Temporally adaptive hierarchical Choquet integrals: A measure-theoretic framework for dynamic non-additive integration in approximate reasoning

Abstract

This paper introduces the Temporally Adaptive Hierarchical Choquet Integral (TAHCI), a novel theoretical framework that extends classical Choquet integration to address temporal dynamics and high-dimensional data through hierarchical structures and time-evolving fuzzy measures. We develop a rigorous measure-theoretic foundation for this framework, including a complete axiomatic characterization, uniqueness results, and preservation properties. The proposed formulation overcomes theoretical limitations of traditional models by offering tractable parameterization (reducing complexity from $O\left(2^n\right)$ to $O\left(Rnd\right)$, where R is the tensor rank, n is the feature dimensionality, and d is the hierarchy depth) while preserving key measure-theoretic properties. We establish fundamental mathematical results including convergence rates, expressivity bounds, and stability under perturbations. Our theoretical analysis reveals several novel classes of non-additive measures with unique properties that emerge from the temporal adaptation process. We characterize TAHCI's well-defined fixed points and convergence behavior under various conditions. This work advances non-additive measure theory by providing a mathematically principled approach to integrating temporal dynamics with hierarchical measure structures, with direct implications for reasoning under uncertainty and imprecision.