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Robust and Conjugate Spatio-Temporal Gaussian Processes
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Adapts robust and conjugate GPs (RCGPs) to spatio-temporal state-space GPs (ST-RCGP), giving outlier robustness at linear-in-time cost, with adaptive IMQ weights and a robust hyperparameter objective.
Problem
Standard spatio-temporal GPs (state-space formulation, linear in time) are not robust to outliers, and existing RCGPs are cubic in data size and have issues with prior mean choice, uncertainty quantification, and selecting the IMQ shrinking parameter c.
Approach
Derive a generalised-Bayes Kalman filter/smoother from the weighted score-matching loss, where IMQ weights are adaptive over time: the centre is the GB predictive mean and the shrinking scale is the GB predictive variance; hyperparameters are fit with a weighted predictive likelihood.
Key results
- On N=46,800 index futures points with induced crash, ST-RCGP achieves RMSE 0.145 and NLPD -0.62 vs STGP RMSE 0.54 and NLPD 13.9, at 9.4 s total vs 8.0 s for STGP.
- In well-specified settings without outliers, ST-RCGP keeps RMSE 0.19 and NLPD 5.7, matching STGP's 6.5 NLPD and RCGP's 5.6.
- With outliers on simulated data, ST-RCGP reduces NLPD to 5.5 from STGP's 30 and RMSE to 0.19 from 0.38.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Use adaptive robust weights and robust hyperparameter optimization instead of the standard STGP | ST-RCGP · Simulated well-specified data with outliers | RMSE: 0.38 \pm 0.02 -> 0.19 \pm 0.01 | Helped |
| Use adaptive robust weights and robust hyperparameter optimization instead of the standard STGP | ST-RCGP · Simulated well-specified data with outliers | NLPD: 30 \pm 2 -> 5.5 \pm 0.2 | Helped |
| Use adaptive robust weights and robust hyperparameter optimization instead of the standard STGP | ST-RCGP · Simulated well-specified data without outliers | RMSE: 0.19 \pm 0.01 -> 0.19\pm0.01 | No clear effect |
| Use adaptive robust weights and robust hyperparameter optimization instead of the standard STGP | ST-RCGP · Simulated well-specified data without outliers | NLPD: 6.5 \pm 0.2 -> 5.7\pm0.1 | Helped |
| Use adaptive robust weights and robust hyperparameter optimization instead of STGP | ST-RCGP · DJIA flash crash, 46800 data points | RMSE: 0.54\pm0.01 -> 0.145\pm0.002 | Helped |
| Use adaptive robust weights and robust hyperparameter optimization instead of STGP | ST-RCGP · DJIA flash crash, 46800 data points | NLPD: 13.9\pm0.3 -> -0.62\pm0.01 | Helped |
| Use ST-RCGP instead of MEP | ST-RCGP · DJIA flash crash, 46800 data points | Total (s): 29.1\pm0.5 -> 9.4\pm0.4 | Helped |
| Use ST-RCGP instead of MEP | ST-RCGP · DJIA flash crash, 46800 data points | RMSE: 0.15\pm0.01 -> 0.145\pm0.002 | No clear effect |
Surprising findings
- Vanilla RCGP performed worse than the non-robust GP on the DJIA crash because its constant prior mean was near the outliers.
Limitations
- Scaling in spatial dimension remains cubic in n_s and is not addressed
- Restricted to Gaussian observation model; non-Gaussian likelihoods left for future work
- Parallel computation not exploited
Experimental setup
Datasets: simulated spatio-temporal GP data with focussed outliers DJIA index futures data (Twitter flash crash, 2013) Climate Research Unit UK temperature data (2022-2023, 571 locations)
Benchmarks: DJIA flash crash data UK temperature forecasting simulated well-specified and outlier data
Metrics: RMSE NLPD expected weighted ratio computation time coverage
Baselines: STGP RCGP relevance pursuit MEP MVI MLa
Builds on: RCGP (Altamirano et al. 2024) weighted score matching state-space Gaussian processes relevance pursuit
Closest papers by meaning
- Robust and Conjugate Gaussian Process Regression (ICML 2024 · similarity 0.90)
- Robust and Computation-Aware Gaussian Processes (NeurIPS 2025 · similarity 0.87)
- Physics-Informed Variational State-Space Gaussian Processes (NeurIPS 2024 · similarity 0.85)
- Computation-Aware Gaussian Processes: Model Selection And Linear-Time Inference (NeurIPS 2024 · similarity 0.85)
- How Different from the Past? Spatio-Temporal Time Series Forecasting with Self-Supervised Deviation Learning (NeurIPS 2025 · similarity 0.85)
- Robust Gaussian Processes via Relevance Pursuit (NeurIPS 2024 · similarity 0.85)
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