giniCOP: The Gini Gamma of a Copula
In wasquith/copBasic: General Bivariate Copula Theory and Many Utility Functions
giniCOP R Documentation
The Gini Gamma of a Copula
Description
Compute the measure of association known as the Gini Gamma \gamma_\mathbf{C} (Nelsen, 2006, pp. 180–182), which is defined as
\gamma_\mathbf{C} = \mathcal{Q}(\mathbf{C},\mathbf{M}) + \mathcal{Q}(\mathbf{C},\mathbf{W})\mbox{,}
where \mathbf{C}(u,v) is the copula, \mathbf{M}(u,v) is the M function, and \mathbf{W}(u,v) is the W function. The function \mathcal{Q}(a,b) (concordCOP) is a concordance function (Nelsen, 2006, p. 158). Nelsen also reports that “Gini Gamma measures a concordance relation of “distance” between \mathbf{C}(u,v) and monotone dependence, as represented by the Fréchet–Hoeffding lower bound and Fréchet–Hoeffding upper bound copulas [\mathbf{M}(u,v), M and \mathbf{W}(u,v), W respectively]”
A simpler method of computation and the default for giniCOP is to compute \gamma_\mathbf{C} by
\gamma_\mathbf{C} = 4\biggl[\int_\mathcal{I} \mathbf{C}(u,u)\,\mathrm{d}u +
\int_\mathcal{I} \mathbf{C}(u,1-u)\,\mathrm{d}u\biggr] -
2\mbox{,}
or in terms of the primary diagonal (alt. main diagonal, Genest et al., 2010) \delta(t) and secondary diagonal \delta^\star(t) (see diagCOP) by
\gamma_\mathbf{C} = 4\biggl[\int_\mathcal{I} \mathbf{\delta}(t)\,\mathrm{d}t +
\int_\mathcal{I} \mathbf{\delta^\star
}(t)\,\mathrm{d}t\biggr] -
2\mbox{.}
The simpler method is more readily implemented because single integration is fast. Lastly, Nelsen et al. (2001, p. 281) show that \gamma_\mathbf{C} also is computable by
\gamma_\mathbf{C} = 2\,\mathcal{Q}(\mathbf{C},\mathbf{A})\mbox{,}
where \mathbf{A} is a convex combination (convex2COP, using \alpha = 1/2) of the copulas \mathbf{M} and \mathbf{W} or \mathbf{A} = (\mathbf{M}+\mathbf{W})/2. However, integral convergence errors seem to trigger occasionally, and the first definition by summation \mathcal{Q}(\mathbf{C},\mathbf{M}) + \mathcal{Q}(\mathbf{C},\mathbf{W}) thus is used. The convex combination is demonstrated in the Examples section.
Usage
giniCOP(cop=NULL, para=NULL, by.concordance=FALSE, as.sample=FALSE, ...)
Arguments
cop
A copula function;
para
Vector of parameters or other data structure, if needed, to pass to the copula;
by.concordance
Instead of using the single integrals (Nelsen, 2006, pp. 181–182) to compute \gamma_\mathbf{C}, use the concordance function method implemented through concordCOP;
as.sample
A logical controlling whether an optional R data.frame in para is used to compute the \hat\gamma_\mathbf{C} (see Note); and
...
Additional arguments to pass, which are dispatched to the copula function cop and possibly concordCOP if by.concordance=TRUE, such as delta used by that function.
Value
The value for \gamma_\mathbf{C} is returned.
Note
Conceptually, the sample Gini Gamma (\hat\gamma; Genest et al., 2010) is
\hat\gamma = \frac{1}{\lfloor n^2/2 \rfloor}\sum_{i=1}^n \mid (n+1-R_i) - S_i \mid- \mid R_i - S_i\mid\mbox{,}
where {\lfloor m \rfloor} denotes the integer part of arbitrary m > 0, R_i and S_i are the respective ranks of X and Y and n is sample size. The sampling variance of \hat\gamma under assumption of independence between X and Y is
\mathrm{var}(\hat\gamma)_{n\ \mathrm{even}} = \frac{2}{3}\frac{(n^2 + 2)}{(n-1)n^2}\mbox{\ and}
\mathrm{var}(\hat\gamma)_{n\ \mathrm{odd}} = \frac{2}{3}\frac{(n^2 + 3)}{(n-1)(n^2-1)}\mbox{.}
Genest et al. (2010) present additional equations for estimation of the distribution \hat\gamma variance for conditions of dependence based on copulas. For a copula having continuous partial derivatives, then as n \rightarrow \infty, the quantity (\hat\gamma - \gamma_\mathbf{C})\sqrt{n} \sim \mathrm{Normal}(0, \mathrm{var}(\gamma_\mathbf{C})). Genest et al. (2010) show variance of \hat\gamma for the independence copula (\mathbf{C}(u,v) = \mathbf{\Pi}(u,v)) (P) as \mathrm{var}(\gamma_\mathbf{C}) = 2/3. For comparison, the Spearman Footrule for independence is smaller at \mathrm{var}(\psi_\mathbf{C}) = 2/5 (see footCOP Note). In Genest et al. independence and two examples of dependence (p. 941), \mathrm{var}(\hat\gamma) > \mathrm{var}(\hat\psi), but those authors do not appear to remark on whether this inequality holds for all copula.
Author(s)
W.H. Asquith
References
Genest, C., Nešlehová, J., and Ghorbal, N.B., 2010, Spearman's footrule and Gini's gamma—A review with complements: Journal of Nonparametric Statistics, v. 22, no. 8, pp. 937–954, \Sexpr[results=rd]{tools:::Rd_expr_doi("10.1080/10485250903499667")}.
Nelsen, R.B., 2006, An introduction to copulas: New York, Springer, 269 p.
Nelsen, R.B., Quesada-Molina, J.J., Rodríguez-Lallena, J.A., Úbeda-Flores, M., 2001, Distribution functions of copulas—A class of bivariate probability integral transforms: Statistics and Probability Letters, v. 54, no. 3, pp. 277–282, \Sexpr[results=rd]{tools:::Rd_expr_doi("10.1016/S0167-7152(01)00060-8")}.
See Also
blomCOP, footCOP, hoefCOP,
rhoCOP, tauCOP, wolfCOP,
joeskewCOP, uvlmoms
Examples
giniCOP(cop=PSP) # = 0.3819757
## Not run:
giniCOP( cop=PSP, by.concordance=TRUE) # Q(C,M) + Q(C,W) = 0.3820045
# use convex combination ---triggers integration warning but returns anyway
cxpara <- list(alpha=1/2, cop1=M, cop2=W) # parameters for convex2COP()
2*tauCOP(cop=PSP, cop2=convex2COP, para2=cxpara) # 2*Q(C,A) = 0.3819807
# where the later issued warnings on the integration
## End(Not run)
## Not run:
n <- 2000; UV <- simCOP(n=n, cop=N4212cop, para=9.3, graphics=FALSE)
giniCOP(para=UV, as.sample=TRUE) # 0.9475900 (sample version)
giniCOP(cop=N4212cop, para=9.3) # 0.9479528 (copula integration)
giniCOP(cop=N4212cop, para=9.3, by.concordance=TRUE) # 0.9480267 (concordance function)
# where the later issued warnings on the integration
## End(Not run)
## Not run:
# The canoncial example of theoretical and sample estimators of bivariate
# association for the package: Blomqvist Beta, Spearman Footrule, Gini Gamma,
# Hoeffding Phi, Kendall Tau, Spearman Rho, and Schweizer-Wolff Sigma
# and comparison to L-correlation via lmomco::lcomoms2().
n <- 9000; set.seed(56)
para <- list(cop1=PLACKETTcop, cop2=PLACKETTcop, para1=1.45, para2=21.9,
alpha=0.41, beta=0.08)
D <- simCOP(n=n, cop=composite2COP, para=para, cex=0.5, col=rgb(0,0,0,0.2), pch=16)
blomCOP(cop=composite2COP, para=para) # 0.4037908 (theoretical)
blomCOP(para=D, as.sample=TRUE) # 0.4008889 (sample)
footCOP(cop=composite2COP, para=para) # 0.3721555 (theoretical)
footCOP(para=D, as.sample=TRUE) # 0.3703623 (sample)
giniCOP(cop=composite2COP, para=para) # 0.4334687 (theoretical)
giniCOP(para=D, as.sample=TRUE) # 0.4311698 (sample)
tauCOP(cop=composite2COP, para=para) # 0.3806909 (theoretical)
tauCOP(para=D, as.sample=TRUE) # 0.3788139 (sample)
rhoCOP(cop=composite2COP, para=para) # 0.5257662 (theoretical)
rhoCOP(para=D, as.sample=TRUE) # 0.5242380 (sample)
lmomco::lcomoms2(D)$T2 # 1 # 0.5242388 (sample matrix)
# 0.5245154 1
hoefCOP(cop=composite2COP, para=para) # 0.5082776 (theoretical)
subsample <- D[sample(1:n, n/5),] # subsampling for speed
hoefCOP(para=subsample, as.sample=TRUE) # 0.5033842 (re-sample)
#hoefCOP(para=D, as.sample=TRUE) # major CPU hog, n too big
# because the Ds are already "probabilities" just resample as shown above
wolfCOP(cop=composite2COP, para=para) # 0.5257662 (theoretical)
#wolfCOP(para=D, as.sample=TRUE) # major CPU hog, n too big
wolfCOP(para=subsample, as.sample=TRUE) # 0.5338009 (re-sample)
## End(Not run)
## Not run:
set.seed(1); nsim <- 1000
varGIunderIndpendence <- function(n) {
if(3 %/% 2 == 3 / 2) { # even
(2/3)*( (n^2 + 2) ) / ( (n-1)* n^2 ) # Genest et al. (2010)
} else {
(2/3)*( (n^2 + 3) ) / ( (n-1)*(n^2-1) ) # Genest et al. (2010)
}
}
ns <- c(10, 15, 20, 25, 50, 75, 100)
plot(min(ns):max(ns), varGIunderIndpendence(10:max(ns)), type="l",
xlab="Sample size", ylab="Variance of Sample Estimator", col="darkgreen")
mtext("Sample Gini Gamma Under Independence", col="darkgreen")
for(n in ns) {
sGI <- vector(length=nsim)
for(i in seq_len(nsim)) {
uv <- simCOP(n=n, cop=P, para=2, graphics=FALSE)
sGI[i] <- giniCOP(para=uv, as.sample=TRUE)
}
varGI <- varGIunderIndpendence(n)
zz <- round(c(n, mean(sGI), var(sGI), varGI), digits=6)
names(zz) <- c("n", "mean_sim", "var_sim", "var_Genest")
print(zz)
points(n, zz[3], cex=2, pch=21, col="darkgreen", bg="lightgreen")
} # results show proper implementation and Genest et al. (2010, sec. 3)
## End(Not run)
wasquith/copBasic documentation built on Dec. 13, 2024, 6:39 p.m.
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| giniCOP | R Documentation |
The Gini Gamma of a Copula
Description
Compute the measure of association known as the Gini Gamma \gamma_\mathbf{C} (Nelsen, 2006, pp. 180–182), which is defined as
\gamma_\mathbf{C} = \mathcal{Q}(\mathbf{C},\mathbf{M}) + \mathcal{Q}(\mathbf{C},\mathbf{W})\mbox{,}
where \mathbf{C}(u,v) is the copula, \mathbf{M}(u,v) is the M function, and \mathbf{W}(u,v) is the W function. The function \mathcal{Q}(a,b) (concordCOP) is a concordance function (Nelsen, 2006, p. 158). Nelsen also reports that “Gini Gamma measures a concordance relation of “distance” between \mathbf{C}(u,v) and monotone dependence, as represented by the Fréchet–Hoeffding lower bound and Fréchet–Hoeffding upper bound copulas [\mathbf{M}(u,v), M and \mathbf{W}(u,v), W respectively]”
A simpler method of computation and the default for giniCOP is to compute \gamma_\mathbf{C} by
\gamma_\mathbf{C} = 4\biggl[\int_\mathcal{I} \mathbf{C}(u,u)\,\mathrm{d}u +
\int_\mathcal{I} \mathbf{C}(u,1-u)\,\mathrm{d}u\biggr] -
2\mbox{,}
or in terms of the primary diagonal (alt. main diagonal, Genest et al., 2010) \delta(t) and secondary diagonal \delta^\star(t) (see diagCOP) by
\gamma_\mathbf{C} = 4\biggl[\int_\mathcal{I} \mathbf{\delta}(t)\,\mathrm{d}t +
\int_\mathcal{I} \mathbf{\delta^\star
}(t)\,\mathrm{d}t\biggr] -
2\mbox{.}
The simpler method is more readily implemented because single integration is fast. Lastly, Nelsen et al. (2001, p. 281) show that \gamma_\mathbf{C} also is computable by
\gamma_\mathbf{C} = 2\,\mathcal{Q}(\mathbf{C},\mathbf{A})\mbox{,}
where \mathbf{A} is a convex combination (convex2COP, using \alpha = 1/2) of the copulas \mathbf{M} and \mathbf{W} or \mathbf{A} = (\mathbf{M}+\mathbf{W})/2. However, integral convergence errors seem to trigger occasionally, and the first definition by summation \mathcal{Q}(\mathbf{C},\mathbf{M}) + \mathcal{Q}(\mathbf{C},\mathbf{W}) thus is used. The convex combination is demonstrated in the Examples section.
Usage
giniCOP(cop=NULL, para=NULL, by.concordance=FALSE, as.sample=FALSE, ...)
Arguments
cop |
A copula function; |
para |
Vector of parameters or other data structure, if needed, to pass to the copula; |
by.concordance |
Instead of using the single integrals (Nelsen, 2006, pp. 181–182) to compute |
as.sample |
A logical controlling whether an optional R |
... |
Additional arguments to pass, which are dispatched to the copula function |
Value
The value for \gamma_\mathbf{C} is returned.
Note
Conceptually, the sample Gini Gamma (\hat\gamma; Genest et al., 2010) is
\hat\gamma = \frac{1}{\lfloor n^2/2 \rfloor}\sum_{i=1}^n \mid (n+1-R_i) - S_i \mid- \mid R_i - S_i\mid\mbox{,}
where {\lfloor m \rfloor} denotes the integer part of arbitrary m > 0, R_i and S_i are the respective ranks of X and Y and n is sample size. The sampling variance of \hat\gamma under assumption of independence between X and Y is
\mathrm{var}(\hat\gamma)_{n\ \mathrm{even}} = \frac{2}{3}\frac{(n^2 + 2)}{(n-1)n^2}\mbox{\ and}
\mathrm{var}(\hat\gamma)_{n\ \mathrm{odd}} = \frac{2}{3}\frac{(n^2 + 3)}{(n-1)(n^2-1)}\mbox{.}
Genest et al. (2010) present additional equations for estimation of the distribution \hat\gamma variance for conditions of dependence based on copulas. For a copula having continuous partial derivatives, then as n \rightarrow \infty, the quantity (\hat\gamma - \gamma_\mathbf{C})\sqrt{n} \sim \mathrm{Normal}(0, \mathrm{var}(\gamma_\mathbf{C})). Genest et al. (2010) show variance of \hat\gamma for the independence copula (\mathbf{C}(u,v) = \mathbf{\Pi}(u,v)) (P) as \mathrm{var}(\gamma_\mathbf{C}) = 2/3. For comparison, the Spearman Footrule for independence is smaller at \mathrm{var}(\psi_\mathbf{C}) = 2/5 (see footCOP Note). In Genest et al. independence and two examples of dependence (p. 941), \mathrm{var}(\hat\gamma) > \mathrm{var}(\hat\psi), but those authors do not appear to remark on whether this inequality holds for all copula.
Author(s)
W.H. Asquith
References
Genest, C., Nešlehová, J., and Ghorbal, N.B., 2010, Spearman's footrule and Gini's gamma—A review with complements: Journal of Nonparametric Statistics, v. 22, no. 8, pp. 937–954, \Sexpr[results=rd]{tools:::Rd_expr_doi("10.1080/10485250903499667")}.
Nelsen, R.B., 2006, An introduction to copulas: New York, Springer, 269 p.
Nelsen, R.B., Quesada-Molina, J.J., Rodríguez-Lallena, J.A., Úbeda-Flores, M., 2001, Distribution functions of copulas—A class of bivariate probability integral transforms: Statistics and Probability Letters, v. 54, no. 3, pp. 277–282, \Sexpr[results=rd]{tools:::Rd_expr_doi("10.1016/S0167-7152(01)00060-8")}.
See Also
blomCOP, footCOP, hoefCOP,
rhoCOP, tauCOP, wolfCOP,
joeskewCOP, uvlmoms
Examples
giniCOP(cop=PSP) # = 0.3819757
## Not run:
giniCOP( cop=PSP, by.concordance=TRUE) # Q(C,M) + Q(C,W) = 0.3820045
# use convex combination ---triggers integration warning but returns anyway
cxpara <- list(alpha=1/2, cop1=M, cop2=W) # parameters for convex2COP()
2*tauCOP(cop=PSP, cop2=convex2COP, para2=cxpara) # 2*Q(C,A) = 0.3819807
# where the later issued warnings on the integration
## End(Not run)
## Not run:
n <- 2000; UV <- simCOP(n=n, cop=N4212cop, para=9.3, graphics=FALSE)
giniCOP(para=UV, as.sample=TRUE) # 0.9475900 (sample version)
giniCOP(cop=N4212cop, para=9.3) # 0.9479528 (copula integration)
giniCOP(cop=N4212cop, para=9.3, by.concordance=TRUE) # 0.9480267 (concordance function)
# where the later issued warnings on the integration
## End(Not run)
## Not run:
# The canoncial example of theoretical and sample estimators of bivariate
# association for the package: Blomqvist Beta, Spearman Footrule, Gini Gamma,
# Hoeffding Phi, Kendall Tau, Spearman Rho, and Schweizer-Wolff Sigma
# and comparison to L-correlation via lmomco::lcomoms2().
n <- 9000; set.seed(56)
para <- list(cop1=PLACKETTcop, cop2=PLACKETTcop, para1=1.45, para2=21.9,
alpha=0.41, beta=0.08)
D <- simCOP(n=n, cop=composite2COP, para=para, cex=0.5, col=rgb(0,0,0,0.2), pch=16)
blomCOP(cop=composite2COP, para=para) # 0.4037908 (theoretical)
blomCOP(para=D, as.sample=TRUE) # 0.4008889 (sample)
footCOP(cop=composite2COP, para=para) # 0.3721555 (theoretical)
footCOP(para=D, as.sample=TRUE) # 0.3703623 (sample)
giniCOP(cop=composite2COP, para=para) # 0.4334687 (theoretical)
giniCOP(para=D, as.sample=TRUE) # 0.4311698 (sample)
tauCOP(cop=composite2COP, para=para) # 0.3806909 (theoretical)
tauCOP(para=D, as.sample=TRUE) # 0.3788139 (sample)
rhoCOP(cop=composite2COP, para=para) # 0.5257662 (theoretical)
rhoCOP(para=D, as.sample=TRUE) # 0.5242380 (sample)
lmomco::lcomoms2(D)$T2 # 1 # 0.5242388 (sample matrix)
# 0.5245154 1
hoefCOP(cop=composite2COP, para=para) # 0.5082776 (theoretical)
subsample <- D[sample(1:n, n/5),] # subsampling for speed
hoefCOP(para=subsample, as.sample=TRUE) # 0.5033842 (re-sample)
#hoefCOP(para=D, as.sample=TRUE) # major CPU hog, n too big
# because the Ds are already "probabilities" just resample as shown above
wolfCOP(cop=composite2COP, para=para) # 0.5257662 (theoretical)
#wolfCOP(para=D, as.sample=TRUE) # major CPU hog, n too big
wolfCOP(para=subsample, as.sample=TRUE) # 0.5338009 (re-sample)
## End(Not run)
## Not run:
set.seed(1); nsim <- 1000
varGIunderIndpendence <- function(n) {
if(3 %/% 2 == 3 / 2) { # even
(2/3)*( (n^2 + 2) ) / ( (n-1)* n^2 ) # Genest et al. (2010)
} else {
(2/3)*( (n^2 + 3) ) / ( (n-1)*(n^2-1) ) # Genest et al. (2010)
}
}
ns <- c(10, 15, 20, 25, 50, 75, 100)
plot(min(ns):max(ns), varGIunderIndpendence(10:max(ns)), type="l",
xlab="Sample size", ylab="Variance of Sample Estimator", col="darkgreen")
mtext("Sample Gini Gamma Under Independence", col="darkgreen")
for(n in ns) {
sGI <- vector(length=nsim)
for(i in seq_len(nsim)) {
uv <- simCOP(n=n, cop=P, para=2, graphics=FALSE)
sGI[i] <- giniCOP(para=uv, as.sample=TRUE)
}
varGI <- varGIunderIndpendence(n)
zz <- round(c(n, mean(sGI), var(sGI), varGI), digits=6)
names(zz) <- c("n", "mean_sim", "var_sim", "var_Genest")
print(zz)
points(n, zz[3], cex=2, pch=21, col="darkgreen", bg="lightgreen")
} # results show proper implementation and Genest et al. (2010, sec. 3)
## End(Not run)
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