<![CDATA[The increasing use of artificial intelligence (AI) in mental health prediction highlights the need for models that are not only accurate but also mathematically interpretable and theoretically grounded. This paper presents a mathematical modeling framework for explainable AI that integrates the K-Nearest Neighbors (KNN) algorithm with rule induction based on fuzzy rough set theory. The proposed hybrid framework is formulated to combine statistical classification with symbolic reasoning, providing transparent post hoc explanations through a set of fuzzy linguistic rules. A large-scale mental health dataset is utilized, comprising behavioral, psychological, and lifestyle attributes, with "coping struggles" as the target classification variable. The mathematical formulation of the fuzzy rough set-based rule induction is explicitly defined using fuzzy similarity relations, lower and upper approximations, and soft rule matching with tunable thresholds. Performance evaluation demonstrates that the hybrid model achieves 94.5% accuracy, 87.7% precision, 100% recall, and 93.4% F1-score, while also producing high-coverage fuzzy rules that align closely with the base KNN predictions. Comparative analysis with a traditional fuzzy inference system (FIS) reveals the superior scalability and fidelity of the proposed method, particularly in high-dimensional feature spaces. This work contributes a modular and mathematically rigorous approach to explainable AI, offering potential applications in clinical screening, early intervention, and intelligent decision support for mental health.]]>
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