Short introduction to TLMoments
2022-03-27
TLMoments is a set of functions whose main functionality is the calculation of trimmed L-moments (TL-moments) and resulting estimates of distribution parameters and quantiles.
One of the goals is to reduce computation time compared to existing implementations (in packages like lmomco, Lmoments, Lmom), therefore the core functions are written in C++ using Rcpp (see vignette “comparison of computation time” for speed comparisons). The package expands the combinations of trimmings that can be used to estimate distribution parameters in comparison to existing packages (which currently mainly support parameter estimation with L-moments). To ensure an easy usage, the package only contains a small set of functions. This vignette gives a short introduction to the most important ones and how to use them.
library(TLMoments)
sessionInfo()
## R version 4.1.2 (2021-11-01)
## Platform: x86_64-pc-linux-gnu (64-bit)
## Running under: Ubuntu 21.04
##
## Matrix products: default
## BLAS: /usr/lib/x86_64-linux-gnu/blas/libblas.so.3.9.0
## LAPACK: /usr/lib/x86_64-linux-gnu/lapack/liblapack.so.3.9.0
##
## locale:
## [1] LC_CTYPE=de_DE.UTF-8 LC_NUMERIC=C
## [3] LC_TIME=de_DE.UTF-8 LC_COLLATE=C
## [5] LC_MONETARY=de_DE.UTF-8 LC_MESSAGES=de_DE.UTF-8
## [7] LC_PAPER=de_DE.UTF-8 LC_NAME=C
## [9] LC_ADDRESS=C LC_TELEPHONE=C
## [11] LC_MEASUREMENT=de_DE.UTF-8 LC_IDENTIFICATION=C
##
## attached base packages:
## [1] stats graphics grDevices utils datasets methods base
##
## other attached packages:
## [1] lmom_2.8 Lmoments_1.3-1 lmomco_2.3.7 TLMoments_0.7.5.3
## [5] Rcpp_1.0.7
##
## loaded via a namespace (and not attached):
## [1] digest_0.6.29 MASS_7.3-55 magrittr_2.0.1 evaluate_0.14
## [5] rlang_1.0.2 stringi_1.7.6 cli_3.1.0 rstudioapi_0.13
## [9] goftest_1.2-3 rmarkdown_2.11 tools_4.1.2 stringr_1.4.0
## [13] xfun_0.29 yaml_2.2.1 fastmap_1.1.0 compiler_4.1.2
## [17] htmltools_0.5.2 evd_2.3-4 knitr_1.37
Calculation of empirical TL-moments, parameter and quantile estimates
First we have a look at the basic functionality of calculating TL-moments and parameter and quantile estimates. Let assume we have a simple random data vector generated from a GEV distribution:
xvec <- rgev(100, loc = 10, scale = 5, shape = .2)
TL-moments are calculated by the function TLMoments with arguments leftrim, rightrim, and max.order (generating an object of class TLMoments):
## $lambdas
## L1 L2 L3 L4
## 13.3826607 3.7257087 0.9870101 0.7340864
##
## $ratios
## T1 T2 T3 T4
## NA 0.2783982 0.2649188 0.1970327
TLMoments(xvec, leftrim = 0, rightrim = 1, max.order = 2)
## $lambdas
## L1 L2
## 9.656952 2.054024
##
## $ratios
## T1 T2
## NA 0.212699
We can calculate parameter estimates by putting a TLMoments-object to the function parameters and specifying argument distr:
tlm <- TLMoments(xvec)
parameters(tlm, distr = "gev")
## loc scale shape
## 9.9581721 4.6219480 0.1432724
tlm <- TLMoments(xvec, rightrim = 1)
parameters(tlm, distr = "gev")
## loc scale shape
## 9.9666536 4.6303617 0.1385292
This generates an object of class parameters, which can be transmitted to quantiles to calculate quantile estimations:
tlm <- TLMoments(xvec)
quantiles(parameters(tlm, distr = "gev"), c(.9, .99, .999))
## 0.9 0.99 0.999
## 22.23170 40.05670 64.48409
tlm <- TLMoments(xvec, rightrim = 1)
quantiles(parameters(tlm, distr = "gev"), c(.9, .99, .999))
## 0.9 0.99 0.999
## 22.19364 39.75791 63.56389
Summary functions
Objects of type TLMoments, parameters, or quantiles (i.e. results from the functions of the same name) feature summary-functions, which give confidence intervals and an overview of the data.
tlm <- TLMoments(rgev(100), leftrim = 0, rightrim = 1)
summary(tlm)
## 1 data row(s) with n = 100.
## TL(0,1) calculated.
##
## Approximate 90% confidence interval of TL moments:
## LCL lambda_hat UCL
## L1 -0.21050958 -0.04010728 0.1302950164
## L2 0.37054673 0.42541664 0.4802865562
## L3 -0.05313960 -0.02629817 0.0005432553
## L4 0.02758415 0.04608429 0.0645844198
## Approximate 90% confidence interval of TL moment ratios:
## LCL tau_hat UCL
## T2 -55.56674228 -10.60696730 34.3528076796
## T3 -0.12264872 -0.06181745 -0.0009861685
## T4 0.05924541 0.10832742 0.1574094311
summary(parameters(tlm, "gev"))
## 1 data row(s) with n = 100.
## TL(0,1) used to generate GEV parameters.
##
## Approximate 90% confidence interval of parameters:
## LCL param UCL
## loc -0.04498695 0.1456896 0.33636614
## scale 0.87132263 1.0022373 1.13315207
## shape -0.36898235 -0.2007409 -0.03249951
summary(quantiles(parameters(tlm, "gev"), .99))
## 1 data row(s) with n = 100.
## TL(0,1) used to generate GEV parameters to calculate 0.99 quantile estimates.
##
## Approximate 90% confidence interval of quantiles:
## LCL quantile UCL
## 0.99 2.253144 3.155527 4.05791
The default confidence interval level is 90%, but it can be set using the argument ci.level. The argument select can be used to subset the results, which can be handy when analysing large data matrices.
summary(tlm, ci.level = .95, select = 3:4)
## 1 data row(s) with n = 100.
## TL(0,1) calculated.
##
## Approximate 95% confidence interval of TL moments:
## LCL lambda_hat UCL
## L3 -0.05828170 -0.02629817 0.005685361
## L4 0.02404002 0.04608429 0.068128555
## Approximate 95% confidence interval of TL moment ratios:
## LCL tau_hat UCL
## T3 0.03584248 0.1083274 0.1808124
## T4 0.04984259 0.1083274 0.1668122
summary(parameters(tlm, "gev"), select = "shape")
## 1 data row(s) with n = 100.
## TL(0,1) used to generate GEV parameters.
##
## Approximate 90% confidence interval of parameters:
## LCL param UCL
## shape -0.3689824 -0.2007409 -0.03249951
At the moment, the summary functions do not work for data in lists or data.frames.
Magrittr syntax
TLMoments is built to support the use in magrittr syntax. The nesting of functions can be written more readable as:
library(magrittr)
TLMoments(xvec, leftrim = 0, rightrim = 1) %>%
parameters("gev") %>%
quantiles(c(.99, .999)) %>%
summary()
## 1 data row(s) with n = 100.
## TL(0,1) used to generate GEV parameters to calculate 0.99, 0.999 quantile estimates.
##
## Approximate 90% confidence interval of quantiles:
## LCL quantile UCL
## 0.99 27.55090 39.75791 51.96492
## 0.999 29.65424 63.56389 97.47354
In the following this syntax is used for a clearer presentation.
Support for different data types
The functions TLMoments, parameters, and quantiles provide the main functionality of the package. In the code above we used single data vectors only, but the same functions can be used for data matrices, lists, and data.frames as well. To demonstrate this, let’s generate sample data of these four types:
xmat <- matrix(rgev(100), nc = 4)
xvec <- xmat[, 3]
xlist <- lapply(1L:ncol(xmat), function(i) xmat[, i])
xdat <- data.frame(station = rep(1:4, each = 25), hq = as.vector(xmat))
Note that the type of the dimensions lambdas and ratios returned by TLMoments matches the input type:
TLMoments(xvec, leftrim = 0, rightrim = 1)
## $lambdas
## L1 L2 L3 L4
## 0.03631499 0.39899946 -0.05375175 0.02462370
##
## $ratios
## T1 T2 T3 T4
## NA 10.98718297 -0.13471636 0.06171362
TLMoments(xmat, leftrim = 0, rightrim = 1)
## $lambdas
## [,1] [,2] [,3] [,4]
## L1 -0.18608012 -0.230243027 0.03631499 -0.32978875
## L2 0.31601417 0.495922514 0.39899946 0.41675746
## L3 0.02435559 0.031014442 -0.05375175 -0.03767806
## L4 0.04527736 0.007043843 0.02462370 0.04164978
##
## $ratios
## [,1] [,2] [,3] [,4]
## T1 NA NA NA NA
## T2 -1.69826937 -2.15390894 10.98718297 -1.26371033
## T3 0.07707121 0.06253889 -0.13471636 -0.09040765
## T4 0.14327637 0.01420352 0.06171362 0.09993769
TLMoments(xlist, leftrim = 0, rightrim = 1)
## $lambdas
## $lambdas[[1]]
## L1 L2 L3 L4
## -0.18608012 0.31601417 0.02435559 0.04527736
##
## $lambdas[[2]]
## L1 L2 L3 L4
## -0.230243027 0.495922514 0.031014442 0.007043843
##
## $lambdas[[3]]
## L1 L2 L3 L4
## 0.03631499 0.39899946 -0.05375175 0.02462370
##
## $lambdas[[4]]
## L1 L2 L3 L4
## -0.32978875 0.41675746 -0.03767806 0.04164978
##
##
## $ratios
## $ratios[[1]]
## T1 T2 T3 T4
## NA -1.69826937 0.07707121 0.14327637
##
## $ratios[[2]]
## T1 T2 T3 T4
## NA -2.15390894 0.06253889 0.01420352
##
## $ratios[[3]]
## T1 T2 T3 T4
## NA 10.98718297 -0.13471636 0.06171362
##
## $ratios[[4]]
## T1 T2 T3 T4
## NA -1.26371033 -0.09040765 0.09993769
TLMoments(xdat, hq ~ station, leftrim = 0, rightrim = 1)
## $lambdas
## station L1 L2 L3 L4
## 1 1 -0.18608012 0.3160142 0.02435559 0.045277362
## 2 2 -0.23024303 0.4959225 0.03101444 0.007043843
## 3 3 0.03631499 0.3989995 -0.05375175 0.024623702
## 4 4 -0.32978875 0.4167575 -0.03767806 0.041649778
##
## $ratios
## station T2 T3 T4
## 1 1 -1.698269 0.07707121 0.14327637
## 2 2 -2.153909 0.06253889 0.01420352
## 3 3 10.987183 -0.13471636 0.06171362
## 4 4 -1.263710 -0.09040765 0.09993769
This holds when parameter and quantile estimations are calculated:
TLMoments(xvec, leftrim = 0, rightrim = 1) %>%
parameters("gev")
## loc scale shape
## 0.2716187 0.9327093 -0.3916404
TLMoments(xmat, leftrim = 0, rightrim = 1) %>%
parameters("gev")
## [,1] [,2] [,3] [,4]
## loc -0.1354520 -0.13712395 0.2716187 -0.1229062
## scale 0.7143313 1.12937085 0.9327093 0.9814387
## shape 0.1274538 0.09509625 -0.3916404 -0.2739527
TLMoments(xlist, leftrim = 0, rightrim = 1) %>%
parameters("gev")
## [[1]]
## loc scale shape
## -0.1354520 0.7143313 0.1274538
##
## [[2]]
## loc scale shape
## -0.13712395 1.12937085 0.09509625
##
## [[3]]
## loc scale shape
## 0.2716187 0.9327093 -0.3916404
##
## [[4]]
## loc scale shape
## -0.1229062 0.9814387 -0.2739527
TLMoments(xdat, hq ~ station, leftrim = 0, rightrim = 1) %>%
parameters("gev")
## station loc scale shape
## 1 1 -0.1354520 0.7143313 0.12745382
## 2 2 -0.1371239 1.1293708 0.09509625
## 3 3 0.2716187 0.9327093 -0.39164040
## 4 4 -0.1229062 0.9814387 -0.27395273
TLMoments(xvec, leftrim = 0, rightrim = 1) %>%
parameters("gev") %>%
quantiles(c(.99, .999))
## 0.99 0.999
## 2.260128 2.493935
TLMoments(xmat, leftrim = 0, rightrim = 1) %>%
parameters("gev") %>%
quantiles(c(.99, .999))
## [,1] [,2] [,3] [,4]
## 0.99 4.333331 6.380038 2.260128 2.443638
## 0.999 7.776929 10.892373 2.493935 2.919611
TLMoments(xlist, leftrim = 0, rightrim = 1) %>%
parameters("gev") %>%
quantiles(c(.99, .999))
## [[1]]
## 0.99 0.999
## 4.333331 7.776929
##
## [[2]]
## 0.99 0.999
## 6.380038 10.892373
##
## [[3]]
## 0.99 0.999
## 2.260128 2.493935
##
## [[4]]
## 0.99 0.999
## 2.443638 2.919611
TLMoments(xdat, hq ~ station, leftrim = 0, rightrim = 1) %>%
parameters("gev") %>%
quantiles(c(.99, .999))
## station 0.99 0.999
## 1 1 4.333331 7.776929
## 2 2 6.380038 10.892373
## 3 3 2.260128 2.493935
## 4 4 2.443638 2.919611
Distributions
TLMoments offers distribution functions (cdf, pdf, quantile, random number generation) for the generalized extreme value distribution (gev), Gumbel distribution (gum), generalized Pareto distribution (gpd), and three-parameter lognormal distribution (ln3) in the common p|d|q|r-syntax. The parameter (and quantile) estimation functionality works for all of them, but more complex functionality like estimation of the covariance matrix of parameter or quantile estimators only works for GEV by now.
TL-moment ratio diagram
Version 0.7.4 added functionality to plot TL-moment ratio diagrams of arbitrary trimming orders. Simply plot an object of TLMoments. Argument distr can be used to specify displayed theoretical distributions. Note that ggplot2 is used. Therefore changes or additions have to be made by adding ggplot2-specific functions.
data <- matrix(rgev(25 * 10, shape = .2), ncol = 10)
plot(TLMoments(data))
plot(TLMoments(data)) + ggplot2::theme_minimal()
plot(TLMoments(data, rightrim = 1), distr = c("gev", "gpd", "exp", "gum"))
Calculations using theoretical/given TL-moments and parameters
The functions as.TLMoments and as.parameters can be used to construct TLMoments- or parameters-objects of given values (not calculated from data). These objects can be used in the same way like before (to convert between TL-moments and their parameters or to calculate the corresponding quantiles):
(tlm <- as.TLMoments(c(14.1, 4.3, 1.32)))
## $lambdas
## L1 L2 L3
## 14.10 4.30 1.32
##
## $ratios
## T1 T2 T3
## NA 0.3049645 0.3069767
parameters(tlm, distr = "gev")
## loc scale shape
## 10.0134305 4.9448851 0.2034746
quantiles(parameters(tlm, distr = "gev"), c(.9, .99, .999))
## 0.9 0.99 0.999
## 24.12668 47.67693 84.80024
(param <- as.parameters(loc = 10, scale = 5, shape = .2, distr = "gev"))
## loc scale shape
## 10.0 5.0 0.2
quantiles(param, c(.9, .99, .999))
## 0.9 0.99 0.999
## 24.21069 47.73413 84.51684
## $lambdas
## L1 L2 L3 L4
## 14.1057429 4.3279754 1.3204343 0.9436158
##
## $ratios
## T1 T2 T3 T4
## NA 0.3068236 0.3050928 0.2180271
TLMoments(param, rightrim = 1)
## $lambdas
## L1 L2 L3 L4
## 9.7777681 2.2556564 0.2512127 0.2553529
##
## $ratios
## T1 T2 T3 T4
## NA 0.2306924 0.1113701 0.1132056
Note, that we can simply use the TLMoments-function to calculate TL-moments corresponding to a parameters-object.