# Simulate a Bivariate Empirical Copula For a Fixed Value of U **EXPERIMENTAL**---Perform a simulation on a **bivariate empirical copula** to extract the random variates $V$ from a given and fixed value for $u=$ constant. The purpose of this function is to return a simple vector of the $V$ simulations. This behavior is similar to `simCOPmicro` but differs from the general 2-D simulation implemented in the other functions: `EMPIRsim` and `simCOP`---these two functions generate `data.frame`s of simulated random variates $U$ and $V$ and optional graphics as well. For the usual situation in which $u$ is not a value aligned on the grid, then the bounding conditional quantile functions are solved for each of the $n$ simulations and the following interpolation is made by [REMOVE_ME]$$ v = \frac{v_1/w_1 + v_2/w_2}{1/w_1 + 1/w_2}\mbox{,} [REMOVE_ME_2]$$ which states that that the weighted mean is computed. The values $v_1$ and $v_2$ are ordinates of the conditional quantile function for the respective grid lines to the left and right of the $u$ value. The values $w_1$ $=$ $u - u^\mathrm{left}_\mathrm{grid}$ and $w_2$ $=$ $u^\mathrm{right}_\mathrm{grid} - u$. utf8 ## Description **EXPERIMENTAL**---Perform a simulation on a **bivariate empirical copula** to extract the random variates $V$ from a given and fixed value for $u=$ constant. The purpose of this function is to return a simple vector of the $V$ simulations. This behavior is similar to `simCOPmicro` but differs from the general 2-D simulation implemented in the other functions: `EMPIRsim` and `simCOP`---these two functions generate `data.frame`s of simulated random variates $U$ and $V$ and optional graphics as well. For the usual situation in which $u$ is not a value aligned on the grid, then the bounding conditional quantile functions are solved for each of the $n$ simulations and the following interpolation is made by $$ v = \frac{v_1/w_1 + v_2/w_2}{1/w_1 + 1/w_2}\mbox{,} $$ which states that that the weighted mean is computed. The values $v_1$ and $v_2$ are ordinates of the conditional quantile function for the respective grid lines to the left and right of the $u$ value. The values $w_1$ $=$ $u - u^\mathrm{left}_\mathrm{grid}$ and $w_2$ $=$ $u^\mathrm{right}_\mathrm{grid} - u$. ```r EMPIRsimv(u, n=1, empgrid=NULL, kumaraswamy=FALSE, ...) ``` ## Arguments - `u`: The fixed probability $u$ on which to perform conditional simulation for a sample of size $n$; - `n`: A sample size, default is 1; - `empgrid`: Gridded empirical copula from `EMPIRgrid`; - `kumaraswamy`: A logical to trigger Kumaraswamy distribution smoothing of the conditional quantile function that is passed to `EMPIRgridderinv`. The Kumaraswamy distribution is a distribution having support $[0,1]$ with an explicit quantile function and takes the place of a Beta distribution (see `lmomco` function `quakur()` for more details); and - `...`: Additional arguments to pass. ## Returns A vector of simulated $V$ values is returned. ## Author(s) W.H. Asquith ## See Also `EMPIRgrid`, `EMPIRsim` ## Examples ```r ## Not run: nsim <- 3000 para <- list(alpha=0.15, beta=0.65, cop1=PLACKETTcop, cop2=PLACKETTcop, para1=.005, para2=1000) set.seed(10) uv <- simCOP(n=nsim, cop=composite2COP, para=para, pch=16, col=rgb(0,0,0,0.2)) uv.grid <- EMPIRgrid(para=uv, deluv=.1) set.seed(1) V1 <- EMPIRsimv(u=0.6, n=nsim, empgrid=uv.grid) set.seed(1) V2 <- EMPIRsimv(u=0.6, n=nsim, empgrid=uv.grid, kumaraswamy=TRUE) plot(V1,V2) abline(0,1) invgrid1 <- EMPIRgridderinv(empgrid=uv.grid) invgrid2 <- EMPIRgridderinv(empgrid=uv.grid, kumaraswamy=TRUE) att <- attributes(invgrid2); kur <- att$kumaraswamy # Now draw random variates from the Kumaraswamy distribution using # rlmomco() and vec2par() provided by the lmomco package. set.seed(1) kurpar <- lmomco::vec2par(c(kur$Alpha[7], kur$Beta[7]), type="kur") Vsim <- lmomco::rlmomco(nsim, kurpar) print(summary(V1)) # Kumaraswamy not core in QDF reconstruction print(summary(V2)) # Kumaraswamy core in QDF reconstruction print(summary(Vsim)) # Kumaraswamy use of the kumaraswamy # Continuing with a conditional quantile 0.74 that will not land along one of the # grid lines, a weighted interpolation will be done. set.seed(1) # try not resetting the seed nsim <- 5000 V <- EMPIRsimv(u=0.74, n=nsim, empgrid=uv.grid) # It is important that the uv.grid used to make V is the same grid used in inversion # with kumaraswamy=TRUE to guarantee that the correct Kumaraswamy parameters are # available if a user is doing cut and paste and exploring these examples. set.seed(1) V1 <- lmomco::rlmomco(nsim, lmomco::vec2par(c(kur$Alpha[8], kur$Beta[8]), type="kur")) set.seed(1) V2 <- lmomco::rlmomco(nsim, lmomco::vec2par(c(kur$Alpha[9], kur$Beta[9]), type="kur")) plot( lmomco::pp(V), sort(V), type="l", lwd=4, col=8) # GREY is empirical from grid lines(lmomco::pp(V1), sort(V1), col=2, lwd=2) # Kumaraswamy at u=0.7 # RED lines(lmomco::pp(V2), sort(V2), col=3, lwd=2) # Kumaraswamy at u=0.8 # GREEN W1 <- 0.74 - 0.7; W2 <- 0.8 - 0.74 Vblend <- (V1/W1 + V2/W2) / sum(1/W1 + 1/W2) lines(lmomco::pp(Vblend), sort(Vblend), col=4, lwd=2) # BLUE LINE # Notice how the grey line and the blue diverge for about F < 0.1 and F > 0.9. # These are the limits of the grid spacing and linear interpolation within the # grid intervals is being used and not direct simulation from the Kumaraswamy. ## End(Not run) ```

EMPIRsimv function