copBasic (version 1.7.1)

composite2COP: Composition of Two Copulas with Two Compositing Parameters

Description

The composition of two copulas (Salvadori et al., 2007, p. 266, Prop. C.3) provides for more sophisticated structures of dependence between variables than many single parameter copula can provide. Further, asymmetrical copulas are readily obtained from symmetrical copulas. Let $\mathbf{A}$ and $\mathbf{B}$ be copulas with respective parameters $\Theta_\mathbf{A}$ and $\Theta_\mathbf{B}$, then

$$\mathbf{C}_{\alpha,\beta}(u,v) = \mathbf{A}(u^\alpha, v^\beta) \cdot \mathbf{B}(u^{1-\alpha},v^{1-\beta})\mbox{,}$$

defines a family of copulas $\mathbf{C}_{\alpha,\beta; \Theta_\mathbf{A}, \Theta_\mathbf{B}}$ with two compositing parameters $\alpha,\beta \in \mathcal{I}:[0,1]$. In particular, if $\alpha = \beta = 1$, then $\mathbf{C}_{1,1} = \mathbf{A}$, and, if $\alpha = \beta = 0$, then $\mathbf{C}_{0,0} = \mathbf{B}$. For $\alpha \ne \beta$, the $\mathbf{C}_{\alpha,\beta}$ is in general asymmetric that is $\mathbf{C}(u,v) \ne \mathbf{C}(v,u)$ for some $(u,v) \in \mathcal{I}^2$.

It is important to stress that copulas $\mathbf{A}_{\Theta_A}$ and $\mathbf{B}_{\Theta_B}$ can be of different families and each parameterized accordingly by the values $\Theta_A$ and $\Theta_B$. This is an interesting feature in the context of building complex copulatic structures in pursuit of fitting asymmetric measures of dependency such as the L-comoments available in the lmomco package. The author does not know whether the copulas $\mathbf{A}$ and $\mathbf{B}$ need be symmetric as the reference makes no stated restriction to that effect. (Symmetry of the copula $\mathbf{C}$ is required for the situation that follows.)

It is possible to simplify the construction of an asymmetric copula for a single copula by the following. Let $\mathbf{C}(u,v)$ be a symmetric copula, $\mathbf{C} \ne \mathbf{\Pi}$ (for $\mathbf{\Pi}$ see P). A family of asymmetric copulas $\mathbf{C}_{\alpha,\beta}$ with two composition parameters $0 < \alpha,\beta < 1, \alpha \ne \beta$ that also includes $\mathbf{C}(u,v)$ as a limiting case and is given by

$$\mathbf{C}_{\alpha,\beta}(u,v) = u^\alpha v^\beta \cdot \mathbf{C}(u^{1-\alpha},v^{1-\beta})\mbox{.}$$

The composite2COP function is based on the more general result given in the former rather than the later mathematical definition to provide more flexibility. For simpler case of composition involving only one copula, the composite1COP function is available. A more complex (extended) composition in composite3COP is available.

Usage

composite2COP(u,v, para, ...)

Arguments

u

Nonexceedance probability $u$ in the $X$ direction;

v

Nonexceedance probability $v$ in the $Y$ direction;

para

A special parameter list (see Note); and

...

Additional arguments to pass to the copula.

Value

• Value(s) for the composited 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

COP, composite1COP, composite3COP

Examples

alpha <- 0.24; beta <- 0.23; Theta1 <- NA; Theta2 <- NA
# The W() and PSP() copulas do not take parameters, but show how the parameters (ThetaX)
# would be set should either or both of the copulas require parameters.
para <- list(alpha=alpha, beta=beta, cop1=W, cop2=PSP, para1=Theta1, para2=Theta2)
t <- composite2COP(0.4,0.6,para)
print(t)


# In this example, the N4212cop uses "3" as its parameter value.
para <- list(alpha=alpha, beta=beta, cop1=W, cop2=N4212cop, para1=Theta1, para2=3)
t <- composite2COP(0.4,0.6,para)
print(t)
# This example does a great job of showing a composited copula with a near singularity,
# but with leakage of chance to the upper left. The example is also critical because
# it shows that gridCOP is returning a matrix in the proper orientation relative to
# the level.curvesCOP and simCOP functions. Example is cross-ref'ed from gridCOP() docs.
layout(matrix(1:2,byrow=TRUE))
para <- list(alpha=0.5, beta=0.90, cop1=M, cop2=N4212cop, para1=NA, para2=1.4)
image(gridCOP(cop=composite2COP, para=para, delta=0.01), col=terrain.colors(30),
      xlab="U, NONEXCEEDANCE PROBABILITY", ylab="V, NONEXCEEDANCE PROBABILITY")
D <- simCOP(n=2000, cop=composite2COP, para=para, ploton=FALSE, pch=4, col=4, cex=0.75)
level.curvesCOP(cop=composite2COP, para=para, ploton=FALSE, delt = 0.05)
mtext("Theoretical composited copula, level curves, and simulation")

emp <- EMPIRgrid(para=D, deluv = 0.05) # CPU heavy
image(emp$empcop, col=terrain.colors(30) ) # orientation is correct!
# Depending on balance between sample size, deluv, delu, and delt, one or more:
# Error in uniroot(func, interval = c(0, 1), u = u, LHS = t, cop = cop,  :
#   f() values at end points not of opposite sign
# warnings might be triggered. This is particularly true because of the flat derivative
# above the near singularity in this example composited copula.
points(D$U, D$V, pch=4, col=4, cex=0.75)
level.curvesCOP(cop=EMPIRcop, para=D, ploton=FALSE, delu=0.02, delt = 0.1)
mtext("Empirical copula from n=2000 simulation")