A Time Fractal-Based Complex Belief Entropy in Complex Evidence Theory for Pattern Classification

In the era of complex data environments, accurately measuring uncertainty is crucial for effective decision making. Complex evidence theory (CET) provides a framework for handling uncertainty reasoning in the complex plane. Within CET, complex basic belief assignment (CBBA) aims to tackle the uncertainty and imprecision inherent in data coinciding with phase or periodic changes. However, measuring the uncertainty of CBBA over time remains an open issue. This study introduces a novel entropy model, the complex belief (CB) entropy, within the framework of CET, designed to tackle the inherent uncertainty and imprecision in data with phase or periodic changes. The model is developed by integrating concepts of interference and fractal theory to extend the understanding of uncertainty over time. Methodologically, the CB entropy is constructed to include discord, nonspecificity, and an interaction term for focal elements, defined as interference. In addition, thanks to the concept of the fractal, the model is further generalized to time fractal-based CB (TFCB) entropy for forecasting future uncertainties. We furthermore analyze the properties of the entropy models. Findings demonstrate that the proposed entropy models provide a more comprehensive measure of uncertainty in complex scenarios. Finally, a decision-making method based on the proposed entropy is proposed.