Quantile Regression, Variational Autoencoders, and Diffusion Models for Uncertainty Quantification: A Spatial Analysis of Sub-seasonal Wind Speed Prediction

TL;DR

This study addresses the need for spatially coherent uncertainty representation in sub-seasonal wind speed downscaling from large-scale predictors such as $Z_{500}$ to surface wind speed $U_{100}$. It evaluates four probabilistic approaches—QNN, VNN, DNN—and a baseline SNN within a two-stage ERA5–ECMWF framework to produce probabilistic wind ensembles. The results show that diffusion-based downscaling preserves multi-scale spatial dependencies and energy spectra more faithfully than quantile regression or variational autoencoding approaches, which tend to smooth or under-disperse, while grid-wise assessments alone can be misleading. The findings have direct implications for wind-energy planning and risk management, underscoring the importance of spatially aware uncertainty representations in operational sub-seasonal forecasts.

Abstract

This study aims to improve the spatial representation of uncertainties when regressing surface wind speeds from large-scale atmospheric predictors for sub-seasonal forecasting. Sub-seasonal forecasting often relies on large-scale atmospheric predictors such as 500 hPa geopotential height (Z500), which exhibit higher predictability than surface variables and can be downscaled to obtain more localised information. Previous work by Tian et al. (2024) demonstrated that stochastic perturbations based on model residuals can improve ensemble dispersion representation in statistical downscaling frameworks, but this method fails to represent spatial correlations and physical consistency adequately. More sophisticated approaches are needed to capture the complex relationships between large-scale predictors and local-scale predictands while maintaining physical consistency. Probabilistic deep learning models offer promising solutions for capturing complex spatial dependencies. This study evaluates three probabilistic methods with distinct uncertainty quantification mechanisms: Quantile Regression Neural Network that directly models distribution quantiles, Variational Autoencoders that leverage latent space sampling, and Diffusion Models that utilise iterative denoising. These models are trained on ERA5 reanalysis data and applied to ECMWF sub-seasonal hindcasts to regress probabilistic wind speed ensembles. Our results show that probabilistic downscaling approaches provide more realistic spatial uncertainty representations compared to simpler stochastic methods, with each probabilistic model offering different strengths in terms of ensemble dispersion, deterministic skill, and physical consistency. These findings establish probabilistic downscaling as an effective enhancement to operational sub-seasonal wind forecasts for renewable energy planning and risk assessment.

Figures (5)

• Figure 1: Architectures of four uncertainty quantification neural networks. (a) Stochastic Neural Network (SNN): SNN incorporates uncertainty by adding gaussian noise $\mathcal{N}(\mu, \sigma^2)$ to deterministic CNN outputs; (b) Quantile Regression Neural Network (QNN): QNN predicts ten specific quantiles ($(\hat{q}_{\tau_1}, \hat{q}_{\tau_2}, ..., \hat{q}_{\tau_{10}})$) simultaneously to characterise conditional distributions; (c) Variational Autoencoder-based Neural Network (VNN): VNN encodes inputs into a latent probability distribution by outputting mean $\mu_{z}$ and log-variance $\log\sigma^2_{z}$, then samples from $\mathcal{N}(\mu_{z}, \sigma^2_{z})$ to generate diverse outputs; (d) Diffusion-based Neural Network (DNN): DNN implements a bidirectional diffusion framework with two complementary processes: a forward process ($f_2$) that progressively adds noise to the clean target field $\mathbf{Y}_0$, and a reverse denoising process ($f_1$) that iteratively removes noise starting from a pure noise state $\hat{\mathbf{Y}}_T \sim \mathcal{N}(0,\mathbf{I})$. During inference, the model predicts noise components $\hat{\boldsymbol{\epsilon}}_t(\hat{\mathbf{Y}}_t,t,\mathbf{X})$ conditioned on both the noisy state and input $\mathbf{X}$ at each timestep, gradually transforming random noise into physically consistent predictions $\hat{\mathbf{Y}}_0$.
• Figure 2: Spatially averaged MSE, CRPS and SSR of different statistical downscaling models and ECMWF ensemble hindcasts across lead weeks. [id=Ganglin]Black arrows indicate optimal directions: MSE and CRPS (lower is better), SSR (closer to 1, shown as black dashed line, is better).
• Figure 3: [id=Ganglin] Spatial distribution of the relative differences in MSE and CRPS ($\Delta_r$ MSE and $\Delta_r$ CRPS) between SNN, QNN, VNN and DNN compared to Climatology. Green grids indicate improvements (negative values), while pink grids indicate degradation (positive values). Statistical significance is assessed using bootstrap resampling and indicated by symbols overlaid on grid points: "+" denotes significance at $\alpha$ = 0.01, "Y" at $\alpha$ = 0.05, and "I" at $\alpha$ = 0.1. The $\mu$ value at the top of each subfigure represents the average relative percentage difference across Europe. Same as \ref{['fig:DeltaMapsRA']}, but for hindcasts at lead week 3.
• Figure 4: [id=Ganglin] Skill scores of reconstructed fields by EOF modes for different models on hindcasts. (a) MSSS of ensemble mean, (b) CRPSS, and (c) SSR vary with the number of EOF modes used. Black arrows indicate optimal directions (as in Figure \ref{['fig:CurveHC']}): MSSS and CRPSS (higher is better); SSR (closer to 1, shown as black dashed line, is better).Same as \ref{['fig:EOFRA']}, but for hindcasts at lead week 3, using ECMWF as a benchmark.
• Figure 5: Energy spectrum analysis on hindcasts at lead week 3 compared with reanalysis data. (a) Absolute energy spectra displayed on a logarithmic scale showing similar patterns between solid lines (reanalysis results) and dashed lines (hindcast results) for corresponding models. (b) Relative energy spectrum skill compared to ERA5 reference, where positive values indicate energy under-estimation and negative values indicate over-estimation. [id=Ganglin]The black arrow and black dashed line at zero indicate the optimal direction: RESS values closer to zero represent better spectral reproduction. The horizontal axis represents zonal wavenumbers with corresponding wavelengths (km). [id=Ganglin]While the absolute spectra (a) show minimal visible differences between reanalysis and hindcast environments, the relative skill metric (b) reveals that model-specific characteristics persist but with modified magnitudes in the forecast environment.