A Bayesian Functional Kriging framework for spatial interpolation of PM2.5 concentration profiles

Air pollution, especially fine particulate matter (PM 2.5 ), is a major global public health problem, causing breathing difficulties, lung disease, heart problems, and premature death. Predicting PM 2.5 concentrations at unobserved locations is essential in environmental science, especially with sparse observational data. This study introduces a new Bayesian Functional Kriging (BFK) model to predict PM 2.5 concentration profiles across Pakistan’s most populous province, using MERRA-2 reanalysis data at 17 city locations across Punjab. The BFK framework combines three methodologies: Functional Data Analysis (FDA), where continuous temporal curves model discrete PM 2.5 concentration profiles through B-spline smoothing; Functional Kriging (FK), in which functional data are used for spatial prediction of curve-valued observations; and Bayesian hierarchical modeling using Markov Chain Monte Carlo (MCMC) sampling for parameter estimation. Prior empirical distributions are assumed for all parameters of interest: mean, partial sill, nugget effect, and spatial range, while posterior estimates are derived via a hybrid Gibbs-Metropolis algorithm within Gibbs sampling. Cross-validation results show that BFK is significantly better than the FK method, with RMSE and MAE differences of 30.9% (from 16.88 µg/m 3 to 11.66 µg/m 3 ) and 29.9% (from 13.03 µg/m 3 to 9.14 µg/m 3 ), respectively. In addition, we used the proposed model to estimate temporal variation in PM 2.5 concentration at five locations not sampled during data collection: Vehari, Sialkot, Attock, Khanewal, and Okara. We found that pollution is much higher in the southern areas than in the northern ones. These findings underscore BFK’s utility as a statistically principled and practically effective tool for air quality monitoring and environmental policy.