MODA*-G: A Robust Mixed-Data Outlier Detection Framework with Attention-based Softmax Gating

We introduce MODA*-G (Mixed-Data Outlier Divergence Analysis with Softmax Gating), anovel unsupervised anomaly detection framework for tabular data containing both numericaland categorical variables. MODA*-G combines four complementary statistical engines—a robustMCD-based Mahalanobis distance (SDNmcd), a directional kurtosis score with robust MADevaluation (SDNpe˜na), an original Gower-MAD distance score (SDGmad), and a categoricalentropy score (SDC)—through a Softmax Gating mechanism with temperature parameter k,yielding observation-level adaptive weights without heuristic rules.We establish five formal theorems (consistency, 50% breakdown point, partial affine equivariance, score comparability, and stochastic dominance) and two convergence propositions: oneestablishing that the gating weight of the dominant engine converges to one as k → ∞, and oneestablishing that the MODA*-G score converges at rate OP(n−1/2).Monte Carlo simulation (n=300, p=4, q=2, 100 replications per cell) across four canonicalcontamination types confirms that MODA*-G with k=3 achieves mean AUC-ROC of 0.978,outperforming Isolation Forest (0.911) and LOF (0.623) in all four scenarios simultaneously. Onthe Annthyroid real-data benchmark (Campos et al., 2016) (n=7,200, p=6, q=15), MODA*-Gachieves AUC-ROC = 0.939, outperforming IF (0.632), EIF (0.496), COPOD (0.553), CatBoostAD (0.554), LOF (0.620), and surpassing the best result of Campos et al. (2016) by +0.189 AUC.The framework operates natively in X = Rp ×Cq without encoding categorical variables, andproduces an interpretable per-engine diagnostic decomposition. Applications to epidemiologicalbiosurveillance are discussed