LMRvuCOP: Conditional L-moments of the Distribution of V given U
In copBasic: General Bivariate Copula Theory and Many Utility Functions
LMRvuCOP R Documentation
Conditional L-moments of the Distribution of V given U
Description
Compute the conditional L-moments of V given a U (the X direction) through the conditional distribution function F(X) using the appropriate partial derivative of a copula (\mathbf{C}(u,v)) with respect to U. The conditional L-moments are computed for \lambda_1 (mean) from EvuCOP (refer to mathematics therein) and for r \ge 2 are computed by
\lambda_r = \int_0^1 F(k) \times (1-F(k)) \times L_r(F(k))\,\mathrm{d}k\mbox{,}
where F(k) becomes replaced in copBasic with the conditional distribution function of a copula by Nelsen (2006, pp. 13, 40–41) with respect to u:
0 \le \frac{\delta}{\delta u} \mathbf{C}(u,v) \le 1\mbox{,}
which is derCOP and L_r(u) is defined (Jones, 2004; Asquith, 2011) as
L_r(z) = \frac{1}{r-1}\sum_{j=0}^{r-2} (-1)^j {r-1 \choose j}{r-1 \choose j+1}z^{r-2-j}(1-z)^j\mbox{.}
Asquith (2011) mistakenly lists the leading multiplier 1/(1-r) in lieu of 1/(1-r) (correct) as shown above; the lmomco package contains a suitable errata as discovered in copBasic development for LMRvuCOP and LMRuvCOP.
Usage
LMRvuCOP(u=seq(0.01, 0.99, by=0.01), cop=NULL, para=NULL, nsim=1E5,
subdivisions=200L, rel.tol=.Machine$double.eps^0.25, abs.tol=rel.tol, ...)
Arguments
u
Nonexceedance probability u in the X direction;
cop
A copula function with vectorization as in asCOP;
para
Vector of parameters or other data structures, if needed, to pass to the copula;
nsim
Number of simulations for Monte Carlo integration if numerical integration were to fail;
subdivisions
Argument of same name passed to integrate();
rel.tol
Argument of same name passed to integrate();
abs.tol
Argument of same name passed to integrate(); and
...
Additional arguments to pass to derCOP.
Value
Value(s) in a list for the median and conditional L-moments given U=u are returned including
median
The conditional median from med.regressCOP;
L1
The conditional mean \lambda_1|U=u from EvuCOP;
L2
The conditional L-scale \lambda_2|U=u;
T3
The conditional L-skew \tau_3|U=u;
T4
The conditional L-kurtosis \tau_4|U=u;
Author(s)
W.H. Asquith
References
Asquith, W.H., 2011, Distributional analysis with L-moment statistics using the R environment for statistical computing: Createspace Independent Publishing Platform, ISBN 978–146350841–8.
Jones, M.C., 2004, On some expressions for variance, covariance, skewness and L-moments:
Journal of Statistical Planning and Inference, v.126, pp. 97–106, \Sexpr[results=rd]{tools:::Rd_expr_doi("10.1016/j.jspi.2003.09.001")}.
Nelsen, R.B., 2006, An introduction to copulas: New York, Springer, 269 p.
See Also
EvuCOP, LMRuvCOP, derCOP
Examples
## Not run:
para <- list(cop=breveCOP, para=list(cop=CLcop, para=c(10), breve=0.2, reflect=2))
uv <- simCOP(500, cop=COP, para=para, graphics=FALSE, set.seed=1)
ustar <- 0.96; set.seed(1)
v_special <- replicate(10000, simCOPv(ustar, cop=COP, para=para))
lmr <- lmomco::lmoms(v_special)
u <- v <- seq(0,1, by=0.001)
LMRvu <- LMRvuCOP(u, cop=COP, para=para)
# LMRuv <- LMRuvCOP(v, cop=COP, para=para)
# LMRvu <- LMRuv # swap here to make a quick study of U given V
# uv <- uv[,2:1] # swap here to make a quick study of U given V
# pdf("LMRvuCOP.pdf", height=7, width=7, useDingbats=FALSE)
par(xpd=NA, las=1, lend=1, col.axis="black", cex.axis=0.9,
mgp=c(2,0.6,0), mai=c(0.65,0.65,0.40,0.65))
plot( uv[,1], uv[,2], type="n", lwd=0.8, col="grey50", bty="n", xlim=c(0,1),
ylim=c(0,1), xlab="", ylab="", xaxs="i", yaxs="i", xaxt="n", yaxt="n")
lines(rep(ustar, 2), par()$usr[3:4], lty=4, lwd=1.2, col="seagreen4")
par(mgp=c(2,0.3,0), las=0, col.axis="seagreen4", cex.axis=0.8)
txt <- latex2exp::TeX(r'($u^\star$)')
axis(1, at=ustar, label=txt, lwd=0, lwd.ticks=1.2, tcl=-0.25, col.ticks="seagreen4")
axis(3, at=ustar, label=txt, lwd=0, lwd.ticks=1.2, tcl=-0.25, col.ticks="seagreen4")
axis(1, at=ustar, label=FALSE,lwd=0, lwd.ticks=1.2, tcl=+0.25, col.ticks="seagreen4")
axis(3, at=ustar, label=FALSE,lwd=0, lwd.ticks=1.2, tcl=+0.25, col.ticks="seagreen4")
par(mgp=c(2,0.6,0), las=1, col.axis="black", cex.axis=0.9)
points(uv[,1], uv[,2], lwd=0.9, pch=21, cex=0.8, col="wheat4", bg="wheat1")
axis(1, at=axTicks(1), label=TRUE, lwd=1, lwd.ticks=1, col.ticks="black")
axis(3, at=axTicks(1), label=FALSE, lwd=1, lwd.ticks=1, col.ticks="black")
tix <- seq(0.1, 0.9, by=0.2)
axis(1, at=tix, label=FALSE, lwd=0, lwd.ticks=1, col.ticks="black", tcl=-0.5)
axis(2, at=tix, label=FALSE, lwd=0, lwd.ticks=1, col.ticks="black", tcl=-0.5)
axis(3, at=tix, label=FALSE, lwd=0, lwd.ticks=1, col.ticks="black", tcl=-0.5)
tix <- seq(0.05, 0.95, by=0.05); tix <- tix[grep("5$", as.character(tix))]
axis(1, at=tix, label=FALSE, lwd=0, lwd.ticks=1, col.ticks="black", tcl=-0.2)
axis(2, at=tix, label=FALSE, lwd=0, lwd.ticks=1, col.ticks="black", tcl=-0.2)
axis(3, at=tix, label=FALSE, lwd=0, lwd.ticks=1, col.ticks="black", tcl=-0.2)
mtext("U, nonexceedance probability", side=1, line=1.6)
par(mgp=c(2,0.8,0))
axis(2, at=axTicks(2), label=TRUE, lwd=1, lwd.ticks=1, col.ticks="black")
par(las=0)
mtext("V, nonexceedance probability", side=2, line=2.1)
par(las=1)
lines(u, LMRvu$L1, lwd=2, col="black" ) # mean probability
lines(u, LMRvu$median, lwd=2, col="red1" ) # median probability
lines(u, LMRvu$L2, lwd=2, col="deepskyblue3") # conditional L-scale
points(ustar, mean(v_special), cex=1.7, pch=1, col="black" )
points(ustar, median(v_special), cex=1.7, pch=1, col="red1" )
points(ustar, lmr$lambdas[1], cex=1.3, pch=16, col="black" )
points(ustar, lmr$lambdas[2], cex=1.3, pch=16, col="deepskyblue3")
par(new=TRUE)
plot(u, LMRvu$T3, type="n", bty="n",xlim=c(0,1), ylim=c(-1, 1),
xlab="", ylab="", xaxs="i", yaxs="i", xaxt="n", yaxt="n")
par(col.axis="orchid3", las=0, lend=1); colo <- "orchid3"
mtext("Lskew and Lkurtosis, dimensionless", 4, line=2.15, col=colo)
par(las=1)
axis(4, at=axTicks(2), label=TRUE, lwd=2, lwd.ticks=1, col=colo, col.ticks=colo)
tix <- seq(-0.9, 0.9, by=0.1)
tix <- tix[grep("[05]$", as.character(tix), invert=TRUE)]
axis(4, at=tix, label=FALSE, lwd=0, lwd.ticks=1, tcl=-0.5, col.ticks=colo)
lines(u, LMRvu$T3, col=colo, lwd=2) # conditional L-skew
lines(u, LMRvu$T4, col=colo, lty=2, lwd=2) # conditional L-kurtosis
points(ustar, lmr$ratios[ 3], cex=1.2, pch=24, col=colo, bg="orchid1")
points(ustar, lmr$ratios[ 4], cex=1.2, pch=25, col=colo, bg="orchid1")
txt <- c(paste0("Conditional (Cond.) mean at ustar = ", ustar, ", EvuCOP(...)"),
"Cond. median at ustar, med.regressCOP(...)",
"Cond. Lscale (variation) at ustar, LMRvuCOP(...)",
"Cond. Lskew at ustar, LMRvuCOP(...)",
"Cond. Lkurtosis at ustar, LMRvuCOP(...)",
"Demonstration of n=500 sim. values for the copula",
"Cond. simulated (sim.) (n=10,000) sample mean, mean(...)",
"Cond. sim. sample median, median(...)",
"Cond. sim. sample mean, lmoms(...)$lambdas[1]",
"Cond. sim. sample Lscale, lmoms(...)$lambdas[2]",
"Cond. sim. sample Lskew, lmoms(...)$ratios[3]",
"Cond. sim. sample Lkurtosis, lmoms(...)$ratios[4]")
legend("topleft", txt, bty="n", cex=0.6, seg.len=4, y.intersp=1.3,
lwd=c(2,2,2,2,2, NA, rep(NA, 6)), lty=c(1,1,1,1,2,0,rep(0, 6)),
pch=c(rep(NA, 5), 21, 1, 1, 16, 16, 24, 25),
pt.lwd=c(rep(NA,5), 0.9, rep(1,6)),
pt.cex=c(rep(NA,5), 0.8, 1.7, 1.7, 1.3, 1.3, 1.2, 1.2),
pt.bg=c(rep(NA,5), "wheat1", rep(NA,4), "orchid1", "orchid1"),
col=c("black", "red1", "deepskyblue3", colo, colo,
"wheat4", "black", "red1", "black", "deepskyblue3", colo, colo))
par(xpd=NA, las=1, lend=1, col.axis="black", cex.axis=0.9,
mgp=c(2,0.6,0), mai=c(0.65,0.65,0.40,0.65))
# dev.off() #
## End(Not run)
copBasic documentation built on July 23, 2026, 1:07 a.m.
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| LMRvuCOP | R Documentation |
Conditional L-moments of the Distribution of V given U
Description
Compute the conditional L-moments of V given a U (the X direction) through the conditional distribution function F(X) using the appropriate partial derivative of a copula (\mathbf{C}(u,v)) with respect to U. The conditional L-moments are computed for \lambda_1 (mean) from EvuCOP (refer to mathematics therein) and for r \ge 2 are computed by
\lambda_r = \int_0^1 F(k) \times (1-F(k)) \times L_r(F(k))\,\mathrm{d}k\mbox{,}
where F(k) becomes replaced in copBasic with the conditional distribution function of a copula by Nelsen (2006, pp. 13, 40–41) with respect to u:
0 \le \frac{\delta}{\delta u} \mathbf{C}(u,v) \le 1\mbox{,}
which is derCOP and L_r(u) is defined (Jones, 2004; Asquith, 2011) as
L_r(z) = \frac{1}{r-1}\sum_{j=0}^{r-2} (-1)^j {r-1 \choose j}{r-1 \choose j+1}z^{r-2-j}(1-z)^j\mbox{.}
Asquith (2011) mistakenly lists the leading multiplier 1/(1-r) in lieu of 1/(1-r) (correct) as shown above; the lmomco package contains a suitable errata as discovered in copBasic development for LMRvuCOP and LMRuvCOP.
Usage
LMRvuCOP(u=seq(0.01, 0.99, by=0.01), cop=NULL, para=NULL, nsim=1E5,
subdivisions=200L, rel.tol=.Machine$double.eps^0.25, abs.tol=rel.tol, ...)
Arguments
u |
Nonexceedance probability |
cop |
A copula function with vectorization as in |
para |
Vector of parameters or other data structures, if needed, to pass to the copula; |
nsim |
Number of simulations for Monte Carlo integration if numerical integration were to fail; |
subdivisions |
Argument of same name passed to |
rel.tol |
Argument of same name passed to |
abs.tol |
Argument of same name passed to |
... |
Additional arguments to pass to |
Value
Value(s) in a list for the median and conditional L-moments given U=u are returned including
median |
The conditional median from |
L1 |
The conditional mean |
L2 |
The conditional L-scale |
T3 |
The conditional L-skew |
T4 |
The conditional L-kurtosis |
Author(s)
W.H. Asquith
References
Asquith, W.H., 2011, Distributional analysis with L-moment statistics using the R environment for statistical computing: Createspace Independent Publishing Platform, ISBN 978–146350841–8.
Jones, M.C., 2004, On some expressions for variance, covariance, skewness and L-moments: Journal of Statistical Planning and Inference, v.126, pp. 97–106, \Sexpr[results=rd]{tools:::Rd_expr_doi("10.1016/j.jspi.2003.09.001")}.
Nelsen, R.B., 2006, An introduction to copulas: New York, Springer, 269 p.
See Also
EvuCOP, LMRuvCOP, derCOP
Examples
## Not run:
para <- list(cop=breveCOP, para=list(cop=CLcop, para=c(10), breve=0.2, reflect=2))
uv <- simCOP(500, cop=COP, para=para, graphics=FALSE, set.seed=1)
ustar <- 0.96; set.seed(1)
v_special <- replicate(10000, simCOPv(ustar, cop=COP, para=para))
lmr <- lmomco::lmoms(v_special)
u <- v <- seq(0,1, by=0.001)
LMRvu <- LMRvuCOP(u, cop=COP, para=para)
# LMRuv <- LMRuvCOP(v, cop=COP, para=para)
# LMRvu <- LMRuv # swap here to make a quick study of U given V
# uv <- uv[,2:1] # swap here to make a quick study of U given V
# pdf("LMRvuCOP.pdf", height=7, width=7, useDingbats=FALSE)
par(xpd=NA, las=1, lend=1, col.axis="black", cex.axis=0.9,
mgp=c(2,0.6,0), mai=c(0.65,0.65,0.40,0.65))
plot( uv[,1], uv[,2], type="n", lwd=0.8, col="grey50", bty="n", xlim=c(0,1),
ylim=c(0,1), xlab="", ylab="", xaxs="i", yaxs="i", xaxt="n", yaxt="n")
lines(rep(ustar, 2), par()$usr[3:4], lty=4, lwd=1.2, col="seagreen4")
par(mgp=c(2,0.3,0), las=0, col.axis="seagreen4", cex.axis=0.8)
txt <- latex2exp::TeX(r'($u^\star$)')
axis(1, at=ustar, label=txt, lwd=0, lwd.ticks=1.2, tcl=-0.25, col.ticks="seagreen4")
axis(3, at=ustar, label=txt, lwd=0, lwd.ticks=1.2, tcl=-0.25, col.ticks="seagreen4")
axis(1, at=ustar, label=FALSE,lwd=0, lwd.ticks=1.2, tcl=+0.25, col.ticks="seagreen4")
axis(3, at=ustar, label=FALSE,lwd=0, lwd.ticks=1.2, tcl=+0.25, col.ticks="seagreen4")
par(mgp=c(2,0.6,0), las=1, col.axis="black", cex.axis=0.9)
points(uv[,1], uv[,2], lwd=0.9, pch=21, cex=0.8, col="wheat4", bg="wheat1")
axis(1, at=axTicks(1), label=TRUE, lwd=1, lwd.ticks=1, col.ticks="black")
axis(3, at=axTicks(1), label=FALSE, lwd=1, lwd.ticks=1, col.ticks="black")
tix <- seq(0.1, 0.9, by=0.2)
axis(1, at=tix, label=FALSE, lwd=0, lwd.ticks=1, col.ticks="black", tcl=-0.5)
axis(2, at=tix, label=FALSE, lwd=0, lwd.ticks=1, col.ticks="black", tcl=-0.5)
axis(3, at=tix, label=FALSE, lwd=0, lwd.ticks=1, col.ticks="black", tcl=-0.5)
tix <- seq(0.05, 0.95, by=0.05); tix <- tix[grep("5$", as.character(tix))]
axis(1, at=tix, label=FALSE, lwd=0, lwd.ticks=1, col.ticks="black", tcl=-0.2)
axis(2, at=tix, label=FALSE, lwd=0, lwd.ticks=1, col.ticks="black", tcl=-0.2)
axis(3, at=tix, label=FALSE, lwd=0, lwd.ticks=1, col.ticks="black", tcl=-0.2)
mtext("U, nonexceedance probability", side=1, line=1.6)
par(mgp=c(2,0.8,0))
axis(2, at=axTicks(2), label=TRUE, lwd=1, lwd.ticks=1, col.ticks="black")
par(las=0)
mtext("V, nonexceedance probability", side=2, line=2.1)
par(las=1)
lines(u, LMRvu$L1, lwd=2, col="black" ) # mean probability
lines(u, LMRvu$median, lwd=2, col="red1" ) # median probability
lines(u, LMRvu$L2, lwd=2, col="deepskyblue3") # conditional L-scale
points(ustar, mean(v_special), cex=1.7, pch=1, col="black" )
points(ustar, median(v_special), cex=1.7, pch=1, col="red1" )
points(ustar, lmr$lambdas[1], cex=1.3, pch=16, col="black" )
points(ustar, lmr$lambdas[2], cex=1.3, pch=16, col="deepskyblue3")
par(new=TRUE)
plot(u, LMRvu$T3, type="n", bty="n",xlim=c(0,1), ylim=c(-1, 1),
xlab="", ylab="", xaxs="i", yaxs="i", xaxt="n", yaxt="n")
par(col.axis="orchid3", las=0, lend=1); colo <- "orchid3"
mtext("Lskew and Lkurtosis, dimensionless", 4, line=2.15, col=colo)
par(las=1)
axis(4, at=axTicks(2), label=TRUE, lwd=2, lwd.ticks=1, col=colo, col.ticks=colo)
tix <- seq(-0.9, 0.9, by=0.1)
tix <- tix[grep("[05]$", as.character(tix), invert=TRUE)]
axis(4, at=tix, label=FALSE, lwd=0, lwd.ticks=1, tcl=-0.5, col.ticks=colo)
lines(u, LMRvu$T3, col=colo, lwd=2) # conditional L-skew
lines(u, LMRvu$T4, col=colo, lty=2, lwd=2) # conditional L-kurtosis
points(ustar, lmr$ratios[ 3], cex=1.2, pch=24, col=colo, bg="orchid1")
points(ustar, lmr$ratios[ 4], cex=1.2, pch=25, col=colo, bg="orchid1")
txt <- c(paste0("Conditional (Cond.) mean at ustar = ", ustar, ", EvuCOP(...)"),
"Cond. median at ustar, med.regressCOP(...)",
"Cond. Lscale (variation) at ustar, LMRvuCOP(...)",
"Cond. Lskew at ustar, LMRvuCOP(...)",
"Cond. Lkurtosis at ustar, LMRvuCOP(...)",
"Demonstration of n=500 sim. values for the copula",
"Cond. simulated (sim.) (n=10,000) sample mean, mean(...)",
"Cond. sim. sample median, median(...)",
"Cond. sim. sample mean, lmoms(...)$lambdas[1]",
"Cond. sim. sample Lscale, lmoms(...)$lambdas[2]",
"Cond. sim. sample Lskew, lmoms(...)$ratios[3]",
"Cond. sim. sample Lkurtosis, lmoms(...)$ratios[4]")
legend("topleft", txt, bty="n", cex=0.6, seg.len=4, y.intersp=1.3,
lwd=c(2,2,2,2,2, NA, rep(NA, 6)), lty=c(1,1,1,1,2,0,rep(0, 6)),
pch=c(rep(NA, 5), 21, 1, 1, 16, 16, 24, 25),
pt.lwd=c(rep(NA,5), 0.9, rep(1,6)),
pt.cex=c(rep(NA,5), 0.8, 1.7, 1.7, 1.3, 1.3, 1.2, 1.2),
pt.bg=c(rep(NA,5), "wheat1", rep(NA,4), "orchid1", "orchid1"),
col=c("black", "red1", "deepskyblue3", colo, colo,
"wheat4", "black", "red1", "black", "deepskyblue3", colo, colo))
par(xpd=NA, las=1, lend=1, col.axis="black", cex.axis=0.9,
mgp=c(2,0.6,0), mai=c(0.65,0.65,0.40,0.65))
# dev.off() #
## End(Not run)
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