# Signature Kernel Scoring Rule: A Spatio-Temporal Diagnostic for Probabilistic Weather Forecasting

arXiv: https://arxiv.org/abs/2510.19110 · HTML: https://mlsanity.com/paper/2510.19110/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 15 of its 36 references that are in the map.
- First posted on arXiv 21 Oct 2025.

## Builds on (most cited first, 15 of 15)

- [Improved Techniques for Training GANs](https://mlsanity.com/paper/1606.03498/) (2016)
- [FourCastNet: A Global Data-driven High-resolution Weather Model using Adaptive Fourier Neural Operators](https://mlsanity.com/paper/2202.11214/) (2022)
- [WeatherBench: A benchmark dataset for data-driven weather forecasting](https://mlsanity.com/paper/2002.00469/) (2020)
- [FuXi: A cascade machine learning forecasting system for 15-day global weather forecast](https://mlsanity.com/paper/2306.12873/) (2023)
- [Neural General Circulation Models for Weather and Climate](https://mlsanity.com/paper/2311.07222/) (2023)
- [BayesFlow: Learning complex stochastic models with invertible neural networks](https://mlsanity.com/paper/2003.06281/) (2020)
- [Scaling transformer neural networks for skillful and reliable medium-range weather forecasting](https://mlsanity.com/paper/2312.03876/) (2023)
- [A Primer on the Signature Method in Machine Learning](https://mlsanity.com/paper/1603.03788/) (2016)
- [Kernels for sequentially ordered data](https://mlsanity.com/paper/1601.08169/) (2016)
- [Deep Signature Transforms](https://mlsanity.com/paper/1905.08494/) (2019)
- [The Signature Kernel is the solution of a Goursat PDE](https://mlsanity.com/paper/2006.14794/) (2020)
- [The iisignature library: efficient calculation of iterated-integral signatures and log signatures](https://mlsanity.com/paper/1802.08252/) (2018)
- [Probabilistic Forecasting with Generative Networks via Scoring Rule Minimization](https://mlsanity.com/paper/2112.08217/) (2021)
- [Non-adversarial training of Neural SDEs with signature kernel scores](https://mlsanity.com/paper/2305.16274/) (2023)
- [Strictly proper kernel scores and characteristic kernels on compact spaces](https://mlsanity.com/paper/1712.05279/) (2017)

## Built on by (most cited first, 2 of 2)

- [Signature-Kernel Based Evaluation Metrics for Robust Probabilistic and Tail-Event Forecasting](https://mlsanity.com/paper/2602.10182/) (2026)
- [Stochastically Perturbed Weights: Ensembles from Deterministic Machine-Learning Weather Models](https://mlsanity.com/paper/2609.08412/) (2026)

## Abstract (arXiv)

Modern weather forecasting has increasingly transitioned from numerical weather prediction (NWP) to data-driven machine learning forecasting techniques. While these new models produce probabilistic forecasts to quantify uncertainty, their training and evaluation may remain hindered by conventional scoring rules, primarily MSE, which are designed for single time point predictions and ignore the highly correlated data structures present in weather behaviour. This work introduces the signature kernel scoring rule to the domain of weather forecasting, which reframes weather variables as continuous paths to encode temporal and spatial dependencies through iterated integrals. Validated as strictly proper through the use of path augmentations to guarantee uniqueness, the signature kernel provides a theoretically robust metric for forecast verification and model training. Empirical evaluations through weather scorecards on WeatherBench 2 models demonstrate the signature kernel scoring rule's high discriminative power and unique capacity to capture path-dependent interactions. Following previous demonstration of successful adversarial-free probabilistic training, we train sliding window generative neural networks using a predictive-sequential scoring rule on ERA5 reanalysis weather data. Using a lightweight model, we demonstrate that signature kernel based training outperforms climatology for forecast paths of up to fifteen timesteps.