Positional Encoding Enhanced Graph Quantile Learning for Uncertainty-Aware Spatial Prediction

Abstract

Spatial prediction is instrumental in environmental monitoring, urban studies, and geostatistics where accuracy and dependable uncertainty bounds are mandatory. This work introduces the Graph Quantile Neural Network (GQNN) with Positional Encoding (PE), which integrates graph-based spatial representation learning, the sinusoidal positional encoding, and ordered multi-quantile regression. The controlled benchmark is randomized to 70/ 15/ 15 train- validation- test splits and includes 100–1000 irregular nodes with k-nearest-neighbor graph construction (k = 8). Concerning practical relevance, the revised evaluation also specifies out-of-sample tests with public U.S. EPA AirData 2024 annual PM2.5 monitor data and the California Housing geospatial data set, with the same split for GQNN-PE, Ordinary Kriging, and Gaussian Process Regression (GPR). GQNN-PE results in MAE = 0.653, RMSE = 0.972 and determination coefficient (R 2) of 0.89, with PICP = 94.3% and MPIW = 2.11, which provides a decrease of RMSE by 24.3% and 17.0% with respect to Kriging and GPR respectively. For practical relevance, an explicit inverse z-score transformation is also provided so that normalized MAE, RMSE, and interval width can be reported in physical units on real data. Five-seed statistical testing gives p < 0.01.