“The zero-degree of freedom chi-squared distribution in a gradient-boosting framework for probabilistic precipitation forecasting” in Proceedings of the 40th International Workshop on Statistical Modelling!

Read, Cite

Precipitation forecasts are often biased and underdispersive and therefore require correction through statistical postprocessing models. In particular, the distribution of precipitation requires special attention due to its mixed nature, which is characterized by a large probability of zero-precipitation (no precipitation) and a continuous component for positive precipitation amounts. In order to account for this setting, Groß and Möller (2024) proposed the zero-degree of freedom chi-squared distribution as postprocessing distribution. Its generalized probability density function is defined as \[\begin{align} f(y;\lambda, \sigma):= \begin{cases} \mathrm{e}^{-\lambda/2},& y=0,\\ \frac{1}{2\sigma} \cdot \mathrm{e}^{-\frac12 \left(y/\sigma + \lambda \right)} \cdot \sqrt{\sigma \lambda/y}\cdot I_{-1}\left(\sqrt{\lambda y/\sigma}\right),& y>0, \end{cases} \end{align}\] where \(I_{\nu}\) is the modified Bessel function of the first kind of order \(\nu\).

Figure 1: Generalized probability density function of the zero-degree of freedom chi-squared distribution for different values of \(\lambda\) and \(\sigma\).

Key Contributions

In our new conference paper, “The zero-degree of freedom chi-squared distribution in a gradient-boosting framework for probabilistic precipitation forecasting”, we extend the work of Groß and Möller (2024) and employ this distribution for the Ensemble Model Output Statistics (EMOS; Gneiting et al. (2005)) approach in a gradient-boosting estimation framework. In particular, we consider the empirical ensemble mean and empirical ensemble standard deviation of eleven different weather variables as covariates in the linear regression functions for the gradient-boosted EMOS (Chi0-EMOS-GB). We evaluate the Chi0-EMOS-GB against an EMOS version which only uses the empirical ensemble mean and empirical ensemble standard deviation of total precipitation as covariates in the linear regression functions (Chi0-EMOS). Both model variants are estimated separately for each of the 339 locations in Germany, using the available observations and ensemble forecasts and optimizing the negative log-likelihood.

Key Findings

• Chi0-EMOS and Chi0-EMOS-GB improve the forecast skill in comparison to the raw ensemble.
• Chi0-EMOS-GB slightly outperforms Chi0-EMOS w.r.t. the mean continuous ranked probability score and mean absolute error.
• Chi0-EMOS produces marginally sharper and better-calibrated forecasts than Chi0-EMOS-GB.

References

Gneiting, Tilmann, Adrian E. Raftery, Anton H. Westveld, and Tom Goldman. 2005. “Calibrated Probabilistic Forecasting Using Ensemble Model Output Statistics and Minimum CRPS Estimation.” Monthly Weather Review 133 (5): 1098–118. https://doi.org/10.1175/mwr2904.1.

Groß, Jürgen, and Annette Möller. 2024. The Zero Degree of Freedom Non-Central Chi Squared Distribution for Ensemble Postprocessing. https://doi.org/10.48550/ARXIV.2404.04964.