R Under development (unstable) (2023-12-04 r85659 ucrt) -- "Unsuffered Consequences"
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> ## Copyright (C) 2012 Marius Hofert, Ivan Kojadinovic, Martin Maechler, and Jun Yan
> ##
> ## This program is free software; you can redistribute it and/or modify it under
> ## the terms of the GNU General Public License as published by the Free Software
> ## Foundation; either version 3 of the License, or (at your option) any later
> ## version.
> ##
> ## This program is distributed in the hope that it will be useful, but WITHOUT
> ## ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or FITNESS
> ## FOR A PARTICULAR PURPOSE. See the GNU General Public License for more
> ## details.
> ##
> ## You should have received a copy of the GNU General Public License along with
> ## this program; if not, see <http://www.gnu.org/licenses/>.
> 
> require(copula)
Loading required package: copula
> source(system.file("Rsource", "utils.R", package="copula", mustWork=TRUE))
Loading required package: tools
> ##-> assertError()
> 
> ## fCopulae: don't do on CRAN, and really "can not" suggest fCopulae
> (tryfCop <- if(interactive())
+                TRUE # for interactive convenience
+             else ## when run as BATCH:
+                nzchar(Sys.getenv("R_copula_check_fCop")) ||
+                    identical("true", unname(Sys.getenv("R_MM_PKG_CHECKING"))))
[1] FALSE
> 
> if(tryfCop) { ## will only "work" if not "--as-cran"
+     .r <- require
+     tryfCop <- suppressWarnings(.r(fCopulae, quietly=TRUE))
+ }
> tryfCop
[1] FALSE
> 
> numTailIndexLower <- function(copula, u) {
+   ## u is a vector approaching 0
+   pCopula(cbind(u, u, deparse.level = 0), copula) / u
+ }
> 
> numTailIndexUpper <- function(copula, u) {
+   # u is a vector approaching 1
+   (1 - 2 * u + pCopula(cbind(u, u, deparse.level = 0), copula)) / (1 - u)
+ }
> 
> (u.0 <- sort(outer(c(1,2,5), 10^-(1:5)), decreasing=TRUE)[-(1:2)])
 [1] 1e-01 5e-02 2e-02 1e-02 5e-03 2e-03 1e-03 5e-04 2e-04 1e-04 5e-05 2e-05
[13] 1e-05
> ## 0.1, 0.05, 0.02, 0.01, ..... 1e-5
> u.1 <- 1 - u.0
> 
> ### Upper Tail Dependence ---------------------------
> 
> # R/Copula:
> gumbC3  <- gumbelCopula(param= 3, dim = 2)
> gumbC20 <- gumbelCopula(param=20, dim = 2)
> gumbC40 <- gumbelCopula(param=40, dim = 2)
> 
> ut20 <- numTailIndexUpper(gumbC20, u.1)
> (ut40 <- numTailIndexUpper(gumbC40, u.1))
 [1] 0.9834402 0.9829724 0.9826993 0.9826095 0.9825648 0.9825381 0.9825292
 [8] 0.9825248 0.9825221 0.9825212 0.9825208 0.9825205 0.9825204
> 
> stopifnot(
+  all.equal(lambda(gumbC20)[["upper"]],
+            numTailIndexUpper(gumbC20, 1 - 1e-7), tolerance=1e-8)
+  ,
+  all.equal(lambda(gumbC40)[["upper"]],
+            numTailIndexUpper(gumbC40, 1 - 1e-7), tolerance=1e-8)
+ )
> 
> if(tryfCop) { ## Rmetrics
+     C <- parchmCopula(u.1,u.1, alpha=40, type = "4", alternative = TRUE)
+     stopifnot(all.equal(ut40,     (1-2*u.1+C)/(1-u.1),
+                         check.attributes=FALSE, tolerance= 1e-14))
+ }
> 
> 
> ### Lower Tail Dependence-------------------------
> 
> S <- cbind(u.0,u.0)
> ## R/Copula:
> ## C  <- pCopula(dim = 2, copula = gumbelCopula(param=20), S)
> ## (C1  <- C/u.0)
> (lt20 <- numTailIndexLower(gumbC20, u.0))
 [1] 0.9220088 0.8997447 0.8711360 0.8501003 0.8295725 0.8031951 0.7837999
 [8] 0.7648732 0.7405529 0.7226705 0.7052198 0.6827963 0.6663085
> 
> if(tryfCop) { ## Rmetrics
+     C <-  parchmCopula(S, alpha=20, type = "4", alternative = FALSE)
+     stopifnot(all.equal(lt20, C/u.0, check.attributes=FALSE, tolerance= 1e-14))
+ }
> 
> signif(numTailIndexLower(gumbC3, 10^-(5*(1:40))),   3)#--> 0
 [1] 5.02e-02 2.52e-03 1.26e-04 6.33e-06 3.18e-07 1.59e-08 7.99e-10 4.01e-11
 [9] 2.01e-12 1.01e-13 5.06e-15 2.54e-16 1.27e-17 6.39e-19 3.21e-20 1.61e-21
[17] 8.07e-23 4.05e-24 2.03e-25 1.02e-26 5.11e-28 2.56e-29 1.29e-30 6.45e-32
[25] 3.23e-33 1.62e-34 8.14e-36 4.08e-37 2.05e-38 1.03e-39 5.16e-41 2.59e-42
[33] 1.30e-43 6.51e-45 3.26e-46 1.64e-47 8.21e-49 4.12e-50 2.07e-51 1.04e-52
> ## but for large theta, the convergence (to 0) is *MUCH* slower:
> signif(numTailIndexLower(gumbC20, 10^-(5*(1:40))),  3)
 [1] 6.66e-01 4.44e-01 2.96e-01 1.97e-01 1.31e-01 8.75e-02 5.83e-02 3.89e-02
 [9] 2.59e-02 1.72e-02 1.15e-02 7.66e-03 5.10e-03 3.40e-03 2.27e-03 1.51e-03
[17] 1.01e-03 6.70e-04 4.47e-04 2.98e-04 1.98e-04 1.32e-04 8.80e-05 5.86e-05
[25] 3.91e-05 2.60e-05 1.73e-05 1.16e-05 7.70e-06 5.13e-06 3.42e-06 2.28e-06
[33] 1.52e-06 1.01e-06 6.74e-07 4.49e-07 2.99e-07 1.99e-07 1.33e-07 8.85e-08
> 
> 
> ###-------------- Frank --------------------------
> Frank2 <- frankCopula(param=2, dim = 2)
> lambda(Frank2) # 0 0
lower upper 
    0     0 
> 
> ## Upper and lower tail dependence
> (tl <- numTailIndexLower(Frank2, u.0))
 [1] 1.937118e-01 1.052856e-01 4.449229e-02 2.267824e-02 1.145085e-02
 [6] 4.607652e-03 2.308420e-03 1.155362e-03 4.624221e-04 2.312573e-04
[11] 1.156402e-04 4.625886e-05 2.312989e-05
> stopifnot(all.equal(tl, numTailIndexUpper(Frank2, u.1), tolerance=1e-10))
> 
> stopifnot(
+   (tu1 <- numTailIndexUpper(Frank2, .99999)) < .00003
+ ,
+   all.equal(tu1, numTailIndexLower(Frank2, .00001), tolerance=1e-6)
+ ,
+   (tu2 <- numTailIndexUpper(Frank2, 1-1e-6)) < 3e-6
+ ,
+   all.equal(tu2, numTailIndexLower(Frank2, 1e-6), tolerance= 1e-4)
+ )
> 
> 
> 
> ###-------------- Elliptic --------------------------
> 
> u2 <- cbind(u.0,u.1)
> 
> (t.7.3 <- tCopula(0.7, df=3, dim = 2))
t-copula, dim. d = 2 
Dimension:  2 
Parameters:
  rho.1   = 0.7
  df      = 3.0
> (t.9.2 <- tCopula(0.9, df=2, dim = 2))
t-copula, dim. d = 2 
Dimension:  2 
Parameters:
  rho.1   = 0.9
  df      = 2.0
> 
> t.frac <- tCopula(0.9, df=2.5, dim = 2)
> ## fractional df  currently (must) *fail* for pCopula
> assertError(pCopula(cbind(u.0,u.1), t.frac))
> 
> ft <- dCopula(u2, t.frac)
> stopifnot(
+     all.equal(ft, dCopula(u2[,2:1], t.frac), tolerance= 8e-15)
+  ,
+  !is.unsorted(ft)
+  ,
+  all.equal(lambda(t.7.3)[["upper"]],
+            numTailIndexUpper(t.7.3, 1 - 1e-8), tolerance=1e-5)
+  ,
+  all.equal(lambda(t.9.2)[["upper"]],
+            numTailIndexUpper(t.9.2, 1 - 1e-8), tolerance=1e-7)
+  ,
+  all.equal(lambda(t.7.3)[["lower"]],
+            numTailIndexLower(t.7.3, 1e-8), tolerance=1e-5)
+  ,
+  all.equal(lambda(t.9.2)[["lower"]],
+            numTailIndexLower(t.9.2, 1e-8), tolerance=1e-7)
+ )
> 
> (ut. <- numTailIndexUpper(t.7.3, u.1))
 [1] 0.5326067 0.4993520 0.4751115 0.4648960 0.4585944 0.4537609 0.4516561
 [8] 0.4503362 0.4493123 0.4488632 0.4485805 0.4483607 0.4482642
> 
> if(tryfCop && .r(fCopulae)) { ## Rmetrics
+     p.fC <- pellipticalCopula(u = u.1, v = u.1, rho = 0.7, param = c(nu=3))
+     p. <- pCopula(u = cbind(u.1, u.1), t.7.3)
+     ## they are really not "so equal"
+     stopifnot(
+     all.equal(p.fC, p., check.attributes=FALSE, tolerance= 0.002)
+     )
+ }
> 
> ###----------------- Compare with fitLambda() , both methods: --------------
> 
> set.seed(101)
> U.7.3 <- rCopula(n = 2^15, t.7.3) # pretty large n .. still
> U.9.2 <- rCopula(n = 2^15, t.9.2)
> 
> showSys.time(fL.7.3 <- fitLambda(U.7.3))              # 0.03 sec
Time    user  system elapsed 
Time    0.05    0.00    0.05 
> showSys.time(fL.9.2 <- fitLambda(U.9.2))              #  "
Time    user  system elapsed 
Time    0.04    0.00    0.05 
> showSys.time(fLt7.3 <- fitLambda(U.7.3, method="t"))  # 2.25 sec
Time    user  system elapsed 
Time    3.38    0.02    3.39 
> fL.7.3
          [,1]      [,2]
[1,] 1.0000000 0.4404536
[2,] 0.4404536 1.0000000
> stopifnot(
+     all.equal(fL.7.3[1,2], 0.440453579986)
+     ,
+     all.equal(fL.9.2[1,2], 0.838509809998)
+     ,
+     names(fLt7.3) == c("Lambda", "P", "Nu")
+     ,
+     all.equal(fLt7.3$Lambda[1,2], 0.447410146067)
+ )
> (doExtras <- copula:::doExtras() && getRversion() >= "3.4") # so have withAutoprint(.)
[1] FALSE
> if(doExtras) withAutoprint({
+     showSys.time(fLt9.2 <- fitLambda(U.9.2, method="t"))
+     fLt9.2
+     stopifnot(all.equal(fLt9.2$Lambda[1,2], 0.719807333))
+ })
> 
> 
> cat('Time elapsed: ', proc.time(),'\n') # for ''statistical reasons''
Time elapsed:  5.29 0.14 5.42 NA NA 
> 
> ## Note: R CMD BATCH tail-pcopula.R => tail-pcopula.Rout => tail-pcopula.Rout.save
> 
> proc.time()
   user  system elapsed 
   5.29    0.14    5.42