footCOP: The Spearman Footrule of a Copula
In copBasic: General Bivariate Copula Theory and Many Utility Functions
footCOP R Documentation
The Spearman Footrule of a Copula
Description
Compute the measure of association known as the Spearman Footrule \psi_\mathbf{C} (Nelsen et al., 2001, p. 281), which is defined as
\psi_\mathbf{C} = \frac{3}{2}\mathcal{Q}(\mathbf{C},\mathbf{M}) - \frac{1}{2}\mbox{,}
where \mathbf{C}(u,v) is the copula, \mathbf{M}(u,v) is the Fréchet–Hoeffding upper bound (M), \mathcal{Q}(a,b) is a concordance function (concordCOP) (Nelsen, 2006, p. 158), and -0.5 \le \psi_\mathbf{C} \le 1. The \psi_\mathbf{C} in terms of a single integration pass on the copula (Genest et al., 2010, p. 940) is
\psi_\mathbf{C} = 1 - \int_{\mathcal{I}^2} |u-v|\,\mathrm{d}\mathbf{C}(u,v) = 6 \int_0^1 \mathbf{C}(u,u)\,\mathrm{d}u - 2\mbox{.}
Note, Nelsen et al. (2001) use \phi_\mathbf{C} but that symbol is taken in copBasic for the Hoeffding Phi (hoefCOP), and Spearman Footrule does not seem to appear in Nelsen (2006). From the definition, Spearman Footrule only depends on the primary diagnonal (alt. main diagonal, Genest et al., 2010) of the copula, \mathbf{C}(t,t) (diagCOP). The function here supports the first \psi_\mathbf{C} computation by as.concordance=TRUE, but the default is the later definition by the direct integration on the diagonal.
Usage
footCOP(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 integral to compute \psi_\mathbf{C}, use the concordance function method implemented through concordCOP; and
as.sample
A logical controlling whether an optional R data.frame in para is used to compute the \hat\psi (see Note); and
...
Additional arguments to pass, which are dispatched to the copula function cop and possibly concordCOP, such as brute or delta used by that function.
Value
The value for \psi_\mathbf{C} is returned.
Note
Conceptually, the sample Spearman Footrule is a standardized sum of the absolute differences in the ranks (Genest et al., 2010, eq. 1). The sample \hat\psi is
\hat\psi = 1 - \frac{3}{n^2 - 1}\sum_{i=1}^n |R_i - S_i|\mbox{,}
where R_i and S_i are the respective ranks of X and Y, n is sample size, and {-}0.5 \lessapprox \hat\psi \le 1. Genest et al. (2010, p. 937) also remark that other normalizations have been used in the literature. The approximate \hat\psi lower bounds can be see by the following where -0.5 is achieved always for odd sample sizes and rapidly approached as sample size grows:
sams <- seq_len(25) # samples from 1 to 25 in size
minFoot <- sapply(sams, function(i) 1 - (3/(i^2-1))*sum(abs((1:i)-(i:1))))
plot(sams, minFoot, type="b", col="brown4")
lines(par()$usr[1:2], rep(-0.5, 2), lty=2)
The sample \hat\psi also can be written (Genest et al., 2010, eq. 3) as
\hat\psi = \frac{6}{n-1} \biggl[ \sum_i^n \mathrm{min}\biggl(\frac{R_i}{n+1},\, \frac{S_i}{n+1}\biggr)\biggr] - \frac{2n+1}{n-1}\mbox{,}
which is not implemented here because the first sample version is modestly faster. Also, the sampling variance of \hat\psi under assumption of independence between X and Y (\mathrm{E}[\hat\psi] = 0) is
\mathrm{var}(\hat\psi) = \frac{2n^2 + 7}{5(n+1)(n-1)^2}\mbox{.}
Genest et al. (2010, p. 938) say that prior literature shows that in small samples, Spearman Footrule is less variable than the well-known Spearman Rho (rhoCOP). For a copula having continuous partial derivatives, then as n \rightarrow \infty, the quantity (\hat\psi - \psi_\mathbf{C})\sqrt{n} \sim \mathrm{Normal}(0, \mathrm{var}(\gamma_\mathbf{C})). Genest et al. (2010) show variance of \hat\psi for the independence copula (\mathbf{C}(u,v) = \mathbf{\Pi}(u,v), P) as \mathrm{var}(\psi_\mathbf{C}) = 2/5. For comparison, the Gini Gamma for independence is larger at \mathrm{var}(\gamma_\mathbf{C}) = 2/3 (see giniCOP Note). Genest et al. (2010) also present additional material for estimation of the distribution \hat\psi variance for conditions of dependence based on copulas. In Genest et al. (2010, p. 941), independence and two examples of dependence, \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
Beliakov, G., de Amo, E., Juan Fernández-Sánchez, J., Úbeda-Flores, M., 2022, Best-possible bounds on the set of copulas with a given value of Spearman's footrule, Fuzzy Sets and Systems, v. 428, pp. 138–152, \Sexpr[results=rd]{tools:::Rd_expr_doi("10.1016/j.fss.2020.11.011")}.
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, giniCOP, hoefCOP,
rhoCOP, tauCOP, wolfCOP
Examples
footCOP(cop=PSP) # 0.3177662
# footCOP(cop=PSP, by.concordance=TRUE) # 0.3178025 # slower, likely giving up accuracy
## Not run:
set.seed(1); n <- 2000; UV <- simCOP(n=n, cop=GHcop, para=2.3, graphics=FALSE)
footCOP(para=UV, as.sample=TRUE) # 0.5356089 (sample version)
footCOP(cop=GHcop, para=2.3) # 0.5513380 (copula integration)
footCOP(cop=GHcop, para=2.3, by.concordance=TRUE) # 0.5513562 (concordance function)
# where the later issued warnings on the integration
## End(Not run)
## Not run:
# Sweep through the Plackett as means to show -0.5 <= Footrule <= 1, and to show
for(p in seq(-4,4, by=0.5)) { # that the sample version is in agreement.
p <- round(10^p, digits=4); UV <- simCOP(10000, cop=PLcop, para=p, lwd=0.5)
footS <- round(footCOP(as.sample=TRUE, para=UV), digits=4)
footT <- round(footCOP(cop=PLcop, para=p ), digits=4)
mtext(paste0("Plackett(theta=", p, ")"), line=1)
mtext(paste0("Spearman Footrule : theoretical=", footT, ", sample=", footS))
} #
## End(Not run)
## Not run:
set.seed(1); nsim <- 1000
varFTunderIndepend <- function(n) {
(2*n^2 + 7) / (5*(n+1)*(n-1)^2) # Genest et al. (2010)
}
ns <- c(10, 15, 20, 25, 50, 75, 100)
plot(min(ns):max(ns), varFTunderIndepend(10:max(ns)), type="l",
xlab="Sample size", ylab="Variance of Sample Estimator", col="salmon4")
mtext("Sample Spearman Footrule Under Independence", col="salmon4")
for(n in ns) {
sFT <- vector(length=nsim)
for(i in seq_len(nsim)) {
uv <- simCOP(n=n, cop=P, para=2, graphics=FALSE)
sFT[i] <- footCOP(para=uv, as.sample=TRUE)
}
varFT <- varFTunderIndepend(n)
zz <- round(c(n, mean(sFT), var(sFT), varFT), digits=6)
names(zz) <- c("n", "mean_sim", "var_sim", "var_Genest")
print(zz)
points(n, zz[3], cex=2, pch=21, col="salmon4", bg="salmon1")
} # results show proper implementation and Genest et al. (2010, sec. 3)
## End(Not run)
copBasic documentation built on July 23, 2026, 1:07 a.m.
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| footCOP | R Documentation |
The Spearman Footrule of a Copula
Description
Compute the measure of association known as the Spearman Footrule \psi_\mathbf{C} (Nelsen et al., 2001, p. 281), which is defined as
\psi_\mathbf{C} = \frac{3}{2}\mathcal{Q}(\mathbf{C},\mathbf{M}) - \frac{1}{2}\mbox{,}
where \mathbf{C}(u,v) is the copula, \mathbf{M}(u,v) is the Fréchet–Hoeffding upper bound (M), \mathcal{Q}(a,b) is a concordance function (concordCOP) (Nelsen, 2006, p. 158), and -0.5 \le \psi_\mathbf{C} \le 1. The \psi_\mathbf{C} in terms of a single integration pass on the copula (Genest et al., 2010, p. 940) is
\psi_\mathbf{C} = 1 - \int_{\mathcal{I}^2} |u-v|\,\mathrm{d}\mathbf{C}(u,v) = 6 \int_0^1 \mathbf{C}(u,u)\,\mathrm{d}u - 2\mbox{.}
Note, Nelsen et al. (2001) use \phi_\mathbf{C} but that symbol is taken in copBasic for the Hoeffding Phi (hoefCOP), and Spearman Footrule does not seem to appear in Nelsen (2006). From the definition, Spearman Footrule only depends on the primary diagnonal (alt. main diagonal, Genest et al., 2010) of the copula, \mathbf{C}(t,t) (diagCOP). The function here supports the first \psi_\mathbf{C} computation by as.concordance=TRUE, but the default is the later definition by the direct integration on the diagonal.
Usage
footCOP(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 integral 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 \psi_\mathbf{C} is returned.
Note
Conceptually, the sample Spearman Footrule is a standardized sum of the absolute differences in the ranks (Genest et al., 2010, eq. 1). The sample \hat\psi is
\hat\psi = 1 - \frac{3}{n^2 - 1}\sum_{i=1}^n |R_i - S_i|\mbox{,}
where R_i and S_i are the respective ranks of X and Y, n is sample size, and {-}0.5 \lessapprox \hat\psi \le 1. Genest et al. (2010, p. 937) also remark that other normalizations have been used in the literature. The approximate \hat\psi lower bounds can be see by the following where -0.5 is achieved always for odd sample sizes and rapidly approached as sample size grows:
sams <- seq_len(25) # samples from 1 to 25 in size minFoot <- sapply(sams, function(i) 1 - (3/(i^2-1))*sum(abs((1:i)-(i:1)))) plot(sams, minFoot, type="b", col="brown4") lines(par()$usr[1:2], rep(-0.5, 2), lty=2)
The sample \hat\psi also can be written (Genest et al., 2010, eq. 3) as
\hat\psi = \frac{6}{n-1} \biggl[ \sum_i^n \mathrm{min}\biggl(\frac{R_i}{n+1},\, \frac{S_i}{n+1}\biggr)\biggr] - \frac{2n+1}{n-1}\mbox{,}
which is not implemented here because the first sample version is modestly faster. Also, the sampling variance of \hat\psi under assumption of independence between X and Y (\mathrm{E}[\hat\psi] = 0) is
\mathrm{var}(\hat\psi) = \frac{2n^2 + 7}{5(n+1)(n-1)^2}\mbox{.}
Genest et al. (2010, p. 938) say that prior literature shows that in small samples, Spearman Footrule is less variable than the well-known Spearman Rho (rhoCOP). For a copula having continuous partial derivatives, then as n \rightarrow \infty, the quantity (\hat\psi - \psi_\mathbf{C})\sqrt{n} \sim \mathrm{Normal}(0, \mathrm{var}(\gamma_\mathbf{C})). Genest et al. (2010) show variance of \hat\psi for the independence copula (\mathbf{C}(u,v) = \mathbf{\Pi}(u,v), P) as \mathrm{var}(\psi_\mathbf{C}) = 2/5. For comparison, the Gini Gamma for independence is larger at \mathrm{var}(\gamma_\mathbf{C}) = 2/3 (see giniCOP Note). Genest et al. (2010) also present additional material for estimation of the distribution \hat\psi variance for conditions of dependence based on copulas. In Genest et al. (2010, p. 941), independence and two examples of dependence, \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
Beliakov, G., de Amo, E., Juan Fernández-Sánchez, J., Úbeda-Flores, M., 2022, Best-possible bounds on the set of copulas with a given value of Spearman's footrule, Fuzzy Sets and Systems, v. 428, pp. 138–152, \Sexpr[results=rd]{tools:::Rd_expr_doi("10.1016/j.fss.2020.11.011")}.
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, giniCOP, hoefCOP,
rhoCOP, tauCOP, wolfCOP
Examples
footCOP(cop=PSP) # 0.3177662
# footCOP(cop=PSP, by.concordance=TRUE) # 0.3178025 # slower, likely giving up accuracy
## Not run:
set.seed(1); n <- 2000; UV <- simCOP(n=n, cop=GHcop, para=2.3, graphics=FALSE)
footCOP(para=UV, as.sample=TRUE) # 0.5356089 (sample version)
footCOP(cop=GHcop, para=2.3) # 0.5513380 (copula integration)
footCOP(cop=GHcop, para=2.3, by.concordance=TRUE) # 0.5513562 (concordance function)
# where the later issued warnings on the integration
## End(Not run)
## Not run:
# Sweep through the Plackett as means to show -0.5 <= Footrule <= 1, and to show
for(p in seq(-4,4, by=0.5)) { # that the sample version is in agreement.
p <- round(10^p, digits=4); UV <- simCOP(10000, cop=PLcop, para=p, lwd=0.5)
footS <- round(footCOP(as.sample=TRUE, para=UV), digits=4)
footT <- round(footCOP(cop=PLcop, para=p ), digits=4)
mtext(paste0("Plackett(theta=", p, ")"), line=1)
mtext(paste0("Spearman Footrule : theoretical=", footT, ", sample=", footS))
} #
## End(Not run)
## Not run:
set.seed(1); nsim <- 1000
varFTunderIndepend <- function(n) {
(2*n^2 + 7) / (5*(n+1)*(n-1)^2) # Genest et al. (2010)
}
ns <- c(10, 15, 20, 25, 50, 75, 100)
plot(min(ns):max(ns), varFTunderIndepend(10:max(ns)), type="l",
xlab="Sample size", ylab="Variance of Sample Estimator", col="salmon4")
mtext("Sample Spearman Footrule Under Independence", col="salmon4")
for(n in ns) {
sFT <- vector(length=nsim)
for(i in seq_len(nsim)) {
uv <- simCOP(n=n, cop=P, para=2, graphics=FALSE)
sFT[i] <- footCOP(para=uv, as.sample=TRUE)
}
varFT <- varFTunderIndepend(n)
zz <- round(c(n, mean(sFT), var(sFT), varFT), digits=6)
names(zz) <- c("n", "mean_sim", "var_sim", "var_Genest")
print(zz)
points(n, zz[3], cex=2, pch=21, col="salmon4", bg="salmon1")
} # results show proper implementation and Genest et al. (2010, sec. 3)
## End(Not run)
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