## The Annals of Applied Statistics

- Ann. Appl. Stat.
- Volume 13, Number 3 (2019), 1564-1589.

### Distributional regression forests for probabilistic precipitation forecasting in complex terrain

Lisa Schlosser, Torsten Hothorn, Reto Stauffer, and Achim Zeileis

#### Abstract

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 nonhomogeneous Gaussian regression (NGR) models both mean and variance of a Gaussian response and is particularly popular in weather forecasting. Moreover, generalized additive models for location, scale and shape (GAMLSS) provide a framework where each distribution parameter is modeled separately capturing smooth linear or nonlinear effects. However, when variable selection is required and/or there are nonsmooth 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.

#### Article information

**Source**

Ann. Appl. Stat., Volume 13, Number 3 (2019), 1564-1589.

**Dates**

Received: November 2018

Revised: February 2019

First available in Project Euclid: 17 October 2019

**Permanent link to this document**

https://projecteuclid.org/euclid.aoas/1571277764

**Digital Object Identifier**

doi:10.1214/19-AOAS1247

**Mathematical Reviews number (MathSciNet)**

MR4019150

**Zentralblatt MATH identifier**

07145968

**Keywords**

Parametric models regression trees random forests recursive partitioning probabilistic forecasting GAMLSS

#### Citation

Schlosser, Lisa; Hothorn, Torsten; Stauffer, Reto; Zeileis, Achim. Distributional regression forests for probabilistic precipitation forecasting in complex terrain. Ann. Appl. Stat. 13 (2019), no. 3, 1564--1589. doi:10.1214/19-AOAS1247. https://projecteuclid.org/euclid.aoas/1571277764

#### Supplemental materials

- Supplement A: Different response distributions. To assess the goodness of fit of the Gaussian distribution, left-censored at zero, this supplement employs the same evaluations as in the main manuscript but based on two other distributional assumptions: A logistic distribution, left-censored at zero, is employed to potentially better capture heavy tails—and a two-part hurdle model combining a binary model for zero vs. positive precipitation and a Gaussian model, truncated at zero, for the positive precipitation observations.Digital Object Identifier: doi:10.1214/19-AOAS1247SUPPASupplemental files are immediately available to subscribers. Non-subscribers gain access to supplemental files with the purchase of the article.
- Supplement B: Stationwise evaluation. To show that Axams is a fairly typical station and similar insights can be obtained for other stations as well, this supplement presents the same analysis as in Section 3.3 of the main manuscript for 14 further meteorological stations.Digital Object Identifier: doi:10.1214/19-AOAS1247SUPPBSupplemental files are immediately available to subscribers. Non-subscribers gain access to supplemental files with the purchase of the article.