MHTFPE_2D: two-dimensional multi-scale hierarchical time–frequency permutation entropy for complexity measurement

As a nonlinear dynamic index, hierarchical permutation entropy (HPE) can effectively represent the complexity change of time series. However, HPE only focuses on extracting time domain information and ignoring the rich information in frequency domain. Meanwhile, HPE is greatly influenced by the length of time series and has poor stability. To address these limitations, a two-dimensional hierarchical time–frequency permutation entropy (HTFPE_2D) is proposed based on the definition of two-dimensional permutation entropy, and its purpose is to combine time-domain and frequency-domain information. To consider the time–frequency information of the multi-scale low-frequency sequences, the two-dimensional multi-scale hierarchical time–frequency permutation entropy (MHTFPE_2D) is further established. MHTFPE_2D allows for the synthesis of multidimensional effective information and leads to better feature extraction. Based on the advantages of the MHTFPE_2D, a new fault diagnosis method of rolling bearing is developed by combining the MHTFPE_2D and GOA-SVM. The proposed fault diagnosis method is validated by using the public rolling bearing datasets of CRWU and our rolling bearing datasets of Anhui University of Technology. The comparison results demonstrate that the proposed method achieves high fault identification accuracy, stability and robustness.