A novel method to measure static and dynamic complexity of time series based on visualization curves

In this paper, reverse transition entropy (RTE) is proposed and combined with refined composite multi-scale analysis and generalized fractional-order entropy to construct the refined composite multi-scale reverse transition generalized fractional-order complexity-entropy curve (RCMS-RT-GFOCEC). This measure aims to characterize and identify different complex time series. First, RTE is used to extract the static and dynamic transition probabilities of the temporal structure. Then, the distribution area and variation law of the visualization curves are adopted to characterize different time series. Finally, the time series are identified by the multi-scale curves of RTE, H α min, and C α max. The characteristic curves ( H q min and C q max) of the refined composite multi-scale q complexity-entropy curves (RCMS-q-CECs) for the comparative analysis are irregular. The experimental results indicate that the RCMS-RT-GFOCEC method could effectively characterize both artificial and empirical temporal series. Moreover, this method can effectively track the dynamical changes of rolling bearing and turbine gearbox time series. The accuracies of the proposed method reach 99.3% and 98.8%, while the recognition rates based on the RCMS-q-CEC method are only 95.7% and 97.8%, suggesting that the proposed method can effectively characterize and identify different complex temporal systems.