Log-Likelihood Visualization for Archimedean Copulas

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Marius Hofert and Martin Mächler

2024-08-12

Intro

This vignette visualizes (log) likelihood functions of Archimedean copulas, some of which are numerically challenging to compute. Because of this computational challenge, we also check for equivalence of some of the several computational methods, testing for numerical near-equality using all.equal(L1, L2).

Auxiliary functions

We start by defining the following auxiliary functions.

##' @title [m]inus Log-Likelihood for Archimedean Copulas ("fast version")
##' @param theta parameter (length 1 for our current families)
##' @param acop Archimedean copula (of class "acopula")
##' @param u data matrix n x d
##' @param n.MC if > 0 MC is applied with sample size equal to n.MC; otherwise,
##'        the exact formula is used
##' @param ... potential further arguments, passed to <acop> @dacopula()
##' @return negative log-likelihood
##' @author Martin Maechler (Marius originally)
mLogL <- function(theta, acop, u, n.MC=0, ...) { # -(log-likelihood)
    -sum(acop@dacopula(u, theta, n.MC=n.MC, log=TRUE, ...))
}

##' @title Plotting the Negative Log-Likelihood for Archimedean Copulas
##' @param cop an outer_nacopula (currently with no children)
##' @param u n x d  data matrix
##' @param xlim x-range for curve() plotting
##' @param main title for curve()
##' @param XtrArgs a list of further arguments for mLogL()
##' @param ... further arguments for curve()
##' @return invisible()
##' @author Martin Maechler
curveLogL <- function(cop, u, xlim, main, XtrArgs=list(), ...) {
    unam <- deparse(substitute(u))
    stopifnot(is(cop, "outer_nacopula"), is.list(XtrArgs),
              (d <- ncol(u)) >= 2, d == dim(cop),
              length(cop@childCops) == 0# not yet *nested* A.copulas
              )
    acop <- cop@copula
    th. <- acop@theta # the true theta
    acop <- setTheta(acop, NA) # so it's clear, the true theta is not used below
    if(missing(main)) {
        tau. <- cop@copula@tau(th.)
        main <- substitute("Neg. Log Lik."~ -italic(l)(theta, UU) ~ TXT ~~
               FUN(theta['*'] == Th) %=>% tau['*'] == Tau,
               list(UU = unam,
                TXT= sprintf("; n=%d, d=%d;  A.cop",
                         nrow(u), d),
                FUN = acop@name,
                Th = format(th.,digits=3),
                Tau = format(tau., digits=3)))
    }
    r <- curve(do.call(Vectorize(mLogL, "theta"), c(list(x, acop, u), XtrArgs)),
               xlim=xlim, main=main,
               xlab = expression(theta),
               ylab = substitute(- log(L(theta, u, ~~ COP)), list(COP=acop@name)),
               ...)
    if(is.finite(th.))
        axis(1, at = th., labels=expression(theta["*"]),
             lwd=2, col="dark gray", tck = -1/30)
    else warning("non-finite cop@copula@theta = ", th.)
    axis(1, at = initOpt(acop@name),
         labels = FALSE, lwd = 2, col = 2, tck = 1/20)
    invisible(r)
}

Ensure that we are told about it, if the numerical algorithms choose methods using Rmpfr (R package interfacing to multi precision arithmetic MPFR):

op <- options("copula:verboseUsingRmpfr"=TRUE) # see when "Rmpfr" methods are chosen automatically

Joe’s family

Easy case (\(\tau=0.2\))

n <- 200
d <- 100
tau <- 0.2
theta <- copJoe@iTau(tau)
cop <- onacopulaL("Joe", list(theta,1:d))
theta

## [1] 1.443824

Here, the three different methods work “the same”:

set.seed(1)
U1 <- rnacopula(n,cop)
enacopula(U1, cop, "mle") # 1.432885 --  fine

## [1] 1.432898

th4 <- 1 + (1:4)/4
mL.tr <- c(-3558.5, -3734.4, -3299.5, -2505.)
mLt1 <- sapply(th4, function(th) mLogL(th, cop@copula, U1, method="log.poly")) # default
mLt2 <- sapply(th4, function(th) mLogL(th, cop@copula, U1, method="log1p"))
mLt3 <- sapply(th4, function(th) mLogL(th, cop@copula, U1, method="poly"))
stopifnot(all.equal(mLt1, mL.tr, tolerance=5e-5),
          all.equal(mLt2, mL.tr, tolerance=5e-5),
          all.equal(mLt3, mL.tr, tolerance=5e-5))
system.time(r1l  <- curveLogL(cop, U1, c(1, 2.5), X=list(method="log.poly")))

##    user  system elapsed 
##   0.528   0.000   0.521

mtext("all three polyJ() methods on top of each other")
system.time({
    r1J <- curveLogL(cop, U1, c(1, 2.5), X=list(method="poly"),
                     add=TRUE, col=adjustcolor("red", .4))
    r1m  <- curveLogL(cop, U1, c(1, 2.5), X=list(method="log1p"),
                      add=TRUE, col=adjustcolor("blue",.5))
})

##    user  system elapsed 
##   0.613   0.000   0.615

U2 <- rnacopula(n,cop)
summary(dCopula(U2, cop)) # => density for the *correct* parameter looks okay

##       Min.    1st Qu.     Median       Mean    3rd Qu.       Max. 
##  0.000e+00  4.900e+01  6.430e+02 2.777e+175  1.932e+04 5.553e+177

## hmm: max = 5.5e177
if(doExtras)
    system.time(r2 <- curveLogL(cop, U2, c(1, 2.5)))
stopifnot(all.equal(enacopula(U2, cop, "mle"), 1.43992755, tolerance=1e-5),
          all.equal(mLogL(1.8, cop@copula, U2), -4070.1953,tolerance=1e-5)) # (was -Inf)

U3 <- rnacopula(n,cop)
(th. <- enacopula(U3, cop, "mle")) # 1.4495

## [1] 1.449569

system.time(r3 <- curveLogL(cop, U3, c(1, 2.5)))

##    user  system elapsed 
##   0.309   0.002   0.313

axis(1, at = th., label = quote(hat(theta)))

U4 <- rnacopula(n,cop)
enacopula(U4, cop, "mle") # 1.4519  (prev. was 2.351 : "completely wrong")

## [1] 1.451916

summary(dCopula(U4, cop)) # ok (had one Inf)

##       Min.    1st Qu.     Median       Mean    3rd Qu.       Max. 
##  0.000e+00  7.500e+01  9.080e+02 1.981e+259  1.434e+04 3.961e+261

if(doExtras)
    system.time(r4 <- curveLogL(cop, U4, c(1, 2.5)))
mLogL(2.2351, cop@copula, U4)

## [1] -1789.59

mLogL(1.5,    cop@copula, U4)

## [1] -3882.819

mLogL(1.2,    cop@copula, U4)

## [1] -3517.366

if(doExtras) # each curve takes almost 2 sec
    system.time({
        curveLogL(cop, U4, c(1, 1.01))
        curveLogL(cop, U4, c(1, 1.0001))
        curveLogL(cop, U4, c(1, 1.000001))
    })
## --> limit goes *VERY* steeply up to  0
## --> theta 1.164 is about the boundary:
stopifnot(identical(setTheta(cop, 1.164), onacopula(cop@copula, C(1.164, 1:100))),
      all.equal(600.59577,
            cop@copula@dacopula(U4[118,,drop=FALSE],
                    theta=1.164, log = TRUE), tolerance=1e-5)) # was "Inf"

Harder case (\(d=150\), \(\tau=0.3\))

n <- 200
d <- 150
tau <- 0.3
(theta <- copJoe@iTau(tau))

## [1] 1.772108

cop <- onacopulaL("Joe",list(theta,1:d))

set.seed(47)
U. <- rnacopula(n,cop)
enacopula(U., cop, "mle") # 1.784578

## [1] 1.78459

system.time(r. <- curveLogL(cop, U., c(1.1, 3)))

##    user  system elapsed 
##   0.406   0.000   0.408

## => still looks very good

Even harder case (\(d=180\), \(\tau=0.4\))

d <- 180
tau <- 0.4
(theta <- copJoe@iTau(tau))

## [1] 2.219066

cop <- onacopulaL("Joe",list(theta,1:d))

U. <- rnacopula(n,cop)
enacopula(U., cop, "mle") # 2.217582

## [1] 2.217593

if(doExtras)
system.time(r. <- curveLogL(cop, U., c(1.1, 4)))
## => still looks very good

Gumbel’s family

Easy case (\(\tau=0.2\))

n <- 200
d <- 50 # smaller 'd' -- so as to not need 'Rmpfr' here
tau <- 0.2
(theta <- copGumbel@iTau(tau))

## [1] 1.25

cop <- onacopulaL("Gumbel",list(theta,1:d))

set.seed(1)
U1 <- rnacopula(n,cop)
if(doExtras) {
    U2 <- rnacopula(n,cop)
    U3 <- rnacopula(n,cop)
}
enacopula(U1, cop, "mle") # 1.227659 (was 1.241927)

## [1] 1.227659

##--> Plots with "many" likelihood evaluations
system.time(r1 <- curveLogL(cop, U1, c(1, 2.1)))

##    user  system elapsed 
##   0.440   0.001   0.442

if(doExtras) system.time({
    mtext("and two other generated samples")
    r2 <- curveLogL(cop, U2, c(1, 2.1), add=TRUE)
    r3 <- curveLogL(cop, U3, c(1, 2.1), add=TRUE)
})

Harder case (\(d=150\), \(\tau=0.6\))

d <- 150
tau <- 0.6
(theta <- copGumbel@iTau(tau))

## [1] 2.5

cG.5 <- onacopulaL("Gumbel",list(theta,1:d))

set.seed(17)
U4 <- rnacopula(n,cG.5)
U5 <- rnacopula(n,cG.5)
U6 <- rnacopula(n,cG.5)
if(doExtras) { ## "Rmpfr" is used {2012-06-21}: -- therefore about 18 seconds!
 tol <- if(interactive()) 1e-12 else 1e-8
 print(system.time(
 ee. <- c(enacopula(U4, cG.5, "mle", tol=tol),
          enacopula(U5, cG.5, "mle", tol=tol),
          enacopula(U6, cG.5, "mle", tol=tol))))
dput(ee.)# in case the following fails
## tol=1e-12 Linux nb-mm3 3.2.0-25-generic x86_64 (2012-06-23):
##   c(2.47567251789004, 2.48424484287686, 2.50410767129408)
##   c(2.475672518,      2.484244763,      2.504107671),
stopifnot(all.equal(ee., c(2.475672518, 2.484244763, 2.504107671),
            tolerance= max(1e-7, 16*tol)))
}
## --> Plots with "many" likelihood evaluations
th. <- seq(1, 3, by= 1/4)
if(doExtras) # "default2012" (polyG default) partly uses Rmpfr here:
system.time(r4   <- sapply(th., mLogL, acop=cG.5@copula, u=U4))## 25.6 sec
## whereas this (polyG method) is very fast {and still ok}:
system.time(r4.p <- sapply(th., mLogL, acop=cG.5@copula, u=U4, method="pois"))

##    user  system elapsed 
##   0.099   0.000   0.100

r4. <- c(0, -18375.33, -21948.033, -24294.995, -25775.502,
         -26562.609, -26772.767, -26490.809, -25781.224)
stopifnot(!doExtras ||
          all.equal(r4,   r4., tolerance = 8e-8),
          all.equal(r4.p, r4., tolerance = 8e-8))
## --> use fast method here as well:
system.time(r5.p <- sapply(th., mLogL, acop=cG.5@copula, u=U5, method="pois"))

##    user  system elapsed 
##   0.098   0.000   0.099

system.time(r6.p <- sapply(th., mLogL, acop=cG.5@copula, u=U6, method="pois"))

##    user  system elapsed 
##     0.1     0.0     0.1

if(doExtras) {
    if(FALSE) # for speed analysis, etc
        debug(copula:::polyG)
    mLogL(1.65, cG.5@copula, U4) # -23472.96
}
dd <- dCopula(U4, setTheta(cG.5, 1.64), log = TRUE,
              method = if(doExtras)"default" else "pois")
summary(dd)

##    Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
##   41.59   53.30   81.09  116.91  137.54  707.13

stopifnot(!is.na(dd), # no NaN's anymore
      40 < range(dd), range(dd) < 710)

Frank’s family (an already hard case)

n <- 64
d <- 5
tau <- 0.8
(theta <- copFrank@iTau(tau))

## [1] 18.19154

cop <- onacopulaL("Frank", list(theta,1:d))

set.seed(11) # these seeds give no problems: 101, 41, 21
U. <- rnacopula(n,cop)
cop@copula <- setTheta(cop@copula, NA) # forget the true theta
system.time(f.ML <- emle(U., cop)); f.ML # --> fine: theta = 18.033, Log-lik = 314.01

##    user  system elapsed 
##   0.014   0.000   0.013

## 
## Call:
## bbmle::mle2(minuslogl = nLL, start = start, optimizer = "optimize", 
##     lower = interval[1], upper = interval[2])
## 
## Coefficients:
##   theta 
## 18.0333 
## 
## Log-likelihood: 314.01

if(doExtras)
    system.time(f.mlMC <- emle(U., cop, n.MC = 1e4)) # with MC
stopifnot(all.equal(unname(coef(f.ML)), 18.03331, tolerance= 1e-6),
      all.equal(f.ML@min, -314.0143, tolerance=1e-6),
      !doExtras || ## Simulate MLE (= SMLE) is "extra" random,  hmm...
      all.equal(unname(coef(f.mlMC)), 17.8, tolerance= 0.01)
      ##           64-bit ubuntu: 17.817523
      ##         ? 64-bit Mac:    17.741
     )

cop@copula <- setTheta(cop@copula, theta)
r. <- curveLogL(cop, U., c(1, 200)) # => now looks fine

tail(as.data.frame(r.), 15)

##          x        y
## 87  172.14 2105.690
## 88  174.13 2143.642
## 89  176.12 2181.637
## 90  178.11 2219.675
## 91  180.10 2257.754
## 92  182.09 2295.874
## 93  184.08 2334.034
## 94  186.07 2372.232
## 95  188.06 2410.468
## 96  190.05 2448.742
## 97  192.04 2487.051
## 98  194.03 2525.396
## 99  196.02 2563.776
## 100 198.01 2602.189
## 101 200.00 2640.636

stopifnot( is.finite( r.$y ),
      ## and is convex (everywhere):
      diff(r.$y, d=2) > 0)
options(op) # revert to previous state

Session information

## R version 4.4.1 (2024-06-14)
## Platform: x86_64-pc-linux-gnu
## Running under: Fedora Linux 40 (Forty)
## 
## Matrix products: default
## BLAS:   /r/app/R/R-4.4.1-isg/lib64/R/lib/libRblas.so 
## LAPACK: /usr/lib64/liblapack.so.3.12.0
## 
## attached base packages:
##  [1] parallel  grid      stats4    tools     stats     graphics  grDevices
##  [8] utils     datasets  methods   base     
## 
## other attached packages:
## [1] rugarch_1.5-1  gsl_2.1-8      mev_1.17       lattice_0.22-6 bbmle_1.0.25.1
## [6] copula_1.1-4  
## 
## loaded via a namespace (and not attached):
##  [1] gmp_0.7-4                   ks_1.14.2                  
##  [3] sass_0.4.9                  KernSmooth_2.23-24         
##  [5] SkewHyperbolic_0.4-2        pracma_2.4.4               
##  [7] digest_0.6.36               evaluate_0.24.0            
##  [9] nleqslv_3.3.5               mvtnorm_1.2-5              
## [11] fastmap_1.2.0               jsonlite_1.8.8             
## [13] Matrix_1.7-0                mclust_6.1.1               
## [15] truncnorm_1.0-9             stabledist_0.7-1           
## [17] spd_2.0-1                   numDeriv_2016.8-1.1        
## [19] jquerylib_0.1.4             Rdpack_2.6.1               
## [21] cli_3.6.3                   rlang_1.1.4                
## [23] rbibutils_2.2.16            pspline_1.0-20             
## [25] cachem_1.1.0                yaml_2.3.10                
## [27] polynom_1.4-1               nloptr_2.1.1               
## [29] bdsmatrix_1.3-7             mathjaxr_1.6-0             
## [31] Runuran_0.38                partitions_1.10-7          
## [33] R6_2.5.1                    zoo_1.8-12                 
## [35] lifecycle_1.0.4             ADGofTest_0.3              
## [37] MASS_7.3-60.2               Rsolnp_1.16                
## [39] pcaPP_2.0-4-1               bslib_0.8.0                
## [41] GeneralizedHyperbolic_0.8-6 Rcpp_1.0.13                
## [43] xfun_0.46                   highr_0.11                 
## [45] knitr_1.48                  htmltools_0.5.8.1          
## [47] rmarkdown_2.27              xts_0.14.0                 
## [49] compiler_4.4.1              alabama_2023.1.0           
## [51] DistributionUtils_0.6-1

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