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Permutation entropy (PE), as one of the powerful complexity measures for analyzing time series, has advantages of easy implementation and high efficiency. In order to improve the performance of PE, some improved PE methods have been proposed through introducing amplitude information and distance information in recent years. Weighted-permutation entropy (W-PE) weight each arrangement pattern by using variance information, which has good robustness and stability in the case of high noise level and can extract complexity information from data with spike feature or abrupt amplitude change. Dispersion entropy (DE) introduces amplitude information by using the normal cumulative distribution function (NCDF); it not only can detect the change of simultaneous frequency and amplitude, but also is superior to the PE method in distinguishing different data sets. Reverse permutation entropy (RPE) is defined as the distance to white noise in the opposite trend with PE and W-PE, which has high stability for time series with varying lengths. To further improve the performance of PE, we propose a new complexity measure for analyzing time series, and term it as reverse dispersion entropy (RDE). RDE takes PE as its theoretical basis and combines the advantages of DE and RPE by introducing amplitude information and distance information. Simulation experiments were carried out on simulated and sensor signals, including mutation signal detection under different parameters, noise robustness testing, stability testing under different signal-to-noise ratios (SNRs), and distinguishing real data for different kinds of ships and faults. The experimental results show, compared with PE, W-PE, RPE, and DE, that RDE has better performance in detecting abrupt signal and noise robustness testing, and has better stability for simulated and sensor signal. Moreover, it also shows higher distinguishing ability than the other four kinds of PE for sensor signals.

Permutation entropy has been used as a powerful nonlinear dynamic tool for randomness measurement of time series and has been used in the area of condition monitoring and early failure fault detection of rolling bearing. However, the detail size relationship between adjacent amplitudes of signal is ignored in the calculation process of the original permutation entropy algorithms. The reverse permutation entropy was developed as a new nonlinear dynamic parameter through introducing distance information to time series with different lengths to improve the performance and stability of permutation entropy. Since the single-scale permutation entropy or reverse permutation entropy cannot completely reflect the complexity features of time series, in this paper, the phase reverse permutation entropy is proposed by introducing phase information into reverse permutation entropy to improve the detection ability of signal dynamic changes as much as possible. Based on phase reverse permutation entropy, the composite multi-scale phase reverse permutation entropy is proposed to extract the complexity information hidden in different time scales and overcome the defects of traditional coarse-grained multi-scale. Also, phase reverse permutation entropy is compared with reverse permutation entropy through simulation data and the result shows that the introduced phase information can increase the sensitivity of phase reverse permutation entropy in mutation characteristics detection of signal. After that, a new fault diagnosis method of rolling bearing was proposed based on composite multi-scale phase reverse permutation entropy for fault feature extraction and the whale optimization algorithm support vector machine for failure mode identification. Finally, the proposed fault diagnosis method was applied to the experimental data analysis of rolling bearing by comparing it with the composite multi-scale permutation entropy, the multi-scale permutation entropy, as well as multi-scale phase reverse permutation entropy based fault diagnosis approaches. The comparison results shows that the proposed method can effectively the fault location and severity of rolling bearings and reaches the highest fault recognition rate among the mentioned methods above.

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.

An original multivariate multi-scale methodology for assessing the complexity of physiological signals is proposed. The technique is able to incorporate the simultaneous analysis of multi-channel data as a unique block within a multi-scale framework. The basic complexity measure is done by using Permutation Entropy, a methodology for time series processing based on ordinal analysis. Permutation Entropy is conceptually simple, structurally robust to noise and artifacts, computationally very fast, which is relevant for designing portable diagnostics. Since time series derived from biological systems show structures on multiple spatial-temporal scales, the proposed technique can be useful for other types of biomedical signal analysis. In this work, the possibility of distinguish among the brain states related to Alzheimer’s disease patients and Mild Cognitive Impaired subjects from normal healthy elderly is checked on a real, although quite limited, experimental database.

Bearing fault diagnosis has attracted significant attention over the past few decades. It consists of two major parts: vibration signal feature extraction and condition classification for the extracted features. In this paper, multiscale permutation entropy (MPE) was introduced for feature extraction from faulty bearing vibration signals. After extracting feature vectors by MPE, the support vector machine (SVM) was applied to automate the fault diagnosis procedure. Simulation results demonstrated that the proposed method is a very powerful algorithm for bearing fault diagnosis and has much better performance than the methods based on single scale permutation entropy (PE) and multiscale entropy (MSE).

Defective rotating machinery usually exhibits complex dynamic behavior. Therefore, feature representation of machinery vibration signals is always critical for condition monitoring of rotating machinery. Permutation entropy (PeEn), an adaptive symbolic description, can measure complexities of signals. However, PeEn, which compresses all the information into a single parameter, may lack the capability to fully describe the dynamics of complex signals. Afterward, multiscale PeEn (MPeEn) is put forward for coping with nonstationarity, outliers and artifacts emerging in complex signals. In MPeEn, a set of parameters serves to describe the dynamics of complex signals in different time scales. Nonetheless, an average procedure in MPeEn may withhold local information of complex signals and destroy internal structures of complex signals. To overcome deficiencies of PeEn and MPeEn, this paper proposes generalized PeEn (GPeEn) by introducing different orders and time lags into PeEn. In GPeEn, a complex signal is converted into a PeEn matrix rather than a single parameter. Moreover, minimal, maximal and average values of the PeEn matrix serve to briefly describe conditions of rotating machinery. Next, a numerical experiment proves that the proposed method in this paper performs better than skewness, kurtosis, PeEn and MPeEn in characterizing conditions of a Lorenz model. Subsequently, the proposed method in this paper is compared with skewness, kurtosis, PeEn and MPeEn by investigating gear and roll-bearing vibration signals containing different types and severity of faults. The results show that the proposed method in this paper outperforms the other four methods in distinguishing between different types and severity of faults of rotating machinery.

The complex and changeable marine environment surrounded by a variety of noise, including sounds of marine animals, industrial noise, traffic noise and the noise formed by molecular movement, not only interferes with the normal life of residents near the port, but also exerts a significant influence on feature extraction of ship-radiated noise (S-RN). In this paper, a novel feature extraction technique for S-RN signals based on optimized variational mode decomposition (OVMD), permutation entropy (PE), and normalized Spearman correlation coefficient (NSCC) is proposed. Firstly, with the mode number determined by reverse weighted permutation entropy (RWPE), OVMD decomposes the target signal into a set of intrinsic mode functions (IMFs). The PE of all the IMFs and SCC between each IMF with the raw signal are then calculated, respectively. Subsequently, feature parameters are extracted through the sum of PE weighted by NSCC for the IMFs. Lastly, the obtained feature vectors are input into the support vector machine multi-class classifier (SVM) to discriminate various types of ships. Experimental results indicate that five kinds of S-RN samples can be accurately identified with a recognition rate of 94% by the proposed scheme, which is higher than other previously published methods. Hence, the proposed method is more advantageous in practical applications.

As a powerful tool for measuring complexity and randomness, multivariate multi-scale permutation entropy (MMPE) has been widely applied to the feature representation and extraction of multi-channel signals. However, MMPE still has some intrinsic shortcomings that exist in the coarse-grained procedure, and it lacks the precise estimation of entropy value. To address these issues, in this paper a novel non-linear dynamic method named composite multivariate multi-scale permutation entropy (CMMPE) is proposed, for optimizing insufficient coarse-grained process in MMPE, and thus to avoid the loss of information. The simulated signals are used to verify the validity of CMMPE by comparing it with the often-used MMPE method. An intelligent fault diagnosis method is then put forward on the basis of CMMPE, Laplacian score (LS), and bat optimization algorithm-based support vector machine (BA-SVM). Finally, the proposed fault diagnosis method is utilized to analyze the test data of rolling bearings and is then compared with the MMPE, multivariate multi-scale multiscale entropy (MMFE), and multi-scale permutation entropy (MPE) based fault diagnosis methods. The results indicate that the proposed fault diagnosis method of rolling bearing can achieve effective identification of fault categories and is superior to comparative methods.

Electroencephalogram (EEG) signal is a time-varying and nonlinear spatial discrete signal, which has been widely used in the field of emotion recognition. Up to now, a large number of studies have chosen time–frequency domain features or extracted features through brain networks. However, partial spatial or time–frequency information of EEG signals will be lost when analyzing from a single point of view. At the same time, the network analysis based on EEG is largely affected by the inherent volume effect of EEG. Therefore, how to eliminate the influence of volume effect on brain network analysis and extract the features that can reflect both time–frequency information and spatial information is the problem we need to solve at present. In this paper, a feature fusion method that can better reflect the emotional state is proposed. This method uses multichannel weighted multiscale permutation entropy (MC-WMPE) as the feature. It not only takes into account the time–frequency and spatial information of EEG signals but also eliminates the inherent volume effect of EEG signals. We first calculate the multiscale permutation entropy (MPE) of the EEG signals in each channel and construct the brain functional network by calculating the Pearson correlation coefficient (PCC) between each channel. PageRank algorithm is used to sort the importance of nodes in the brain functional network, and the weight of each node is obtained to screen out the important channels in emotion recognition. Then the weights of each channel and the MPE are weighted combined to obtain MC-WMPE as the feature. The research shows that both temporal information and spatial information are of great significance in processing EEG signals. Moreover, the analysis of the frontal, parietal and occipital lobes is necessary for studying the activity state of the cerebral cortex under emotional stimulation. Finally, we carried out experiments on the DEAP and SEED database, and the highest accuracy rate of emotion recognition with this combination feature is 85.28% and 87.31%.

In this paper, we propose multivariate multiscale permutation entropy (MMPE) and multivariate weighted multiscale permutation entropy (MWMPE) to explore the complexity of the multivariate time series over multiple different time scales. First, we apply these methods to the simulated trivariate time series which is compose of white noise and 1/ f noise to test the validity of multivariate methods. The standard deviations of weighted methods are bigger because of containing more amplitude information, while the standard deviations of multivariate method are smaller than the method for single channel. Hence, it can be found that MWMPE shows a better distinguish capacity, while it is able to measure the complexity of the multichannel data accurately and reflect more information about the multivariate time series as well as holds a better robustness. Then MMPE and MWMPE methods are employed to the financial time series: closing prices and trade volume, from different area. It can be verified that the methods for multichannel data analyze the properties of multivariate time series comprehensively. The entropy values taking the weight into account for both the multichannel and single channel amplify the local fluctuation and reflect more amplitude information. The MWMPE maintain the fluctuation characteristic of SCWMPE of both price and volume. The MWMPE results of these stock markets can be divided into three groups: (1) S&P500, FTSE, and HSI, (2) KOSPI, and (3) ShangZheng. The weighted contingency also shows the difference of inhomogenity of the distributions of ordinal patterns between these groups. Thus, MWMPE method is capable of differentiating these stock markets, detecting their multiscale structure and reflects more information containing in the financial time series.