Interval multiscale sample entropy: A novel tool for interval-valued time series complexity analysis

To analyze the complexity of interval-valued time series (ITSs), a novel interval multiscale sample entropy (IMSE) methodology is proposed in this paper. To validate the effectiveness and feasibility of IMSE in characterizing ITS complexity, the method is initially implemented on simulated time series. The experimental results demonstrate that IMSE not only successfully identifies series complexity and long-range autocorrelation patterns but also effectively captures the intrinsic relationships between interval boundaries. Furthermore, the test results show that IMSE can also be applied to measure the complexity of multivariate time series of equal length. Subsequently, IMSE is applied to investigate interval temperature series (2000–2023) from four Chinese cities: Shanghai, Kunming, Chongqing, and Nagqu. The results show that IMSE not only distinctly differentiates temperature patterns across cities but also effectively quantifies complexity and long-term autocorrelation in ITSs. All the results indicate that IMSE is an alternative and effective method for studying the complexity of ITSs.