Adaptive Drift Aware Statistical Learning via Decaying Memory Operators

Modern data-driven systems operate in environments where the underlying data distribution changes over time. These non-stationary conditions arise in domains such as energy markets, financial systems, online services, and large-scale cyber-physical infrastructures. Classical statistical learning assumes stationary distributions. Under concept drift this assumption breaks, and model performance degrades. Many adaptive techniques attempt to address this challenge through sliding windows, exponential forgetting, or drift detection. These strategies often rely on heuristic update rules rather than explicit statistical formulations. This study introduces a mathematically grounded framework for adaptive statistical learning based on decaying memory operators. The approach embeds temporal memory directly inside the empirical risk functional. Memory decay becomes part of the learning objective rather than an external algorithmic mechanism. This formulation unifies sliding window estimation, exponential forgetting, and adaptive reweighting within a single analytical structure. Theoretical analysis is conducted under bounded loss and bounded drift assumptions. The analysis provides interpretable guarantees on convergence behavior and excess risk in evolving environments. The results show how the decay parameter governs the balance between stability and responsiveness. Experimental evaluation uses both synthetic drift benchmarks and a real-world electricity demand dataset containing more than forty-five thousand observations. Results demonstrate improved predictive accuracy and smoother adaptation compared with static empirical risk minimization and sliding window baselines. Sensitivity analysis further illustrates the impact of the decay parameter on learning dynamics. The proposed framework offers a principled foundation for adaptive statistical learning in non-stationary data environments. By modeling temporal memory explicitly, the approach improves interpretability, theoretical clarity, and practical robustness in streaming analytics applications.