A Markov Reward Process-Based Approach to Spatial Interpolation

The interpolation of spatial data can be of tremendous value in various applications, such as forecasting weather from only a few measurements of meteorological or remote sensing data. Existing methods for spatial interpolation, such as variants of kriging and spatial autoregressive models, tend to suffer from at least one of the following limitations: (a) the assumption of stationarity, (b) the assumption of isotropy, and (c) the trade-off between modelling local or global spatial interaction. Addressing these issues in this work, we propose the use of Markov reward processes (MRPs) as a spatial interpolation method, and we introduce three variants thereof: (i) a basic static discount MRP (SD-MRP), (ii) an accurate but mostly theoretical optimised MRP (O-MRP), and (iii) a transferable weight prediction MRP (WP-MRP). All variants of MRP interpolation operate locally, while also implicitly accounting for global spatial relationships in the entire system through recursion. Additionally, O-MRP and WP-MRP no longer assume stationarity and are robust to anisotropy. We evaluated our proposed methods by comparing the mean absolute errors of their interpolated grid cells to those of 7 common baselines, selected from models based on spatial autocorrelation, (spatial) regression, and deep learning. We performed detailed evaluations on two publicly available datasets (local GDP values, and COVID-19 patient trajectory data). The results from these experiments clearly show the competitive advantage of MRP interpolation, which achieved significantly lower errors than the existing methods in 23 out of 40 experimental conditions, or 35 out of 40 when including O-MRP.