lwd=2) med.wrtu <- EMPIRqua.regress(f=0.5, empinv=uv.inv1, lowess=TRUE, f.lowess=0.1) lines(med.wrtu, col=2, lwd=4) qua.wrtu <- EMPIRqua.regress(f=0.2, empinv=uv.inv1, lowess=TRUE, f.lowess=0.1) lines(qua.wrtu, col=2, lwd=2, lty=2) qua.wrtu <- EMPIRqua.regress(f=0.7, empinv=uv.inv1, lowess=TRUE, f.lowess=0.1) lines(qua.wrtu, col=2, lwd=2, lty=2) qua.wrtu <- EMPIRqua.regress(f=0.1, empinv=uv.inv1, lowess=TRUE, f.lowess=0.1) lines(qua.wrtu, col=2, lwd=2, lty=4) qua.wrtu <- EMPIRqua.regress(f=0.9, empinv=uv.inv1, lowess=TRUE, f.lowess=0.1) lines(qua.wrtu, col=2, lwd=2, lty=4) library(quantreg) # Quantile Regression by quantreg U <- seq(0.01, 0.99, by=0.01) rqlm <- rq(V~U, data=uv, tau=0.1) rq.1 <- rqlm$coefficients[1] + rqlm$coefficients[2]*U rqlm <- rq(V~U, data=uv, tau=0.2) rq.2 <- rqlm$coefficients[1] + rqlm$coefficients[2]*U rqlm <- rq(V~U, data=uv, tau=0.5) rq.5 <- rqlm$coefficients[1] + rqlm$coefficients[2]*U rqlm <- rq(V~U, data=uv, tau=0.7) rq.7 <- rqlm$coefficients[1] + rqlm$coefficients[2]*U rqlm <- rq(V~U, data=uv, tau=0.9) rq.9 <- rqlm$coefficients[1] + rqlm$coefficients[2]*U lines(U, rq.1, col=4, lwd=2, lty=4) lines(U, rq.2, col=4, lwd=2, lty=2) lines(U, rq.5, col=4, lwd=4) lines(U, rq.7, col=4, lwd=2, lty=2) lines(U, rq.9, col=4, lwd=2, lty=4)# ## End(Not run) ## Not run: # EXAMPLE 3 uv <- simCOP(n=10000, cop=PSP, ploton=FALSE, points=FALSE) fakeU <- lmomco::pp(uv[,1], sort=FALSE) fakeV <- lmomco::pp(uv[,2], sort=FALSE) uv <- data.frame(U=fakeU, V=fakeV) uv.grid <- EMPIRgrid(para=uv, deluv=0.1) # CPU hungry uv.inv1 <- EMPIRgridderinv(empgrid=uv.grid) uv.inv2 <- EMPIRgridderinv2(empgrid=uv.grid) plot(uv, pch=16, col=rgb(0,0,0,.1), xlim=c(0,1), ylim=c(0,1), xlab="U, NONEXCEEDANCE PROBABILITY", ylab="V, NONEXCEEDANCE PROBABILITY") lines(qua.regressCOP(f=0.5, cop=PSP), col=2) lines(qua.regressCOP(f=0.2, cop=PSP), col=2) lines(qua.regressCOP(f=0.7, cop=PSP), col=2) lines(qua.regressCOP(f=0.1, cop=PSP), col=2) lines(qua.regressCOP(f=0.9, cop=PSP), col=2) med.wrtu <- EMPIRqua.regress(f=0.5, empinv=uv.inv1) lines(med.wrtu, col=2, lwd=4) qua.wrtu <- EMPIRqua.regress(f=0.2, empinv=uv.inv1) lines(qua.wrtu, col=2, lwd=2, lty=2) qua.wrtu <- EMPIRqua.regress(f=0.7, empinv=uv.inv1) lines(qua.wrtu, col=2, lwd=2, lty=2) qua.wrtu <- EMPIRqua.regress(f=0.1, empinv=uv.inv1) lines(qua.wrtu, col=2, lwd=2, lty=4) qua.wrtu <- EMPIRqua.regress(f=0.9, empinv=uv.inv1) lines(qua.wrtu, col=2, lwd=2, lty=4) lines(qua.regressCOP2(f=0.5, cop=PSP), col=4) lines(qua.regressCOP2(f=0.2, cop=PSP), col=4) lines(qua.regressCOP2(f=0.7, cop=PSP), col=4) lines(qua.regressCOP2(f=0.1, cop=PSP), col=4) lines(qua.regressCOP2(f=0.9, cop=PSP), col=4) med.wrtv <- EMPIRqua.regress2(f=0.5, empinv=uv.inv2) lines(med.wrtv, col=4, lwd=4) qua.wrtv <- EMPIRqua.regress2(f=0.2, empinv=uv.inv2) lines(qua.wrtv, col=4, lwd=2, lty=2) qua.wrtv <- EMPIRqua.regress2(f=0.7, empinv=uv.inv2) lines(qua.wrtv, col=4, lwd=2, lty=2) qua.wrtv <- EMPIRqua.regress2(f=0.1, empinv=uv.inv2) lines(qua.wrtv, col=4, lwd=2, lty=4) qua.wrtv <- EMPIRqua.regress2(f=0.9, empinv=uv.inv2) lines(qua.wrtv, col=4, lwd=2, lty=4)# ## End(Not run) ## Not run: # EXAMPLE 4 # Now try a much more complex shape # lowess smoothing on quantile regression is possible, # see next example para <- list(alpha=0.15, beta=0.65, cop1=PLACKETTcop, cop2=PLACKETTcop, para1=0.005, para2=1000) uv <- simCOP(n=20000, cop=composite2COP, para=para) fakeU <- lmomco::pp(uv[,1], sort=FALSE) fakeV <- lmomco::pp(uv[,2], sort=FALSE) uv <- data.frame(U=fakeU, V=fakeV) uv.grid <- EMPIRgrid(para=uv, deluv=0.025) # CPU hungry uv.inv1 <- EMPIRgridderinv(empgrid=uv.grid) uv.inv2 <- EMPIRgridderinv2(empgrid=uv.grid) plot(uv, pch=16, col=rgb(0,0,0,0.1), xlim=c(0,1), ylim=c(0,1), xlab="U, NONEXCEEDANCE PROBABILITY", ylab="V, NONEXCEEDANCE PROBABILTIY") lines(qua.regressCOP(f=0.5, cop=composite2COP, para=para), col=2) lines(qua.regressCOP(f=0.2, cop=composite2COP, para=para), col=2) lines(qua.regressCOP(f=0.7, cop=composite2COP, para=para), col=2) lines(qua.regressCOP(f=0.1, cop=composite2COP, para=para), col=2) lines(qua.regressCOP(f=0.9, cop=composite2COP, para=para), col=2) med.wrtu <- EMPIRqua.regress(f=0.5, empinv=uv.inv1) lines(med.wrtu, col=2, lwd=4) qua.wrtu <- EMPIRqua.regress(f=0.2, empinv=uv.inv1) lines(qua.wrtu, col=2, lwd=2, lty=2) qua.wrtu <- EMPIRqua.regress(f=0.7, empinv=uv.inv1) lines(qua.wrtu, col=2, lwd=2, lty=2) qua.wrtu <- EMPIRqua.regress(f=0.1, empinv=uv.inv1) lines(qua.wrtu, col=2, lwd=2, lty=4) qua.wrtu <- EMPIRqua.regress(f=0.9, empinv=uv.inv1) lines(qua.wrtu, col=2, lwd=2, lty=4) lines(qua.regressCOP2(f=0.5, cop=composite2COP, para=para), col=4) lines(qua.regressCOP2(f=0.2, cop=composite2COP, para=para), col=4) lines(qua.regressCOP2(f=0.7, cop=composite2COP, para=para), col=4) lines(qua.regressCOP2(f=0.1, cop=composite2COP, para=para), col=4) lines(qua.regressCOP2(f=0.9, cop=composite2COP, para=para), col=4) med.wrtv <- EMPIRqua.regress2(f=0.5, empinv=uv.inv2) lines(med.wrtv, col=4, lwd=4) qua.wrtv <- EMPIRqua.regress2(f=0.2, empinv=uv.inv2) lines(qua.wrtv, col=4, lwd=2, lty=2) qua.wrtv <- EMPIRqua.regress2(f=0.7, empinv=uv.inv2) lines(qua.wrtv, col=4, lwd=2, lty=2) qua.wrtv <- EMPIRqua.regress2(f=0.1, empinv=uv.inv2) lines(qua.wrtv, col=4, lwd=2, lty=4) qua.wrtv <- EMPIRqua.regress2(f=0.9, empinv=uv.inv2) lines(qua.wrtv, col=4, lwd=2, lty=4)# ## End(Not run) ## Not run: # EXAMPLE 5 plot(uv, pch=16, col=rgb(0,0,0,.1), xlim=c(0,1), ylim=c(0,1), xlab="U, NONEXCEEDANCE PROBABILITY", ylab="V, NONEXCEEDANCE PROBABILITY") lines(qua.regressCOP(f=0.5, cop=composite2COP, para=para), col=2) lines(qua.regressCOP(f=0.2, cop=composite2COP, para=para), col=2) lines(qua.regressCOP(f=0.7, cop=composite2COP, para=para), col=2) lines(qua.regressCOP(f=0.1, cop=composite2COP, para=para), col=2) lines(qua.regressCOP(f=0.9, cop=composite2COP, para=para), col=2) med.wrtu <- EMPIRqua.regress(f=0.5, empinv=uv.inv1, lowess=TRUE) lines(med.wrtu, col=2, lwd=4) qua.wrtu <- EMPIRqua.regress(f=0.2, empinv=uv.inv1, lowess=TRUE) lines(qua.wrtu, col=2, lwd=2, lty=2) qua.wrtu <- EMPIRqua.regress(f=0.7, empinv=uv.inv1, lowess=TRUE) lines(qua.wrtu, col=2, lwd=2, lty=2) qua.wrtu <- EMPIRqua.regress(f=0.1, empinv=uv.inv1, lowess=TRUE) lines(qua.wrtu, col=2, lwd=2, lty=4) qua.wrtu <- EMPIRqua.regress(f=0.9, empinv=uv.inv1, lowess=TRUE) lines(qua.wrtu, col=2, lwd=2, lty=4) lines(qua.regressCOP2(f=0.5, cop=composite2COP, para=para), col=4) lines(qua.regressCOP2(f=0.2, cop=composite2COP, para=para), col=4) lines(qua.regressCOP2(f=0.7, cop=composite2COP, para=para), col=4) lines(qua.regressCOP2(f=0.1, cop=composite2COP, para=para), col=4) lines(qua.regressCOP2(f=0.9, cop=composite2COP, para=para), col=4) med.wrtv <- EMPIRqua.regress2(f=0.5, empinv=uv.inv2, lowess=TRUE) lines(med.wrtv, col=4, lwd=4) qua.wrtv <- EMPIRqua.regress2(f=0.2, empinv=uv.inv2, lowess=TRUE) lines(qua.wrtv, col=4, lwd=2, lty=2) qua.wrtv <- EMPIRqua.regress2(f=0.7, empinv=uv.inv2, lowess=TRUE) lines(qua.wrtv, col=4, lwd=2, lty=2) qua.wrtv <- EMPIRqua.regress2(f=0.1, empinv=uv.inv2, lowess=TRUE) lines(qua.wrtv, col=4, lwd=2, lty=4) qua.wrtv <- EMPIRqua.regress2(f=0.9, empinv=uv.inv2, lowess=TRUE) lines(qua.wrtv, col=4, lwd=2, lty=4)# ## End(Not run) ## Not run: # EXAMPLE 6 (changing the smoothing on the lowess) plot(uv, pch=16, col=rgb(0,0,0,0.1), xlim=c(0,1), ylim=c(0,1), xlab="U, NONEXCEEDANCE PROBABILITY", ylab="V, NONEXCEEDANCE PROBABILTIY") lines(qua.regressCOP(f=0.5, cop=composite2COP, para=para), col=2) lines(qua.regressCOP(f=0.2, cop=composite2COP, para=para), col=2) lines(qua.regressCOP(f=0.7, cop=composite2COP, para=para), col=2) lines(qua.regressCOP(f=0.1, cop=composite2COP, para=para), col=2) lines(qua.regressCOP(f=0.9, cop=composite2COP, para=para), col=2) med.wrtu <- EMPIRqua.regress(f=0.5, empinv=uv.inv1, lowess=TRUE, f.lowess=0.1) lines(med.wrtu, col=2, lwd=4) qua.wrtu <- EMPIRqua.regress(f=0.2, empinv=uv.inv1, lowess=TRUE, f.lowess=0.1) lines(qua.wrtu, col=2, lwd=2, lty=2) qua.wrtu <- EMPIRqua.regress(f=0.7, empinv=uv.inv1, lowess=TRUE, f.lowess=0.1) lines(qua.wrtu, col=2, lwd=2, lty=2) qua.wrtu <- EMPIRqua.regress(f=0.1, empinv=uv.inv1, lowess=TRUE, f.lowess=0.1) lines(qua.wrtu, col=2, lwd=2, lty=4) qua.wrtu <- EMPIRqua.regress(f=0.9, empinv=uv.inv1, lowess=TRUE, f.lowess=0.1) lines(qua.wrtu, col=2, lwd=2, lty=4) lines(qua.regressCOP2(f=0.5, cop=composite2COP, para=para), col=4) lines(qua.regressCOP2(f=0.2, cop=composite2COP, para=para), col=4) lines(qua.regressCOP2(f=0.7, cop=composite2COP, para=para), col=4) lines(qua.regressCOP2(f=0.1, cop=composite2COP, para=para), col=4) lines(qua.regressCOP2(f=0.9, cop=composite2COP, para=para), col=4) med.wrtv <- EMPIRqua.regress2(f=0.5, empinv=uv.inv2, lowess=TRUE, f.lowess=0.1) lines(med.wrtv, col=4, lwd=4) qua.wrtv <- EMPIRqua.regress2(f=0.2, empinv=uv.inv2, lowess=TRUE, f.lowess=0.1) lines(qua.wrtv, col=4, lwd=2, lty=2) qua.wrtv <- EMPIRqua.regress2(f=0.7, empinv=uv.inv2, lowess=TRUE, f.lowess=0.1) lines(qua.wrtv, col=4, lwd=2, lty=2) qua.wrtv <- EMPIRqua.regress2(f=0.1, empinv=uv.inv2, lowess=TRUE, f.lowess=0.1) lines(qua.wrtv, col=4, lwd=2, lty=4) qua.wrtv <- EMPIRqua.regress2(f=0.9, empinv=uv.inv2, lowess=TRUE, f.lowess=0.1) lines(qua.wrtv, col=4, lwd=2, lty=4)# ## End(Not run) ```

EMPIRqua.regress function