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

Abstract

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), R 2 (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.

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