copBasic (version 1.5.1)

EMPIRcop: The Bivariate Empirical Copula

Description

The Bivariate Empirical Copula for a bivariate sample of length $n$ is

$$\mathbf{C}_{n}(u,v) = \frac{1}{n}\sum_{i=1}^n \mathbf{1}\bigl(u_{obs} \le u, v_{obs} \le v \bigr)$$ where $u_{obs}$ and $v_{obs}$ are the estimated nonexceedance probabilities in unranked form as computed by Weibull plotting positions ($i/(n+1)$, for example) or other plotting position formula. The example shown in this documentation sufficiently clarifies this concept. The sort=FALSE of the function pp of the lmomco package is present just for the very operation needed here for $u_{obs}$ and $v_{obs}$.

Usage

EMPIRcop(u, v, para=NULL, ...)

Arguments

u

A nonexceedance probability in X direction,

v

A nonexceedance probability in Y direction,

para

A vector (single element) of parameters---the U-statistics of the data (see example), and

...

Additional arguments to pass.

Value

• The value for the copula is returned.

References

Salvadori, G., De Michele, C., Kottegoda, N.T., and Rosso, R., 2007, Extremes in Nature---An approach using copulas: Springer, 289 p.

See Also

simCOP, PSP, level.curvesCOP

Examples

psp <- simCOP(n=50, cop=PSP, ploton=FALSE, points=FALSE) *150;
# Pretend psp is real data, the use of *150 is to clearly get the
# probabilities from simCOP into some other arbitrary unit system.

# The sort=FALSE is critical in the following two calls
fakeU <- pp(psp[,1], sort=FALSE); # Weibull plotting position i/(n+1)
fakeV <- pp(psp[,2], sort=FALSE); # Weibull plotting position i/(n+1)

uv <- data.frame(U=fakeU, V=fakeV); # our U-statistics

# these next two values should be REAL close with n=1000
print(EMPIRcop(0.4,0.6,para=uv));
print(PSP(0.4,0.6));

level.curvesCOP(cop=PSP); # parametric, fast,  BLACK CURVES

# The level curves of the empirical copula consume lots of CPU.
# The next operation is slow, so change delu to a larger value
# from the default (0.02) works ok for a "modern" computer.
# RED CURVES
level.curvesCOP(cop=EMPIRcop, para=uv, delu=0.02, col=2, ploton=FALSE)

diagCOP(cop=EMPIRcop, para=uv)

# Experimental for author

# From R Graphics by Murrell (2005, p.112)
"trans3d" <- function(x,y,z, pmat) {
   tmat <- cbind(x,y,z,1)   return(tmat[,1:2] / tmat[,4])
}

the.grid <- EMPIRgrid(para=uv)
the.diag <- diagCOP(cop=EMPIRcop, para=uv, ploton=FALSE, lines=FALSE)

the.persp <- persp(the.grid$z, theta=-25, phi=20,
          xlab="U VARIABLE", ylab="V VARIABLE", zlab="COPULA C(u,v)")

the.trace <- trans3d(the.diag$t, the.diag$t, the.diag$diagcop, the.persp)
lines(the.trace, lwd=2, col=2)


the.persp <- persp(x=the.grid$x, y=the.grid$y, z=the.grid$z,
                   theta=-25, phi=20,
          xlab="U VARIABLE", ylab="V VARIABLE", zlab="COPULA C(u,v)")

the.trace <- trans3d(the.diag$t, the.diag$t, the.diag$diagcop, the.persp)
lines(the.trace, lwd=2, col=2)