Lisa Schlosser, Torsten Hothorn, Reto Stauffer, Achim Zeileis (2018). “Distributional Regression Forests for Probabilistic Precipitation Forecasting in Complex Terrain.” arXiv.org E-Print Archive arXiv:1804.02921 [stat.ME]. https://arxiv.org/abs/1804.02921
To obtain a probabilistic model for a dependent variable based on some set of explanatory variables, a distributional approach is often adopted where the parameters of the distribution are linked to regressors. In many classical models this only captures the location of the distribution but over the last decade there has been increasing interest in distributional regression approaches modeling all parameters including location, scale, and shape. Notably, so-called non-homogenous Gaussian regression (NGR) models both mean and variance of a Gaussian response and is particularly popular in weather forecasting. More generally, the GAMLSS framework allows to establish generalized additive models for location, scale, and shape with smooth linear or nonlinear effects. However, when variable selection is required and/or there are non-smooth dependencies or interactions (especially unknown or of high-order), it is challenging to establish a good GAMLSS. A natural alternative in these situations would be the application of regression trees or random forests but, so far, no general distributional framework is available for these. Therefore, a framework for distributional regression trees and forests is proposed that blends regression trees and random forests with classical distributions from the GAMLSS framework as well as their censored or truncated counterparts. To illustrate these novel approaches in practice, they are employed to obtain probabilistic precipitation forecasts at numerous sites in a mountainous region (Tyrol, Austria) based on a large number of numerical weather prediction quantities. It is shown that the novel distributional regression forests automatically select variables and interactions, performing on par or often even better than GAMLSS specified either through prior meteorological knowledge or a computationally more demanding boosting approach.
Distributional trees as part of the parametric and recursive partitioning modeling toolbox.
Total precipitation predictions by a distributional forest at station Axams for July 24 in 2009, 2010, 2011 and 2012 learned on data from 1985-2008. Observations are left-censored at 0.
Map of Tyrol coding the best-performing model for each station (type of symbol). The color codes whether the distributional forest had higher (green) or lower (red) CRPS compared to the best of the other three models. Station Axams is highlighted in bold.