lcomCOP {copBasic}

L-comoments and Bivariate L-moments of a Copula

Description

Compute the L-comoments (Serfling and Xiao, 2007; Asquith, 2011) through the bivariate L-moments (ratios) (\delta^{[\ldots]}_{k;\mathbf{C}}) of a copula \mathbf{C}(u,v; \Theta) The L-comoments include L-correlation (Spearman Rho), L-coskew, and L-cokurtosis. As described by Brahimi et al. (2015), the first four bivariate L-moments \delta^{[12]}_k for random variable X^{(1)} or U with respect to (wrt) random variable X^{(2)} or V are defined as

\delta^{[12]}_{1;\mathbf{C}} = 2\int\!\!\int_{\mathcal{I}^2} \mathbf{C}(u,v)\,\mathrm{d}u\mathrm{d}v - \frac{1}{2}\mbox{,}

\delta^{[12]}_{2;\mathbf{C}} = \int\!\!\int_{\mathcal{I}^2} (12v - 6) \mathbf{C}(u,v)\,\mathrm{d}u\mathrm{d}v - \frac{1}{2}\mbox{,}

\delta^{[12]}_{3;\mathbf{C}} = \int\!\!\int_{\mathcal{I}^2} (60v^2 - 60v + 12) \mathbf{C}(u,v)\,\mathrm{d}u\mathrm{d}v - \frac{1}{2}\mbox{, and}

\delta^{[12]}_{4;\mathbf{C}} = \int\!\!\int_{\mathcal{I}^2} (280v^3 - 420v^2 + 180v - 20) \mathbf{C}(u,v)\,\mathrm{d}u\mathrm{d}v - \frac{1}{2}\mbox{,}

where the bivariate L-moments are related to the L-comoment ratios by

6\delta^{[12]}_k = \tau^{[12]}_{k+1}\mbox{\quad and \quad}6\delta^{[21]}_k = \tau^{[21]}_{k+1}\mbox{,}

where in otherwords, “the third bivariate L-moment \delta^{[12]}_3 is one sixth the L-cokurtosis \tau^{[12]}_4.” The first four bivariate L-moments yield the first five L-comoments. The terms and nomenclature are not easy and also the English grammar adjective “ratios” is not always consistent in the literature. The \delta^{[\ldots]}_{k;\mathbf{C}} are ratios. The sample L-comoments are supported by the lmomco package, and in particular for the bivariate case, they are supported by the lcomoms2() function of that package.

Similarly, the \delta^{[21]}_k are computed by switching u \rightarrow v in the polynomials within the above integrals multiplied to the copula in the system of equations with u. In general, \delta^{[12]}_k \not= \delta^{[21]}_k for k > 1 unless in the case of permutation symmetric (isCOP.permsym) copulas. By theory, \delta^{[12]}_1 = \delta^{[21]}_1 = \rho_\mathbf{C}/6 where \rho_\mathbf{C} is the Spearman Rho rhoCOP.

The integral for \delta^{[12]}_{4;\mathbf{C}} does not appear in Brahimi et al. (2015) but this and the other forms are verified in the Examples and discussion in Note. The four k \in (1,2,3,4) for U wrt V and V wrt U comprise a full spectrum of system of seven (not eight) equations. One equation is lost because \delta^{[12]}_1 = \delta^{[21]}_1.

Chine and Benatia (2017) describe trimmed L-comoments as the multivariate extensions of the univariate trimmed L-moments (Elamir and Seheult, 2003) that are implemented in lmomco. These are not yet implemented in copBasic.

Usage

lcomCOP(cop=NULL, para=NULL, as.bilmoms=FALSE, orders=2:5, ...)

Arguments

cop

A copula function;

para

Vector of parameters or other data structure, if needed, to pass to the copula;

as.bilmoms

A logical to trigger return of the \delta_k and the return vectors will be named differently;

orders

The orders of the L-comoments to return, which is internally adjusted if the argument as.bilmoms is set. There is no first order L-comoment and the first index on returned values is set to NA to remain index consistent with the lmomco package. An order greater than 5 is not supported; and

...

Additional arguments to pass to the densityCOP function.

Value

An R list of the L-comoments or bivariate L-moments is returned depending on as.bilmoms setting.

bilmomUV

The bivariate L-moments \delta^{[12]}_k of U with respect to V for k \in [1,2,3,4] if orders is 2:5 and there is no NA index as for the L-comoments;

bilmomVU

The bivariate L-moments \delta^{[21]}_k of V with respect to U for k \in [1,2,3,4] if orders is 2:5 and there is no NA index as for the L-comoments;

lcomUV

The L-comoments \tau^{[12]}_k of V with respect to U for k \in [2,3,4,5] if orders is 2:5 and index 1 is NA; and

lcomVU

The L-comoments \tau^{[21]}_k of V with respect to U for k \in [2,3,4,5] if orders is 2:5 and index 1 is NA.

Note

The documention here is highly parallel to bilmoms for which that function was developed some years before lmomCOP was developed in January 2019. Also, bilmoms is based on gridded or Monte Carlo integration, and bilmoms is to be considered deprecated. However, it is deliberate that related background and various algorithm testing are still documented in bilmoms.

Author(s)

W.H. Asquith

References

Asquith, W.H., 2011, Distributional analysis with L-moment statistics using the R environment for statistical computing: Createspace Independent Publishing Platform, ISBN 978–146350841–8.

Brahimi, B., Chebana, F., and Necir, A., 2015, Copula representation of bivariate L-moments—A new estimation method for multiparameter two-dimensional copula models: Statistics, v. 49, no. 3, pp. 497–521.

Chine, Amel, and Benatia, Fatah, 2017, Bivariate copulas parameters estimation using the trimmed L-moments methods: Afrika Statistika, v. 12, no. 1, pp. 1185–1197.

Elamir, E.A.H, and Seheult, A.H., 2003, Trimmed L-moments: Computational Statistics and Data Analysis, v. 43, p. 299–314.

Nelsen, R.B., 2006, An introduction to copulas: New York, Springer, 269 p.

Serfling, R., and Xiao, P., 2007, A contribution to multivariate L-moments—L-comoment matrices: Journal of Multivariate Analysis, v. 98, pp. 1765–1781.

See Also

bilmoms, lcomCOPpv, uvlmoms

Examples

## Not run: 
para <- list(alpha=0.5, beta=0.93, para1=4.5, cop1=GLcop, cop2=PSP)
copBasic:::lcomCOP(cop=composite2COP, para=para)$lcomUV[3]
# Lcomom:T3[12]=  +0.156
copBasic:::lcomCOP(cop=composite2COP, para=para)$lcomVU[3]
# Lcomom:T3[21]=  -0.0668
bilmoms(cop=composite2COP, n=10000, para=para, sobol=TRUE)$bilcomoms$T3
# Tau3[12]=+0.1566, Tau3[21]=-0.0655
# The numerical default Monte Carlo integration of bilmoms()
# matches the numerical integration of lcomCOP albeit with a
# substantially slower and less elegant means in bilmoms().
## End(Not run)

## Not run: 
# The following Spearman Rho and L-coskew values are predicted for a monitoring
# location of the relation between peak streamflow (V) and time into the
# water year (U) where U and V are "U-statistics."
site_srho  <-  0.15536430
site_T3_12 <-  0.03866056
site_T3_21 <- -0.03090144

# Create an objective function for 3D optimization with some explicit intention that
# transforms are used to keep the parameters in acceptable parameter space for the
# alpha [0,1] and beta [0,1] and theta for the Plackett copula and the composite1COP
# used to add two more parameters to the Plackett.
ofunc <- function(par, srho=NA, T3_12=NA, T3_21=NA) {
  alpha <- pnorm(par[1]) # takes -Inf to +Inf ---> 0 to 1 # compositing domain
  beta  <- pnorm(par[2]) # takes -Inf to +Inf ---> 0 to 1 # compositing domain
  theta <-   exp(par[3]) # takes -Inf to +Inf ---> 0 to +Inf # Plackett domain
  lmr <- lcomCOP(cop=composite1COP,
                 para=list(alpha=alpha, beta=beta, para1=theta, cop1=PLcop))
  return((lmr$lcomUV[2] - srho )^2 + # look carefully, the 2, 3, 3 index
         (lmr$lcomUV[3] - T3_12)^2 + # use on the lmr list are correct, so do not
         (lmr$lcomVU[3] - T3_21)^2)  # expect to see 1, 2, 3 or 2, 3, 4.
}
# initial parameter guess ('middle' [0.5] compositing and independence [1]) and
# showing the transformations involved.
para_init <- c(qnorm(0.5), qnorm(0.5), log(1))
rt <- optim(par=para_init, ofunc,
            srho=site_srho, T3_12=site_T3_12, T3_21=site_T3_21)
lcom_para <- list(alpha=pnorm(rt$par[1]), beta=pnorm(rt$par[2]),
                  para1=exp(rt$par[3]), cop1=PLcop)
sUV <- simCOP(10000, cop=composite1COP, para=lcom_para, col=grey(0, 0.2), pch=16)
# Now as an exercise, consider increasing site_srho or negating it. Consider
# switching the signs on the L-coskews or increasing their magnitudes and study
# the resulting simulation to develop a personal feeling for L-coskew meaning. #
## End(Not run)