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Learning joint probabilistic weather forecasts from station observations alone

Chaeyeon Yi, Yun Am Seo

arXiv:2610.09898v1stat.APcs.LGphysics.ao-ph

TL;DR

Joint weather-risk forecasts need to represent dependence between variables, yet the paper frames station-based joint forecasting as a gap. CLARA learns five-variable Gaussian forecasts from station observations and uses an explicit covariance; it outperforms persistence across ten regions, while the value of learned dependence varies by scoring rule.

  • Problem

    The forecast target is a joint distribution of surface variables, because marginal forecasts alone do not specify the dependence needed for derived quantities and compound events.

    Source: Forecast problem · p. 13

  • Method

    CLARA learns joint five-variable Gaussian means and covariances from station observations, then applies a lead-wise covariance scale that preserves correlations.

    Source: Discussion · p. 12, Lead-wise recalibration of the… · p. 17

  • Results

    CLARA has lower multi-year lead-mean energy scores than persistence and climatology in ten regions, and than similarly sized learned models across the evaluated networks and regions.

    Source: Discussion · p. 12

  • Takeaways & Limitations

    The learned routing coefficient cannot be interpreted as forecast influence by itself; the content scale and forecast interventions also matter.

    Source: Discussion · p. 11

    The Gaussian assumption cannot represent multimodal, asymmetric, or heavy-tailed conditional distributions.

    Source: Discussion · p. 12

Abstract

from arXiv · show

Assessing compound weather risks requires forecasts representing dependence between variables. CLARA (Calibrated Advection-Routing Attention) learns joint Gaussian predictive distributions of five surface variables from station observations alone, without numerical weather prediction or reanalysis; the approximately 28,000-parameter model supports CPU training and prediction. Across six multi-year folds on 96 stations, its lead-mean energy score is 4.9% lower than that of a learned comparator with matched temporal inputs (4.7% with a similar parameter count) and 11-65% lower than those of statistical baselines. Holding marginal variances fixed, removing learned correlations worsens joint negative log-likelihood by 1.0-2.8 nats per station. A covariance-scale estimator, proved consistent under stated assumptions, improves short-lead calibration but over-corrects at long leads. Synthetic interventions show an attention-bias coefficient alone does not measure forecast influence. Retrained in ten regions on six continents, CLARA outperforms persistence in all 60 multi-year region-lead comparisons and a similarly sized learned model in 57 of 60.

Results

CLARA improves joint probabilistic forecast scores across nationwide and regional networks, while its explicit covariance makes calibration and dependence measurable. Its routing coefficient declines with lead, but forecast influence must be assessed against the content scale and through interventions.

Overall forecast accuracy

  • 4.9% lower lead-mean energy score than iTransformer + phase (4.7% with a similar parameter count), and 11.0%, 28.8%, and 64.7% lower than EMOS, persistence, and climatology.CLARA also has lower energy scores than every comparison model at all six leads on 96 ASOS.Source: Overall forecast accuracy · pp. 3–4, Figure 2 · p. 4

Joint uncertainty and recalibration

  • At 1 h, calibration error D falls from 1.89 to 0.52; at 12–48 h, rescaling over-corrects, raising D to 0.90–1.48.The 3 h error falls from 1.17 to 0.43, while at 6 h it changes from 0.63 to 0.66.Source: Figure 3 · p. 5, Joint uncertainty and recalibration · pp. 5–6
  • Removing learned correlations while holding marginal variances fixed worsens joint negative log-likelihood by 1.0–2.8 nats per station.A fixed empirical correlation instead attains lower negative log-likelihood than learned correlations on 96 ASOS; lowering the Cholesky floor improves that ranking at 3–48 h but worsens energy score by 0.8–3.5%.Source: Figure 4 · p. 6, Joint uncertainty and recalibration · p. 6

Derived quantities and compound events

  • At 6–48 h, conditional independence and fixed empirical correlations both score 1.2–5.0% better than learned correlations on the rare hot–dry event.Dependence changes derived quantities by different amounts: removing correlations increases the pressure-difference predictive spread, while effects on discomfort index and wind chill are small.Source: Figure 4 · p. 6, Derived quantities and compound events · pp. 6–7

Ablation and the routing mechanism

  • No-phase raises energy score 15.2% on 96 ASOS and 13.2% on 724 AWS at 6 h; no-advection raises it 0.7–1.3% at 1–12 h on ASOS, while zero-wind effects are mixed.On 724 AWS, the no-advection increase is 2.0–2.8% at 1–6 h.Source: Ablation and the routing mechanism · p. 7

Reading the learned coefficient

Replication across observing networks

  • On 724 AWS, CLARA has a 7.0% lower lead-mean energy score than iTransformer + phase and a 6.2% lower score than its similarly sized version.It also scores lower than all tested baselines at every lead.Source: Replication across observing networks · p. 9

Replication across regional observing networks

Discussion

The discussion situates CLARA’s reported forecast comparisons alongside interpretation of learned dependence, routing coefficient, and covariance recalibration. It also specifies the recalibration assumptions, evaluation and implementation choices, and limits on evaluation scope and statistical interpretation.

Multivariate joint predictive distribution and training loss

  • Learned correlations improve joint log-likelihood by one to three nats per station versus conditional independence, but improve energy score by less than 1%.The multivariate Gaussian negative log-likelihood jointly learns mean error, uncertainty scale and correlation; the energy score is described as less sensitive to dependence than the log score.Source: Discussion · p. 11, Multivariate joint predictive… · p. 17

Lead-conditioned advection-routing attention

  • Routing gains mainly come from distance weighting: zero-wind retraining has small effects, removing the distance term causes losses, and surface-wind displacement adds no consistent benefit.The reported upwind displacement is 5–6 km per hour, small relative to the 31–62 km distance-factor decay length at 1 h on the two nationwide networks.Source: Discussion · p. 11, Lead-conditioned advection-routing… · p. 15

Proof of Lemma 1: scale dependence of the advection coefficient

  • The attention-bias coefficient alone does not establish forecast influence, because its relative weight depends on content-logit scale and synthetic interventions found it nearly constant as generating weight varied eight-fold.Joint scaling on the weather models likewise shows that relative weight alone does not determine forecast influence.Source: Discussion · p. 11, Lead-conditioned advection-routing… · p. 16

Discussion

  • Multi-year evaluations report lower energy scores than statistical baselines and lower lead-mean scores than similarly sized learned models, with regional advantages over persistence and climatology.Outside Korea, evaluations use three variables; NWP references cover two regions, some regional long-lead means favour iTransformer, and Gaussian forecasts cannot represent multimodal, asymmetric or heavy-tailed conditionals.Source: Discussion · pp. 11–12

Related approaches

  • Among compared systems, CLARA issues a joint predictive distribution from station observations alone while varying both the upwind-position bias and covariance magnitude by lead.Its Gaussian parameters provide the joint predictive law in closed form, so one forward pass per lead yields the joint forecast.Source: Related approaches · p. 13

Forecast problem

  • CLARA predicts the five-variable joint distribution at each station, representing dependence among variables but not a spatial joint distribution across stations.The variables are temperature, wind speed, relative humidity, station pressure and sea-level pressure.Source: Forecast problem · p. 13

Model overview

  • CLARA represents each station with an input token, combines neighbouring-station information through lead-conditioned attention, and outputs a joint distribution followed by lead-wise uncertainty recalibration.The output layer produces a mean and Cholesky factor for the five-variable covariance.Source: Model overview · p. 14

Lead-wise recalibration of the covariance scale

  • Lead-wise covariance scaling preserves correlations and is consistent under stated assumptions, but recalibration corrects scale at short leads and over-corrects at long leads.The Cholesky diagonal floor also trades joint likelihood against kernel scores: lowering it reverses the log-likelihood ranking at 3–48 h and reduces 1 h pressure CRPS below EMOS, at the cost of worse energy score.Source: Discussion · p. 11, Lead-wise recalibration of the… · p. 18
  • Under stated assumptions, the lead-wise scale estimator is consistent, and at its population value covariance rescaling recovers the true covariance while preserving correlations.Under (A1)–(A2), (A4) gives estimator variance O(1/B), and (A3) transfers calibration-period scale to evaluation. Under general misspecification, the statistic tracks a sample-averaged pseudo-true scale that absorbs bias if that average converges.Source: Lead-wise recalibration of the… · p. 18

Observations and preprocessing

  • The primary evaluation used 96 ASOS stations across six multi-year and two held-out 2026 seasonal folds, with AWS and ten-region GHCNh evaluations testing replication across networks.CLARA’s configuration was selected using 2024–2025 Korean AWS data overlapping domestic evaluation folds, but not the 2026 folds; network type changes alongside density in the AWS comparison.Source: Observations and preprocessing · p. 19

Baselines and ablation

  • Comparisons included six external baselines, a matched-temporal-input learned model, capacity-matched learned comparisons, and structural ablations retrained on the same folds and seeds.The physical baseline was scorable on 81% of 96-station samples at 1 h and 43% at 48 h, so it served as a physical reference rather than a matched competitor. Ablation contributions were defined as fold-mean energy-score differences relative to the full model, with positive values indicating improvement from the component.Source: Baselines and ablation · pp. 19–20

Evaluation metrics and statistical testing

  • Evaluation used energy score for joint forecasts, RMSE and CRPS for per-variable forecasts, and Mahalanobis-distance and PIT diagnostics for calibration.Dependence was tested by rescoring stored forecasts with fixed marginal variances but altered correlations, and by comparing derived quantities and compound-event probabilities across correlation structures. Fold differences were not treated as independent replicates, so no formal significance test was performed; the two 2026 seasons were held out from model development and settings selection.Source: Evaluation metrics and statistical… · pp. 20–22

Unbiasedness of the energy-score estimator

  • The two-batch Monte Carlo energy-score estimator is unbiased, with variance of order 1/n_s under finite second moments.The paired estimator requires n_s distance evaluations for its second term, rather than one evaluation for each of the n_s(n_s−1)/2 distinct within-batch pairs used by the alternative estimator (Pacchiardi et al., 2024).Source: Unbiasedness of the energy-score… · p. 22

Implementation and training settings

  • CLARA has 27,979 trainable parameters and produced a 96-station, one-base-time-and-lead joint distribution in 0.43 ms on one CPU thread.A 40-epoch run took 2.1 min on one CPU thread or 1.7 min on an NVIDIA GeForce RTX 3080 GPU; CPU-run peak host memory was below 700 MB.Source: Implementation and training settings · p. 23

Limits of the evaluation design

  • The evaluation included only stations reporting all variables jointly; per-variable sensor dropout and extreme observation gaps were not tested.Robustness to removing context stations was tested.Source: Limits of the evaluation design · p. 23

Extended Data

The Extended Data figures and tables describe forecast comparisons, robustness tests, calibration and dependence diagnostics, ablations, and regional replication. The supplied passages establish that CLARA outperforms persistence in every region–lead combination in both 2026 held-out seasons.

Extended Data Fig. 1

  • The forecast comparison covers fold-mean energy scores by lead and lead-mean energy-score reductions against baselines on the 724-station AWS network.It includes a width-60 comparator retrained to give a parameter count similar to CLARA; the OA comparison uses the same OA-covered rows for both methods.Source: Figure 1 · p. 24

Extended Data Figs. 2–3

  • The supplemental figures cover robustness to removing context stations, case forecasts, and diagnostic comparisons with operational NWP references.The NWP comparisons use RMSE against HRRR on shared US Midwest stations and CRPS against the raw ECMWF ensemble on shared Korean Peninsula stations.Source: Figure 2 · p. 24, Figure 3 · p. 25

Extended Data Table 2

  • Extended Data Table 2 reports multi-year fold-mean energy scores by lead for CLARA and comparison models on 96 ASOS stations.The table caption specifies six rolling-origin folds and eight seeds for learned models.Source: Table 2 · p. 26

Extended Data Tables 3–6

  • The extended tables also describe calibration, derived-quantity dependence, structural ablations, and regional energy-score comparisons.Their captions specify calibration before and after lead-conditioned scaling, derived quantities under three correlation structures, ablations on the same backbone, and regional comparisons with persistence and climatology.Source: Table 3 · p. 26, Table 4 · p. 27, Table 5 · p. 27

Extended Data

  • The compound hot–dry event is defined using temperature above the training-split 90th percentile and humidity below the 10th percentile.The supplied passage identifies the relative change in its Brier score as an evaluation measure.Source: Extended Data · p. 27

Extended Data Table 6

  • In both 2026 held-out seasons, CLARA outperforms persistence in every region–lead combination.The regional joint distribution is restricted to three variables because station and sea-level pressure are reported at base hours by fewer than half of qualifying stations in several regions.Source: Extended Data · p. 28

Supplementary Notes

The Supplementary Notes cover additional evaluations, reference comparisons, diagnostics, model variants, implementation, validation, proofs, data processing, calibration, and computational cost.

Supplementary Notes

  • Supplementary Note 1 covers additional evaluations and diagnostic contrasts.The supplied contents list also identifies Supplementary Note 2 as the reference comparison with Corrformer.Source: Supplementary Notes · p. 33
  • Supplementary Notes 3 and 4 address measurements for interpreting the learned coefficient and the effect of the covariance floor.Source: Supplementary Notes · p. 33
  • Supplementary Notes 5 and 6 cover evaluation of the dependence structure and related approaches.Source: Supplementary Notes · p. 33
  • Supplementary Notes 7–9 describe ablation and decomposition variants, network architecture and training, and baseline implementations.Source: Supplementary Notes · p. 33
  • Supplementary Note 10 specifies the rolling-origin validation protocol and calibration split.Source: Supplementary Notes · p. 33
  • Supplementary Notes 11–14 cover Proposition 1 assumptions and proofs, observations and preprocessing, joint calibration diagnostics, and computational cost.Source: Supplementary Notes · p. 33

Supplementary Note 1. Additional evaluations and diagnostic contrasts

Additional evaluations examine held-out-season comparisons, learned dependence, lead conditioning, specialist models, and wind in the distance bias. Results vary by lead and evaluation setting.

Lead-conditioned coupling

  • Replacing lead-conditioned γ(τ) with a constant raises energy score by 0.36% and 0.46% at 1 and 3 h on 96 ASOS, while changes at 6–48 h range from −0.07% to +0.18%.On 724 AWS, the increases are 1.33% and 1.37% at 1 and 3 h and 0.78% at 6 h; they are at most 0.27% at 12–48 h.Source: Supplementary Note 1. Additional… · p. 34

Where the lead enters

  • Using lead only through γ(τ) raises energy score by 5.5% at 1 h and 0.9–1.8% at 3–48 h, and is worse than removing lead from both pathways at every lead.The comparison holds in four or five of the six folds; both variants retain target-hour phase and lead-proportional upwind displacement.Source: Supplementary Note 1. Additional… · p. 34

Per-lead specialists

  • Six single-lead specialists are worse than the unified model at 12, 24, and 48 h in every fold, by 5.0%, 16.0%, and 19.6%, respectively.Together, the specialists require about six times as many optimization steps as the unified model.Source: Supplementary Note 1. Additional… · p. 34

Wind in the distance bias

  • Across wind-speed strata and leads, zero-wind energy-score differences vary in sign across folds on both networks.On 96 ASOS the fold-mean difference ranges from −0.0038 to +0.0038 in every stratum and lead; on 724 AWS it ranges from −0.0039 to +0.0022.Source: Supplementary Note 1. Additional… · pp. 34–35

Dependence on the 724-station network

  • Removing learned correlations worsens log-likelihood by 1.4–2.8 nats per station and energy score by 0.15–0.65% on the 724-station network.The changes occur in all six folds at every lead.Source: Supplementary Note 1. Additional… · p. 35

Evaluation in the two 2026 seasons

  • CLARA beats each of five pre-specified baselines in all 60 season–lead–baseline comparisons on each domestic network in the 2026 held-out seasons.Against iTransformer + phase, lead-mean reductions are 6.1% on 96 ASOS and 8.5% on 724 AWS; against the wider, similarly sized comparator, they are 5.8% and 7.8%.Source: Supplementary Note 1. Additional… · p. 35

Supplementary Note 2. Reference comparison with Corrformer

CLARA has lower issued energy scores than Corrformer at every lead in every multi-year fold. The comparison is not a matched comparison of architectures.

Reference comparison with Corrformer

  • CLARA’s issued energy score is lower than Corrformer’s at every lead in every fold, by 35.3% at 1 h and 7.2–9.6% at 12–48 h.The comparison uses the six multi-year folds of the 96-station network.Source: Supplementary Note 2. Reference… · p. 36
  • CLARA’s predictive-mean RMSE is lower in all six folds at every lead except 12 h, where it is lower in five folds.At 1 h, RMSE is 0.219 for CLARA versus 0.310 for Corrformer; at 48 h, it is 0.592 versus 0.632.Source: Supplementary Note 2. Reference… · p. 36
  • The Corrformer comparison is not architecture-matched because its objective, capacity, and input window differ from CLARA’s, and its hyperparameters were not tuned for this network.It uses the public implementation and was added after the 2026 folds had been scored.Source: Supplementary Note 2. Reference… · pp. 35–36

Supplementary Note 3. Measurements for reading the learned coefficient

The learned distance-bias coefficient declines with lead, but it does not by itself measure either its relative weight against content attention or its influence on forecasts.

Supplementary Note 3. Measurements for reading the learned coefficient

  • On both networks, the distance term’s relative size and attention divergence decrease with lead, while the upwind pull ratio rises.The content scale does not offset the decline in relative distance-term weight.Source: Supplementary Note 3. Measurements for… · p. 36
  • In the synthetic control with dependence, the true effect correlates more strongly with model interventions (0.94–1.00) than with γ (0.14–0.54).When dependence is absent, γ can become large even though mean intervention effects remain below 0.0001 in magnitude.Source: Supplementary Note 3. Measurements for… · p. 37
  • At 6 h, scaling both γ and content logits changes divergence and energy-score effect despite leaving their relative size unchanged.With both unchanged, divergence is 0.207 nats and the energy-score effect is 0.0083; the results show neither γ nor its ratio to content scale determines forecast effect alone.Source: Supplementary Note 3. Measurements for… · p. 37

Supplementary Note 4. Effect of the covariance floor

The covariance floor trades off joint likelihood, pressure marginals, and kernel scores. The related-methods discussion positions CLARA as combining lead-aware spatial information with joint probabilistic prediction from station observations.

Supplementary Note 4. Effect of the covariance floor

  • Lowering the Cholesky diagonal floor from 0.05 to 0.005 improves joint negative log-likelihood by 0.49–0.96 nats per station but worsens energy score at every lead.The original floor is retained because energy score is the primary metric, despite improved 1 h pressure CRPS and pressure-difference calibration with the lower floor.Source: Supplementary Note 4. Effect of the… · pp. 37–38

Supplementary Note 5. Evaluation of the dependence structure

  • The dependence comparison holds predictive means and marginal variances fixed while varying only the correlation structure, and scores distributions before and after lead-wise scaling.It evaluates joint negative log-likelihood, energy and variogram scores, plus derived quantities and a compound hot–dry event.Source: Supplementary Note 5. Evaluation of… · p. 38

Supplementary Note 6. Related approaches

  • Unlike stepwise advection models, CLARA computes each lead directly from the same base-time observations and adjusts upwind-distance bias strength by lead.The coefficient curve is interpreted relative to the content-path scale, not in isolation.Source: Supplementary Note 6. Related approaches · p. 40
  • CLARA combines a five-variable Gaussian mean and covariance with lead-wise covariance scaling that preserves correlations.Its scale estimator is stated to be consistent under the conditions in Proposition 1.Source: Supplementary Note 6. Related approaches · pp. 40, 42
  • The related-work comparison identifies a gap in systems issuing multi-variable joint distributions from station observations alone while adjusting spatial bias and covariance magnitude by lead.Prior approaches include ensemble-based post-processing, marginal or deterministic observation-based forecasts, and lead-varying calibration in other settings.Source: Supplementary Note 6. Related approaches · pp. 39, 41, 42

Supplementary Note 7. Definition of ablation and decomposition variants

The ablations vary one component at a time against the full model, while decomposition variants isolate where explicit lead information enters.

Supplementary Note 7. Definition of ablation and decomposition variants

  • Each ablation variant matches the full model’s architecture, data, folds, seeds, and training settings, differing in one respect.Its contribution is measured as variant minus full-model fold-mean energy score; positive values mean the component improves performance.Source: Supplementary Note 7. Definition of… · p. 42
  • Gateonly removes the lead channel from the token embedding, leaving explicit lead information only in γ(τ); Constonly also fixes γ at a learned constant.Both retain target-hour phase and lead-proportional upwind displacement, so lead remains partly recoverable.Source: Supplementary Note 7. Definition of… · pp. 42–43
  • The per-lead specialist trains a separate model for each lead, while the floor variant changes only the Cholesky diagonal minimum from 0.05 to 0.005.Each specialist receives the same number of optimization steps as the full model and therefore sees its data about six times as often.Source: Supplementary Note 7. Definition of… · p. 43

Supplementary Note 8. Network architecture and training details

CLARA’s issued energy score is lower than the retrained, parameter-matched iTransformer + phase comparator on both domestic networks. The note specifies the model, issuance procedure, baselines, and rolling-origin validation protocol.

Supplementary Note 8. Network architecture and training details

  • The 64-dimensional station token combines current observations, 60-minute tendencies, elevation, normalized lead, and temporal phase before attention and mean and Cholesky output heads.The advection coupling is produced by a 1–16–1 MLP, and the diagonal Cholesky outputs have a 0.05 lower bound.Source: Supplementary Note 8. Network… · p. 43

Training and forecast issuance procedure of CLARA

  • Training uses base-time observations and related features; issuance produces a Gaussian predictive law for each station and lead, with a lead-wise scale estimated on held-out calibration data.Training uses Adam for 40 epochs; recalibration preserves the covariance correlation structure.Source: Training and forecast issuance… · p. 44

Supplementary Note 9. Baseline implementations

  • The comparison includes statistical baselines and learned alternatives, with iTransformer + phase serving as the matched-temporal-input learned comparator.The widened iTransformer + phase has 27,616 parameters versus CLARA’s 27,979; the physical baseline is scored separately on rows where it issues a forecast.Source: Supplementary Note 9. Baseline… · pp. 44–45
  • CLARA’s lead-mean energy score is 4.7% lower on 96 ASOS and 6.2% lower on 724 AWS than iTransformer + phase with a similar parameter count.At every lead the issued fold-mean score favoured CLARA; the comparison used six multi-year folds.Source: Supplementary Note 9. Baseline… · p. 45

Supplementary Note 10. Rolling-origin validation protocol and calibration split

  • Evaluation uses rolling-origin folds with a calibration split reserved for fitting the lead-wise scale; the two 2026 seasons were held out from model development and setting selection.The multi-year folds share training windows and number only six, so year-to-year results are reported through effect sizes and sign counts rather than formal significance tests.Source: Supplementary Note 10. Rolling-origin… · pp. 46–47

Supplementary Note 11. Assumptions and proofs of Proposition 1 and Corollary 1

Supplementary Note 11 establishes when the covariance-scale recalibration estimator is consistent and what it can guarantee. Its results depend on explicit Gaussian, stability, and sampling assumptions.

Supplementary Note 11. Assumptions and proofs of Proposition 1 and Corollary 1

  • Under (A1)–(A4), the note establishes consistency of the lead-wise scale estimator and exact covariance recalibration that preserves correlations.(A1)–(A2) specify the predictive mean and Gaussian covariance; (A3) requires the calibration and evaluation periods to share a scale, and (A4) supplies a sampling condition for the proofs.Source: Supplementary Note 11. Assumptions and… · pp. 47–48
  • Assumption (A3) is testable, and its possible failure at long leads is consistent with the observed over-correction when calibration and evaluation seasons differ.The calibration-split scale over-corrects at 12–48 h in the evaluation season.Source: Supplementary Note 11. Assumptions and… · p. 47
  • The proof derives d2 ∼ c(τ)^2χ²_V, so its calibration-sample mean is c(τ)^2V and supports the scale estimator.The consistency argument groups dependent samples by day and uses the stated weak-dependence condition; its independent-sampling reference rate is not used quantitatively for dependent samples.Source: Supplementary Note 11. Assumptions and… · pp. 47–48
  • The condition E[d2] = V is necessary but not sufficient for joint calibration; the note gives a covariance counterexample.Under the stated scalar assumptions, the pseudo-true scale equals c(τ)^2; under general misspecification, the note derives a pseudo-true scale.Source: Supplementary Note 11. Assumptions and… · pp. 48–49

Supplementary Note 12. Observations, quality control and preprocessing

This note specifies the station records, sampling rules, and preprocessing used across the domestic and regional evaluations. The network comparison changes both station density and network type, so it does not isolate density alone.

Supplementary Note 12. Observations, quality control and preprocessing

  • Samples require at least five stations with all five variables valid at base time, and scoring uses only input stations with all five variables also valid at target time.Inputs include current observations, 60-minute tendencies, elevation, normalized lead time, and temporal phase; standardization uses each fold’s training split only.Source: Supplementary Note 12. Observations,… · p. 49
  • The three datasets serve distinct roles: 96 ASOS stations anchor evaluation and ablations, 724 AWS sites test another domestic network, and GHCNh sites test regional replication.The nominal 724 AWS sites yield 4.6–6.3 times the ASOS count of scored stations per sample in fold means.Source: Supplementary Note 12. Observations,… · p. 49
  • Because the AWS comparison changes network type as well as density, it tests reproduction on another network rather than density’s isolated effect.The comparison concerns both accuracy and operating characteristics.Source: Supplementary Note 12. Observations,… · p. 50
  • When prior-hour observations are absent, tendencies are set to zero; missing wind direction sets both wind components to zero, making the advection bias an ordinary distance bias.Inputs use the latest valid regional observation within 20 minutes before base time, while targets use the nearest report within ±20 minutes of target time.Source: Supplementary Note 12. Observations,… · p. 50
  • Regional sites were selected using 2021–2025 observation criteria and spatial sampling; regional forecasts cover three variables because pressure reporting is sparse in several regions.The ten regions span six continents and Köppen classes A–D; central Brazil, East Africa, and southeastern Australia have fewer than 50 qualifying stations.Source: Supplementary Note 12. Observations,… · p. 50

Supplementary Note 13. Joint calibration diagnostics

The supplementary material defines joint-calibration diagnostics, reports computational timings, and summarizes benchmark comparisons. CLARA’s issued energy score is lower than the listed alternatives across nearly all reported fold–lead cells.

Supplementary Note 13. Joint calibration diagnostics

  • Joint calibration uses mean Mahalanobis distance d2, ideal at V = 5, and a multivariate PIT whose uniformity alone does not establish joint calibration.Issued distributions use a per-lead scalar scale; scalar and per-variable diagonal scaling preserve the correlation matrix.Source: Supplementary Note 13. Joint… · p. 51

Supplementary Note 14. Computational cost

  • On one CPU thread, a forward pass takes 0.43 ms and training takes 2.1 min; power consumption was not measured.The reported setup uses a 27,979-parameter model and reports input assembly separately.Source: Supplementary Note 14. Computational… · p. 51

Supplementary Tables

  • CLARA’s fold-mean energy score is lower than every listed model at all six leads and in 143 of 144 fold–lead cells against iTransformer + phase and statistical baselines.In two held-out 2026 seasons, its seed-mean energy score is lower than five pre-specified baselines in all 60 season–lead–baseline cells.Source: Supplementary Tables · p. 54

96 ASOS

The tables document evaluation-fold dates and station counts, AWS coverage, and the timing protocol. Their notes identify the held-out folds and the calibration split.

Base times (eval)

  • The calibration split is the last 15% of each training window and is excluded from training; base times are sampled every four days at 00, 06, 12, and 18 KST.Winter 2026 and summer 2026 are the two held-out folds.Source: 724 AWS · p. 55

(min)

  • Inference timing uses 200 evaluation samples after 20 warm-up passes; training starts from new weights on 4,508 groups in batches of 18.The listed devices include an Intel Core i9-12900 CPU and an NVIDIA GeForce RTX 3080 GPU.Source: (min) · p. 56

Supplementary Figures

The supplementary figures describe variable-level forecast scores, matched-row comparisons with objective analysis, and retrained CLARA ablations.

Supplementary Figures

  • Per-variable RMSE and CRPS by lead compare CLARA with iTransformer + phase, EMOS, and persistence across 96 ASOS stations and six multi-year folds.Scores are reported in physical units; RMSE is computed from fold-mean squared error.Source: Figure 1 · p. 56
  • Matched-row energy scores compare CLARA with deterministic objective analysis on two networks, alongside the fraction of rows with valid objective-analysis forecasts by lead.The networks are 96 ASOS and 724 AWS; scores and coverage are fold means.Source: Figure 2 · p. 57
  • Retrained ablations report energy-score changes after removing temporal phase or advection, making γ constant, or setting wind in the distance bias to zero.Positive changes indicate higher scores than CLARA; results cover 96 ASOS, with no-advection and zero-wind series also reported for 724 AWS.Source: Figure 3 · p. 58

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