Dual Kriging with a Nonlinear Hybrid Gaussian RBF–Polynomial Trend: The Theory and Application to PM2.5 Estimation in Northern Thailand

Accurate spatial interpolation of environmental data requires utilizing flexible models that can capture complex spatial patterns. In this paper, we present two improved dual kriging (DK) models comprising a nonlinear trend function that combines Gaussian radial basis functions with a first-order polynomial. The proposed model, DK–RBFP, and its extension, DK–RBFPGA, which includes k-means clustering and a genetic algorithm for parameter optimization, respectively, exhibit enhanced performance in capturing spatial variation. The complete monotonicity of the covariance function and the strict positive definiteness of the coefficient matrix provide theoretical support for the uniqueness of the DK solution. When applied to datasets of PM2.5 concentrations for northern Thailand, both models perform better than the conventional DK model using a second-order polynomial trend (DK–POLY), as evidenced by accuracy metrics including the mean absolute percentage error (MAPE), the mean squared error (MSE), and the root mean square error (RMSE). The outcomes indicate that integrating nonlinear trend components with data-driven optimization significantly enhances accuracy and flexibility in environmental spatial predictions.