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Spatial Classification Modelling of Imbalanced Response Using Firth Penalized Class-Weighted Geographically Weighted Logistic Regression
Abstract
Binary regional classification of food security is often complicated by class imbalance and spatially heterogeneous predictor–outcome relationships. This study applies Firth Penalized Class-Weighted Geographically Weighted Logistic Regression (Firth CW-GWLR), which combines geographically weighted estimation, class weighting, and Firth's penalty on each local likelihood to reduce small-sample bias and protect against separation. The model is applied to the 2025 food security status of the 22 regencies/cities of East Nusa Tenggara (NTT), Indonesia (18 food insecure and 4 food secure), using the open unemployment rate (X1), the share of agricultural workers (X2), and the number of markets (X3). VIF values (1.12–2.03) indicate no multicollinearity. The Breusch–Pagan test (p = 0.0375) and Moran's I of the global-model residuals (I = 0.2025, p = 0.0248) suggest that the global model is misspecified and motivate local modelling, but they do not by themselves show that the regression coefficients vary spatially. An adaptive Gaussian kernel with 21 nearest neighbours was selected (AICc = 21.170). GWLR, CW-GWLR, Firth-GWLR, and Firth CW-GWLR were compared by leave-one-out cross-validation (LOOCV), in which the class weights, bandwidth, and coefficients were re-estimated in each training fold; because the kernel type was selected on the full data, the LOOCV results are conditional on that choice. All local estimates converged. CW-GWLR achieved the highest balanced accuracy (0.764) and F1 (0.545), Firth-GWLR the highest specificity (0.944), and Firth CW-GWLR identified three of the four food-secure regencies (recall 0.750) but classified five food-insecure regencies as food secure. The fully local model had a slightly lower LOOCV Brier score than models with one global coefficient, which is descriptive rather than formal evidence of spatial variation. With only four food-secure regencies, these results are indicative rather than conclusive.
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DOI: https://doi.org/10.18860/cauchy.v11i2.45533
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