On Uncertainty of Dynamic Systems via State Aggregation Coarse-Graining and State Decomposition Fine-Graining Ways

TL;DR

This paper investigates how state aggregation and decomposition affect the entropy of dynamic systems, providing proofs and analysis for over 20 entropy measures to understand their behavior in uncertainty quantification.

Contribution

It offers a comprehensive analysis of how state aggregation reduces and decomposition increases entropy across multiple entropy measures, with formal proofs for key cases.

Findings

State aggregation decreases entropy in dynamic systems.

State decomposition increases entropy in dynamic systems.

Most popular entropy measures adhere to these properties.

Abstract

Uncertainty is an important feature of dynamic systems, and entropy has been widely used to measure this attribute. In this Letter, we prove that state aggregation and decomposition can decrease and increase the entropy, respectively, of dynamic systems. More than 20 popular entropies in the literature are summarized and analyzed, and it is noted that none of them breaks this property. Finally, pertinent proofs are given for four cases.