Compute W-ordinal sum of the summands (Klement et al., 2017) within \(\mathcal{I}^2\) into \(n\) partitions (possibly infinite) within \(\mathcal{I}^2\). 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
$$\mathbf{C}\bigl(u,v; \mathcal{J}_i, \mathbf{C}_i, \Theta_i, i \in 1,2,\cdots,n\bigr) = a_i + (b_i-a_i)\mathbf{C}_i\biggl(\frac{u-a_i}{b_i-a_i},\, \frac{v-1+b_i}{b_i-a_i}; \Theta_i\biggr)\ \mbox{for}\ (u,v) \in \mathcal{J}^2\mbox{,}$$
for points within the partitions, and for points otherwise outside the partitions the coupla is given by
$$\mathbf{C}\bigl(u,v; \mathcal{J}_i, \mathbf{C}_i, i \in 1,2,\cdots,n\bigr) = \mathbf{W}(u,v)\ \mathrm{for}\ (u,v) \ni \mathcal{J}^2\mbox{, and}$$
let \(\mathbf{C}_\mathcal{J}(u,v)\) be a convenient abbreviation for the copula. (See ORDSUMcop, M-ordinal sum of the summands.)
ORDSUWcop(u,v, para=list(cop=M, para=NA, part=c(0,1)), ...)Value(s) for the copula are returned.
Klement, E.P., Kolesárová, 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., Sánchez, J.F.: Springer, Cham, Switzerland, ISBN 978--3--319--64220--9, tools:::Rd_expr_doi("10.1007/978-3-319-64221-5").
copBasic-package, W_N5p12a, ORDSUMcop