A Unified Dynamic Soft Vector Space Framework for Time-Varying Uncertainty Modeling

Abstract

Soft vector spaces provide an algebraic framework for representing parameterized uncertainty; however, existing formulations are inherently static, assuming time-independent parameters and soft values. Such restriction poses an obstacle for real-life applications in which uncertainty changes dynamically with time. Although the notion of time-dependent uncertainty has found its place in fuzzy sets, intuitionistic fuzzy sets, neutrosophic sets, and dynamic soft sets, there is still no general algebraic vector space approach allowing linear combinations and transformations with the time dependency taken into account. In this regard, this work is focused on developing a new theory called dynamic soft vector spaces (DSVS) that extends classical soft vector spaces with a time dimension, which becomes an integral part of dynamic vectors. The main contributions of the DSVS theory include defining such important concepts as dynamic linear combinations, dynamic spans, dynamic bases, dynamic linear independence, as well as dynamic soft linear transformations. The time-dependent continuity of dynamic structures is guaranteed by using the Hausdorff metric. The benefits of the proposed approach are demonstrated by means of several practical examples, including rotations in subspaces, discrete-time dynamic evolution, weather prediction models, health monitoring of patients, and mobile robot navigation.