Spatial Conformal Inference through Localized Quantile Regression

Reliable uncertainty quantification at unobserved spatial locations is a key challenge in spatial statistics, particularly for complex and heterogeneous datasets. While traditional methods such as Kriging rely on strong distributional assumptions, conformal prediction (CP) offers a distribution-free alternative. However, although non-i.i.d. CP theory is well established for time-series data, a significant gap remains for spatial data, where the lack of a natural ordering and discrete index complicates theoretical guarantees. Existing CP theory for spatial data often relies on exchangeability. We propose Localized Spatial Conformal Prediction (LSCP), a model-agnostic framework that bridges this gap by coupling local quantile regression with conformal calibration. LSCP conditions on spatial neighborhoods to capture local heterogeneity. We show that LSCP retains finite-sample marginal coverage under spatial exchangeability and attains asymptotic conditional coverage under stationarity and spatial mixing. Across synthetic and real datasets, LSCP consistently achieves near-nominal coverage with tighter and more stable prediction intervals than existing methods that fail to capture these spatial dependencies.

Many important decisions rely on predictions at places where no direct measurement is available, such as estimating mobile network quality, weather-related quantities, crop yield, or housing prices. A useful prediction system should not only give a best guess, but also say how uncertain it is. This is difficult for spatial data because nearby locations tend to be related, different regions can behave differently, and many standard statistical methods depend on assumptions that may not hold. We introduce Localized Spatial Conformal Prediction, a method that builds prediction intervals for new locations by learning from nearby prediction errors. The method is flexible: it can wrap around many existing prediction models and does not require a fully specified probability model for the spatial data. Our theory shows when the method can provide reliable uncertainty estimates even when the data are spatially dependent. In experiments on simulated data and real datasets involving mobile signal strength, agriculture, and housing prices, the intervals contained the true value about as often as intended while being narrower and more stable than several alternatives. This can help practitioners make better calibrated decisions when predictions vary across space and uncertainty matters.