# Convert L-comoments to Parameters of Alpha-Beta Compositions of Two One-Parameter Copulas **EXPERIMENTAL**---This function converts the **L-comoments** of a bivariate sample to the four parameters of a composition of two one-parameter copulas. Critical inputs are of course the first three dimensionless L-comoments: **L-correlation**, **L-coskew**, and **L-cokurtosis**. The most complex input is the `solutionenvir`, which is an `environment` containing arbitrarily long, but individual tables, of L-comoment and parameter pairings. These pairings could be computed from the examples in `simcompositeCOP`. The individual tables are prescanned for potentially acceptable solutions and the absolute additive error of both L-comoments for a given order is controlled by the `tNeps` arguments. The default values seem acceptable. The purpose of the prescanning is to reduce the computation space from perhaps millions of solutions to a few orders of magnitude. The computation of the solution error can be further controlled by $X$ or $u$ with respect to $Y$ or $v$ using the `comptNerrXY` arguments, but experiments thus far indicate that the defaults are likely the most desired. A solution matching the L-correlation is always sought; thus there is no `uset2err` argument. The arguments `uset3err` and `uset4err` provide some level of granular control on addition error minimization; the defaults seek to match L-coskew and ignore L-cokurtosis. The `setreturn` controls which rank of computed solution is returned; users might want to manually inspect a few of the most favorable solutions, which can be done by the `setreturn` or inspection of the returned object from the `lcomoms2.cop2parameter` function. The examples are detailed and self-contained to the `copBasic` package; curious users are asked to test these. utf8 ```r lcomoms2.ABcop2parameter(solutionenvir=NULL, T2.12=NULL, T2.21=NULL, T3.12=NULL, T3.21=NULL, T4.12=NULL, T4.21=NULL, t2eps=0.1, t3eps=0.1, t4eps=0.1, compt2erruv=TRUE, compt2errvu=TRUE, compt3erruv=TRUE, compt3errvu=TRUE, compt4erruv=TRUE, compt4errvu=TRUE, uset3err=TRUE, uset4err=FALSE, setreturn=1, maxtokeep=1e5) ``` ## Arguments - `solutionenvir`: The environment containing solutions; - `T2.12`: L-correlation $\tau_2^{[12]}$; - `T2.21`: L-correlation $\tau_2^{[21]}$; - `T3.12`: L-coskew $\tau_3^{[12]}$; - `T3.21`: L-coskew $\tau_3^{[21]}$; - `T4.12`: L-cokurtosis $\tau_4^{[12]}$; - `T4.21`: L-cokurtosis $\tau_4^{[21]}$; - `t2eps`: An error term in which to pick a potential solution as close enough on preliminary processing for $\tau_2^{[1 \leftrightarrow 2]}$; - `t3eps`: An error term in which to pick a potential solution as close enough on preliminary processing for $\tau_3^{[1 \leftrightarrow 2]}$; - `t4eps`: An error term in which to pick a potential solution as close enough on preliminary processing for $\tau_4^{[1 \leftrightarrow 2]}$; - `compt2erruv`: Compute an L-correlation error using the 1 with respect to 2 (or $u$ wrt $v$); - `compt2errvu`: Compute an L-correlation error using the 2 with respect to 1 (or $v$ wrt $u$); - `compt3erruv`: Compute an L-coskew error using the 1 with respect to 2 (or $u$ wrt $v$); - `compt3errvu`: Compute an L-coskew error using the 2 with respect to 1 (or $v$ wrt $u$); - `compt4erruv`: Compute an L-cokurtosis error using the 1 with respect to 2 (or $u$ wrt $v$); - `compt4errvu`: Compute an L-cokurtosis error using the 2 with respect to 1 (or $v$ wrt $u$); - `uset3err`: Use the L-coskew error in the determination of the solution. The L-correlation error is always used; - `uset4err`: Use the L-cokurtosis error in the determination of the solution. The L-correlation error is always used; - `setreturn`: Set (index) number of the solution to return. The default of 1 returns the preferred solutions based on the controls for the minimization; and - `maxtokeep`: The value presets the number of rows in the solution matrix. This matrix is filled with potential solutions as the various subfiles of the `solutionenvir` are scanned. The matrix is trimmed of `NA`s and error trapping is in place for too small values of `maxtokeep`. The default value appears appropriate for the feeding of massively large simulated parameter spaces. ## Returns An `data.frame` is returned. ## 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. Salvadori, G., De Michele, C., Kottegoda, N.T., and Rosso, R., 2007, Extremes in Nature---An approach using copulas: Springer, 289 p. ## Author(s) W.H. Asquith ## See Also `simCOP`, `simcompositeCOP`, `composite2COP` ## Examples ```r ## Not run: # Build an initial parameter to L-comoment mapping table. mainpara <- list(cop1=PLACKETTcop, cop2=PLACKETTcop, para1gen=function() { return(10^runif(1, min=-5, max=0)) }, para2gen=function() { return(10^runif(1, min=0, max=5)) }) nsim <- 1E4 sample.size.for.estimation <- 1000 # really use vastly larger sample size PlackettPlackettNP <- simcompositeCOP(n=sample.size.for.estimation, nsim=nsim, parent=mainpara) save(PlackettPlackettNP, file="PlackettPlackettNP.RData", compress="xz") # Plackett-Plackett composited copula from the copBasic package # Then create an environment to house the "table." PlackettPlackett <- new.env() assign("NeedToCreateForDemo", PlackettPlackettNP, envir=PlackettPlackett) # Now that the table is assigned into the environment, the parameter # estimation function can be used. In reality, a much much larger # solution set is needed, but this effort is experimental. # Now grab the closest Plackett-Plackett solution having the following six # arbitrary L-comoments. Then simulate 1000 values and plot them to show # the underlying bivariate distribution. PPcop <- lcomoms2.ABcop2parameter(solutionenvir=PlackettPlackett, T2.12=-0.5059, T2.21=-0.5110, T3.12= 0.1500, T3.21= 0.1700, T4.12=-0.0500, T4.21= 0.0329, uset3err=TRUE, uset4err=TRUE) # A user in encouraged to inspect the contents of PPcop to "assess" the # solution by a method of L-comoments, we will now proceed with showing the # copula via a simulation of the fitted version. para <- list(cop1=PLACKETTcop, cop2=PLACKETTcop, alpha=PPcop$alpha, beta=PPcop$beta, para1=PPcop$Cop1Thetas, para2=PPcop$Cop2Thetas) D <- simCOP(n=5000, cop=composite2COP, para=para, col=rgb(0,0,0,0.1), pch=16) # The sample L-comoments of the fitted Plackett-Plackett may be found by lmomco::lcomoms2(D, nmom=4) # from the lmomco package, and six sample values shown T2.12 <- -0.5151547; T2.21 <- -0.5139863 T3.12 <- 0.1502336; T3.21 <- 0.1721355 T4.12 <- -0.0326277; T4.21 <- 0.0233979 PPcop <- lcomoms2.ABcop2parameter(solutionenvir=PlackettPlackett, T2.12=T2.12, T2.21=T2.21, T3.12=T3.12, T3.21=T3.21, T4.12=T4.12, T4.21=T4.21, uset4err=TRUE) para <- list(cop1=PLACKETTcop, cop2=PLACKETTcop, alpha=PPcop$alpha, beta=PPcop$beta, para1=PPcop$Cop1Thetas, para2=PPcop$Cop2Thetas) D <- simCOP(n=5000, cop=composite2COP, para=para, col=rgb(0,0,0,0.1), pch=16) level.curvesCOP(cop=composite2COP, para=para, delt=.1, ploton=FALSE) qua.regressCOP.draw(cop=composite2COP, para=para, ploton=FALSE, f=seq(0.05, 0.95, by=0.05)) qua.regressCOP.draw(cop=composite2COP, para=para, wrtV=TRUE, ploton=FALSE, f=seq(0.05, 0.95, by=0.05), col=c(3,2)) diag <- diagCOP(cop=composite2COP, para=para, ploton=FALSE, lwd=4) image(gridCOP(cop=composite2COP, para=para), col=terrain.colors(20)) # One can inspect alternative solutions like this # S <- PPcop$solutions$solutions[,1:16] # B <- S[abs(S$t2.12res) < 0.02 & abs(S$t2.21res) < 0.02 & # abs(S$t3.12res) < 0.02 & abs(S$t3.21res) < 0.02, ] #print(B) ## End(Not run) ```

lcomoms2.ABcop2parameter function