Optimized Fuzzy Slope Entropy: A Complexity Measure for Nonlinear Time Series

Entropy has long been a subject that has attracted researchers from a diverse range of fields, including healthcare, finance, and fault detection. Slope entropy (SE) has recently been proposed as a new approach to address the shortcomings of permutation entropy (PE), which ignores magnitude information; however, SE is sensitive to parameters $\boldsymbol {\gamma }$ and $\boldsymbol {\delta }$ , and some information may be lost when segmenting symbols. The $\boldsymbol {\delta }$ , moreover, has only a limited gain on the time series classification performance of SE and increases the algorithm complexity. Considering the aforementioned limitations, this study introduces the concept of fuzzification to the SE and eliminates the $\boldsymbol {\delta }$ to simplify the parameters, resulting in the proposal of fuzzy SE (FuSE); furthermore, we incorporate the artificial rabbit optimization (ARO) algorithm to optimize the parameter $\boldsymbol {\gamma }$ to enhance the effectiveness of FuSE for time series classification and finally proposed an optimized FuSE (OFuSE). OFuSE can greatly reduce the information loss in the mapping process and adaptively search for the optimal parameter. The study evaluated FuSE and OFuSE on several synthetic datasets and concluded that FuSE is more sensitive to changes in signal amplitude and frequency while confirming the advantage of OFuSE in classification. The application of OFuSE on three different real datasets verifies that its classification performance and generalization ability are better than other entropy methods.