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STACI: Spatio-Temporal Aleatoric Conformal Inference
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Proposes STACI, a scalable variational Bayesian neural network (SVGD-trained random-Fourier-feature GP with INR latent dimension expansion) combined with a spatio-temporal local conformal inference step to produce valid aleatoric prediction intervals for millions of ST observations.
Problem
Spatio-temporal field estimation with calibrated uncertainty: exact GPs are cubic in data size, approximate GPs assume stationary separable kernels, and approximations break nominal interval coverage.
Approach
Model a non-stationary spatio-temporal GP as a Bayesian random-Fourier-feature network with an implicit-neural-representation latent warping trained by SVGD, then apply split-style local conformal calibration over K nearest space-time neighbors using standardized residuals to correct the aleatoric interval widths.
Key results
- On MSS, STACI-FFNP (Bayes) achieves the lowest RMSE of 0.161 and NLL of -1.331 among all methods.
- Conformal STACI variants reach 0.958 coverage with interval score 0.514 and width 0.500 on MSS, versus 0.974-0.951 coverage and interval scores around 1.1-2.6 for the competitors.
- On AOD, conformal STACI-FFNP lowers the interval score from 3.944 to 1.850 while keeping coverage at 0.948.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| add spatio-temporal conformal inference | STACI-FFNP · MSS test | interval score | Helped |
| add spatio-temporal conformal inference | STACI-FFNP · AOD test | interval score | Helped |
| use FFNP rather than ResMLP or FFNG as the latent model | STACI · MSS test | RMSE: 0.422, 0.247 -> 0.161 | Helped |
| use FFNP rather than ResMLP or FFNG as the latent model | STACI · AOD test | RMSE: 0.720, 0.675 -> 0.560 | Helped |
| increase latent dimension from 128 to 256 (ablation) | STACI-ResMLP · AOD | RMSE | Helped |
| vary latent dimension from 32 to 256 (ablation) | STACI-FFNP · AOD | RMSE | No clear effect |
| vary latent dimension from 32 to 256 (ablation) | STACI-FFNG · AOD | RMSE | No clear effect |
| increase training sampling percentage from 5% to 25% (ablation) | STACI-FFNP · AOD | RMSE | Helped |
| increase training sampling percentage from 5% to 25% (ablation) | Deep RF · AOD | RMSE | Mixed |
| increase SVGD samples from M = 5 to M = 30 (ablation) | STACI-FFNP · AOD test | RMSE: 0.559 -> 0.561 | No clear effect |
| increase SVGD samples from M = 5 to M = 30 (ablation) | STACI-FFNP · AOD test | NLL: 0.041 -> 0.044 | No clear effect |
| increase SVGD samples from M = 5 to M = 30 (ablation) | STACI-FFNP · AOD test | time per epoch: 31 -> 155 | Hurt |
Surprising findings
- GPSat obtains a better NLL (-0.064) than STACI on AOD despite STACI's better RMSE and interval score.
- Deep RF required fixing the nugget variance to converge on the sparse AOD setting.
Limitations
- SVGD training cost grows with the number of particles M, and small M may not explore the posterior well.
- The conformal procedure weights all selected neighbors equally and relies on approximate exchangeability of neighbors.
- Results are reported for a single run with no uncertainty bounds, and the synthetic MSS data is only a simulation.
Experimental setup
Datasets: simulated Arctic mean sea surface height (MSS) data MODIS Terra aerosol optical depth (AOD) March 2025
Benchmarks: MSS dataset AOD dataset
Metrics: RMSE NLL CRPS coverage interval score interval width time per epoch
Baselines: SVGP Deep GP (doubly stochastic) GPSat Deep RF
Builds on: Bayesian random Fourier features Stein variational gradient descent spatial conformal prediction deep random features dimension expansion for non-stationary GPs
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