First Steps

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 \(30^\circ\text{C}\) 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"))
}

Installing package into '/usr/local/lib/R/site-library'
(as 'lib' is unspecified)

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",
    repos = c("https://gamlss-dev.r-universe.dev",
              "https://cloud.r-project.org"))
}
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 = 65324.7244 eps = 0.572868     
GAMLSS-RS iteration  2: Global Deviance = 65122.8496 eps = 0.003090     
GAMLSS-RS iteration  3: Global Deviance = 64958.3544 eps = 0.002525     
GAMLSS-RS iteration  4: Global Deviance = 64895.5747 eps = 0.000966     
GAMLSS-RS iteration  5: Global Deviance = 64870.287 eps = 0.000389     
GAMLSS-RS iteration  6: Global Deviance = 64859.484 eps = 0.000166     
GAMLSS-RS iteration  7: Global Deviance = 64854.7149 eps = 0.000073     
GAMLSS-RS iteration  8: Global Deviance = 64852.3536 eps = 0.000036     
GAMLSS-RS iteration  9: Global Deviance = 64851.0278 eps = 0.000020     
GAMLSS-RS iteration 10: Global Deviance = 64850.1599 eps = 0.000013     
GAMLSS-RS iteration 11: Global Deviance = 64849.5251 eps = 0.000009     

This model converges in 11 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 = 64914.5444 eps = 0.575550     
GAMLSS-RS iteration  2: Global Deviance = 64722.0558 eps = 0.002965     
GAMLSS-RS iteration  3: Global Deviance = 64649.9977 eps = 0.001113     
GAMLSS-RS iteration  4: Global Deviance = 64618.3532 eps = 0.000489     
GAMLSS-RS iteration  5: Global Deviance = 64603.4491 eps = 0.000230     
GAMLSS-RS iteration  6: Global Deviance = 64595.9065 eps = 0.000116     
GAMLSS-RS iteration  7: Global Deviance = 64591.9297 eps = 0.000061     
GAMLSS-RS iteration  8: Global Deviance = 64589.3226 eps = 0.000040     
GAMLSS-RS iteration  9: Global Deviance = 64587.5186 eps = 0.000027     
GAMLSS-RS iteration 10: Global Deviance = 64585.9935 eps = 0.000023     
GAMLSS-RS iteration 11: Global Deviance = 64584.739 eps = 0.000019     
GAMLSS-RS iteration 12: Global Deviance = 64583.6196 eps = 0.000017     
GAMLSS-RS iteration 13: Global Deviance = 64582.5866 eps = 0.000015     
GAMLSS-RS iteration 14: Global Deviance = 64581.616 eps = 0.000015     
GAMLSS-RS iteration 15: Global Deviance = 64580.6968 eps = 0.000014     
GAMLSS-RS iteration 16: Global Deviance = 64579.8228 eps = 0.000013     
GAMLSS-RS iteration 17: Global Deviance = 64578.9915 eps = 0.000012     
GAMLSS-RS iteration 18: Global Deviance = 64578.2004 eps = 0.000012     
GAMLSS-RS iteration 19: Global Deviance = 64577.4475 eps = 0.000011     
GAMLSS-RS iteration 20: Global Deviance = 64576.737 eps = 0.000011     

This model converges in 20 iterations in about 30 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.

References

Rigby, R. A., and D. M. Stasinopoulos. 2005. “Generalized Additive Models for Location, Scale and Shape.” Journal of the Royal Statistical Society C 54 (3): 507–54. https://doi.org/10.1111/j.1467-9876.2005.00510.x.