Compute the coordinates of the bivariate marginal probabilities for variables \(U\) and \(V\) given selected probabilities levels \(t\) for a copula \(\mathbf{C}(u,v)\) for \(v\) with respect to \(u\). For the case of a joint and probability, symbolically the solution is
$$\mathrm{Pr}[U \le v,\ V \le v] = t = \mathbf{C}(u,v)\mbox{,}$$
where \(U \mapsto [t_i, u_{j}, u_{j+1}, \cdots, 1; \Delta t]\) (an irregular sequence of \(u\) values from the \(i\)th value of \(t_i\) provided through to unity) and thus
$$t_i \mapsto \mathbf{C}(u=U, v)\mbox{,}$$
and solving for the sequence of \(v\). The index \(j\) is to indicate that a separate loop is involved and is distinct from \(i\). The pairings \(\{u(t_i), v(t_i)\}\) for each \(t\) are packaged as an R data.frame. This operation is very similiar to the plotting capabilities in level.curvesCOP for level curves (Nelsen, 2006, pp. 12--13) but implemented in the function joint.curvesCOP for alternative utility.
For the case of a joint or probability, the dual of a copula (function) or \(\tilde{\mathbf{C}}(u,v)\) from a copula (Nelsen, 2006, pp. 33--34) is used and symbolically the solution is:
$$\mathrm{Pr}[U \le v \mathrm{\ or\ } V \le v] = t = \tilde{\mathbf{C}}(u,v) = u + v - \mathbf{C}(u,v)\mbox{,}$$
where \(U \mapsto [0, u_j, u_{j+1}, \cdots, t_i; \Delta t]\) (an irregular sequence of \(u\) values from zero through to the \(i\)th value of \(t\)) and thus
$$t_i \mapsto \tilde{\mathbf{C}}(u=U, v)\mbox{,}$$
and solving for the sequence of \(v\). The index \(j\) is to indicate that a separate loop is involved and is distinct from \(i\). The pairings \(\{u(t_i), v(t_i)\}\) for each \(t\) are packaged as an R data.frame.
joint.curvesCOP(cop=NULL, para=NULL, type=c("and", "or"),
probs=c(0.5, 0.8, 0.90, 0.96, 0.98, 0.99, 0.995, 0.998),
zero2small=TRUE, small=1E-6, divisor=100, delu=0.001, ...)An R
list is returned with elements each of the given probs.
A copula function;
Vector of parameters or other data structure, if needed, to pass to the copula;
What type of joint probability is to be computed;
The joint probabilities \(t_i\) from which to compute the coordinates. The default values represent especially useful annual return period equivalents that are useful in hydrologic risk analyses;
A logical controlling whether exactly zero value for probability are converted to a small value and exactly unity values for probability are converted to the value 1 - small; this logical is useful if transformation from probability space into standard normal variates or Gumbel reduced variates (see function prob2grv() in package lmomco) is later desired by the user for attendant graphics (see Examples section);
The value for small described for zero2small;
A divisor on a computation of a \(\Delta t\) for incrementing through the irregularly-spaced \(u\) domain as part of the coordinate computation (see source code);
A \(\Delta u\) for setup of the incrementing through the irregularly-space \(u\) domain as part of the coordinate computation (see source code); and
Additional arguments to pass to the duCOP function of copBasic or uniroot() function in R.
W.H. Asquith
Nelsen, R.B., 2006, An introduction to copulas: New York, Springer, 269 p.
diagCOPatf, duCOP, jointCOP, joint.curvesCOP2, level.curvesCOP