Research map / Probabilistic and scientific modeling / Uncertainty and Gaussian processes
Smooth Sailing: Lipschitz-Driven Uncertainty Quantification for Spatial Associations
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Proposes Lipschitz-driven confidence intervals for target-conditional linear associations in spatial settings, using 1-NN weights, a Wasserstein-based bias bound, and a consistent Lipschitz-based noise variance estimate.
Problem
Constructing valid frequentist confidence intervals for covariate-response associations in spatial regression when the model is misspecified and inference is needed at fixed target locations that differ from observed source locations.
Approach
Assume the conditional mean is L-Lipschitz in space and homoskedastic Gaussian noise; estimate target-conditional OLS via 1-nearest-neighbor spatial weights, bound the resulting bias by a Wasserstein-1 linear program (Kantorovich-Rubinstein duality), and combine with a Gaussian interval that widens by the bias bound, with noise variance estimated by a Lipschitz quadratic program.
Key results
- In the single-covariate simulation, the proposed method and GP BCIs achieve nominal 95% coverage across shifts, but the proposed intervals are much narrower than GP BCIs; OLS, sandwich, and GLS reach zero coverage under non-zero shift.
- In the several-covariate simulation (N=10,000, M=100), no competing method exceeds 30% coverage at any shift, while the proposed method maintains coverage.
- In the tree cover experiment, the proposed method is the only one achieving 95% coverage for all three parameters; for Slope, every other method covers at most 54%.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Use Lipschitz-driven confidence intervals instead of alternative methods | Spatial association inference · Single covariate simulation, non-zero shifts | coverage: zero coverage for OLS, sandwich, and GLS; nominal coverage (95%) for the proposed method | Helped |
| Use Lipschitz-driven confidence intervals instead of alternative methods | Spatial association inference · Several covariates simulation | coverage: no other method achieves coverage over 30%; proposed method achieves nominal coverage | Helped |
| Use Lipschitz-driven confidence intervals instead of alternative methods | Spatial association inference · Tree cover linear regression, Slope | coverage: at most 54% for every other method; 95% nominal coverage for the proposed method | Helped |
| Vary L from 0.1 to 10 (ablation) | Lipschitz-driven CI · Single covariate simulation | coverage: coverage of 1.0 for all L values | No clear effect |
| Use L=0.1 instead of the selected L=0.2 (ablation) | Lipschitz-driven CI · Tree cover linear regression, Aridity Index | coverage: 89% instead of 95% | Hurt |
| Use L≤0.1 (ablation) | Lipschitz-driven CI · Tree cover linear regression, Aridity Index | coverage | Hurt |
| Increase L above 0.5 (ablation) | Lipschitz-driven CI · Tree cover linear regression | confidence interval width | Hurt |
| Increase L (ablation) | Lipschitz-driven CI · Single covariate simulation | bias contribution to interval width | Hurt |
| Increase L (ablation) | Lipschitz-driven CI · Single covariate simulation | randomness contribution to interval width | Helped |
Surprising findings
- Standard OLS, sandwich, GLS, and importance-weighted intervals have near-zero coverage under distribution shift despite being valid under correct specification.
- Coverage of the proposed method stays at 1.0 for Lipschitz constants below the theoretical minimum, though theory does not guarantee it there.
Limitations
- Requires a known Lipschitz constant chosen by domain knowledge; coverage guarantees depend on L and the theory needs L at least a minimum value.
- Sigma-squared quadratic program scales poorly with the number of source locations, requiring approximations for large N.
- Intervals can be wide under extrapolation (e.g., tree cover experiment), and the method is developed only for linear models with fixed covariates, not GLMs or local (GWR) coefficients.
Experimental setup
Datasets: synthetic spatial simulations CONUS tree cover data (983 points, Aridity Index, Elevation, Slope covariates)
Benchmarks: synthetic spatial simulation with single covariate synthetic simulation with several covariates CONUS tree cover regression
Metrics: empirical coverage confidence interval width
Baselines: OLS sandwich estimator importance weighted least squares (KDEIW) generalized least squares Gaussian process Bayesian credible intervals
Builds on: ordinary least squares sandwich estimator covariate shift importance weighting KNN imputation restricted spatial regression
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