derCOPinv {copBasic}

Numerical Derivative Inverse of a Copula for V with respect to U

Description

Compute the inverse of a numerical partial derivative for V with respect to U of a copula, which is a conditional quantile function for nonexceedance probability t, or

t = c_u(v) = \mathbf{C}^{(-1)}_{2 \mid 1}(v \mid u) = \frac{\delta \mathbf{C}(u,v)}{\delta u}\mbox{,}

and solving for v. Nelsen (2006, pp. 13, 40–41) shows that this inverse is quite important for random variable generation using the conditional distribution method. This function is not vectorized and will not be so.

Usage

derCOPinv(cop=NULL, u, t, trace=FALSE,
          delu=.Machine$double.eps^0.50, para=NULL, ...)

Arguments

cop

A copula function;

u

A single nonexceedance probability u in the X direction;

t

A single nonexceedance probability level t;

trace

A logical controlling a message on whether the signs on the uniroot are the same—this is helpful in exploring the numerical derivative limits of a given implementation of a copula.

delu

The \Delta u interval for the derivative;

para

Vector of parameters or other data structures, if needed, to pass to cop; and

...

Additional arguments to pass to the copula.

Value

Value(s) for the derivative inverse are returned.

Note

AN EDUCATIONAL OPPORTUNITY—The Farlie-Gumbel-Morgenstern copula \mathbf{FGM}(u,v)
(FGMcop) (Joe, 2014, p. 213) is

\mathbf{FGM}(u,v; \Theta) = uv[1+\Theta(1-u)(1-v)]\mbox{,}

where -1 \le \Theta \le 1 has analytical solutions to the conditional cumulative distribution function (CDF) \mathbf{C}_{2 \mid 1}(v \mid u) as

\mathbf{C}_{2 \mid 1}(v \mid u) = v[1 + \Theta(1-v)(1-2u)]\mbox{,}

and the inverse of the conditional CDF as

\mathbf{C}_{2 \mid 1}(v \mid u) = \frac{[1 + \Theta(1-2u)] - \sqrt{[1+\Theta(1-2u)]^2 - 4t(1-2u)}}{2\Theta(1-2u)}\mbox{.}

These three functions for the copula can be defined in R by

  "FGMcop"       <- function(u,v, para=NULL, ...) u*v*(1 + para*(1-u)*(1-v)  )
  "joeFGMder"    <- function(u,v, para=NULL, ...)   v*(1 + para*(1-v)*(1-2*u))
  "joeFGMderinv" <- function(u,t, para=NULL, ...) {
      K <- (1-2*u)
      ((1 + para*K) - sqrt((1 + para*K)^2 - 4*t*K))/(2*para*K)
  }

The \mathbf{C}^{(-1)}_{2 \mid 1}(v \mid u) is critical for simulation by the conditional simulation method. Although exclusively for simulation, copBasic uses inversion of the numerical derivative, the \mathbf{FGM} copula has three representations of supposedly the same analytical algorithm for simulation in the literature (Durante, 2007; Johnson, 1987; Nelsen, 2006). An opportunity for comparison is thus available.

The three analytical algorithms for nonexceedance probability t given u by mathematics and code, following Durante (2007, p. 245), are

A = \Theta(1-2u) - 1\mbox{,}

B = \sqrt{A^2 - 4t(A+1)}\mbox{, and}

v = 2t/(B-A)\mbox{,}

and in R, this “Durante algorithm” is

  "durFGMderinv" <- function(u,t, para=NULL, ...) { # Durante (2007, p. 245)
      A <- para*(1-2*u) - 1; B <- sqrt(A^2 - 4*t*(A+1)); return(2*t/(B - A))
  }

and, letting K = (2u - 1), following Johnson (1987, p. 185)

A = K\Theta - 1

B = \sqrt{1 - 2K\Theta + (K\Theta)^2 + 4tK\Theta}

v = 2t/(B - A)

and in R, this “Johnson algorithm” is

  "jonFGMderinv" <- function(u,t, para=NULL, ...) { # Johnson (1987, p. 185)
      K <- (2*u - 1)
      A <- K*para - 1; B <- sqrt(1 - 2*K*para + (K*para)^2 + 4*t*K*para)
      2*t/(B - A)
  }

and finally following Nelsen (2006, p. 87)

A = 1 + \theta(1 - 2u)\mbox{,}

B = \sqrt{A^2 - 4t(A-1)}\mbox{, and}

v = 2t/(B+A)\mbox{,}

and in R, this “Nelsen algorithm” is

  "nelFGMderinv" <- function(u,t, para=NULL, ...) { # Nelsen (2006, p. 87)
      A <- 1 + para*(1-2*u); B <- sqrt(A^2 - 4*t*(A-1)); return(2*t/(B + A))
  }

With appropriate code now available, two comparisons can be made in the following sections.

CONDITIONAL DISTRIBUTION FUNCTION—A comparison of the analytical \mathbf{FGM}(u,v) derivative shows that Joe's equation is congruent with the numerical derivative of copBasic:

  joeFGMder(0.8, 0.44, para=0.78)             # 0.3246848      (Joe, 2014)
  derCOP(   0.8, 0.44, para=0.78, cop=FGMcop) # 0.3246848      (copBasic )

and the result will be used in the computations that follow.

A comparison for t = 0.3246848 of the analytical inverse and the numerical optimization of the numerical derivative of copBasic is

  joeFGMderinv(0.8, 0.3246848, para=0.78)             # 0.5327603
  derCOPinv(   0.8, 0.3246848, para=0.78, cop=FGMcop) # 0.4399934 --> 0.44

where obviously, the two results are not in agreement—so something is amiss. Because many examples in this documentation clearly demonstrate numerical reliability, a tentative conclusion is that Joe's listed equation must be in error. Let us check this hypothesis against the three other sources:

  durFGMderinv(0.8, 0.3246848, para=0.78) # 0.2074546          (Durante, 2007)
  jonFGMderinv(0.8, 0.3246848, para=0.78) # 0.44               (Johnson, 1987)
  nelFGMderinv(0.8, 0.3246848, para=0.78) # 0.44               (Nelsen,  2006)

The result from Durante (2007) is different from both Joe (2014) and from copBasic. However, the Johnson (1987) and Nelsen (2006) versions are equivalent and congruent to copBasic with the distinctly different numerical methods of derCOPinv. These incongruent results demonstrate that care is needed when navigating the copula literature and the usefulness of the copBasic-style implementation of copula theory. In words, these computations show that the t \approx 32nd percentile of the \mathbf{FGM} copula given that the 80th percentile in U is about the 44th percentile of V.

Author(s)

W.H. Asquith

References

Durante, F., 2007, Families of copulas, Appendix C, in Salvadori, G., De Michele, C., Kottegoda, N.T., and Rosso, R., 2007, Extremes in Nature—An approach using copulas: Springer, 289 p.

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

Johnson, M.E., 1987, Multivariate statistical simulation: New York, John Wiley, 230 p.

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

Zhang, L., and Singh, V.P., 2019, Copulas and their applications in water resources engineering: Cambridge University Press, ISBN 978–1–108–47425–2.

See Also

derCOP

Examples

u <- runif(1); t <- runif(1)
derCOPinv(u,t, cop=W)   # perfect negative dependence
derCOPinv(u,t, cop=P)   # independence
derCOPinv(u,t, cop=M)   # perfect positive dependence
derCOPinv(u,t, cop=PSP) # a parameterless copula example
## Not run: 
# Simulate 500 values from product (independent) copula
plot(NA,NA, type="n", xlim=c(0,1), ylim=c(0,1), xlab="U", ylab="V")
for(i in 1:500) {
   u <- runif(1); t <- runif(1)
   points(u, derCOPinv(cop=P, u, t), cex=0.5, pch=16) # black dots
}
# Now simulate 500 from the Nelsen 4.2.12 copula.
for(i in 1:500) {
   u <- runif(1); t <- runif(1)
   points(u,derCOPinv(cop=N4212cop,para=9.3,u,t), cex=2, pch=16, col=2) # red dots
} #
## End(Not run)

## Not run: 
# Zhang and Singh (2019) exam. 3.23, p. 105
# show the application of the derivative inversion C2|1
# for u=0.6036 and t=0.6036 ---> v = 0.4719
derCOPinv( cop=CLcop, 0.6036, 0.4028, para=0.5) # 0.4719 for C2|1
derCOPinv2(cop=CLcop, 0.6036, 0.4028, para=0.5) # 0.4719 for C1|2
# and C2|1 and C1|2 are equal because the copula has permutation symmetry
isCOP.permsym(cop=CLcop, para=0.5) # TRUE
## End(Not run)