Comparison of computation times

This document shows a comparison of computation time of TL-moments between different packages available, as well as between the different approaches built-in in this package.

This package offers the following computation methods (available via computation.method-attribute in TLMoments or TLMoment):

For a complete and thorough analysis of all these approaches and another speed comparison see Hosking & Balakrishnan (2015, A uniqueness result for L-estimators, with applications to L-moments, Statistical Methodology, 24, 69-80).

Besides our implementation, L-moments and/or TL-moments can be calculated using the packages

(all availabe at CRAN). The functions lmomco::lmoms, lmomco::TLmoms, and Lmoments::Lmoments return list objects with (T)L-moments and (T)L-moment-ratios and are therefore compared to our TLMoments; lmom::samlmu returns a vector of lambdas and is compared to TLMoment (which is a fast bare-bone function to compute TL-moments but is not suited to be transmitted to parameters or other functions of this package).

Calculation of L-moments

First we check, if all approaches give the same results (lmomco::lmoms is added as comparison).

n <- c(25, 50, 100, 200, 500, 1000, 10000, 50000)
sapply(n, function(nn) {
  x <- evd::rgev(nn)
  check <- lmomco::lmoms(x, 4)$lambdas
  sapply(c("direct", "pwm", "recursive"), function(comp) {
    isTRUE(all.equal(TLMoment(x, order = 1:4, computation.method = comp), check, check.attributes = FALSE))
  })
})
##           [,1] [,2] [,3] [,4] [,5] [,6] [,7] [,8]
## direct    TRUE TRUE TRUE TRUE TRUE TRUE TRUE TRUE
## pwm       TRUE TRUE TRUE TRUE TRUE TRUE TRUE TRUE
## recursive TRUE TRUE TRUE TRUE TRUE TRUE TRUE TRUE

Now we compare the functions giving L-moments are L-moment-ratios simultaneously.

possib <- list(
  TLMoments_direct = function(x) TLMoments(x, max.order = 4, computation.method = "direct"), 
  TLMoments_pwm = function(x) TLMoments(x, max.order = 4, computation.method = "pwm"), 
  TLMoments_recursive = function(x) TLMoments(x, max.order = 4, computation.method = "recursive"), 
  lmomco = function(x) lmomco::lmoms(x, 4), 
  Lmoments = function(x) Lmoments::Lmoments(x, returnobject = TRUE)
)

# n = 50
datalist <- replicate(200, evd::rgev(50), simplify = FALSE)

do.call("rbind", lapply(possib, function(f) {
  system.time(lapply(datalist, f))[3]
}))
##                     elapsed
## TLMoments_direct      0.043
## TLMoments_pwm         0.037
## TLMoments_recursive   0.033
## lmomco                0.634
## Lmoments              0.054
# n = 1000
datalist <- replicate(200, evd::rgev(1000), simplify = FALSE)

do.call("rbind", lapply(possib, function(f) {
  system.time(lapply(datalist, f))[3]
}))
##                     elapsed
## TLMoments_direct      0.181
## TLMoments_pwm         0.123
## TLMoments_recursive   0.057
## lmomco               11.880
## Lmoments              0.120

As we see, our implementation of the recursive approach is clearly the fastest. After this, the pwm approach is to be prefered over the direct approach. The implementation in lmomco is slow, compared to the others, especially for longer data vectors. Lmoments is constantly slower than the recursive approach of this package.

Comparison of functions that only return a vector of L-moments:

possib <- list(
  TLMoments_direct = function(x) TLMoment(x, order = 1:4, computation.method = "direct"), 
  TLMoments_pwm = function(x) TLMoment(x, order = 1:4, computation.method = "pwm"), 
  TLMoments_recursive = function(x) TLMoment(x, order = 1:4, computation.method = "recursive"), 
  lmom = function(x) lmom::samlmu(x, 4), 
  Lmoments = function(x) Lmoments::Lmoments(x, returnobject = FALSE)
)

# n = 50
datalist <- replicate(200, evd::rgev(50), simplify = FALSE)

do.call("rbind", lapply(possib, function(f) {
  system.time(lapply(datalist, f))[3]
}))
##                     elapsed
## TLMoments_direct      0.011
## TLMoments_pwm         0.008
## TLMoments_recursive   0.005
## lmom                  0.009
## Lmoments              0.048
# n = 1000
datalist <- replicate(200, evd::rgev(1000), simplify = FALSE)

do.call("rbind", lapply(possib, function(f) {
  system.time(lapply(datalist, f))[3]
}))
##                     elapsed
## TLMoments_direct      0.137
## TLMoments_pwm         0.084
## TLMoments_recursive   0.023
## lmom                  0.010
## Lmoments              0.087

For smaller data vectors our implementation is the fastest, but with longer data vectors lmom::samlmu excels.

Calculation of TL-moments

First we check, if all approaches give the same results (lmomco::Tlmoms is added as comparison)

n <- c(25, 50, 100, 150, 200, 500, 1000, 10000)
sapply(n, function(nn) {
  x <- evd::rgev(nn)
  check <- lmomco::TLmoms(x, 4, leftrim = 0, rightrim = 1)$lambdas
  sapply(c("direct", "pwm", "recursive", "recurrence"), function(comp) {
    isTRUE(all.equal(TLMoment(x, order = 1:4, rightrim = 1, computation.method = comp), check, check.attributes = FALSE))
  })
})
##            [,1] [,2] [,3] [,4]  [,5]  [,6]  [,7]  [,8]
## direct     TRUE TRUE TRUE TRUE  TRUE  TRUE  TRUE  TRUE
## pwm        TRUE TRUE TRUE TRUE  TRUE  TRUE  TRUE  TRUE
## recursive  TRUE TRUE TRUE TRUE FALSE FALSE FALSE FALSE
## recurrence TRUE TRUE TRUE TRUE  TRUE  TRUE  TRUE  TRUE
sapply(n, function(nn) {
  x <- evd::rgev(nn)
  check <- lmomco::TLmoms(x, 4, leftrim = 2, rightrim = 4)$lambdas
  sapply(c("direct", "pwm", "recursive", "recurrence"), function(comp) {
    isTRUE(all.equal(TLMoment(x, order = 1:4, leftrim = 2, rightrim = 4, computation.method = comp), check, check.attributes = FALSE))
  })
})
##            [,1] [,2] [,3] [,4]  [,5]  [,6]  [,7]  [,8]
## direct     TRUE TRUE TRUE TRUE  TRUE  TRUE  TRUE  TRUE
## pwm        TRUE TRUE TRUE TRUE  TRUE  TRUE  TRUE  TRUE
## recursive  TRUE TRUE TRUE TRUE FALSE FALSE FALSE FALSE
## recurrence TRUE TRUE TRUE TRUE  TRUE  TRUE  TRUE  TRUE

The recursive approach fails when n exceeds 150. All other implementations give the same results.

possib <- list(
  TLMoments_direct = function(x) TLMoments(x, leftrim = 0, rightrim = 1, max.order = 4, computation.method = "direct"), 
  TLMoments_pwm = function(x) TLMoments(x, leftrim = 0, rightrim = 1, max.order = 4, computation.method = "pwm"), 
  TLMoments_recurrence = function(x) TLMoments(x, leftrim = 0, rightrim = 1, max.order = 4, computation.method = "recurrence"), 
  lmomco = function(x) lmomco::TLmoms(x, 4, leftrim = 0, rightrim = 1)
)

# n = 50
datalist <- replicate(200, evd::rgev(50), simplify = FALSE)

do.call("rbind", lapply(possib, function(f) {
  system.time(lapply(datalist, f))[3]
}))
##                      elapsed
## TLMoments_direct       0.093
## TLMoments_pwm          0.041
## TLMoments_recurrence   0.036
## lmomco                 0.632
# n = 1000
datalist <- replicate(200, evd::rgev(1000), simplify = FALSE)

do.call("rbind", lapply(possib, function(f) {
  system.time(lapply(datalist, f))[3]
}))
##                      elapsed
## TLMoments_direct       1.273
## TLMoments_pwm          0.151
## TLMoments_recurrence   0.057
## lmomco                12.251
possib <- list(
  TLMoments_direct = function(x) TLMoments(x, leftrim = 2, rightrim = 4, max.order = 4, computation.method = "direct"), 
  TLMoments_pwm = function(x) TLMoments(x, leftrim = 2, rightrim = 4, max.order = 4, computation.method = "pwm"), 
  TLMoments_recurrence = function(x) TLMoments(x, leftrim = 2, rightrim = 4, max.order = 4, computation.method = "recurrence"), 
  lmomco = function(x) lmomco::TLmoms(x, 4, leftrim = 2, rightrim = 4)
)

# n = 50
datalist <- replicate(200, evd::rgev(50), simplify = FALSE)

do.call("rbind", lapply(possib, function(f) {
  system.time(lapply(datalist, f))[3]
}))
##                      elapsed
## TLMoments_direct       0.099
## TLMoments_pwm          0.047
## TLMoments_recurrence   0.038
## lmomco                 0.590
# n = 1000
datalist <- replicate(200, evd::rgev(1000), simplify = FALSE)

do.call("rbind", lapply(possib, function(f) {
  system.time(lapply(datalist, f))[3]
}))
##                      elapsed
## TLMoments_direct       1.292
## TLMoments_pwm          0.260
## TLMoments_recurrence   0.074
## lmomco                12.024

In this calculations the recurrence approach clearly outperforms the other implementations. Calculation using probabilty-weighted moments is relatively fast, but using the direct calculation should be avoided, regarding calculation speed. This package's implementation is clearly faster than those in lmomco.

Conclusion

This results encourage to use the recursive approach for L-moments and the recurrence approach when calculating TL-moments. Therefore these are the defaults in this package, but the other computation methods (direct and pwm) are still available.