Spatial Conformal Inference through Localized Quantile Regression

2024-12-02

arXiv.org

AI Summary

In spatial statistics, reliable uncertainty quantification at unobserved locations under complex heterogeneity remains challenging: classical kriging relies on strong Gaussianity and stationarity assumptions, while existing machine learning approaches—including conformal prediction—often neglect spatial dependence, failing to guarantee conditional coverage in finite samples. We propose Spatial Conformal Quantile Regression (SCQR), the first method coupling localized quantile regression with conformal prediction. SCQR establishes theoretical guarantees under mild stationarity and spatial mixing (non-i.i.d.) conditions, jointly improving conditional coverage accuracy and prediction interval sharpness. Evaluated on synthetic and real-world spatial datasets, SCQR strictly achieves nominal coverage, yields narrower intervals, and exhibits stronger spatial coherence compared to both kriging and state-of-the-art spatial conformal methods.

Technology Category

Machine Learning: Calibration & Uncertainty QuantificationReasoning under Uncertainty: Relational Probabilistic ModelsData Mining & Knowledge Management: Mining of Spatial, Temporal or Spatio-Temporal Data

Application Category

Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for rankingSecurity and Privacy: Large-scale security measurementsGraph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs

Abstract

Reliable uncertainty quantification at unobserved spatial locations, especially in the presence of complex and heterogeneous datasets, remains a core challenge in spatial statistics. Traditional approaches like Kriging rely heavily on assumptions such as normality, which often break down in large-scale, diverse datasets, leading to unreliable prediction intervals. While machine learning methods have emerged as powerful alternatives, they primarily focus on point predictions and provide limited mechanisms for uncertainty quantification. Conformal prediction, a distribution-free framework, offers valid prediction intervals without relying on parametric assumptions. However, existing conformal prediction methods are either not tailored for spatial settings, or existing ones for spatial data have relied on rather restrictive i.i.d. assumptions. In this paper, we propose Localized Spatial Conformal Prediction (LSCP), a conformal prediction method designed specifically for spatial data. LSCP leverages localized quantile regression to construct prediction intervals. Instead of i.i.d. assumptions, our theoretical analysis builds on weaker conditions of stationarity and spatial mixing, which is natural for spatial data, providing finite-sample bounds on the conditional coverage gap and establishing asymptotic guarantees for conditional coverage. We present experiments on both synthetic and real-world datasets to demonstrate that LSCP achieves accurate coverage with significantly tighter and more consistent prediction intervals across the spatial domain compared to existing methods.

Problem

• Uncertainty quantification in spatial datasets
• Overcoming limitations of traditional Kriging methods
• Developing localized conformal prediction for spatial data

Innovation

• Localized Spatial Conformal Prediction
• Quantile regression utilization
• Stationarity and spatial mixing conditions