copBasic (version 2.1.5)

COP: The Copula

Description

Compute the copula or joint distribution function through a copula as shown by Nelsen (2006, p. 18) is the joint probability

$$\mathrm{Pr}[U \le u, V \le v] = \mathbf{C}(u,v)\mbox{.}$$

The copula is an expression of the joint probability that both \(U \le u\) and \(V \le v\). A radially symmetric or permutation symmetric copula is one such that \(\mathbf{C}(u,v) = \mathbf{C}(v,u)\) otherwise the copula is asymmetric. A copula is a type of dependence function that permits straightforward characterization of dependence from independence. Joe (2014, p. 8) comments that “copula families are usually given as cdfs [cumulative distribution functions.]”

The copula inversions \(t = \mathbf{C}(u{=}U, v)\) or \(t = \mathbf{C}(u, v{=}V)\) for a given \(t\) and \(U\) or \(V\) are provided by COPinv and COPinv2, respectively. A copula exists in the domain of the unit square (\(\mathcal{I}^2 = [0, 1]\times [0,1]\)) and is a grounded function meaning that $$\mathbf{C}(u,0) = 0 = \mathbf{C}(0,v) \mbox{\ and\ thus\ } \mathbf{C}(0,0) = 0\mbox{, }$$ and other properties of a copula are that $$\mathbf{C}(u,1) = u \mbox{\ and\ } \mathbf{C}(1,v) = v\mbox{\ and}$$ $$\mathbf{C}(1,1) = 1\mbox{.}$$

Copulas can be combined with each other (convex2COP, composite1COP, composite2COP, and composite3COP) to form more complex and sophisticated dependence structures. Also copula multiplication---a special product of two copulas---yields another copula (see prod2COP).

Perhaps the one of the more useful features of this function is that in practical applications it can be used to take a copula formula and reflect or rotated it in fashions to attain association structures that the native definition of the copula can not acquire. The terminal demonstration in the Examples demonstrates this for the Raftery copula (RFcop).

Usage

COP(u, v, cop=NULL, para=NULL,
          reflect=c("cop", "surv", "acute", "grave",
                      "1",    "2",     "3",     "4"), ...)

Arguments

u

Nonexceedance probability \(u\) in the \(X\) direction;

v

Nonexceedance probability \(v\) in the \(Y\) direction;

cop

A copula function with vectorization as in asCOP;

para

Vector of parameters or other data structures, if needed, to pass to the copula;

reflect

The reflection of the copula form (see Note) and the default "cop" or "1" is the usual copula definition (also see simCOPmicro). The numbered values correspond, respectively, to the named values; and

...

Additional arguments to pass to the copula.

Value

Value(s) for the copula are returned.

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

coCOP, duCOP, surCOP, surfuncCOP

Examples

# NOT RUN {
u <- runif(1); v <- runif(1)
COP(cop=W,u,v); COP(cop=P,u,v); COP(cop=M,u,v); COP(cop=PSP,u,v)

FF <- 0.75 # 75th percentile, nonexceedance
GG <- 0.20 # 25th percentile, nonexceedance
bF <- 1 - FF; bG <- 1 - GG     # exceedance
# What is the probability that both X and Y are less than
# 75th and 20th percentiles, respectively?
COP(cop=P,FF,GG)    # 0.15
# What is the probability that both X and Y are greater than
# 75th and 20th percentiles, respectively?
surCOP(cop=P,bF,bG) # 0.20
# What is the probability that either X or Y are less than
# the 75th and 20th percentiles, respectively?
duCOP(cop=P,FF,GG)  # 0.8
# What is the probability that either X or Y are greater than
# the 75th and 20th percentiles, respectively?
coCOP(cop=P,bF,bG)  # 0.85

# Repeat for the PSP copula:
# What is the probability that both X and Y are less than
# 75th and 20th percentiles, respectively?
COP(cop=PSP,FF,GG)    # 0.1875
# What is the probability that both X and Y are greater than
# 75th and 20th percentiles, respectively?
surCOP(cop=PSP,bF,bG) # 0.2375
# What is the probability that either X or Y are less than
# the 75th and 20th percentiles, respectively?
duCOP(cop=PSP,FF,GG)  # 0.7625
# What is the probability that either X or Y are greater than
# the 75th and 20th percentiles, respectively?
coCOP(cop=PSP,bF,bG)  # 0.8125
# Both of these summations equal unity
   COP(cop=PSP, FF, GG) + coCOP(cop=PSP, bF, bG) # 1
surCOP(cop=PSP, bF, bG) + duCOP(cop=PSP, FF, GG) # 1

FF <- 0.99 # 99th percentile, nonexceedance
GG <- 0.50 # 50th percentile, nonexceedance
bF <- 1 - FF # nonexceedance
bG <- 1 - GG # nonexceedance
# What is the probability that both X and Y are less than
# 99th and 50th percentiles, respectively?
COP(cop=P,FF,GG)    # 0.495
# What is the probability that both X and Y are greater than
# 99th and 50th percentiles, respectively?
surCOP(cop=P,bF,bG) # 0.005
# What is the probability that either X or Y are less than
# the 99th and 50th percentiles, respectively?
duCOP(cop=P,FF,GG)  # 0.995
# What is the probability that either X or Y are greater than
# the 99th and 50th percentiles, respectively?
coCOP(cop=P,bF,bG)  # 0.505

# }
# NOT RUN {
# MAJOR EXAMPLE FOR QUICKLY MODIFYING INHERENT ASSOCIATION STRUCTURES
p <- .5 # Reasonable strong positive association for the Raftery copula
"RFcop1" <- function(u,v, para) COP(u,v, cop=RFcop, para=p, reflect="1")
"RFcop2" <- function(u,v, para) COP(u,v, cop=RFcop, para=p, reflect="2")
"RFcop3" <- function(u,v, para) COP(u,v, cop=RFcop, para=p, reflect="3")
"RFcop4" <- function(u,v, para) COP(u,v, cop=RFcop, para=p, reflect="4")

d <- 0.01 # Just to speed up the density plots a bit
densityCOPplot(RFcop1, para=p, contour.col=1, deluv=d) # the Raftery in the literature
densityCOPplot(RFcop2, para=p, contour.col=1, deluv=d, ploton=FALSE)
densityCOPplot(RFcop3, para=p, contour.col=1, deluv=d, ploton=FALSE)
densityCOPplot(RFcop4, para=p, contour.col=1, deluv=d, ploton=FALSE)
# Now some text into the converging tail to show the reflection used.
text(-2,-2, "reflect=1", col=2); text(+2,+2, "reflect=2", col=2)
text(+2,-2, "reflect=3", col=2); text(-2,+2, "reflect=4", col=2) #
# }