densityCOPplot: Contour Density Plot of a Copula

In wasquith/copBasic: General Bivariate Copula Theory and Many Utility Functions

Contour Density Plot of a Copula

Description

Generate a contour density plot after the advocation of Joe (2014, pp. 9–15). Such graphics are plots of scaled copula densities (c^\star(u,v), bivariate herein) that are copula densities scaled to the standard normal distribution \sim N(0,1) margins. Joe (2014) repeatedly emphasizes such plots in contrast to the uniform distribution \sim U(0,1) margins. Nelsen (2006) does not discuss such scaling but seemingly Nelsen's objectives for his book were somewhat different.

The density of copula \mathbf{C}(u,v) is numerically estimated by

c(u,v) = \bigl[\mathbf{C}(u_2,v_2) - \mathbf{C}(u_2,v_1) - \mathbf{C}(u_1,v_2) + \mathbf{C}(u_1,v_1)\bigr]\, /\, \bigl[\Delta(uv)\times\Delta(uv)\bigr]\mbox{,}

where c(u,v) \ge 0 (see Nelsen, 2006, p. 10; densityCOP). Given a numerically estimated quantity c^\star(u,v) = c(u,v)\times\phi(\Phi^{(-1)}(u))\times\phi(\Phi^{(-1)}(v)) for copula density c(u,v), a grid of the c^\star(u,v) values can be contoured by the contour() function in R. The density function of the N(0,1) is \phi(z) for standard normal variate z and the quantile function of the N(0,1) is \Phi^{(-1)}(t) for nonexceedance probability t.

A grid (matrix) of c(u,v) or c^\star(u,v) is defined for sequence of u and v probabilities for which each sequence has equal steps that are equal to \Delta(uv). This function has as focus on plotting of the contour lines of c^\star(u,v) but the R matrix of either c(u,v) or c^\star(u,v) can be requested on return. For either matrix, the colnames() and rownames() (the R functions) are set equal to the sequence of u and v, respectively. Neither the column or row names are set to the standard normal variates for the matrix of c^\star(u,v), the names remain in terms of nonexceedance probability.

For plotting and other uses of normal scores of data, Joe (2014, p. 245) advocates that one should use the plotting position formula u_i = (i-1/2)/n (Hazen plotting position) for normal scores z_i = \Phi^{-1}(u_i) in preference to i/(n+1) (Weibull plotting position) because n^{-1}\sum_{i=1}^{n} z^2_i is closer to unity. Other examples of Joe's advocation for the Hazen plotting positions are available (Joe, 2014, pp. 9, 17, 245, 247–248).

Usage

densityCOPplot(cop=NULL, para=NULL, deluv=0.002,
               getmatrix=c("none", "cdenzz", "cden"), n=0,
               ploton=TRUE, snv=TRUE, origins=TRUE,
               contour.col=1, contour.lwd=1.5, ...)

Arguments

cop	A copula function;
para	Vector of parameters or other data structure, if needed, to pass to the copula;
deluv	The change in the sequences \{u, v\} \mapsto \delta, \ldots, 1-\delta; \delta = \Delta(uv) probabilities;
getmatrix	A trigger on whether the density matrix is to be returned. The option cdenzz returns the density scaled by the two standard normal densities (c^\star(u,v)), whereas the option cden returns simply the copula density (c(u,v));
ploton	A logical to toggle on the plot;
snv	A logical to toggle standard normal variates for the axes;
origins	A logical to plot the origin lines, if and only if snv is true;
contour.col	The color of the contour lines, which corresponds to the col argument of the contour function in R;
contour.lwd	The width of the contour lines, which corresponds to the lwd argument of the contour function in R;
n	An optional sample size for which simulation of this many values from the copula will be made by simCOP and drawn; and
...	Additional arguments to pass to the copula function and to the contour function in R (e.g. to turn off labeling of contours add drawlabels=FALSE).

Value

This is a high-level function used for its side effects; an R matrix can be triggered, however, as a returned value.

Note

Joe (2014, p. 244) says “if [density] plots show multimodality, then multivariate techniques of mixture models, cluster analysis[,] and nonparametric functional data analysis might be more appropriate” than relatively straightforward parametric copula models.

Author(s)

W.H. Asquith

References

Joe, H., 2014, Dependence modeling with copulas: Boca Raton, CRC Press, 462 p.

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

See Also

simCOP, densityCOP

Examples

## Not run: 
# Joe (2014, p. 5) names rMTCJ = reflected Mardia-Takahasi-Cook-Johnson copula
"rMTCJ" <- function(u, v, para, ...) {
   u + v - 1 + ((1-u)^(-para) + (1-v)^(-para) - 1)^(-1/para)
} # Survival Copula ("reflected" in Joe's terms)
densityCOPplot(cop=rMTCJ, para=1.0760, n=9000, snv=TRUE)
# The density plot matches that shown by Joe (2014, p. 11, fig. 1.2, lower left plot)
# for a Spearman Rho equaling 0.5. Let us compute then Rho:
rhoCOP(cop=rMTCJ, para=1.076075) # 0.4999958

# Now let us get really wild with a composition with TWO modes!
# This example also proves that the grid orientation is consistent with respect
# to the orientation of the simulations.
para <- list(alpha=0.15, beta=0.90, kappa=0.06, gamma=0.96,
             cop1=GHcop, cop2=PLACKETTcop, para1=5.5, para2=0.07)
densityCOPplot(cop=composite2COP, para=para, n=9000)

# Now, let us hack back to a contour density plot with U(0,1) and not N(0,1) margins
# just so show that capability exists, but emphasis of densityCOPplot is clearly
# on Joe's advocation, because it does not have a default trigger to use U(0,1) margins.
set.seed(12)
H <- densityCOPplot(cop=PLACKETTcop, para=41.25, getmatrix="cdenzz", n=1000, snv=FALSE)
set.seed(12)
UV <- simCOP(cop=PLACKETTcop, para=41.25, n=1000, col=8, snv=FALSE)
U  <- as.numeric(colnames(H)); V <- as.numeric(rownames(H))
contour(x=U, y=V, z=t(H), lwd=1.5, cex=2, add=TRUE, col=2) #
## End(Not run)

## Not run: 
set.seed(12)
UV <- rCOP(400,  cop=PSP, pch=16, col=8, n=400)
CL <- mleCOP(UV, cop=CLcop,   interval=c(1  , 20))
JO <- mleCOP(UV, cop=JOcopB5, interval=c(0.1, 20))
PL <- mleCOP(UV, cop=PLcop,   interval=c(0.1, 20))

AICs <- c(CL$AIC, JO$AIC, PL$AIC)
names(AICs) <- c("Clayton", "Joe(B5)", "Plackett")
print(round(AICs, digits=2))
#  Clayton    Joe(B5)   Plackett
#  -156.77     -36.91    -118.38
# So, we conclude Clayton must be the best fit of the three.

plot(qnorm(UV[,1]), qnorm(UV[,2]), pch=16, col=8, cex=0.5,
        xlab="Standard normal variate of U", xlim=c(-3,3),
        ylab="Standard normal variate of V", ylim=c(-3,3))
densityCOPplot(cop=PSP, contour.col=grey(0.5), lty=2,
               contour.lwd=3.5, ploton=FALSE, snv=TRUE)
densityCOPplot(cop=CLcop,     para=CL$para,
               contour.col=2,   ploton=FALSE, snv=TRUE)
densityCOPplot(cop=JOcopB5,   para=JO$para,
               contour.col=3,   ploton=FALSE, snv=TRUE)
densityCOPplot(cop=PLcop,     para=PL$para,
               contour.col=4,   ploton=FALSE, snv=TRUE) #
## End(Not run)

wasquith/copBasic documentation built on Dec. 13, 2024, 6:39 p.m.