# Ordinal Sums of M-Copula Compute **ordinal sums of a copula** (Nelsen, 2006, p. 63) or **M-ordinal sum of the summands** (Klement **et al.**, 2017) within $\mathcal{I}^2$ into $n$ partitions (possibly infinite) within $\mathcal{I}^2$. According to Nelsen, letting $\mathcal{J}$ denote a **partition** of $\mathcal{I}^2$ and $\mathcal{J}_i = [a_i,\, b_i]$ be the $i$th partition that does not overlap with others and letting also $\mathbf{C}_i$ be a copula for the $i$th partition, then the **ordinal sum** of these $\mathbf{C}_i$ with parameters $\Theta_i$ **with respect to** $\mathcal{J}_i$ is the copula $\mathbf{C}$ given by [REMOVE_ME]$$ \mathbf{C}\bigl(u,v; \mathcal{J}_i, \mathbf{C}_i, \Theta_i, i \in 1,2,\cdots,n\bigr) = a_i + (b_i-a_i)\mathbf{C}_i\biggl(\frac{u-a_i}{b_i-a_i},\, \frac{v-a_i}{b_i-a_i}; \Theta_i\biggr)\ \mbox{for}\ (u,v) \in \mathcal{J}^2\mbox{,} [REMOVE_ME_2]$$ for points within the partitions, and for points otherwise outside the partitions the coupla is given by [REMOVE_ME]$$ \mathbf{C}\bigl(u,v; \mathcal{J}_i, \mathbf{C}_i, i \in 1,2,\cdots,n\bigr) = \mathbf{M}(u,v)\ \mathrm{for}\ (u,v) \ni \mathcal{J}^2\mbox{, and} [REMOVE_ME_2]$$ let $\mathbf{C}_\mathcal{J}(u,v)$ be a convenient abbreviation for the copula. Finally, Nelsen (2006, theorem 3.2.1) states that a copula is an ordinal sum if and only if for a $t$ if $\mathbf{C}(t,t)=t$ for $t \in (0,1)$. The **diagonal of a coupla** can be useful for quick assessment (see Examples ) of this theorem. (See `ORDSUWcop`, **W-ordinal sum of the summands**.) utf8 ## Description Compute **ordinal sums of a copula** (Nelsen, 2006, p. 63) or **M-ordinal sum of the summands** (Klement **et al.**, 2017) within $\mathcal{I}^2$ into $n$ partitions (possibly infinite) within $\mathcal{I}^2$. According to Nelsen, letting $\mathcal{J}$ denote a **partition** of $\mathcal{I}^2$ and $\mathcal{J}_i = [a_i,\, b_i]$ be the $i$th partition that does not overlap with others and letting also $\mathbf{C}_i$ be a copula for the $i$th partition, then the **ordinal sum** of these $\mathbf{C}_i$ with parameters $\Theta_i$ **with respect to** $\mathcal{J}_i$ is the copula $\mathbf{C}$ given by $$ \mathbf{C}\bigl(u,v; \mathcal{J}_i, \mathbf{C}_i, \Theta_i, i \in 1,2,\cdots,n\bigr) = a_i + (b_i-a_i)\mathbf{C}_i\biggl(\frac{u-a_i}{b_i-a_i},\, \frac{v-a_i}{b_i-a_i}; \Theta_i\biggr)\ \mbox{for}\ (u,v) \in \mathcal{J}^2\mbox{,} $$ for points within the partitions, and for points otherwise outside the partitions the coupla is given by $$ \mathbf{C}\bigl(u,v; \mathcal{J}_i, \mathbf{C}_i, i \in 1,2,\cdots,n\bigr) = \mathbf{M}(u,v)\ \mathrm{for}\ (u,v) \ni \mathcal{J}^2\mbox{, and} $$ let $\mathbf{C}_\mathcal{J}(u,v)$ be a convenient abbreviation for the copula. Finally, Nelsen (2006, theorem 3.2.1) states that a copula is an ordinal sum if and only if for a $t$ if $\mathbf{C}(t,t)=t$ for $t \in (0,1)$. The **diagonal of a coupla** can be useful for quick assessment (see Examples ) of this theorem. (See `ORDSUWcop`, **W-ordinal sum of the summands**.) ```r ORDSUMcop(u,v, para=list(cop=W, para=NA, part=c(0,1)), ...) ``` ## Arguments - `u`: Nonexceedance probability $u$ in the $X$ direction; - `v`: Nonexceedance probability $v$ in the $Y$ direction; - `para`: A list of sublists for the coupla, parameters, and partitions (see Examples ) and some attempt for intelligent in-fill of `para` is made within the sources (the default `para` is an example for which `cop` and `para` elements are converted to lists). The user is responsible that `part` element properly canvases by end-point alignment all of $\mathcal{I}^2$; and - `...`: Additional arguments to pass. ## Returns Value(s) for the copula are returned. ## References Nelsen, R.B., 2006, An introduction to copulas: New York, Springer, 269 p. Klement, E.P., , A., Mesiar, R., Saminger-Platz, S., 2017, Copula constructions using ultramodularity (chap. 9) **in** Copulas and dependence models with applications---Contributions in honor of Roger B. Nelsen, **eds.** Flores, U.M., Amo Artero, E., Durante, F., , J.F.: Springer, Cham, Switzerland, ISBN 978--3--319--64220--9, tools:::Rd_expr_doi("10.1007/978-3-319-64221-5") . ## Author(s) W.H. Asquith ## See Also `copBasic-package`, `W_N5p12a`, `ORDSUWcop` ## Examples ```r ## Not run: para <- list(cop=c(CLcop, M, PLcop, GHcop), para=list(4, NA, 0.1, c(3,4)), part=list(c(0,0.25), c(0.25,0.35), c(0.35,0.85), c(0.85,1))) UV <- simCOP(n=100, cop=ORDSUMcop, para=para, ploton=FALSE) plot(c(0,1), c(0,1), xlab="U, NONEXCEEDANCE PROBABILITY", type="n", ylab="V, NONEXCEEDANCE PROBABILITY") for(k in seq_len(length(para$part))) { # to draw the partitions a <- para$part[[k]][1]; b <- para$part[[k]][2] polygon(c(a, b, b, a, a), c(a,a,b,b,a), lty=2, lwd=0.8, col="lightgreen") text((a+b)/2, (a+b)/2, k, cex=3, col="blue") # numbered by partition } points(UV, pch=21, cex=0.8, col=grey(0.1), bg="white") # ## End(Not run) ## Not run: para <- list(cop=c(GHcop), para=list(c(2,3)), # internally replicated part=list(c(0,0.2), c(0.2,0.3), c(0.3,0.5), c(0.5,0.7), c(0.7,1))) UV <- simCOP(n=100, cop=ORDSUMcop, para=para, ploton=FALSE) plot(c(0,1), c(0,1), xlab="U, NONEXCEEDANCE PROBABILITY", type="n", ylab="V, NONEXCEEDANCE PROBABILITY") for(k in seq_len(length(para$part))) { # to draw the partitions a <- para$part[[k]][1]; b <- para$part[[k]][2] polygon(c(a, b, b, a, a), c(a,a,b,b,a), lty=2, lwd=0.8, col="lightgreen") text((a+b)/2, (a+b)/2, k, cex=3, col="blue") # numbered by partition } points(UV, pch=21, cex=0.8, col=grey(0.1), bg="white") # ## End(Not run) ## Not run: # In this example, it is important that the delt is of the resolution # matching the edges of the partitions. para <- list(cop=P, para=list(NULL), part=list(c(0,0.257), c(0.257,0.358), c(0.358,1))) DI <- diagCOP(cop=ORDSUMcop, para=para, delt=0.001) if(sum(DI$diagcop == DI$t) >= 1) { message("The ORDSUMcop() operation is an ordinal sum if there exists\n", "a t=(0,1) exists such that C(t,t)=t by Nelsen (2006, theorem 3.2.1).") } abline(0,1, col="red") # ## End(Not run) ```

ORDSUMcop function