wolfCOP: Schweizer and Wolff Sigma of a Copula
In copBasic: General Bivariate Copula Theory and Many Utility Functions
wolfCOP R Documentation
Schweizer and Wolff Sigma of a Copula
Description
Compute the measure of association known as Schweizer–Wolff Sigma \sigma_\mathbf{C} of a copula according to Nelsen (2006, p. 209) by
\sigma_\mathbf{C} = 12\int\!\!\int_{\mathcal{I}^2} \bigl|\mathbf{C}(u,v) - uv\bigr|\,\mathrm{d}u\mathrm{d}v\mbox{,}
which is 0 \le \sigma_\mathbf{C} \le 1. It is obvious that this measure of association, without the positive sign restriction, is similar to the following form of Spearman Rho (rhoCOP) of a copula:
\rho_\mathbf{C} = 12\int\!\!\int_{\mathcal{I}^2} \bigl[\mathbf{C}(u,v) - uv\bigr]\,\mathrm{d}u\mathrm{d}v\mbox{.}
If a copula is positively quadrant dependent (PQD, see isCOP.PQD), then \sigma_\mathbf{C} = \rho_\mathbf{C}; conversely if a copula is negatively quadrant dependent (NQD), then \sigma_\mathbf{C} = -\rho_\mathbf{C}. However, a feature making \sigma_\mathbf{C} especially attractive is that for random variables X and Y, which are not PQD or NQD—copulas that are neither larger nor smaller than \mathbf{\Pi}—is that “\sigma_\mathbf{C} is often a better measure of [dependency] than \rho_\mathbf{C}” (Nelsen, 2006, p. 209).
Úbeda-Flores (2022) study the maximum difference between \sigma_\mathbf{C} - |\rho_\mathbf{C}| \approx 0.60496 for a special class of singular copula (see Examples), but states that it “remains an open problem to check if, for any copula \mathbf{C}_\Theta,” that the maximum listed here is the maximum difference possible.
Li et al. (2019) compare \sigma_\mathbf{C} (they label it \delta^\mathrm{E}_i, “the extended delta index”) to alternative measures of dependency with emphasis on information content. Those authors also show that dependence can also be measured by
\delta_i = \frac{1}{2}\int\!\!\int_{\mathcal{I}^2} \bigl|c(u,v) - 1\bigr|\,\mathrm{d}u\mathrm{d}v\mbox{,}
where c(u,v) is a copula density (densityCOP). Li et al. (2019, p. 45) “measure how much the model output Y and the input X_i are dependent,” and “importance of the inputs globally and
quantitatively with model free and moment independent properties.”
By definition for two random variables U and V with cumulative distribution functions of F(u) and F(v), the variables are independent, if and only if, for all subsets (conditions) selecting values u and y of U \le u and V \le v, when for all u and v, the combined cumulative distribution function is
H(x, y) = F(x)G(x) = \mathbf{C}(u,v) = uv\mbox{.}
Note, structural similarity between \sigma_\mathbf{C} and the definition for LzCOPpermsym. It seems to follow, by intuition, if therein, the copula on the left were to be replaced with the permutation symmetric (exchangeable symmetric) copula isCOP.permsym, such as \mathbf{\Pi} = uv = vu, then the copula on the right, now being, without loss of generality, \mathbf{C}(u,v) or \mathbf{C}(v,u), then \sigma_\mathbf{C} must respond partly to dependence existing in the presence of permutation asymmetry.
Usage
wolfCOP(cop=NULL, para=NULL, as.sample=FALSE, brute=FALSE, delta=0.002,
nlarge=500, usefastgrid=TRUE, ...)
Arguments
cop
A copula function;
para
Vector of parameters or other data structure, if needed, to pass to the copula;
as.sample
A logical controlling whether an optional R data.frame in para is used to compute the \hat{\sigma}_\mathbf{C} (see Details). If set to -1, then the message concerning CPU effort will be suppressed. This feature of controlling the message is not applicable if the fast computation by usefastgrid is set and sample size is larger than nlarge as the double summation in the Pózos et al. (2015) is bypassed in favor of EMPIRgrid_fast;
brute
Should brute force be used instead of two nested integrate() functions in R to perform the double integration;
delta
The \mathrm{d}u and \mathrm{d}v for the brute force (brute=TRUE) integration;
nlarge
A large sample size for which to consider the fast-grid computation if usefastgrid is true;
usefastgrid
A logical controlling whether the fast-grid computation is acceptable to the user; and
...
Additional arguments to pass to the copula itself, to EMPIRgrid_fast, or the integrate() function.
Details
A natural estimator for \sigma_\mathbf{C} is through the empirical copula (Póczos et al., 2015) and can be computed as
\hat{\sigma}_\mathbf{C} = \frac{12}{n^2 - 1}
\sum_{i=1}^n\sum_{j=1}^n \bigg|\hat{\mathbf{C}}_n\biggl(\frac{i}{n}, \frac{j}{n}\biggr) -
\frac{i}{n}\times\frac{j}{n}\bigg|\mbox{,}
where \hat{\mathbf{C}}_n is the simplest empirical copula of
\hat{\mathbf{C}}_n\biggl(\frac{i}{n}, \frac{j}{n}\biggr) =
\frac{1}{n}\{\# \mathrm{\ of\ } (U_k \le U_i, V_k \le V_j)\}
The sample version of \hat\sigma_\mathbf{C} is triggered by wolfCOP(para=D, as.sample=TRUE) for the double summation if usefastgrid is false and or usefastgrid is true and the same size is smaller than nlarge. A Monte Carlo integration version potentially suitable to very large sample sizes or difficult to integrate copula families is located at wolfCOPsamc, but the function itself is likely of limited general use because of the implementation of EMPIRgrid_fast inside wolfCOP. It is upon the fast estimator listed above that the extensive simulations leading to wolfCOPtest are based. In general, there are several asymptotically equivalent definitions of the empirical copula and some of which are implemented in EMPIRcop.
Consideration of speed performance between the as.sample=TRUE for the sample estimator argued by Póczos et al., 2015) (refer to section Note), the EMPIRgrid_fast is measurably faster by about n \ge 500. The following code shows numerical equivalence between these estimators, but use of other empirical copula will have divergence unless samples are large:
"MYwolfCOPgrid" <- function(uv, ctype="1/n", ...) {
ef <- EMPIRgrid_fast(para=uv, gridonly=FALSE, ctype=ctype)
pf <- ef$empcop; n <- nrow(uv)
gu <- as.numeric( rownames(ef$empcop) ) # gv <- gu (can compute w/o gv)
for(i in seq_len(nrow(pf))) pf[i,] <- gu * gu[i]
return(12 * sum(abs(ef$empcop - pf)) / (n^2 - 1))
}
ns <- as.integer( 10^seq(1, 3.3, by=0.1) ) # some sample sizes
WG <- WS <- vector(mode="numeric", length(ns))
for(i in seq_len(length(ns))) {
uv <- simCOP(n=ns[i], cop=P, graphics=FALSE)
WG[i] <- wolfCOP( para=uv, as.sample=TRUE, usefastgrid=FALSE)
WS[i] <- MYwolfCOPgrid(uv)
}
opts <- par(xpd=NA, no.readonly=FALSE)
df <- data.frame(n=ns, wolf_grid=WG, wolf_sample=WS)
plot(df$n, df$wolf_sample, type="n", xlim=c(10, 10000), ylim=c(0, 0.60),
xaxs="i", yaxs="i", log="x", las=1,
xlab="Sample size of Independence copula",
ylab="Schweizer-Wolff Sigma")
points(df$n, df$wolf_sample, pch=21, bg="white")
points(df$n, df$wolf_grid, pch=16, cex=0.6)
par(opts) #
Value
The value for \sigma_\mathbf{C} is returned, and the sample \hat\sigma_\mathbf{C} computation versions have names() attached to verify to the user which subsystem was used given the nlarge and usefastgrid arguments.
Note
An extended example is informative. First, declare a composite of two different Plackett copulas (PLcop) and simulate a few hundred values. The parameter settings are such that the data will attain a look of independence.:
para <- list(cop1=PLcop, para1=0.145, alpha=0.81,
cop2=PLcop, para2=21.90, beta=0.22)
D <- simCOP(n=300, cop=composite2COP, para=para,
cex=0.5, col=grey(0, 0.5), pch=16, seed=15)
Second, show that this copula is globally PQD (isCOP.PQD), but there is a substantial local NQD part of \mathcal{I}^2 space that clearly is NQD.
cols <- c("red3", "green4")
PQD <- isCOP.PQD(cop=composite2COP, para=para, uv=D)
message(PQD$global.PQD) # TRUE
points(D, col=cols[1+as.numeric(PQD$local.PQD)], lwd=2) # TRUE are green
This composited copula intersects, that is, passes through, the P copula. By the logic of Nelsen (2006), then the \sigma_\mathbf{C} should be larger than \rho_\mathbf{C} as shown below
wolfCOP(cop=composite2COP, para=para) # 0.08373378 (theoretical)
rhoCOP(cop=composite2COP, para=para) # 0.02845131 (theoretical)
hoefCOP(cop=composite2COP, para=para) # 0.08563823 (theoretical)
In fact, output above also shows Schweizer–Wolff Sigma to be larger than Blomqvist Beta (blomCOP), Gini Gamma (giniCOP), and Kendall Tau (tauCOP). Schweizer–Wolff Sigma captures the fact that although the symbols seems to plot randomly on the figure, symbol coloring for PQD and NQD clearly shows local dependency differences.
Author(s)
W.H. Asquith
References
Li, L. Liu, Y., and Lu, Z., 2019, A new dependence measure for importance analysis—Application to an environmental model: Applied Mathematical Modelling, v. 74, pp. 43–61, \Sexpr[results=rd]{tools:::Rd_expr_doi("10.1016/j.apm.2019.04.046")}.
Nelsen, R.B., 2006, An introduction to copulas: New York, Springer, 269 p.
Póczos, B., Krishner, S., Pál, D., Szepesvári, C., and Schneider, J., 2015, Robust nonparametric copula based dependence estimators, accessed on August 11, 2015, at https://www.cs.cmu.edu/~bapoczos/articles/poczos11nipscopula.pdf.
Schweizer, B., and Wolff, E.F., 1981, On nonparametric measures of dependence for random variables: The Annuals of Statistics, v. 9, no. 4, pp. 879–855, https://www.jstor.org/stable/2240856.
Úbeda-Flores, M., 2022, A note on the maximum difference between Schweizer and Wolff's Sigma and the absolute value of Spearman's Rho: Transactions on Fuzzy Sets and Systems, v. 1, no. 2, pp. 1–5, \Sexpr[results=rd]{tools:::Rd_expr_doi("10.30495/tfss.2022.1956899.1024")}.
See Also
wolfCOPsamc, wolfCOPtest
blomCOP, footCOP, giniCOP,
hoefCOP, rhoCOP, tauCOP,
joeskewCOP, uvlmoms, ORDSUMcop
Examples
wolfCOP(cop=PSP) # 0.4784176
## Not run:
# Maximum Difference between Sigma and Absolute Spearman Rho
k <- 1/2 + sqrt(3)/6 # Ubeda-Flores (2022) or 1/2 - sqrt(3)/6
para <- list(cop=c(W, W), para=NULL, part=c(0, k, 1))
rho <- rhoCOP( ORDSUMcop, para=para)
wol <- wolfCOP(ORDSUMcop, para=para)
message("Wolf - |Rho| = ", wol - abs(rho))
# 0.604967382.... that is approx. 0.60496 given by Ubeda-Flores (2022).
ks <- seq(0.01, 0.99, by=0.002); rhos <- wols <- NULL
for(k in ks) {
message(k, "-", appendLF=FALSE)
para <- list(cop=c(W, W), para=NULL, part=c(0, k, 1))
rhos <- c(rhos, abs( rhoCOP( cop=ORDSUMcop, para=para, subdivisions=200) ))
wols <- c(wols, wolfCOP(cop=ORDSUMcop, para=para, subdivisions=200) )
}
message("done")
plot( ks, rhos, type="l", lty=2, main="Ubeda-Flores (2022) Example 2.3",
xlab="Theta of W,W shuffle", ylab="Value")
lines(ks, wols, lty=1); lines(ks, wols - rhos, lwd=3)
legend(0.3, 0.9, c("|Spearman Rho|", "Schweizer and Wolff Sigma",
"Schweizer and Wolff Sigma minus |Spearman Rho|"),
bty="n", lty=c(2,1,1), lwd=c(1,1,2), cex=0.8)
abline(v=1/2 + c(-1, +1) * sqrt(3)/6, lty=4) # Ubeda-Flores (2022)
## End(Not run)
## Not run:
n <- 1000; UV <- simCOP(n=n, cop=N4212cop, para=7.53, graphics=FALSE, seed=1)
wolfCOP(cop=N4212cop, para=7.53) # 0.9884666 (theoretical)
wolfCOP(para=UV, as.sample=TRUE, usefastgrid=TRUE ) # 0.9873274 ( sample )
wolfCOP(para=UV, as.sample=TRUE, usefastgrid=FALSE) # 0.9873274 ( sample )
wolfCOPsamc(para=UV)$estimates[2] # 0.9861096 (Monte Carlo integration)
# Much faster as the sample size gets large to use the usefastgrid as shown
# in lieu of using the Monte Carlo. The wolfCOPsamc() was developed before
# the algorithm used for EMPIRgrid_fast and used by wolfCOP() was found.
## End(Not run)
## Not run:
# Redo D from Note section above
para <- list(cop1 =PLcop, cop2=PLcop,
para1=0.145, para2=21.9, alpha=0.81, beta=0.22)
D <- simCOP(n=300, cop=composite2COP, para=para,
cex=0.5, col=rgb(0, 0, 0, 0.2), pch=16)
PQD <- isCOP.PQD(cop=composite2COP, para=para, uv=D)
the.grid <- EMPIRgrid(para=D)
the.persp <- persp(the.grid$empcop, theta=-25, phi=20, shade=TRUE,
xlab="U VARIABLE", ylab="V VARIABLE", zlab="COPULA C(u,v)")
empcop <- EMPIRcopdf(para=D) # data.frame of all points
points(trans3d(empcop$u, empcop$v, empcop$empcop, the.persp), cex=0.7,
col=rgb(0, 1-sqrt(empcop$empcop), 1, sqrt(empcop$empcop)), pch=16)
points(trans3d(empcop$u, empcop$v, empcop$empcop, the.persp),
col=PQD$local.PQD+1, pch=1)
layout(matrix(c(1,2,3,4), 2, 2, byrow = TRUE), respect = TRUE)
PQD.NQD.cop <- gridCOP(cop=composite2COP, para=para)
Pi <- gridCOP(cop=P)
RHO <- PQD.NQD.cop - Pi; SIG <- abs(RHO)
the.persp <- persp(PQD.NQD.cop, theta=-25, phi=20, shade=TRUE, cex=0.5,
xlab="U VARIABLE", ylab="V VARIABLE", zlab="COPULA C(u,v)")
mtext("The Copula that has local PQD and NQD", cex=0.5)
the.persp <- persp(Pi, theta=-25, phi=20, shade=TRUE, cex=0.5,
xlab="U VARIABLE", ylab="V VARIABLE", zlab="COPULA C(u,v)")
mtext("Independence (Pi)", cex=0.5)
the.persp <- persp(RHO, theta=-25, phi=20, shade=TRUE, cex=0.5,
xlab="U VARIABLE", ylab="V VARIABLE", zlab="COPULA C(u,v)")
mtext("Copula delta: Integrand of Spearman Rho", cex=0.5)
the.persp <- persp(SIG, theta=-25, phi=20, shade=TRUE, cex=0.5,
xlab="U VARIABLE", ylab="V VARIABLE", zlab="COPULA C(u,v)")
mtext("abs(Copula delta): Integrand of Schweizer-Wolff Sigma", cex=0.5) #
## End(Not run)
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| wolfCOP | R Documentation |
Schweizer and Wolff Sigma of a Copula
Description
Compute the measure of association known as Schweizer–Wolff Sigma \sigma_\mathbf{C} of a copula according to Nelsen (2006, p. 209) by
\sigma_\mathbf{C} = 12\int\!\!\int_{\mathcal{I}^2} \bigl|\mathbf{C}(u,v) - uv\bigr|\,\mathrm{d}u\mathrm{d}v\mbox{,}
which is 0 \le \sigma_\mathbf{C} \le 1. It is obvious that this measure of association, without the positive sign restriction, is similar to the following form of Spearman Rho (rhoCOP) of a copula:
\rho_\mathbf{C} = 12\int\!\!\int_{\mathcal{I}^2} \bigl[\mathbf{C}(u,v) - uv\bigr]\,\mathrm{d}u\mathrm{d}v\mbox{.}
If a copula is positively quadrant dependent (PQD, see isCOP.PQD), then \sigma_\mathbf{C} = \rho_\mathbf{C}; conversely if a copula is negatively quadrant dependent (NQD), then \sigma_\mathbf{C} = -\rho_\mathbf{C}. However, a feature making \sigma_\mathbf{C} especially attractive is that for random variables X and Y, which are not PQD or NQD—copulas that are neither larger nor smaller than \mathbf{\Pi}—is that “\sigma_\mathbf{C} is often a better measure of [dependency] than \rho_\mathbf{C}” (Nelsen, 2006, p. 209).
Úbeda-Flores (2022) study the maximum difference between \sigma_\mathbf{C} - |\rho_\mathbf{C}| \approx 0.60496 for a special class of singular copula (see Examples), but states that it “remains an open problem to check if, for any copula \mathbf{C}_\Theta,” that the maximum listed here is the maximum difference possible.
Li et al. (2019) compare \sigma_\mathbf{C} (they label it \delta^\mathrm{E}_i, “the extended delta index”) to alternative measures of dependency with emphasis on information content. Those authors also show that dependence can also be measured by
\delta_i = \frac{1}{2}\int\!\!\int_{\mathcal{I}^2} \bigl|c(u,v) - 1\bigr|\,\mathrm{d}u\mathrm{d}v\mbox{,}
where c(u,v) is a copula density (densityCOP). Li et al. (2019, p. 45) “measure how much the model output Y and the input X_i are dependent,” and “importance of the inputs globally and
quantitatively with model free and moment independent properties.”
By definition for two random variables U and V with cumulative distribution functions of F(u) and F(v), the variables are independent, if and only if, for all subsets (conditions) selecting values u and y of U \le u and V \le v, when for all u and v, the combined cumulative distribution function is
H(x, y) = F(x)G(x) = \mathbf{C}(u,v) = uv\mbox{.}
Note, structural similarity between \sigma_\mathbf{C} and the definition for LzCOPpermsym. It seems to follow, by intuition, if therein, the copula on the left were to be replaced with the permutation symmetric (exchangeable symmetric) copula isCOP.permsym, such as \mathbf{\Pi} = uv = vu, then the copula on the right, now being, without loss of generality, \mathbf{C}(u,v) or \mathbf{C}(v,u), then \sigma_\mathbf{C} must respond partly to dependence existing in the presence of permutation asymmetry.
Usage
wolfCOP(cop=NULL, para=NULL, as.sample=FALSE, brute=FALSE, delta=0.002,
nlarge=500, usefastgrid=TRUE, ...)
Arguments
cop |
A copula function; |
para |
Vector of parameters or other data structure, if needed, to pass to the copula; |
as.sample |
A logical controlling whether an optional R |
brute |
Should brute force be used instead of two nested |
delta |
The |
nlarge |
A large sample size for which to consider the fast-grid computation if |
usefastgrid |
A logical controlling whether the fast-grid computation is acceptable to the user; and |
... |
Additional arguments to pass to the copula itself, to |
Details
A natural estimator for \sigma_\mathbf{C} is through the empirical copula (Póczos et al., 2015) and can be computed as
\hat{\sigma}_\mathbf{C} = \frac{12}{n^2 - 1}
\sum_{i=1}^n\sum_{j=1}^n \bigg|\hat{\mathbf{C}}_n\biggl(\frac{i}{n}, \frac{j}{n}\biggr) -
\frac{i}{n}\times\frac{j}{n}\bigg|\mbox{,}
where \hat{\mathbf{C}}_n is the simplest empirical copula of
\hat{\mathbf{C}}_n\biggl(\frac{i}{n}, \frac{j}{n}\biggr) =
\frac{1}{n}\{\# \mathrm{\ of\ } (U_k \le U_i, V_k \le V_j)\}
The sample version of \hat\sigma_\mathbf{C} is triggered by wolfCOP(para=D, as.sample=TRUE) for the double summation if usefastgrid is false and or usefastgrid is true and the same size is smaller than nlarge. A Monte Carlo integration version potentially suitable to very large sample sizes or difficult to integrate copula families is located at wolfCOPsamc, but the function itself is likely of limited general use because of the implementation of EMPIRgrid_fast inside wolfCOP. It is upon the fast estimator listed above that the extensive simulations leading to wolfCOPtest are based. In general, there are several asymptotically equivalent definitions of the empirical copula and some of which are implemented in EMPIRcop.
Consideration of speed performance between the as.sample=TRUE for the sample estimator argued by Póczos et al., 2015) (refer to section Note), the EMPIRgrid_fast is measurably faster by about n \ge 500. The following code shows numerical equivalence between these estimators, but use of other empirical copula will have divergence unless samples are large:
"MYwolfCOPgrid" <- function(uv, ctype="1/n", ...) {
ef <- EMPIRgrid_fast(para=uv, gridonly=FALSE, ctype=ctype)
pf <- ef$empcop; n <- nrow(uv)
gu <- as.numeric( rownames(ef$empcop) ) # gv <- gu (can compute w/o gv)
for(i in seq_len(nrow(pf))) pf[i,] <- gu * gu[i]
return(12 * sum(abs(ef$empcop - pf)) / (n^2 - 1))
}
ns <- as.integer( 10^seq(1, 3.3, by=0.1) ) # some sample sizes
WG <- WS <- vector(mode="numeric", length(ns))
for(i in seq_len(length(ns))) {
uv <- simCOP(n=ns[i], cop=P, graphics=FALSE)
WG[i] <- wolfCOP( para=uv, as.sample=TRUE, usefastgrid=FALSE)
WS[i] <- MYwolfCOPgrid(uv)
}
opts <- par(xpd=NA, no.readonly=FALSE)
df <- data.frame(n=ns, wolf_grid=WG, wolf_sample=WS)
plot(df$n, df$wolf_sample, type="n", xlim=c(10, 10000), ylim=c(0, 0.60),
xaxs="i", yaxs="i", log="x", las=1,
xlab="Sample size of Independence copula",
ylab="Schweizer-Wolff Sigma")
points(df$n, df$wolf_sample, pch=21, bg="white")
points(df$n, df$wolf_grid, pch=16, cex=0.6)
par(opts) #
Value
The value for \sigma_\mathbf{C} is returned, and the sample \hat\sigma_\mathbf{C} computation versions have names() attached to verify to the user which subsystem was used given the nlarge and usefastgrid arguments.
Note
An extended example is informative. First, declare a composite of two different Plackett copulas (PLcop) and simulate a few hundred values. The parameter settings are such that the data will attain a look of independence.:
para <- list(cop1=PLcop, para1=0.145, alpha=0.81,
cop2=PLcop, para2=21.90, beta=0.22)
D <- simCOP(n=300, cop=composite2COP, para=para,
cex=0.5, col=grey(0, 0.5), pch=16, seed=15)
Second, show that this copula is globally PQD (isCOP.PQD), but there is a substantial local NQD part of \mathcal{I}^2 space that clearly is NQD.
cols <- c("red3", "green4")
PQD <- isCOP.PQD(cop=composite2COP, para=para, uv=D)
message(PQD$global.PQD) # TRUE
points(D, col=cols[1+as.numeric(PQD$local.PQD)], lwd=2) # TRUE are green
This composited copula intersects, that is, passes through, the P copula. By the logic of Nelsen (2006), then the \sigma_\mathbf{C} should be larger than \rho_\mathbf{C} as shown below
wolfCOP(cop=composite2COP, para=para) # 0.08373378 (theoretical) rhoCOP(cop=composite2COP, para=para) # 0.02845131 (theoretical) hoefCOP(cop=composite2COP, para=para) # 0.08563823 (theoretical)
In fact, output above also shows Schweizer–Wolff Sigma to be larger than Blomqvist Beta (blomCOP), Gini Gamma (giniCOP), and Kendall Tau (tauCOP). Schweizer–Wolff Sigma captures the fact that although the symbols seems to plot randomly on the figure, symbol coloring for PQD and NQD clearly shows local dependency differences.
Author(s)
W.H. Asquith
References
Li, L. Liu, Y., and Lu, Z., 2019, A new dependence measure for importance analysis—Application to an environmental model: Applied Mathematical Modelling, v. 74, pp. 43–61, \Sexpr[results=rd]{tools:::Rd_expr_doi("10.1016/j.apm.2019.04.046")}.
Nelsen, R.B., 2006, An introduction to copulas: New York, Springer, 269 p.
Póczos, B., Krishner, S., Pál, D., Szepesvári, C., and Schneider, J., 2015, Robust nonparametric copula based dependence estimators, accessed on August 11, 2015, at https://www.cs.cmu.edu/~bapoczos/articles/poczos11nipscopula.pdf.
Schweizer, B., and Wolff, E.F., 1981, On nonparametric measures of dependence for random variables: The Annuals of Statistics, v. 9, no. 4, pp. 879–855, https://www.jstor.org/stable/2240856.
Úbeda-Flores, M., 2022, A note on the maximum difference between Schweizer and Wolff's Sigma and the absolute value of Spearman's Rho: Transactions on Fuzzy Sets and Systems, v. 1, no. 2, pp. 1–5, \Sexpr[results=rd]{tools:::Rd_expr_doi("10.30495/tfss.2022.1956899.1024")}.
See Also
wolfCOPsamc, wolfCOPtest
blomCOP, footCOP, giniCOP,
hoefCOP, rhoCOP, tauCOP,
joeskewCOP, uvlmoms, ORDSUMcop
Examples
wolfCOP(cop=PSP) # 0.4784176
## Not run:
# Maximum Difference between Sigma and Absolute Spearman Rho
k <- 1/2 + sqrt(3)/6 # Ubeda-Flores (2022) or 1/2 - sqrt(3)/6
para <- list(cop=c(W, W), para=NULL, part=c(0, k, 1))
rho <- rhoCOP( ORDSUMcop, para=para)
wol <- wolfCOP(ORDSUMcop, para=para)
message("Wolf - |Rho| = ", wol - abs(rho))
# 0.604967382.... that is approx. 0.60496 given by Ubeda-Flores (2022).
ks <- seq(0.01, 0.99, by=0.002); rhos <- wols <- NULL
for(k in ks) {
message(k, "-", appendLF=FALSE)
para <- list(cop=c(W, W), para=NULL, part=c(0, k, 1))
rhos <- c(rhos, abs( rhoCOP( cop=ORDSUMcop, para=para, subdivisions=200) ))
wols <- c(wols, wolfCOP(cop=ORDSUMcop, para=para, subdivisions=200) )
}
message("done")
plot( ks, rhos, type="l", lty=2, main="Ubeda-Flores (2022) Example 2.3",
xlab="Theta of W,W shuffle", ylab="Value")
lines(ks, wols, lty=1); lines(ks, wols - rhos, lwd=3)
legend(0.3, 0.9, c("|Spearman Rho|", "Schweizer and Wolff Sigma",
"Schweizer and Wolff Sigma minus |Spearman Rho|"),
bty="n", lty=c(2,1,1), lwd=c(1,1,2), cex=0.8)
abline(v=1/2 + c(-1, +1) * sqrt(3)/6, lty=4) # Ubeda-Flores (2022)
## End(Not run)
## Not run:
n <- 1000; UV <- simCOP(n=n, cop=N4212cop, para=7.53, graphics=FALSE, seed=1)
wolfCOP(cop=N4212cop, para=7.53) # 0.9884666 (theoretical)
wolfCOP(para=UV, as.sample=TRUE, usefastgrid=TRUE ) # 0.9873274 ( sample )
wolfCOP(para=UV, as.sample=TRUE, usefastgrid=FALSE) # 0.9873274 ( sample )
wolfCOPsamc(para=UV)$estimates[2] # 0.9861096 (Monte Carlo integration)
# Much faster as the sample size gets large to use the usefastgrid as shown
# in lieu of using the Monte Carlo. The wolfCOPsamc() was developed before
# the algorithm used for EMPIRgrid_fast and used by wolfCOP() was found.
## End(Not run)
## Not run:
# Redo D from Note section above
para <- list(cop1 =PLcop, cop2=PLcop,
para1=0.145, para2=21.9, alpha=0.81, beta=0.22)
D <- simCOP(n=300, cop=composite2COP, para=para,
cex=0.5, col=rgb(0, 0, 0, 0.2), pch=16)
PQD <- isCOP.PQD(cop=composite2COP, para=para, uv=D)
the.grid <- EMPIRgrid(para=D)
the.persp <- persp(the.grid$empcop, theta=-25, phi=20, shade=TRUE,
xlab="U VARIABLE", ylab="V VARIABLE", zlab="COPULA C(u,v)")
empcop <- EMPIRcopdf(para=D) # data.frame of all points
points(trans3d(empcop$u, empcop$v, empcop$empcop, the.persp), cex=0.7,
col=rgb(0, 1-sqrt(empcop$empcop), 1, sqrt(empcop$empcop)), pch=16)
points(trans3d(empcop$u, empcop$v, empcop$empcop, the.persp),
col=PQD$local.PQD+1, pch=1)
layout(matrix(c(1,2,3,4), 2, 2, byrow = TRUE), respect = TRUE)
PQD.NQD.cop <- gridCOP(cop=composite2COP, para=para)
Pi <- gridCOP(cop=P)
RHO <- PQD.NQD.cop - Pi; SIG <- abs(RHO)
the.persp <- persp(PQD.NQD.cop, theta=-25, phi=20, shade=TRUE, cex=0.5,
xlab="U VARIABLE", ylab="V VARIABLE", zlab="COPULA C(u,v)")
mtext("The Copula that has local PQD and NQD", cex=0.5)
the.persp <- persp(Pi, theta=-25, phi=20, shade=TRUE, cex=0.5,
xlab="U VARIABLE", ylab="V VARIABLE", zlab="COPULA C(u,v)")
mtext("Independence (Pi)", cex=0.5)
the.persp <- persp(RHO, theta=-25, phi=20, shade=TRUE, cex=0.5,
xlab="U VARIABLE", ylab="V VARIABLE", zlab="COPULA C(u,v)")
mtext("Copula delta: Integrand of Spearman Rho", cex=0.5)
the.persp <- persp(SIG, theta=-25, phi=20, shade=TRUE, cex=0.5,
xlab="U VARIABLE", ylab="V VARIABLE", zlab="COPULA C(u,v)")
mtext("abs(Copula delta): Integrand of Schweizer-Wolff Sigma", cex=0.5) #
## End(Not run)
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