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# Covariance Estimation

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arXiv Nov 9, 2026
GraphCast takes over only ensemble perturbations: a month of real-observation cycling keeps background errors at least level with a physics model
Placing the machine-learning weather model GraphCast inside a real data-assimilation cycle, with the mean still supplied by the physics-based MPAS model and GraphCast evolving only ensemble perturbations, gives ensemble spread and covariance structure comparable to an all-physics baseline over one month; the EnVar background RMSE against GFS analyses is consistently lower below model level 40, while degradation appears near the model top from interpolation and extrapolation.
Placing the machine-learning weather model GraphCast inside a real data-assimilation cycle, with the mean still supplied by the physics-based MPAS model and GraphCast evolving only ensemble perturbations, gives ensemble spread and covariance structure comparable to an all-physics baseline over one month; the EnVar background RMSE against GFS analyses is consistently lower below model level 40, while degradation appears near the model top from interpolation and extrapolation.
Simon Toedtli（First author）
GraphCast Data Assimilation Covariance Estimation Machine learning weather forecasting
arXiv Oct 7, 2026
Hierarchical Spectral Shrinkage (HSS) beats separate, fully pooled, and shared-subspace estimators on synthetic experiments and multi-study gene-expression data
The authors introduce Hierarchical Spectral Shrinkage (HSS), an empirical-Bayes framework that shrinks each task's leading spectral directions toward a data-adaptively learned common basis with shrinkage varying across tasks and spectral components, yielding the posterior mean of singular vectors in a surrogate Bayesian regression; across synthetic experiments and three immune-cell gene-expression datasets (sample sizes 156, 628, 146), HSS improves estimation accuracy and out-of-sample performance over separate, fully pooled, and shared-subspace approaches.
The authors introduce Hierarchical Spectral Shrinkage (HSS), an empirical-Bayes framework that shrinks each task's leading spectral directions toward a data-adaptively learned common basis with shrinkage varying across tasks and spectral components, yielding the posterior mean of singular vectors in a surrogate Bayesian regression; across synthetic experiments and three immune-cell gene-expression datasets (sample sizes 156, 628, 146), HSS improves estimation accuracy and out-of-sample performance over separate, fully pooled, and shared-subspace approaches.
Duke University
Lorenzo Mauri（First author, Corresponding author）
Hierarchical Spectral Shrinkage Empirical Bayes Covariance Estimation Factor Models Singular Value Decomposition

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