Positional Encoder Graph Quantile Neural Networks for Geographic Data

TL;DR

The paper tackles reliable uncertainty quantification for large-scale geospatial data by introducing PE-GQNN, a single end-to-end framework that merges Positional Encoder Graph Neural Networks with Quantile Neural Network techniques. It employs a transformed tau input, Simultaneous Quantile Regression loss, and Lipschitz Monotonic Networks to produce non-crossing, calibrated conditional quantiles, while reengineering the PE-GNN to utilize neighbours' mean targets without leakage. The authors demonstrate substantial improvements in predictive accuracy and calibration across multiple geospatial datasets (California Housing, Air Temperature, and 3D Road) compared to GNN, PE-GNN, and SMACNP baselines, highlighting the practicality of scalable, fully probabilistic spatial predictions without extra computational burden. The work advances geospatial machine learning by enabling flexible, inherently calibrated density estimation with a scalable neural architecture suitable for broad spatial and non-spatial applications.

Abstract

Positional Encoder Graph Neural Networks (PE-GNNs) are among the most effective models for learning from continuous spatial data. However, their predictive distributions are often poorly calibrated, limiting their utility in applications that require reliable uncertainty quantification. We propose the Positional Encoder Graph Quantile Neural Network (PE-GQNN), a novel framework that combines PE-GNNs with Quantile Neural Networks, partially monotonic neural blocks, and post-hoc recalibration techniques. The PE-GQNN enables flexible and robust conditional density estimation with minimal assumptions about the target distribution, and it extends naturally to tasks beyond spatial data. Empirical results on benchmark datasets show that the PE-GQNN outperforms existing methods in both predictive accuracy and uncertainty quantification, without incurring additional computational cost. We also provide theoretical insights and identify important special cases arising from our formulation, including the PE-GNN.

Figures (4)

• Figure 1: PE-GQNN compared to PE-GNN and GNN
• Figure 2: Diagnostics for the California Housing dataset. (a) Validation MSE curves. (b) PE-GQSAGE predicted densities of 3 observations of the test set. (c) Calibration Plot.
• Figure 3: (a) For each quantile of interest, a separate NN is trained. (b) rodrigues2020: one NN outputs $d+1$ predictions: one for the expectation and $d$ for the quantiles. (c) si2022: a single NN trained to predict any generic quantile of the conditional distribution. (d) kuleshov2022: two-step procedure: the first model outputs a low-dimensional representation of the conditional distribution, which a recalibrator then uses to produce calibrated predictions.
• Figure 4: Geographical maps of the predicted results on the California Housing test dataset.