Stop using root-mean-square error as a precipitation target!

Abstract

Root-mean-square error (RMSE) remains the default training loss for data-driven precipitation models, despite precipitation being semi-continuous, zero-inflated, strictly non-negative, and heavy-tailed. This Gaussian-implied objective misspecifies the data-generating process because it tolerates negative predictions, underpenalises rare heavy events, and ignores the mass at zero. We propose replacing RMSE with the Tweedie deviance, a likelihood-based and differentiable loss from the exponential--dispersion family with variance function $V(\mu)=\mu^p$. For $1<p<2$ it yields a compound Poisson--Gamma distribution with a point mass at zero and a continuous density for $y>0$, matching observed precipitation characteristics. We (i) estimate $p$ from the variance--mean power law and show that precipitation across temporal aggregations is far from Gaussian, with the Tweedie power $p$ increasing with accumulation length towards a Gamma limit; and (ii) demonstrate consistent skill gains when training deep data-driven models with Tweedie deviance in place of RMSE. In diffusion-model downscaling over Beijing, Tweedie loss improves wet-pixel MAE and extreme recall ($\sim0.60$ vs $0.50$ at the 99th percentile). In ConvLSTM nowcasting over Kolkata, Tweedie loss yields improved wet-pixel MAE and dry-pixel hit rates, with improvements that compound autoregressively with lead time (for MAE, $\sim2%$ at $t{+}1$ growing to $\sim16%$ at $t{+}4$). Because the Tweedie deviance is continuous in $p$, it adapts smoothly across scales, offering a statistically justified, practical replacement for RMSE in precipitation-based learning tasks.