gamlss2: Infrastructure for Flexible Distributional Regression

Overview

The primary purpose of this package is to facilitate the creation of advanced infrastructures designed to enhance the GAMLSS modeling framework. Notably, the gamlss2 package represents a significant overhaul of its predecessor, gamlss, with a key emphasis on improving estimation speed and incorporating more flexible infrastructures. These enhancements enable the seamless integration of various algorithms into GAMLSS, including gradient boosting, Bayesian estimation, regression trees, and forests, fostering a more versatile and powerful modeling environment.

Moreover, the package expands its compatibility by supporting all model terms from the base R mgcv package. Additionally, the gamlss2 package introduces the capability to accommodate more than four parameter families. Essentially, this means that users can now specify any type of model using these new infrastructures, making the package highly flexible and accommodating to a wide range of modeling requirements.

• The main model function is gamlss2().
• The default optimizer functions is RS(). Optimizer functions can be exchanged.
• Most important methods: summary(), plot(), predict().
• Easy development of new family objects, see ?gamlss2,family.
• User-specific “special” terms are possible, see ?special_terms.

For examples, please visit the manual pages.

help(package = "gamlss2")

Installation

The development version of gamlss2 can be installed via

install.packages("gamlss2",
  repos = c("https://gamlss-dev.R-universe.dev",
            "https://cloud.R-project.org"))

Licence

The package is available under the General Public License version 3 or version 2

Illustration

The package is designed to follow the workflow of well-established model fitting functions like lm() or glm(), i.e., the step of estimating full distributional regression models is actually not very difficult.

We illustrate how gamlss2 builds on the established gamlss framework by modeling daily maximum temperature (Tmax) at Munich Airport (MUC) to estimate the probability of “heat days” (Tmax $\geq 30^\circ\text{C}$). Heat days can have serious impacts by stressing highways and railways, increasing the load on healthcare facilities, and affecting airport operations. Using 30 years of historical Tmax data, we fit a flexible distributional regression model that captures the full conditional distribution of daily temperatures. By evaluating this fitted distribution at the 30$^\circ$C threshold, we obtain heat-day probabilities. Required packages can be loaded by

if(!("gamlss" %in% installed.packages())) {
  install.packages("gamlss")
}
library("gamlss")
library("gamlss2")

The data comes from the same R-universe as gamlss2 and is loaded with

if(!("WeatherGermany" %in% installed.packages())) {
  install.packages('WeatherGermany',
    repos = c("https://gamlss-dev.r-universe.dev",
              "https://cloud.r-project.org"))
}
data("WeatherGermany", package = "WeatherGermany")
MUC <- subset(WeatherGermany, id == 1262)

We find that the four-parameter SEP family fits the marginal distribution of Tmax quite well. To estimate a full distributional model, we specify the following additive predictor $$ \eta = \beta_0 + f_1(\texttt{year}) + f_2(\texttt{yday}) + f_3(\texttt{year}, \texttt{yday}) $$ for each parameter. Here, $f_1( \cdot )$ captures the long-term trend, $f_2( \cdot )$ models seasonal variation, and $f_3( \cdot, \cdot )$ represents a time-varying seasonal effect. The required variables can be added to the data by

MUC$year <- as.POSIXlt(MUC$date)$year + 1900
MUC$yday <- as.POSIXlt(MUC$date)$yday

In gamlss, model estimation is performed via

if(!("gamlss.add" %in% installed.packages())) {
  install.packages("gamlss.add")
}
library("gamlss.add")

f1 <- Tmax ~ ga(~ s(year) + s(yday, bs = "cc") +
  te(year, yday, bs = c("cr", "cc")))
b1 <- gamlss(f1, family = SEP,
  data = MUC[, c("Tmax", "year", "yday")])

GAMLSS-RS iteration 1: Global Deviance = 65081.31 
GAMLSS-RS iteration 2: Global Deviance = 64953.19 
GAMLSS-RS iteration 3: Global Deviance = 64893.26 
GAMLSS-RS iteration 4: Global Deviance = 64869.21 
GAMLSS-RS iteration 5: Global Deviance = 64859.01 
GAMLSS-RS iteration 6: Global Deviance = 64854.42 
GAMLSS-RS iteration 7: Global Deviance = 64852.19 
GAMLSS-RS iteration 8: Global Deviance = 64850.94 
GAMLSS-RS iteration 9: Global Deviance = 64850.14 
GAMLSS-RS iteration 10: Global Deviance = 64849.54 
GAMLSS-RS iteration 11: Global Deviance = 64849.04 
GAMLSS-RS iteration 12: Global Deviance = 64848.58 
GAMLSS-RS iteration 13: Global Deviance = 64848.13 
GAMLSS-RS iteration 14: Global Deviance = 64847.73 
GAMLSS-RS iteration 15: Global Deviance = 64847.33 
GAMLSS-RS iteration 16: Global Deviance = 64846.94 
GAMLSS-RS iteration 17: Global Deviance = 64846.56 
GAMLSS-RS iteration 18: Global Deviance = 64846.19 
GAMLSS-RS iteration 19: Global Deviance = 64845.83 
GAMLSS-RS iteration 20: Global Deviance = 64845.48 

Warning in RS(): Algorithm RS has not yet converged

This setup requires loading the gamlss.add package to access mgcv-based smooth terms. Estimation takes 20 iterations of the backfitting algorithm (without full convergence) and about 44 seconds on a 64-bit Linux system. Moreover, gamlss() requires that the input data contains no NA values. In gamlss2 the model can be specified directly, following mgcv syntax

f2 <- Tmax ~ s(year) + s(yday, bs = "cc") +
  te(year, yday, bs = c("cr", "cc"))
b2 <- gamlss2(f2, family = SEP, data = MUC)

GAMLSS-RS iteration  1: Global Deviance = 65102.5733 eps = 0.574320     
GAMLSS-RS iteration  2: Global Deviance = 64949.999 eps = 0.002343     
GAMLSS-RS iteration  3: Global Deviance = 64891.6543 eps = 0.000898     
GAMLSS-RS iteration  4: Global Deviance = 64868.3595 eps = 0.000358     
GAMLSS-RS iteration  5: Global Deviance = 64858.5604 eps = 0.000151     
GAMLSS-RS iteration  6: Global Deviance = 64854.1832 eps = 0.000067     
GAMLSS-RS iteration  7: Global Deviance = 64852.0539 eps = 0.000032     
GAMLSS-RS iteration  8: Global Deviance = 64850.8608 eps = 0.000018     
GAMLSS-RS iteration  9: Global Deviance = 64850.0759 eps = 0.000012     
GAMLSS-RS iteration 10: Global Deviance = 64849.4948 eps = 0.000008     

This model converges in 10 iterations and requires only about 2 seconds of computation time, yielding a similar deviance (small differences arise due to differences in smoothing parameter optimization). In many applications, it is desirable to use the same predictor structure for all distribution parameters. In gamlss, this requires specifying identical formulas separately via sigma.formula, nu.formula, and tau.formula, which can be tedious. In gamlss2, this is simplified using “.”

f3 <- Tmax ~ s(year) + s(yday, bs = "cc") +
  te(year, yday, bs = c("cr", "cc")) | . | . | .
b3 <- gamlss2(f3, family = SEP, data = MUC)

GAMLSS-RS iteration  1: Global Deviance = 64940.3003 eps = 0.575381     
GAMLSS-RS iteration  2: Global Deviance = 64721.0181 eps = 0.003376     
GAMLSS-RS iteration  3: Global Deviance = 64634.6688 eps = 0.001334     
GAMLSS-RS iteration  4: Global Deviance = 64597.9348 eps = 0.000568     
GAMLSS-RS iteration  5: Global Deviance = 64582.5418 eps = 0.000238     
GAMLSS-RS iteration  6: Global Deviance = 64574.6097 eps = 0.000122     
GAMLSS-RS iteration  7: Global Deviance = 64570.3663 eps = 0.000065     
GAMLSS-RS iteration  8: Global Deviance = 64567.5089 eps = 0.000044     
GAMLSS-RS iteration  9: Global Deviance = 64565.4716 eps = 0.000031     
GAMLSS-RS iteration 10: Global Deviance = 64563.7761 eps = 0.000026     
GAMLSS-RS iteration 11: Global Deviance = 64562.4298 eps = 0.000020     
GAMLSS-RS iteration 12: Global Deviance = 64561.2848 eps = 0.000017     
GAMLSS-RS iteration 13: Global Deviance = 64560.2679 eps = 0.000015     
GAMLSS-RS iteration 14: Global Deviance = 64559.3577 eps = 0.000014     
GAMLSS-RS iteration 15: Global Deviance = 64558.5393 eps = 0.000012     
GAMLSS-RS iteration 16: Global Deviance = 64557.797 eps = 0.000011     
GAMLSS-RS iteration 17: Global Deviance = 64557.1061 eps = 0.000010     
GAMLSS-RS iteration 18: Global Deviance = 64556.4892 eps = 0.000009     

This model converges in 18 iterations in about 15 seconds. After estimation, results can be inspected using the summary() method for both packages. Using plot() in gamlss produces standard residual diagnostic plots, whereas in gamlss2

plot(b3)

displays all estimated covariate effects. For residual diagnostics, gamlss2 leverages the topmodels package, which provides infrastructures for probabilistic model assessment. E.g., a PIT histogram can be created by

if(!("topmodels" %in% installed.packages())) {
  install.packages("topmodels", repos = "https://zeileis.R-universe.dev")
}
library("topmodels")

pithist(b3)

showing good model calibration. Finally, we compute the probability of a heat day for 2025. First, the procast() function from `topmodels predicts the fitted distributions

nd <- data.frame("year" = 2025, "yday" = 0:365)
pf <- procast(b3, newdata = nd, drop = TRUE)

This yields a distribution vector pf using the infrastructure from the distributions3 package. Probabilities of a heat day can then be calculated with the corresponding cdf() method.

if(!("distributions3" %in% installed.packages())) {
  install.packages("distributions3")
}
library("distributions3")
probs <- 1 - cdf(pf, 30)

and visualized, for example, by

par(mar = c(4, 4, 1, 1))
plot(probs, type = "l", xlab = "Day of Year",
  ylab = "Prob(Tmax > 30)")

Note that a predict() method is available for both gamlss and gamlss2, allowing direct prediction of distribution parameters. However, in gamlss, predict() may not fully support new data in all cases.