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arxiv: 2605.30122 · v2 · pith:VXMJI2GLnew · submitted 2026-05-28 · 💻 cs.LG · cs.AI

Beyond MSE: Improving Precipitation Nowcasting with Multi-Quantile Regression

Gijs van Nieuwkoop , Siamak Mehrkanoon This is my paper

Pith reviewed 2026-06-29 08:42 UTC · model grok-4.3

classification 💻 cs.LG cs.AI
keywords precipitation nowcastingmulti-quantile regressionpinball lossradar nowcastingheavy precipitationdeterministic forecastSmaAt-UNetdeep learning
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The pith

Reformulating precipitation nowcasting training as multi-quantile regression improves the central deterministic forecast by 8.6% MSE while also producing upper-quantile outputs for heavy rain risk.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper examines whether an established nowcasting model can be trained more effectively by treating the task as simultaneous prediction of multiple quantiles rather than a single point estimate. It compares standard MSE and MAE losses against a multi-quantile pinball loss on radar data over the Netherlands and reports that the quantile approach yields a lower test-set MSE for the central forecast. The same training run also supplies upper-quantile fields that directly support risk assessment for intense rainfall without requiring a separate generative model. A sympathetic reader would care because current pointwise losses often produce overly smooth outputs that under-represent extremes, and a simple loss swap could improve both accuracy and utility in operational settings.

Core claim

Using SmaAt-UNet as the base architecture, the study shows that multi-quantile training with pinball loss improves the central deterministic forecast, decreasing test-set MSE by 8.6% relative to a model trained with MSE, while simultaneously generating upper-quantile outputs that are useful for risk-sensitive prediction of heavy precipitation.

What carries the argument

The multi-quantile pinball loss, which jointly optimizes predictions at several quantile levels so that the median quantile serves as the deterministic forecast and the upper quantiles supply tail estimates.

If this is right

  • An existing deterministic nowcasting architecture can be retrained with the new loss to obtain both a stronger central forecast and explicit upper-tail estimates.
  • Upper-quantile outputs can be used directly for threshold-based warnings without post-processing or ensemble generation.
  • The same training procedure offers a lightweight alternative to generative or sampling-based uncertainty methods in precipitation nowcasting.
  • The approach requires no change to model architecture or inference procedure beyond the loss function.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • Operational nowcasting pipelines could replace separate uncertainty modules with a single multi-quantile model if the gain generalizes to other regions and lead times.
  • The method may extend to other meteorological regression tasks where both central accuracy and tail behavior matter, such as wind or temperature extremes.
  • If the improvement holds under stricter controls, it suggests that quantile-based training can mitigate the smoothing bias common in MSE-optimized convolutional nowcasters.

Load-bearing premise

The measured MSE reduction and the usefulness of the upper quantiles result from the choice of multi-quantile loss rather than from differences in hyperparameter choices, preprocessing, or random seed effects.

What would settle it

Re-running the MSE and multi-quantile experiments with identical hyperparameters, data splits, and random seeds across multiple independent trials and finding that the 8.6% MSE gap disappears or reverses.

Figures

Figures reproduced from arXiv: 2605.30122 by Gijs van Nieuwkoop, Siamak Mehrkanoon.

Figure 1
Figure 1. Figure 1: Schematic overview of the multi-quantile training setup, with pinball losses 𝜌𝑞 and corresponding weights 𝑤𝑞 for quantiles 𝑞 ∈ 𝑄 [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: MSE, MAE, and pinball losses, visualized as functions of the error 𝑒 = 𝑦 − ̂𝑦. error 𝑒 = 𝑦 − ̂𝑦 (𝑞) , the pinball loss is defined as follows: 𝜌𝑞 (𝑦, ̂𝑦(𝑞) ) = max(𝑞𝑒,(𝑞 − 1)𝑒) = { 𝑞|𝑒|, 𝑒 ≥ 0, (1 − 𝑞)|𝑒|, 𝑒 < 0. (1) This loss is asymmetric unless 𝑞 = 0.50, as illustrated in [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: One-dimensional grid search over the shared upper￾quantile loss weight 𝑤0.90 = 𝑤0.95, with 𝑤0.50 fixed at 1.0. output maps but also influence the shared representation used by the median forecast. Multi-quantile training can therefore be interpreted as a form of auxiliary training, where the model is encouraged to learn features that support both central precipitation prediction and risk-sensitive upper-ta… view at source ↗
Figure 4
Figure 4. Figure 4: CSI as a function of forecast lead time for precipitation thresholds of 0.5, 10 and 20 mm/h [PITH_FULL_IMAGE:figures/full_fig_p005_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Qualitative comparison of ground-truth precipitation and model predictions at selected forecast lead times. upper quantiles are not merely more sensitive forecasts, but better operating points for risk-sensitive heavy-precipitation prediction, where missed events may be more costly than additional false alarms [PITH_FULL_IMAGE:figures/full_fig_p006_5.png] view at source ↗
read the original abstract

Deep-learning precipitation nowcasting models are often optimized using pointwise losses such as mean squared error or mean absolute error, which can lead to overly smooth forecasts and poor representation of heavy rainfall. This study investigates whether the predictive performance of an established deterministic nowcasting architecture can be improved by reformulating training as a multi-quantile regression problem. Using SmaAt-UNet as a core model, we compare MSE, MAE, and multi-quantile pinball-loss training on radar precipitation nowcasting over the Netherlands. The results show that multi-quantile training improves the central deterministic forecast, decreasing test-set MSE by 8.6\% compared to a model trained using MSE, while also producing upper-quantile outputs that are useful for risk-sensitive prediction of heavy precipitation. These findings suggest that quantile regression provides a simple alternative to standard pointwise losses without requiring a new architecture or generative sampling procedure. The implementation of our models and training setup is available on \href{https://github.com/gijsvn/Multi-Quantile-Precipitation-Nowcasting}{GitHub}.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

1 major / 0 minor

Summary. The paper claims that reformulating training of the SmaAt-UNet architecture as a multi-quantile regression problem with pinball loss improves the central deterministic forecast on Dutch radar precipitation nowcasting data, yielding an 8.6% reduction in test-set MSE relative to MSE training while also producing useful upper-quantile outputs for heavy-precipitation risk assessment; the work positions this as a simple alternative to pointwise losses that requires no new architecture or generative sampling.

Significance. If the improvement is attributable to the loss formulation rather than uncontrolled experimental factors, the result would show that multi-quantile training can simultaneously boost deterministic accuracy and supply calibrated risk-sensitive predictions without architectural innovation. The public GitHub release of code and training setup is a clear strength that aids reproducibility and allows direct verification of the empirical protocol.

major comments (1)
  1. [Abstract] Abstract: the headline claim of an 8.6% test-set MSE reduction is presented without any description of whether the MSE-trained and multi-quantile models used identical hyperparameter search procedures, learning-rate schedules, data-augmentation policies, early-stopping criteria, random seeds, or validation-based model selection. Because the central claim is an empirical head-to-head comparison that attributes the gain specifically to the multi-quantile pinball loss, the absence of these controls is load-bearing; single-run point estimates without matched protocols cannot support the attribution.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the constructive feedback on the need for explicit experimental controls to support the central empirical claim. We address the major comment below and will revise the manuscript to strengthen clarity.

read point-by-point responses

  1. Referee: [Abstract] Abstract: the headline claim of an 8.6% test-set MSE reduction is presented without any description of whether the MSE-trained and multi-quantile models used identical hyperparameter search procedures, learning-rate schedules, data-augmentation policies, early-stopping criteria, random seeds, or validation-based model selection. Because the central claim is an empirical head-to-head comparison that attributes the gain specifically to the multi-quantile pinball loss, the absence of these controls is load-bearing; single-run point estimates without matched protocols cannot support the attribution.

    Authors: We agree that the abstract does not explicitly describe the matched protocols and that this detail is important for attributing the MSE improvement to the pinball loss. The Methods section of the manuscript specifies that both models were trained using the same hyperparameter search, learning-rate schedules, data-augmentation policies, early-stopping criteria, random seeds, and validation-based selection. To make this explicit at the level of the headline claim, we will revise the abstract to include a concise statement confirming the identical experimental setup for the two training regimes. This change will be incorporated in the revised version. revision: yes

Circularity Check

0 steps flagged

No circularity: purely empirical loss-function comparison on fixed architecture

full rationale

The manuscript reports an experimental head-to-head evaluation of three training losses (MSE, MAE, multi-quantile pinball) applied to the identical SmaAt-UNet architecture on the same radar dataset. The headline 8.6% test-MSE reduction is presented as an observed outcome of that comparison, not as the output of any derivation, uniqueness theorem, or self-referential fitting step. No equations are shown that define a quantity in terms of itself, no parameter is fitted on a subset and then relabeled a prediction, and no load-bearing claim rests on a self-citation chain. The work is therefore self-contained against external benchmarks and receives the default non-circularity finding.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on the assumption that the SmaAt-UNet architecture and the Netherlands radar dataset are appropriate test beds, plus standard supervised-learning assumptions about i.i.d. train/test splits and that pinball loss is correctly implemented for multiple quantiles. No new entities are postulated.

axioms (2)
  • domain assumption SmaAt-UNet is a suitable base architecture for the nowcasting task
    The paper selects it as the core model without additional justification or ablation.
  • domain assumption The radar precipitation dataset over the Netherlands is representative for evaluating nowcasting performance
    All experiments are conducted on this single regional dataset.

pith-pipeline@v0.9.1-grok · 5717 in / 1396 out tokens · 28871 ms · 2026-06-29T08:42:59.469942+00:00 · methodology

discussion (0)

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Reference graph

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