# L-comoments and Bivariate L-moments of a Copula Compute the **L-comoments** (Serfling and Xiao, 2007; Asquith, 2011) through the **bivariate L-moments (ratios)** ($\delta^{[\ldots]}_{k;\mathbf{C}}$) of a copula $\mathbf{C}(u,v; \Theta)$ The L-comoments include **L-correlation** (**Spearman Rho**), **L-coskew**, and **L-cokurtosis**. As described by Brahimi **et al.** (2015), the first four bivariate L-moments $\delta^{[12]}_k$ for random variable $X^{(1)}$ or $U$ with respect to (**wrt**) random variable $X^{(2)}$ or $V$ are defined as [REMOVE_ME]$$ \delta^{[12]}_{1;\mathbf{C}} = 2\int\!\!\int_{\mathcal{I}^2}\mathbf{C}(u,v)\,\mathrm{d}u\mathrm{d}v - \frac{1}{2}\mbox{,} [REMOVE_ME_2]$$ [REMOVE_ME]$$ \delta^{[12]}_{2;\mathbf{C}} = \int\!\!\int_{\mathcal{I}^2}(12v - 6)\mathbf{C}(u,v)\,\mathrm{d}u\mathrm{d}v - \frac{1}{2}\mbox{,} [REMOVE_ME_2]$$ [REMOVE_ME]$$ \delta^{[12]}_{3;\mathbf{C}} = \int\!\!\int_{\mathcal{I}^2}(60v^2 - 60v + 12)\mathbf{C}(u,v)\,\mathrm{d}u\mathrm{d}v - \frac{1}{2}\mbox{, and} [REMOVE_ME_2]$$ [REMOVE_ME]$$ \delta^{[12]}_{4;\mathbf{C}} = \int\!\!\int_{\mathcal{I}^2}(280v^3 - 420v^2 + 180v - 20)\mathbf{C}(u,v)\,\mathrm{d}u\mathrm{d}v - \frac{1}{2}\mbox{,} [REMOVE_ME_2]$$ where the bivariate L-moments are related to the L-comoment ratios by [REMOVE_ME]$$ 6\delta^{[12]}_k = \tau^{[12]}_{k+1}\mbox{\quad and \quad}6\delta^{[21]}_k = \tau^{[21]}_{k+1}\mbox{,} [REMOVE_ME_2]$$ where in otherwords, the third bivariate L-moment delta^{[12]}_3 is one sixth theL-cokurtosis tau^{[12]}_4. The first four bivariate L-moments yield the first five L-comoments. The terms and nomenclature are not easy and also the English grammar adjective ratios is not always consistent in the literature. The $\delta^{[\ldots]}_{k;\mathbf{C}}$ are ratios . The sample L-comoments are supported by the `lmomco` package, and in particular for the bivariate case, they are supported by the `lcomoms2()` function of that package. Similarly, the $\delta^{[21]}_k$ are computed by switching $u \rightarrow v$ in the polynomials within the above integrals multiplied to the copula in the system of equations with $u$. In general, $\delta^{[12]}_k \not= \delta^{[21]}_k$ for $k > 1$ unless in the case of **permutation symmetric** (`isCOP.permsym`) copulas. By theory, $\delta^{[12]}_1 = \delta^{[21]}_1 = \rho_\mathbf{C}/6$ where $\rho_\mathbf{C}$ is the **Spearman Rho** `rhoCOP`. The integral for $\delta^{[12]}_{4;\mathbf{C}}$ does not appear in Brahimi **et al.** (2015) but this and the other forms are verified in the Examples and discussion in Note . The four $k \in (1,2,3,4)$ for $U$ **wrt** $V$ and $V$ **wrt** $U$ comprise a full spectrum of system of seven (not eight) equations. One equation is lost because $\delta^{[12]}_1 = \delta^{[21]}_1$. Chine and Benatia (2017) describe **trimmed L-comoments** as the multivariate extensions of the univariate **trimmed L-moments** (Elamir and Seheult, 2003) that are implemented in `lmomco`. These are not yet implemented in `copBasic`. utf8 ## Description Compute the **L-comoments** (Serfling and Xiao, 2007; Asquith, 2011) through the **bivariate L-moments (ratios)** ($\delta^{[\ldots]}_{k;\mathbf{C}}$) of a copula $\mathbf{C}(u,v; \Theta)$ The L-comoments include **L-correlation** (**Spearman Rho**), **L-coskew**, and **L-cokurtosis**. As described by Brahimi **et al.** (2015), the first four bivariate L-moments $\delta^{[12]}_k$ for random variable $X^{(1)}$ or $U$ with respect to (**wrt**) random variable $X^{(2)}$ or $V$ are defined as $$ \delta^{[12]}_{1;\mathbf{C}} = 2\int\!\!\int_{\mathcal{I}^2}\mathbf{C}(u,v)\,\mathrm{d}u\mathrm{d}v - \frac{1}{2}\mbox{,} $$ $$ \delta^{[12]}_{2;\mathbf{C}} = \int\!\!\int_{\mathcal{I}^2}(12v - 6)\mathbf{C}(u,v)\,\mathrm{d}u\mathrm{d}v - \frac{1}{2}\mbox{,} $$ $$ \delta^{[12]}_{3;\mathbf{C}} = \int\!\!\int_{\mathcal{I}^2}(60v^2 - 60v + 12)\mathbf{C}(u,v)\,\mathrm{d}u\mathrm{d}v - \frac{1}{2}\mbox{, and} $$ $$ \delta^{[12]}_{4;\mathbf{C}} = \int\!\!\int_{\mathcal{I}^2}(280v^3 - 420v^2 + 180v - 20)\mathbf{C}(u,v)\,\mathrm{d}u\mathrm{d}v - \frac{1}{2}\mbox{,} $$ where the bivariate L-moments are related to the L-comoment ratios by $$ 6\delta^{[12]}_k = \tau^{[12]}_{k+1}\mbox{\quad and \quad}6\delta^{[21]}_k = \tau^{[21]}_{k+1}\mbox{,} $$ where in otherwords, the third bivariate L-moment delta^{[12]}_3 is one sixth theL-cokurtosis tau^{[12]}_4. The first four bivariate L-moments yield the first five L-comoments. The terms and nomenclature are not easy and also the English grammar adjective ratios is not always consistent in the literature. The $\delta^{[\ldots]}_{k;\mathbf{C}}$ are ratios . The sample L-comoments are supported by the `lmomco` package, and in particular for the bivariate case, they are supported by the `lcomoms2()` function of that package. Similarly, the $\delta^{[21]}_k$ are computed by switching $u \rightarrow v$ in the polynomials within the above integrals multiplied to the copula in the system of equations with $u$. In general, $\delta^{[12]}_k \not= \delta^{[21]}_k$ for $k > 1$ unless in the case of **permutation symmetric** (`isCOP.permsym`) copulas. By theory, $\delta^{[12]}_1 = \delta^{[21]}_1 = \rho_\mathbf{C}/6$ where $\rho_\mathbf{C}$ is the **Spearman Rho** `rhoCOP`. The integral for $\delta^{[12]}_{4;\mathbf{C}}$ does not appear in Brahimi **et al.** (2015) but this and the other forms are verified in the Examples and discussion in Note . The four $k \in (1,2,3,4)$ for $U$ **wrt** $V$ and $V$ **wrt** $U$ comprise a full spectrum of system of seven (not eight) equations. One equation is lost because $\delta^{[12]}_1 = \delta^{[21]}_1$. Chine and Benatia (2017) describe **trimmed L-comoments** as the multivariate extensions of the univariate **trimmed L-moments** (Elamir and Seheult, 2003) that are implemented in `lmomco`. These are not yet implemented in `copBasic`. ```r lcomCOP(cop=NULL, para=NULL, as.bilmoms=FALSE, orders=2:5, ...) ``` ## Arguments - `cop`: A copula function; - `para`: Vector of parameters or other data structure, if needed, to pass to the copula; - `as.bilmoms`: A logical to trigger return of the $\delta_k$ and the return vectors will be named differently; - `orders`: The orders of the L-comoments to return, which is internally adjusted if the argument `as.bilmoms` is set. There is no first order L-comoment and the first index on returned values is set to `NA` to remain index consistent with the `lmomco` package. An order greater than 5 is not supported; and - `...`: Additional arguments to pass to the `densityCOP` function. ## Returns An `list` of the L-comoments or bivariate L-moments is returned depending on `as.bilmoms` setting. - **bilmomUV**: The bivariate L-moments $\delta^{[12]}_k$ of $U$ with respect to $V$ for $k \in [1,2,3,4]$ if `orders` is `2:5` and there is no `NA` index as for the L-comoments; - **bilmomVU**: The bivariate L-moments $\delta^{[21]}_k$ of $V$ with respect to $U$ for $k \in [1,2,3,4]$ if `orders` is `2:5` and there is no `NA` index as for the L-comoments; - **lcomUV**: The L-comoments $\tau^{[12]}_k$ of $V$ with respect to $U$ for $k \in [2,3,4,5]$ if `orders` is `2:5` and index 1 is `NA`; and - **lcomVU**: The L-comoments $\tau^{[21]}_k$ of $V$ with respect to $U$ for $k \in [2,3,4,5]$ if `orders` is `2:5` and index 1 is `NA`. ## Note The documention here is highly parallel to `bilmoms` for which that function was developed some years before `lmomCOP` was developed in January 2019. Also, `bilmoms` is based on gridded or Monte Carlo integration, and `bilmoms` is to be considered deprecated . However, it is deliberate that related background and various algorithm testing are still documented in `bilmoms`. ## 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. Brahimi, B., Chebana, F., and Necir, A., 2015, Copula representation of bivariate L-moments---A new estimation method for multiparameter two-dimensional copula models: Statistics, v. 49, no. 3, pp. 497--521. Chine, Amel, and Benatia, Fatah, 2017, Bivariate copulas parameters estimation using the trimmed L-moments methods: Afrika Statistika, v. 12, no. 1, pp. 1185--1197. Elamir, E.A.H, and Seheult, A.H., 2003, Trimmed L-moments: Computational Statistics and Data Analysis, v. 43, p. 299--314. Nelsen, R.B., 2006, An introduction to copulas: New York, Springer, 269 p. Serfling, R., and Xiao, P., 2007, A contribution to multivariate L-moments---L-comoment matrices: Journal of Multivariate Analysis, v. 98, pp. 1765--1781. ## Author(s) W.H. Asquith ## See Also `bilmoms`, `lcomCOPpv`, `uvlmoms` ## Examples ```r ## Not run: para <- list(alpha=0.5, beta=0.93, para1=4.5, cop1=GLcop, cop2=PSP) copBasic:::lcomCOP(cop=composite2COP, para=para)$lcomUV[3] # Lcomom:T3[12]= +0.156 copBasic:::lcomCOP(cop=composite2COP, para=para)$lcomVU[3] # Lcomom:T3[21]= -0.0668 bilmoms(cop=composite2COP, n=10000, para=para, sobol=TRUE)$bilcomoms$T3 # Tau3[12]=+0.1566, Tau3[21]=-0.0655 # The numerical default Monte Carlo integration of bilmoms() # matches the numerical integration of lcomCOP albeit with a # substantially slower and less elegant means in bilmoms(). ## End(Not run) ## Not run: # The following Spearman Rho and L-coskew values are predicted for a monitoring # location of the relation between peak streamflow (V) and time into the # water year (U) where U and V are "U-statistics." site_srho <- 0.15536430 site_T3_12 <- 0.03866056 site_T3_21 <- -0.03090144 # Create an objective function for 3D optimization with some explicit intention that # transforms are used to keep the parameters in acceptable parameter space for the # alpha [0,1] and beta [0,1] and theta for the Plackett copula and the composite1COP # used to add two more parameters to the Plackett. ofunc <- function(par, srho=NA, T3_12=NA, T3_21=NA) { alpha <- pnorm(par[1]) # takes -Inf to +Inf ---> 0 to 1 # compositing domain beta <- pnorm(par[2]) # takes -Inf to +Inf ---> 0 to 1 # compositing domain theta <- exp(par[3]) # takes -Inf to +Inf ---> 0 to +Inf # Plackett domain lmr <- lcomCOP(cop=composite1COP, para=list(alpha=alpha, beta=beta, para1=theta, cop1=PLcop)) return((lmr$lcomUV[2] - srho )^2 + # look carefully, the 2, 3, 3 index (lmr$lcomUV[3] - T3_12)^2 + # use on the lmr list are correct, so do not (lmr$lcomVU[3] - T3_21)^2) # expect to see 1, 2, 3 or 2, 3, 4. } # initial parameter guess ('middle' [0.5] compositing and independence [1]) and # showing the transformations involved. para_init <- c(qnorm(0.5), qnorm(0.5), log(1)) rt <- optim(par=para_init, ofunc, srho=site_srho, T3_12=site_T3_12, T3_21=site_T3_21) lcom_para <- list(alpha=pnorm(rt$par[1]), beta=pnorm(rt$par[2]), para1=exp(rt$par[3]), cop1=PLcop) sUV <- simCOP(10000, cop=composite1COP, para=lcom_para, col=grey(0, 0.2), pch=16) # Now as an exercise, consider increasing site_srho or negating it. Consider # switching the signs on the L-coskews or increasing their magnitudes and study # the resulting simulation to develop a personal feeling for L-coskew meaning. # ## End(Not run) ```

lcomCOP function