# The Sections or Derivative of the Sections of a Copula Compute the **copula sections** or the **(partial) derivatives of copula sections** of a copula (Nelsen, 2006, pp. 12--14). The **horizontal section** at $V=a$ (a constant) is [REMOVE_ME]$$ t \mapsto \mathbf{C}(t,a)\mbox{, and} [REMOVE_ME_2]$$ the **vertical section** at $U=a$ (a constant, with respect to $V$ or `wrtV=TRUE`) is [REMOVE_ME]$$ t \mapsto \mathbf{C}(a,t)\mbox{.} [REMOVE_ME_2]$$ The partial derivatives of the copula sections are **conditional cumulative distribution functions** (see `derCOP` and `derCOP2`). The derivatives are constrained as [REMOVE_ME]$$ 0 \le \frac{\delta}{\delta u}\mathbf{C}(u,v) \le 1\mbox{, and} [REMOVE_ME_2]$$ [REMOVE_ME]$$ 0 \le \frac{\delta}{\delta v}\mathbf{C}(u,v) \le 1\mbox{.} [REMOVE_ME_2]$$ utf8 ## Description Compute the **copula sections** or the **(partial) derivatives of copula sections** of a copula (Nelsen, 2006, pp. 12--14). The **horizontal section** at $V=a$ (a constant) is $$ t \mapsto \mathbf{C}(t,a)\mbox{, and} $$ the **vertical section** at $U=a$ (a constant, with respect to $V$ or `wrtV=TRUE`) is $$ t \mapsto \mathbf{C}(a,t)\mbox{.} $$ The partial derivatives of the copula sections are **conditional cumulative distribution functions** (see `derCOP` and `derCOP2`). The derivatives are constrained as $$ 0 \le \frac{\delta}{\delta u}\mathbf{C}(u,v) \le 1\mbox{, and} $$ $$ 0 \le \frac{\delta}{\delta v}\mathbf{C}(u,v) \le 1\mbox{.} $$ ```r sectionCOP(f, cop=NULL, para=NULL, wrtV=FALSE, dercop=FALSE, delt=0.005, ploton=TRUE, lines=TRUE, xlab="NONEXCEEDANCE PROBABILITY", ...) ``` ## Arguments - `f`: A single value of nonexceedance probability $u$ or $v$ along the horizontal $U$ axis or vertical $V$ axis of the unit square $\mathcal{I}^2$; - `cop`: A copula function; - `para`: Vector of parameters, if needed, to pass to the copula; - `wrtV`: A logical to toggle between with respect to $v$ or $u$ (default). The default provides the vertical section whereas the horizontal comes from `wrtV = TRUE`; - `dercop`: A logical that triggers the derivative of the section; - `delt`: The increment of the level curves to plot, defaults to 5-percent intervals; - `ploton`: A logical to toggle on the plot; - `lines`: Draw the lines of diagonal to the current device; - `xlab`: A label for the x-axis title passed to `plot()` in ; and - `...`: Additional arguments to pass to the `plot()` and `lines()` functions in . ## Returns An `list` is returned. - **t**: The nonexceedance probability along the section. The nomenclature $t$ mimics Nelsen (2006) and is **not** the same as the $u$ or $v$; - **seccop**: The section of the copula or its derivative; - **wrt**: A text string declaring what the setting for `wrtV` was; - **fvalue**: The provided value of nonexceedance probability; and - **isderivative**: A logical stating whether the derivative of the section is `seccop`. ## References Nelsen, R.B., 2006, An introduction to copulas: New York, Springer, 269 p. ## Author(s) W.H. Asquith ## See Also `COP`, `diagCOP` ## Examples ```r ## Not run: # EXAMPLE 1, plot the v=0.55 section and then u=0.55 section, which will overlay # the other because the PSP is a symmetrical copula tmp <- sectionCOP(0.55, cop=PSP, ylab="COPULA SECTIONS", lwd=5, col=2) tmp <- sectionCOP(0.55, cop=PSP, wrtV=TRUE, ploton=FALSE, lwd=2, col=3) # now add the v=0.85 section and the u=0.85, again overlay each other tmp <- sectionCOP(0.85, cop=PSP, ploton=FALSE, lwd=5, col=2, lty=2) tmp <- sectionCOP(0.85, cop=PSP, wrtV=TRUE, ploton=FALSE, lwd=2, col=3, lty=2)# ## End(Not run) ## Not run: # EXAMPLE 2, v=0.35 section and derivative (the conditional distribution) function tmp <- sectionCOP(0.35, cop=PSP, ylab="COPULA SECTIONS OR DERIV.", lwd=5, col=3) tmp <- sectionCOP(0.35, cop=PSP, dercop=TRUE, ploton=FALSE, col=3) # The thin green line represents the cumulative distribution function conditional # on u = 0.35 from the derCOP function. Then see Example 3 ## End(Not run) ## Not run: # EXAMPLE 3 (random selection commented out) #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)) para <- list(cop1=PLACKETTcop, cop2=PLACKETTcop, alpha=0.7, beta=0.22, para1=0.0155, para2=214.4) 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)) layout(matrix(1:2,byrow=TRUE)) D <- simCOP(n=1000, cop=composite2COP, para=para, cex=0.5, col=rgb(0,0,0,0.2), pch=16) mtext(paste(txts,collapse="")) #f <- c(runif(1),runif(1)) f <- c(0.2,0.9) # RED is the horizontal section and BLACK is the vertical section segments(f[1],0,f[1],1, col=2, lwd=2); segments(0,f[2],1,f[2], lwd=2) ftxt <- c("Sections (thick) and derivatives (thin) at ", f, " nonexceed. prob.") tmp <- sectionCOP(f[1],cop=composite2COP,para=para, col=2, lwd=4) tmp <- sectionCOP(f[1],cop=composite2COP,para=para, dercop=TRUE, ploton=FALSE, col=2) tmp <- sectionCOP(f[2],cop=composite2COP,para=para,wrtV=TRUE,ploton=FALSE,lwd=4) tmp <- sectionCOP(f[2],cop=composite2COP,para=para,wrtV=TRUE,ploton=FALSE,dercop=TRUE) mtext(paste(ftxt, collapse="")) # The thin lines are the CDFs conditional on the respective values of "f". Carefully # compare the point densities along and near the sections in the top plot to the # respective shapes of the CDFs in the bottom plot. If the bottom plot were rotated # 90 degrees clockwise and then reflected top to bottom, the conditional quantile # function QDF results. Reflection is needed because, by convention, QDFs are monotonic # increasing to right---functions derCOPinv() and derCOPinv2() provide the CDF inversion. ## End(Not run) ```

sectionCOP function