Evaluating the use of a coordinate-based tensor-product for addressing spatial autocorrelation in shallow landslide susceptibility modelling

This study develops two statistical models for assessing shallow landslide susceptibility at the slope-unit scale in the Aosta Valley (Italy), explicitly accounting for spatial autocorrelation. A shallow landslide inventory was compiled by integrating the Italian Landslide Inventory (IFFI) with the Regional Inventory of Instabilities of the Aosta Valley and used as a binary response variable. Geo-environmental predictors were optimised through a structured workflow combining multicollinearity analysis, stepwise selection, Random Forest classification, and Generalised Additive Models (GAMs), which were used to explore predictor–response relationships. Spatial autocorrelation was addressed by including slope-unit coordinates through a tensor-product smooth, resulting in two models: model_A, excluding the tensor term, and model_B, including it. Model performance was evaluated using spatial and non-spatial k-fold cross-validation based on mean Decrease in Deviance explained (mDD%), Effective Degrees of Freedom (EDF), and AUROC. In addition, on the final maps, Global Moran’s I was calculated on prediction residuals to evaluate spatial autocorrelation. Both models are statistically significant and show high discriminatory power (AUROC>0.85). Including the tensor term improved training performance as suggested by increasing deviance explained (39.0 vs 35.9), R2 (0.42 vs 0.39), and decreasing AIC (714.2 vs 724.5), and helped remove residual spatial autocorrelation. Distributions of mDD% and EDF indicate greater stability for model_B, with constrained variability in predictor contributions, whereas model_A shows wider dispersion, reflecting sensitivity to training data partitioning. However, improvements in testing performance under spatial cross-validation are modest, indicating that the spatial tensor captures local spatial structure but does not substantially enhance spatial generalisation.