copBasic (version 2.1.4)

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 $$\mathrm{Pr}[U\le v,V\le v]=t=\mathbf{C}(u,v)\text{,}$$ where $V\mapsto [t_i,t_j,t_{j+1},\cdots ,1;\mathrm{\Delta }]$ (an irregular sequence of $v$ values from the $i$th value of $t_i$ provided through to unity) and thus $$t_i\mapsto \mathbf{C}(u,v=V)\text{,}$$ 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: $$\mathrm{Pr}[U\le v\mathrm{or}V\le v]=t=\tilde{\mathbf{C}}(u,v)=u+v-\mathbf{C}(u,v)\text{,}$$ where $V\mapsto [0,v_j,v_{j+1},\cdots ,t_i;\mathrm{\Delta }]$ (an irregular sequence of $v$ values from zero through to the $i$th value of $t$) and thus $$t_i\mapsto \tilde{\mathbf{C}}(u,v=V)\text{,}$$ 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, ...)

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 $\mathrm{\Delta }$ for incrementing through the $v$ domain as part of the coordinate computation (see source code);

delv

A $\mathrm{\Delta }v$ 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.

Value

An R list is returned with elements each of the given probs.

References

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

See Also

diagCOPatf, duCOP, jointCOP, joint.curvesCOP, level.curvesCOP2

Examples

# NOT RUN {
# Approach the joint curves from both "with respect two" perspectives---results same.
JCvwrtu <- joint.curvesCOP( cop=PSP, prob=0.98)$"0.98"
JCuwrtv <- joint.curvesCOP2(cop=PSP, prob=0.98)$"0.98"; lim <- c(2,5)
plot(qnorm(JCvwrtu$U), qnorm(JCvwrtu$V), type="l", lwd=6, col=8, xlim=lim, ylim=lim,
     xlab="STANDARD NORMAL VARIATE IN U", ylab="STANDARD NORMAL VARIATE IN V")
lines(qnorm(JCuwrtv$U), qnorm(JCuwrtv$V), col=2, lwd=2)
mtext("98th Joint Percentile Level Curve for PSP Copula")#
# }