isCOP.radsym: Is a Copula Radially Symmetric
In copBasic: General Bivariate Copula Theory and Many Utility Functions
isCOP.radsym R Documentation
Is a Copula Radially Symmetric
Description
Numerically set a logical whether a copula is radially symmetric (Nelsen, 2006, p. 37) [reflection symmetric, Joe (2014, p. 64)] (refer also to permutation symmetric, isCOP.permsym). A copula \mathbf{C}(u,v) is radially symmetric if and only if for any \{u,v\} \in [0,1] either of the following hold
\mathbf{C}(u,v) = u + v - 1 + \mathbf{C}(1-u, 1-v)
or
u + v - 1 + \mathbf{C}(1-u, 1-v) - \mathbf{C}(u,v) \equiv 0\mbox{.}
Thus, if the equality of the copula \mathbf{C}(u,v) = \hat{\mathbf{C}}(u,v) (the survival copula), then radial symmetry exists: COP = surCOP or \mathbf{C}(u,v) = \hat{\mathbf{C}}(1-u,1-v). The computation is (can be) CPU intensive.
Usage
isCOP.radsym(cop=NULL, para=NULL, delta=0.005, tol=1e-4, ...)
Arguments
cop
A copula function;
para
Vector of parameters, if needed, to pass to the copula;
delta
The increments of \{u,v\} \mapsto [0+\Delta\delta, 1-\Delta\delta, \Delta\delta];
tol
A tolerance on the check for symmetry, default 1 part in 10,000, which is the test for the \equiv 0 (zero equivalence, see source code); and
...
Additional arguments to pass to the copula or derivative of a copula function.
Value
A logical TRUE or FALSE is returned.
Note
L-comoments and Radial and Permutation Symmetry — A natural question exists: Is a radially symmetric copula characterized by the L-comoments for orders r{\ge}3 as having values of zero? Apparently, yes, and review of copula literature does not seem to directly address this question. Let us consider the two symmetrical copulas: the parameterless \mathbf{PSP}(u,v) (see PSP) and the single parameter \mathbf{PL}(u,v; \Theta) (see PLcop) with the \Theta_\mathbf{PL} = 4.708664 (rhoCOP). The two copulas have different radial symmetries as shown below.
J <- PLACKETTpar(rho=rhoCOP(cop=PSP)) # Spearman Rho = 0.4784176
isCOP.radsym( cop=PSP) # FALSE
isCOP.radsym( cop=PLcop, para=J) # TRUE
UV <- simCOP(300, cop=PSP, para=J, ploton=TRUE, pch= 1, col="blue", seed=1)
UV <- simCOP(300, cop=PLcop, para=J, ploton=FALSE, pch=16, col="red" , seed=1)
Now, let us compute the L-comoments (\tau_k^{[12]|[21]}) (lcomCOP) from each copula in the code that follows. The L-correlations (Spearman Rho) are each about 0.478, which agree with the given \rho_\mathbf{C}. The L-coskews (T3_PSP[12,21]) for the \mathbf{PSP} are about -0.129 with those for the \mathbf{PL} copula at zero. The \mathbf{PSP} L-cokurtoses (T4_PSP[12,21]) are about 0.047, whereas those for the \mathbf{PL} are zero. The [12], [21] equalities result from permutation symmetry (isCOP.permsym), but the nonzero L-cokurtoses for the \mathbf{PSP} result from radial asymmetry.
lcmPSP <- lcomCOP(cop=PSP)
lcmPLA <- lcomCOP(cop=PLcop, para=J)
lcmPSP <- lapply(lcmPSP, function(i) round(i, digits=5))
lcmPLA <- lapply(lcmPLA, function(i) round(i, digits=5))
message( # L-correlations (T2), L-coskews (T3), L-cokurtoses (T4)
"T2[12]_PSP == T2_PSP[21] =", lcmPSP$lcomUV[2], "=", lcmPSP$lcomVU[2], "\n",
"T2[12]_PLA == T2_PLA[21] =", lcmPLA$lcomUV[2], "=", lcmPLA$lcomVU[2], "\n",
"T3[12]_PSP == T3_PSP[21] =", lcmPSP$lcomUV[3], "=", lcmPSP$lcomVU[3], "\n",
"T3[12]_PLA == T3_PLA[21] =", lcmPLA$lcomUV[3], "=", lcmPLA$lcomVU[3], "\n",
"T4[12]_PSP == T4_PSP[21] =", lcmPSP$lcomUV[4], "=", lcmPSP$lcomVU[4], "\n",
"T4[12]_PLA == T4_PLA[21] =", lcmPLA$lcomUV[4], "=", lcmPLA$lcomVU[4], "\n")
Now, isCOP.radsym() is false for all rotations (reflections) of the \mathbf{PSP}, which requires that in magnitude that the L-coskews are equal but with reflections 3 and 4 rotating the coupla to a \rho_\mathbf{C} = \tau_2^{[12]} = \tau_2^{[21]} = -0.478. The L-cokurtoses are all equal with a change in sign with reflections 3 and 4. The \mathbf{PSP} inherently has a bilateral symmetry about any diagonal, which makes the r{\ge}3 L-comoments as mutual equals in magnitude.
lcmPSP1 <- lcomCOP(cop=COP, para=list(cop=PSP, reflect=1)) # default
lcmPSP2 <- lcomCOP(cop=COP, para=list(cop=PSP, reflect=2)) # 180 degrees
lcmPSP3 <- lcomCOP(cop=COP, para=list(cop=PSP, reflect=3)) # -90 degrees
lcmPSP4 <- lcomCOP(cop=COP, para=list(cop=PSP, reflect=4)) # +90 degrees
lcmPSP1 <- lapply(lcmPSP1, function(i) round(i, digits=5))
lcmPSP2 <- lapply(lcmPSP2, function(i) round(i, digits=5))
lcmPSP3 <- lapply(lcmPSP3, function(i) round(i, digits=5))
lcmPSP4 <- lapply(lcmPSP4, function(i) round(i, digits=5))
message( # L-coskews (T3), reflections 1~, 2~, 3~, 4~
"1~T3[12] == T3[21] = ", lcmPSP1$lcomUV[3], " = ", lcmPSP1$lcomVU[3], "\n",
"2~T3[12] == T3[21] = ", lcmPSP2$lcomUV[3], " = ", lcmPSP2$lcomVU[3], "\n",
"3~T3[12] != T3[21] = ", lcmPSP3$lcomUV[3], " = ", lcmPSP3$lcomVU[3], "\n",
"4~T3[12] != T3[21] = ", lcmPSP4$lcomUV[3], " = ", lcmPSP4$lcomVU[3], "\n")
# 1*T3[12] == T3[21] = -0.12949 = -0.12949
# 2*T3[12] == T3[21] = 0.12949 = 0.12949
# 3*T3[12] != T3[21] = 0.12949 = -0.12949
# 4*T3[12] != T3[21] = -0.12949 = 0.12949
message( # L-cokurtoses (T4), reflections 1~, 2~, 3~, 4~
"1~T4[12] == T3[21] =", lcmPSP1$lcomUV[4], " = ", lcmPSP1$lcomVU[4], "\n",
"2~T4[12] == T3[21] =", lcmPSP2$lcomUV[4], " = ", lcmPSP2$lcomVU[4], "\n",
"3~T4[12] == T3[21] =", lcmPSP3$lcomUV[4], " = ", lcmPSP3$lcomVU[4], "\n",
"4~T4[12] == T3[21] =", lcmPSP4$lcomUV[4], " = ", lcmPSP4$lcomVU[4], "\n")
# 1~T4[12] == T3[21] = 0.04677 = 0.04677
# 2~T4[12] == T3[21] = 0.04677 = 0.04677
# 3~T4[12] == T3[21] = -0.04677 = -0.04677
# 4~T4[12] == T3[21] = -0.04677 = -0.04677
Author(s)
W.H. Asquith
References
Nelsen, R.B., 2006, An introduction to copulas: New York, Springer, 269 p.
Salvadori, G., De Michele, C., Kottegoda, N.T., and Rosso, R., 2007, Extremes in Nature—An approach using copulas: Springer, 289 p.
See Also
isCOP.permsym, lcomCOP
Examples
# Radially symmetry is computationally intensive and relies on a gridded [0,1]x[0,1]
# space and laborious check on equality. Thus these examples are commented out for the
# R --timings check. Note, proof of radial symmetry absent of algebraic manipulation
# or verification is difficult and subject to the grid fineness to find a nonequality
# from which to immediately conclude FALSE. L-comoments could be used for simultaneous
# radial and permutation symmetry assessment and even beyond those by detection of
# no bilateral symmetry from any perspective (refer to the Note section above).
## Not run:
isCOP.radsym(cop=P) # TRUE
para <- list(cop1=PLcop, cop2=M, para1=c(.03), para2=NA, alpha=0.8, beta=0.5)
isCOP.radsym( composite2COP, para=para) # FALSE
isCOP.permsym( composite2COP, para=para) # FALSE
lcm <- lcomCOP(cop=composite2COP, para=para)
lcm <- lapply(lcm, function(i) round(i, digits=5))
message(" SpearmanRho, Lcoskew, Lcokurt\n",
"Tk[12] ", paste(lcm$lcomUV[2:4], collapse=", "), "\n",
"Tk[21] ", paste(lcm$lcomVU[2:4], collapse=", "))
# SpearmanRho, Lcoskew, Lcokurt # Magitudes for L-comoment k>=3
# Tk[12] -0.25236, 0.22828, 0.01796 # are not equal because there is
# Tk[21] -0.25236, 0.17612, -0.05165 # no bilateral symmetry.
## End(Not run)
## Not run:
set.seed(120); theta <- 2
gh <- simCOP(n=34, cop=GHcop, para=theta, ploton=FALSE, points=FALSE) * 150
# Pretend gh is real data, the * 150 is to clearly get into an arbitrary unit system.
# The sort=FALSE is critical in the following two calls
fakeU <- lmomco::pp(gh[,1], sort=FALSE) # Weibull plotting position i/(n+1)
fakeV <- lmomco::pp(gh[,2], sort=FALSE) # Weibull plotting position i/(n+1)
uv <- data.frame(U=fakeU, V=fakeV); # our U-statistics
isCOP.radsym(cop=EMPIRcop, para=uv) # FALSE
isCOP.LTD(cop=EMPIRcop, para=uv) # TRUE
isCOP.RTI(cop=EMPIRcop, para=uv) # FALSE
isCOP.PQD(cop=EMPIRcop, para=uv, # TRUE
empirical=TRUE) # need to set this true to avoid integrate()
# Blomqvist's Beta = 0.2941
# Gini's Gamma = 0.5606
# Spearman's Rho = 0.6584
# Kendall's Tau = 0.5045
isCOP.radsym(cop=GHcop, para=theta) # FALSE
isCOP.LTD( cop=GHcop, para=theta) # TRUE
isCOP.RTI( cop=GHcop, para=theta) # TRUE
isCOP.PQD( cop=GHcop, para=theta) # TRUE
# Blomqvist's Beta = 0.5009
# Gini's Gamma = 0.5591
# Spearman's Rho = 0.6822
# Kendall's Tau = 0.5000
# Notice that isCOP.RTI is not the same for empirical and theoretical.
# This shows the difficulty in tail dependence parameter estimation for
# small samples (refer also to Salvadori et al., 2007 p. 175).
## End(Not run)
copBasic documentation built on July 23, 2026, 1:07 a.m.
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| isCOP.radsym | R Documentation |
Is a Copula Radially Symmetric
Description
Numerically set a logical whether a copula is radially symmetric (Nelsen, 2006, p. 37) [reflection symmetric, Joe (2014, p. 64)] (refer also to permutation symmetric, isCOP.permsym). A copula \mathbf{C}(u,v) is radially symmetric if and only if for any \{u,v\} \in [0,1] either of the following hold
\mathbf{C}(u,v) = u + v - 1 + \mathbf{C}(1-u, 1-v)
or
u + v - 1 + \mathbf{C}(1-u, 1-v) - \mathbf{C}(u,v) \equiv 0\mbox{.}
Thus, if the equality of the copula \mathbf{C}(u,v) = \hat{\mathbf{C}}(u,v) (the survival copula), then radial symmetry exists: COP = surCOP or \mathbf{C}(u,v) = \hat{\mathbf{C}}(1-u,1-v). The computation is (can be) CPU intensive.
Usage
isCOP.radsym(cop=NULL, para=NULL, delta=0.005, tol=1e-4, ...)
Arguments
cop |
A copula function; |
para |
Vector of parameters, if needed, to pass to the copula; |
delta |
The increments of |
tol |
A tolerance on the check for symmetry, default 1 part in 10,000, which is the test for the |
... |
Additional arguments to pass to the copula or derivative of a copula function. |
Value
A logical TRUE or FALSE is returned.
Note
L-comoments and Radial and Permutation Symmetry — A natural question exists: Is a radially symmetric copula characterized by the L-comoments for orders r{\ge}3 as having values of zero? Apparently, yes, and review of copula literature does not seem to directly address this question. Let us consider the two symmetrical copulas: the parameterless \mathbf{PSP}(u,v) (see PSP) and the single parameter \mathbf{PL}(u,v; \Theta) (see PLcop) with the \Theta_\mathbf{PL} = 4.708664 (rhoCOP). The two copulas have different radial symmetries as shown below.
J <- PLACKETTpar(rho=rhoCOP(cop=PSP)) # Spearman Rho = 0.4784176 isCOP.radsym( cop=PSP) # FALSE isCOP.radsym( cop=PLcop, para=J) # TRUE UV <- simCOP(300, cop=PSP, para=J, ploton=TRUE, pch= 1, col="blue", seed=1) UV <- simCOP(300, cop=PLcop, para=J, ploton=FALSE, pch=16, col="red" , seed=1)
Now, let us compute the L-comoments (\tau_k^{[12]|[21]}) (lcomCOP) from each copula in the code that follows. The L-correlations (Spearman Rho) are each about 0.478, which agree with the given \rho_\mathbf{C}. The L-coskews (T3_PSP[12,21]) for the \mathbf{PSP} are about -0.129 with those for the \mathbf{PL} copula at zero. The \mathbf{PSP} L-cokurtoses (T4_PSP[12,21]) are about 0.047, whereas those for the \mathbf{PL} are zero. The [12], [21] equalities result from permutation symmetry (isCOP.permsym), but the nonzero L-cokurtoses for the \mathbf{PSP} result from radial asymmetry.
lcmPSP <- lcomCOP(cop=PSP) lcmPLA <- lcomCOP(cop=PLcop, para=J) lcmPSP <- lapply(lcmPSP, function(i) round(i, digits=5)) lcmPLA <- lapply(lcmPLA, function(i) round(i, digits=5)) message( # L-correlations (T2), L-coskews (T3), L-cokurtoses (T4) "T2[12]_PSP == T2_PSP[21] =", lcmPSP$lcomUV[2], "=", lcmPSP$lcomVU[2], "\n", "T2[12]_PLA == T2_PLA[21] =", lcmPLA$lcomUV[2], "=", lcmPLA$lcomVU[2], "\n", "T3[12]_PSP == T3_PSP[21] =", lcmPSP$lcomUV[3], "=", lcmPSP$lcomVU[3], "\n", "T3[12]_PLA == T3_PLA[21] =", lcmPLA$lcomUV[3], "=", lcmPLA$lcomVU[3], "\n", "T4[12]_PSP == T4_PSP[21] =", lcmPSP$lcomUV[4], "=", lcmPSP$lcomVU[4], "\n", "T4[12]_PLA == T4_PLA[21] =", lcmPLA$lcomUV[4], "=", lcmPLA$lcomVU[4], "\n")
Now, isCOP.radsym() is false for all rotations (reflections) of the \mathbf{PSP}, which requires that in magnitude that the L-coskews are equal but with reflections 3 and 4 rotating the coupla to a \rho_\mathbf{C} = \tau_2^{[12]} = \tau_2^{[21]} = -0.478. The L-cokurtoses are all equal with a change in sign with reflections 3 and 4. The \mathbf{PSP} inherently has a bilateral symmetry about any diagonal, which makes the r{\ge}3 L-comoments as mutual equals in magnitude.
lcmPSP1 <- lcomCOP(cop=COP, para=list(cop=PSP, reflect=1)) # default lcmPSP2 <- lcomCOP(cop=COP, para=list(cop=PSP, reflect=2)) # 180 degrees lcmPSP3 <- lcomCOP(cop=COP, para=list(cop=PSP, reflect=3)) # -90 degrees lcmPSP4 <- lcomCOP(cop=COP, para=list(cop=PSP, reflect=4)) # +90 degrees lcmPSP1 <- lapply(lcmPSP1, function(i) round(i, digits=5)) lcmPSP2 <- lapply(lcmPSP2, function(i) round(i, digits=5)) lcmPSP3 <- lapply(lcmPSP3, function(i) round(i, digits=5)) lcmPSP4 <- lapply(lcmPSP4, function(i) round(i, digits=5)) message( # L-coskews (T3), reflections 1~, 2~, 3~, 4~ "1~T3[12] == T3[21] = ", lcmPSP1$lcomUV[3], " = ", lcmPSP1$lcomVU[3], "\n", "2~T3[12] == T3[21] = ", lcmPSP2$lcomUV[3], " = ", lcmPSP2$lcomVU[3], "\n", "3~T3[12] != T3[21] = ", lcmPSP3$lcomUV[3], " = ", lcmPSP3$lcomVU[3], "\n", "4~T3[12] != T3[21] = ", lcmPSP4$lcomUV[3], " = ", lcmPSP4$lcomVU[3], "\n") # 1*T3[12] == T3[21] = -0.12949 = -0.12949 # 2*T3[12] == T3[21] = 0.12949 = 0.12949 # 3*T3[12] != T3[21] = 0.12949 = -0.12949 # 4*T3[12] != T3[21] = -0.12949 = 0.12949 message( # L-cokurtoses (T4), reflections 1~, 2~, 3~, 4~ "1~T4[12] == T3[21] =", lcmPSP1$lcomUV[4], " = ", lcmPSP1$lcomVU[4], "\n", "2~T4[12] == T3[21] =", lcmPSP2$lcomUV[4], " = ", lcmPSP2$lcomVU[4], "\n", "3~T4[12] == T3[21] =", lcmPSP3$lcomUV[4], " = ", lcmPSP3$lcomVU[4], "\n", "4~T4[12] == T3[21] =", lcmPSP4$lcomUV[4], " = ", lcmPSP4$lcomVU[4], "\n") # 1~T4[12] == T3[21] = 0.04677 = 0.04677 # 2~T4[12] == T3[21] = 0.04677 = 0.04677 # 3~T4[12] == T3[21] = -0.04677 = -0.04677 # 4~T4[12] == T3[21] = -0.04677 = -0.04677
Author(s)
W.H. Asquith
References
Nelsen, R.B., 2006, An introduction to copulas: New York, Springer, 269 p.
Salvadori, G., De Michele, C., Kottegoda, N.T., and Rosso, R., 2007, Extremes in Nature—An approach using copulas: Springer, 289 p.
See Also
isCOP.permsym, lcomCOP
Examples
# Radially symmetry is computationally intensive and relies on a gridded [0,1]x[0,1]
# space and laborious check on equality. Thus these examples are commented out for the
# R --timings check. Note, proof of radial symmetry absent of algebraic manipulation
# or verification is difficult and subject to the grid fineness to find a nonequality
# from which to immediately conclude FALSE. L-comoments could be used for simultaneous
# radial and permutation symmetry assessment and even beyond those by detection of
# no bilateral symmetry from any perspective (refer to the Note section above).
## Not run:
isCOP.radsym(cop=P) # TRUE
para <- list(cop1=PLcop, cop2=M, para1=c(.03), para2=NA, alpha=0.8, beta=0.5)
isCOP.radsym( composite2COP, para=para) # FALSE
isCOP.permsym( composite2COP, para=para) # FALSE
lcm <- lcomCOP(cop=composite2COP, para=para)
lcm <- lapply(lcm, function(i) round(i, digits=5))
message(" SpearmanRho, Lcoskew, Lcokurt\n",
"Tk[12] ", paste(lcm$lcomUV[2:4], collapse=", "), "\n",
"Tk[21] ", paste(lcm$lcomVU[2:4], collapse=", "))
# SpearmanRho, Lcoskew, Lcokurt # Magitudes for L-comoment k>=3
# Tk[12] -0.25236, 0.22828, 0.01796 # are not equal because there is
# Tk[21] -0.25236, 0.17612, -0.05165 # no bilateral symmetry.
## End(Not run)
## Not run:
set.seed(120); theta <- 2
gh <- simCOP(n=34, cop=GHcop, para=theta, ploton=FALSE, points=FALSE) * 150
# Pretend gh is real data, the * 150 is to clearly get into an arbitrary unit system.
# The sort=FALSE is critical in the following two calls
fakeU <- lmomco::pp(gh[,1], sort=FALSE) # Weibull plotting position i/(n+1)
fakeV <- lmomco::pp(gh[,2], sort=FALSE) # Weibull plotting position i/(n+1)
uv <- data.frame(U=fakeU, V=fakeV); # our U-statistics
isCOP.radsym(cop=EMPIRcop, para=uv) # FALSE
isCOP.LTD(cop=EMPIRcop, para=uv) # TRUE
isCOP.RTI(cop=EMPIRcop, para=uv) # FALSE
isCOP.PQD(cop=EMPIRcop, para=uv, # TRUE
empirical=TRUE) # need to set this true to avoid integrate()
# Blomqvist's Beta = 0.2941
# Gini's Gamma = 0.5606
# Spearman's Rho = 0.6584
# Kendall's Tau = 0.5045
isCOP.radsym(cop=GHcop, para=theta) # FALSE
isCOP.LTD( cop=GHcop, para=theta) # TRUE
isCOP.RTI( cop=GHcop, para=theta) # TRUE
isCOP.PQD( cop=GHcop, para=theta) # TRUE
# Blomqvist's Beta = 0.5009
# Gini's Gamma = 0.5591
# Spearman's Rho = 0.6822
# Kendall's Tau = 0.5000
# Notice that isCOP.RTI is not the same for empirical and theoretical.
# This shows the difficulty in tail dependence parameter estimation for
# small samples (refer also to Salvadori et al., 2007 p. 175).
## End(Not run)
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