Spatial modelling with Euclidean distance fields and machine learning

This study introduces a hybrid spatial modelling framework, which accounts for spatial non-stationarity, spatial autocorrelation and environmental correlation. A set of geographic spatially autocorrelated Euclidean distance fields (EDF) was used to provide additional spatially relevant predictors to the environmental covariates commonly used for mapping. The approach was used in combination with machine-learning methods, so we called the method Euclidean distance fields in machine-learning (EDM). This method provides advantages over other prediction methods that integrate spatial dependence and state factor models, for example, regression kriging (RK) and geographically weighted regression (GWR). We used seven generic (EDFs) and several commonly used predictors with different regression algorithms in two digital soil mapping (DSM) case studies and compared the results to those achieved with ordinary kriging (OK), RK and GWR as well as the multiscale methods ConMap, ConStat and contextual spatial modelling (CSM). The algorithms tested in EDM were a linear model, bagged multivariate adaptive regression splines (MARS), radial basis function support vector machines (SVM), Cubist, random forest (RF) and a neural network (NN) ensemble. The study demonstrated that DSM with EDM provided results comparable to RK and to the contextual multiscale methods. Best results were obtained with Cubist, RF and bagged MARS. Because the tree-based approaches produce discontinuous response surfaces, the resulting maps can show visible artefacts when only the EDFs are used as predictors (i.e. no additional environmental covariates). Artefacts were not obvious for SVM and NN and to a lesser extent bagged MARS. An advantage of EDM is that it accounts for spatial non-stationarity and spatial autocorrelation when using a small set of additional predictors. The EDM is a new method that provides a practical alternative to more conventional spatial modelling and thus it enhances the DSM toolbox. Highlights: We present a hybrid mapping approach that accounts for spatial dependence and environmental correlation. The approach is based on a set of generic Euclidean distance fields (EDF). Our Euclidean distance fields in machine learning (EDM) can model non-stationarity and spatial autocorrelation. The EDM approach eliminates the need for kriging of residuals and produces accurate digital soil maps