Estimation of temperature and precipitation uncertainties using quantile neural networks

TL;DR

The paper tackles uncertainty quantification for geophysical variables conditioned on observable states, a problem where nonlinear dynamics and non-Gaussian statistics complicate traditional Gaussian-based approaches. It introduces the ReLU-bias loss quantile neural network (RBLQNN), a multi-quantile regression model that enforces monotonicity and balanced quantile learning via a ReLU-based penalty and a quantile-balancing scheme, trained with a composite loss. Across synthetic datasets and observational data from GSOD and TRMM, the RBLQNN consistently outperforms linear quantile regression and mean-variance estimation baselines, especially when conditional distributions are non-Gaussian or highly nonlinear; temperature distributions tend to be near-Gaussian with nonlinear drivers, while precipitation exhibits strong non-Gaussian behavior necessitating flexible quantile estimation. The results demonstrate the method’s flexibility, robustness, and utility for climate risk assessment, offering a practical tool to constrain uncertainties in geophysical fields without strict parametric distributional assumptions.

Abstract

Extreme events pose significant risks and are challenging to predict. Assessing climate hazards requires placing quantitative constraints on geophysical fields under observable but fluctuating conditions. We propose a framework for estimating uncertainties -- a ReLU-bias loss quantile neural network (RBLQNN) -- with two novel modifications to the loss function to enforce uniform quantile accuracy and reduce degenerate predicted probability distributions. We evaluate the RBLQNN against other probabilistic baselines on a suite of datasets: synthetic datasets, observed daily temperature maxima from 1,501 NOAA Global Surface Summary of the Day (GSOD) weather stations, and altimetry-observed precipitation from the Tropical Rainfall Measuring Mission (TRMM). On synthetic datasets, the RBLQNN accurately predicts conditional distributions where more restrictive methods like linear quantile regression (LQR) or mean-variance estimation (MVE) neural networks fail, mitigates shortcomings of some other quantile neural networks, and converges stably under a range of hyperparameters. When applied to daily temperature maxima, the RBLQNN reveals that temperature distributions are relatively well described by Gaussian statistics, though nonlinear dependencies on local sea level pressure and geopotential heights appear important. For precipitation statistics, the RBLQNN strongly outperforms both LQR and MVE baselines, demonstrating its capacity to capture highly nonlinear and non-Gaussian conditional distributions. The RBLQNN's performance across varied datasets demonstrates it is a flexible and general approach for constraining uncertainties in geophysical quantities with nonlinear or non-Gaussian conditional dependencies.

Figures (10)

• Figure 1: Non-Gaussianity of daily maximum surface air temperature (TMAX) seasonal anomalies, as measured by the sample's (a) skewness, (b) kurtosis, and (c) Kullback-Leibler divergence between the sample and a Gaussian estimated with the approach of hyvarinen2000independent using contrast function $G(u) = \log \cosh{u}$ (higher values indicate greater deviations from Gaussianity).
• Figure 2: Predicted conditional quantiles on Synthetic Dataset 1 (a--d), Dataset 2 (e--h), and Dataset 3 (i--l). Scatterplot shows the test sample generated from the ground truth distribution, while colored lines indicate predicted quantiles at specified probability levels $q$. (a, e, i) True quantiles. (b, f, j) Predictions made using linear quantile regression. (c, g, k) Predictions made using MVE networks. (d, h, l) Predictions made using the RBLQNN.
• Figure 3: Mean absolute errors of predicted quantiles for the RBLQNN (red), linear quantile regression (light green), and MVE neural network (blue) for (a) Synthetic Dataset 1 and (b) Synthetic Dataset 2.
• Figure 4: Performance of different quantile regression neural network techniques on Synthetic Dataset 1 (a, b), Dataset 2 (c, d), and Dataset 3 (e, f). (a, c, e): Mean absolute errors for quantile neural network predictions, averaged over 100 ensembles created by initializing network weights with different random seeds. Red: RBLQNN (Eq. \ref{['eq:net_quantile_loss']}). Blue: Unweighted quantile regression neural network of xu2017composite. Light green: No-bias quantile regression neural network framework implementing the inverse-expectation weighting but without the ReLU bias loss (Eq. \ref{['eq:relu_bias_loss']}). Purple: cumulative increment network of padilla2022quantile. (b, d, f) Boxplots showing the fraction of nondegenerate probability distributions predicted by each quantile neural network approach. Conditional probability distributions are considered degenerate for a given sample if the predicted quantiles are not monotonic.
• Figure 5: Training stability of the RBLQNN and MVE network evaluated over 144 hyperparameter combinations for the three synthetic datasets. (a) Strip plots of the lowest validation loss attained during training for the RBLQNN and MVE for the three different datasets, with gray shading showing a kernel density estimate. Losses are normalized to the $[0, 1]$ range. Colors indicate the learning rate. (b, c, d) Empirical CDFs of the normalized validation losses for each of the three different synthetic datasets for the RBLQNN (red) and MVE network (blue). Boldface percentages in the top left corner indicate proportion of neural networks which converge within 5% of the best loss from all hyperparameter configurations (indicated by dark gray shading). Plain typeface percentages indicate proportions of neural networks within 20% of the best loss (light gray shading).