Distributional regression using generalized additive models for location, scale and shape

Traditional regression analysis has long been dominated by a mean-centered perspective, which treats variability as noise. However, in many natural and social systems, explanatory variables influence not just the average outcome, but also the spread, skewness, and tail behavior of the response. Distributional regression (DR) shifts the focus from conditional means to modeling the entire conditional probability distribution of a response variable.

The Generalized Additive Model for Location, Scale and Shape (GAMLSS) is the most prominent framework for DR. It extends traditional linear and additive models by allowing every parameter of a chosen probability distribution—such as the mean (location), variance (scale), and skewness or kurtosis (shape)—to be modeled as a function of explanatory variables. These functions can include linear terms, non-parametric smoothers, or even machine learning architectures like neural networks.

By capturing the full conditional distribution, GAMLSS provides a robust foundation for risk assessment and uncertainty quantification. It is particularly valuable for:

• Extreme Event Prediction: Directly modeling exceedance probabilities for critical thresholds.
• Health and Growth Standards: Creating centile charts that account for natural variability across populations.
• Ecological Modeling: Identifying how environmental drivers affect species flexibility and regime shifts.
• Probabilistic Forecasting: Improving weather and climate predictions by capturing uncertainty rather than just point estimates.

Because GAMLSS models are more flexible, they require rigorous validation. Standard mean-based diagnostics are insufficient. Instead, researchers use normalized quantile residuals (z-scores) to assess whether the model correctly captures the assumed distribution. Tools like worm plots, bucket plots, and QQ plots are essential for identifying misspecification in specific regions of the data or across the range of explanatory variables.