ORDSUMcop {copBasic}

Ordinal Sums of M-Copula

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 ith partition that does not overlap with others and letting also \mathbf{C}_i be a copula for the ith 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.)

Usage

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.

Value

Value(s) for the copula are returned.

Author(s)

W.H. Asquith

References

Nelsen, R.B., 2006, An introduction to copulas: New York, Springer, 269 p.

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, doi:10.1007/978-3-319-64221-5.

See Also

copBasic-package, W_N5p12a, ORDSUWcop

Examples

## 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)