joint.curvesCOP2: Compute Coordinates of the Marginal Probabilities given joint AND or OR Probability

Description

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 $u$ with respect to $v$. 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 $V \mapsto [t_i, t_{j}, t_{j+1}, \cdots, 1; \Delta]$ (an irregular sequence of $v$ values from the $i$th value of $t_i$ provided through to unity) and thus $$t_i \mapsto \mathbf{C}(u, v=V)\mbox{,}$$ and solving for the sequence of $u$. 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.curvesCOP2 for level curves (Nelsen, 2006, pp. 12--13) but implemented in the function joint.curvesCOP2 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 $V \mapsto [0, v_j, v_{j+1}, \cdots, t_i; \Delta]$ (an irregular sequence of $v$ values from zero through to the $i$th value of $t$) and thus $$t_i \mapsto \tilde{\mathbf{C}}(u, v=V)\mbox{,}$$ and solving for the sequence of $u$. 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.

Usage

joint.curvesCOP2(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, delv=0.001, ...)

Value

An R

list is returned with elements each of the given probs.

Arguments

cop: A copula function;
para: Vector of parameters or other data structure, if needed, to pass to the copula;
type: What type of joint probability is to be computed;
probs: The joint probabilities for which to compute the coordinates. The default values represent especially useful annual return period equivalents that are useful in hydrologic risk analyses;
zero2small: A logical controlling whether precise zero value for probability are converted to a small value and precise 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 (GRV; see function prob2grv() in package lmomco) is later desired by the user for attendant graphics (see Examples section);
small: The value for small described for zero2small;
divisor: A divisor on a computation of a $\Delta$ for incrementing through the $v$ domain as part of the coordinate computation (see source code);
delv: A $\Delta v$ for setup of the incrementing through the $v$ domain as part of the coordinate computation (see source code); and
...: Additional arguments to pass to the duCOP function of copBasic or uniroot() function.

Author

W.H. Asquith

References