| rhoCOP {copBasic} | R Documentation |
The Spearman Rho of a Copula
Description
Compute the measure of association known as the Spearman Rho \rho_\mathbf{C} of a copula according to Nelsen (2006, pp. 167–170, 189, 208) by
\rho_\mathbf{C} = 12\int\!\!\int_{\mathcal{I}^2} \mathbf{C}(u,v)\, \mathrm{d}u\mathrm{d}v - 3\mbox{,}
or
\rho_\mathbf{C} = 12\int\!\!\int_{\mathcal{I}^2} [\mathbf{C}(u,v) - uv]\, \mathrm{d}u\mathrm{d}v\mbox{,}
where the later equation is implemented by rhoCOP as the default method (method="default"). This equation, here having p = 1 and k_p(1) = 12, is generalized under hoefCOP. The absence of the 12 in the above equation makes it equal to the covariance defined by the Hoeffding Identity (Joe, 2014, p. 54):
\mathrm{cov}(U, V) = \int\!\!\int_{\mathcal{I}^2} [\mathbf{C}(u,v) - uv]\, \mathrm{d}u\mathrm{d}v\mbox{ or}
\mathrm{cov}(U, V) = \int\!\!\int_{\mathcal{I}^2} [\hat{\mathbf{C}}(u,v) - uv]\, \mathrm{d}u\mathrm{d}v\mbox{, which is}
\mathrm{cov}(U, V) = \int\!\!\int_{\mathcal{I}^2} [u+v-1+\mathbf{C}(1-u,1-v) - uv]\, \mathrm{d}u\mathrm{d}v\mbox{.}
Depending on copula family (Joe, 2014, pp. 56 and 267), the alternative formulation for \rho_\mathbf{C} could be used
\rho_\mathbf{C} = 3 - 12\int\!\!\int_{\mathcal{I}^2} u \frac{\delta\mathbf{C}(u,v)}{\delta u} \, \mathrm{d}u\mathrm{d}v = 3 - 12\int\!\!\int_{\mathcal{I}^2} v\frac{\delta\mathbf{C}(u,v)}{\delta v} \, \mathrm{d}u\mathrm{d}v\mbox{,}
where the first integral form corresponds to Joe (2014, eq. 248, p. 56) and is the method="joe21", and the second integral form is the method="joe12".
The integral
\int\!\!\int_{\mathcal{I}^2} \mathbf{C}(u,v)\,\mathrm{d}u\mathrm{d}v\mbox{,}
represents the “volume under the graph of the copula and over the unit square” (Nelsen, 2006, p. 170) and therefore \rho_\mathbf{C} is simple a rescaled volume under the copula. The second equation for \rho_\mathbf{C} expresses the “average distance” between the joint distribution and statistical independence \mathbf{\Pi} = uv. Nelsen (2006, pp. 175–176) shows that the following relation between \rho_\mathbf{C} and \tau_\mathbf{C} (tauCOP) exists
-1 \le 3\tau - 2\rho \le 1\mbox{.}
Usage
rhoCOP(cop=NULL, para=NULL, method=c("default", "joe21", "joe12"),
as.sample=FALSE, brute=FALSE, delta=0.002, ...)
Arguments
cop |
A copula function; |
para |
Vector of parameters or other data structure, if needed, to pass to the copula; |
method |
The form of integration used to compute (see above); |
as.sample |
A logical controlling whether an optional R |
brute |
Should brute force be used instead of two nested |
delta |
The |
... |
Additional arguments to pass. |
Value
The value for \rho_\mathbf{C} is returned.
Note
Technically, Nelsen (2006) also shows that these definitions are a form of call to a concordance function \mathcal{Q}(\mathbf{C}_1,\mathbf{C}_2) of two copulas that involve \mathbf{C}_1{=}\mathbf{C}(u,v) and \mathbf{C}_2{=}\mathbf{\Pi}. As such in order to keep rhoCOP a small function when brute=TRUE, \rho_\mathbf{C} is computed by a special call to tauCOP, which by itself and although titled for computation of Kendall Tau, does support the concordance function \mathcal{Q}(\mathbf{C}_1, \mathbf{C}_2) [see Nelsen (2006, pp. 158–159)] when given two different copulas and respective parameters as arguments. The well-known Pearson correlation coefficient equals Spearman rho value if random variables X and Y are both uniformly distributed on [0,1].
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
blomCOP, footCOP, giniCOP,
hoefCOP, tauCOP, wolfCOP,
joeskewCOP, uvlmoms
Examples
rhoCOP(cop=PSP) # 0.4784176
## Not run:
rhoCOP(cop=PSP, brute=TRUE) # 0.4684063
# CPU heavy example showing that the dual-integration (fast) results in
# a Spearman Rho that mimics a sample version
do_rho <- function(n) {
uv <- simCOP(n=n, cop=PSP, ploton=FALSE, points=FALSE)
return(cor(uv$U, uv$V, method="spearman"))
}
rhos <- replicate(100, do_rho(1000))
rho_sample <- mean(rhos); print(rho_sample) # 0.472661
## End(Not run)
## Not run:
para <- list(cop1=PLACKETTcop, cop2=PLACKETTcop,
para1=0.00395, para2=4.67, alpha=0.9392, beta=0.5699)
rhoCOP(cop=composite2COP, para=para) # -0.5924796
para <- list(cop1=PLACKETTcop, cop2=PLACKETTcop,
para1=0.14147, para2=20.96, alpha=0.0411, beta=0.6873)
rhoCOP(cop=composite2COP, para=para) # 0.2818874
para <- list(cop1=PLACKETTcop, cop2=PLACKETTcop,
para1=0.10137, para2=4492.87, alpha=0.0063, beta=0.0167)
rhoCOP(cop=composite2COP, para=para) # 0.9812919
rhoCOP(cop=composite2COP, para=para, brute=TRUE) # 0.9752155
## End(Not run)
## Not run:
# This is the same composited copula used in a highly asymmetric multi-modal
# plotting example under densityCOPplot(). Let us use that copula as a means to
# check on the Spearman Rho from the alternative formulations from Joe (2014).
para <- list(alpha=0.15, beta=0.90, kappa=0.06, gamma=0.96,
cop1=GHcop, cop2=PLACKETTcop, para1=5.5, para2=0.07)
"rhoCOPbyJoe21" <- function(cop=NULL, para=NULL, ...) { # Joe (2014, eq. 2.48)
myint <- NULL
try(myint <- integrate(function(u) {
sapply(u,function(u) { integrate(function(v) {
u * derCOP( u, v, cop=cop, para=para, ...)}, 0, 1)$value })}, 0, 1))
ifelse(is.null(myint), return(NA), return(3 - 12*myint$value))
}
"rhoCOPbyJoe12" <- function(cop=NULL, para=NULL, ...) { # Not in Joe (2014)
myint <- NULL
try(myint <- integrate(function(u) {
sapply(u,function(u) { integrate(function(v) {
v * derCOP2( u, v, cop=cop, para=para, ...)}, 0, 1)$value })}, 0, 1))
ifelse(is.null(myint), return(NA), return(3 - 12*myint$value))
}
rhoCOP( cop=composite2COP, para=para) # 0.1031758
rhoCOPbyJoe21(cop=composite2COP, para=para) # 0.1031803
rhoCOPbyJoe12(cop=composite2COP, para=para) # 0.1031532
## End(Not run)