Refined Composite Multi-Scale Reverse Weighted Permutation Entropy and Its Applications in Ship-Radiated Noise.

A comparative study of four multi-scale entropies combined with grey relational degree in classification of ship-radiated noise

Financial time series analysis using the relation between MPE and MWPE

In order to solve the shortcoming of multi-scale weighted permutation entropy ignoring high-frequency components, this paper proposes a new nonlinear dynamics analysis method, which is called hierarchical weighted permutation entropy. This method can fully and accurately reflect the low-frequency and high-frequency information of the time series. Based on this, a new intelligent fault diagnosis method for common rail injector based on hierarchical weighted permutation entropy and LSSVM is proposed. Through simulation signal and experimental data analysis, the proposed method is compared with the fault diagnosis method based on multi-scale permutation entropy and multi-scale weighted permutation entropy. The results show that HWPE can more comprehensively reflect the fault information and the proposed diagnosis method has a higher fault recognition rate and stronger robustness.

Due to the complicated engineering operation of the check valve in a high−pressure diaphragm pump, its vibration signal tends to show non−stationary and non−linear characteristics. These leads to difficulty extracting fault features and, hence, a low accuracy for fault diagnosis. It is difficult to extract fault features accurately and reliably using the traditional MPE method, and the ELM model has a low accuracy rate in fault classification. Multi−scale weighted permutation entropy (MWPE) is based on extracting multi−scale fault features and arrangement pattern features, and due to the combination of extracting a sequence of amplitude features, fault features are significantly enhanced, which overcomes the deficiency of the single−scale permutation entropy characterizing the complexity of vibration signals. It establishes the check valve fault diagnosis model from the twin extreme learning machine (TELM). The TELM fault diagnosis model established, based on MWPE, aims to find a pair of non−parallel classification hyperplanes in the equipment state space to improve the model’s applicability. Experiments show that the proposed method effectively extracts the characteristics of the vibration signal, and the fault diagnosis model effectively identifies the fault state of the check valve with an accuracy rate of 97.222%.

The slope fitting line between MPE (multi-scale permutation entropy) and MWPE (multi-scale weighted permutation entropy) is recently proposed as a discriminant statistic for testing the nonlinearity of a time series. The main objective of this paper is to demonstrate that the selection of the optimal parameters of the non-uniform embedding is essential for the proposed discriminant statistic. In particular, the presented case studies indicate that the modified discriminant statistic based on non-uniform embedding can detect differences between such time series which remain indistinguishable if the original approach is used.

A new denoising algorithm and feature extraction algorithm that combine a new kind of permutation entropy (NPE) and variational mode decomposition (VMD) are put forward in this paper. VMD is a new self-adaptive signal processing algorithm, which is more robust to sampling and noise, and also can overcome the problem of mode mixing in empirical mode decomposition (EMD) and ensemble EMD (EEMD). Permutation entropy (PE), as a nonlinear dynamics parameter, is a powerful tool that can describe the complexity of a time series. NPE, a new version of PE, is interpreted as distance to white noise, which shows a reverse trend to PE and has better stability than PE. In this paper, three kinds of ship-radiated noise (SN) signal are decomposed by VMD algorithm, and a series of intrinsic mode functions (IMF) are obtained. The NPEs of all the IMFs are calculated, the noise IMFs are screened out according to the value of NPE, and the process of denoising can be realized by reconstructing the rest of IMFs. Then the reconstructed SN signal is decomposed by VMD algorithm again, and one IMF containing the most dominant information is chosen to represent the original SN signal. Finally, NPE of the chosen IMF is calculated as a new complexity feature, which constitutes the input of the support vector machine (SVM) for pattern recognition of SN. Compared with the existing denoising algorithms and feature extraction algorithms, the effectiveness of proposed algorithms is validated using the numerical simulation signal and the different kinds of SN signal.

Although the technical requirements for the feature extraction of ship radiated noise (SRN) in the fields of national defense and economy increase with each passing day, the complexity of the marine environment makes the feature extraction of SRN difficult. The traditional feature extraction method based on variational mode decomposition (VMD) is widely used in the feature extraction of SRN. Nevertheless, the use of VMD is greatly affected by parameters. In this paper, the butterfly optimization algorithm (BOA) is introduced to optimize VMD, which is called BOA-VMD algorithm, and realizes the optimal selection of VMD parameters K and α. To further improve the efficiency of feature extraction method, combined with slope entropy (SE), a feature extraction method of SRN based on BOA-VMD and SE is proposed. The experimental results of the simulated signal show that the BOA-VMD algorithm has a smaller envelope entropy value and better decomposition effect than the genetic algorithm (GA) and particle swarm optimization (PSO). The experimental results of feature extraction of SRN show that the highest recognition rate of the four entropy values improve with the increase of the number of extracted features, compared with the three entropy values of dispersion entropy (DE), fluctuation dispersion entropy (FDE) and permutation entropy (PE), the SRN feature extraction method based on BOA-VMD and SE has the highest recognition rate under different quantity features, and the recognition rate has reached 100% under three features.

The aim of the present work was to investigate the performance of entropic tools to classify, subjects with normal hearing and those with pathologies in the auditory pathway, using short-latency records of auditory evoked potentials. To accomplish with this objective, traditional permutation entropy, weighted permutation entropy and a modified version of the original permutation entropy, correcting the count of the missing or forbidden patterns. The database used consisted of two age groups, one of minors aged from a few months to four years and the other of older aged over 18 years. For both minor and older subjects, thirty samples were randomly selected from both normal and hearing-impaired subjects. Once the different varieties of entropy had been calculated, the difference in mean values was analyzed using statistical tests to check whether the difference between them was significant or not. The differences between means of the groups with and without pathology were found to be significantly different at a level of 99.9 %. Finally, the choice of the most appropriate entropy was made based on the calculation of the specificity, selectivity, precision and area under the operation reception curve. The results obtained showed that for the differentiation of healthy and pathological records in the case of minors, the weighted permutation entropy was more appropriate and the modified version of the permutation in the case of older adults.

Quantifying the complexity of the EEG signal during prolonged wakefulness and during sleep is gaining interest as an additional mean to characterize the mechanisms associated with sleep and wakefulness regulation. Here, we characterized how EEG complexity, as indexed by Multiscale Permutation Entropy (MSPE), changed progressively in the evening prior to light off and during the transition from wakefulness to sleep. We further explored whether MSPE was able to discriminate between wakefulness and sleep around sleep onset and whether MSPE changes were correlated with spectral measures of the EEG related to sleep need during concomitant wakefulness (theta power-Ptheta: 4-8 Hz). To address these questions, we took advantage of large datasets of several hundred of ambulatory EEG recordings of individual of both sexes aged 25-101 years. Results show that MSPE significantly decreases before light off (i.e. before sleep time) and in the transition from wakefulness to sleep onset. Furthermore, MSPE allows for an excellent discrimination between pre-sleep wakefulness and early sleep. Finally, we show that MSPE is correlated with concomitant Ptheta. Yet, the direction of the latter correlation changed from before light-off to the transition to sleep. Given the association between EEG complexity and consciousness, MSPE may track efficiently putative changes in consciousness preceding sleep onset. An MSPE stands as a comprehensive measure that is not limited to a given frequency band and reflects a progressive change brain state associated with sleep and wakefulness regulation. It may be an effective mean to detect when the brain is in a state close to sleep onset.

A method based on compound multiscale permutation entropy, support vector machine, and particle swarm optimization for roller bearings fault diagnosis was presented in this study. Firstly, the roller bearings vibration signals under different conditions were decomposed into permutation entropy values by the multiscale permutation entropy and compound multiscale permutation entropy methods. The compound multiscale permutation entropy model combined the different graining sequence information under each scale factor. The average value of each scale factor was regarded as the final entropy value in the compound multiscale permutation entropy model. The compound multiscale permutation entropy model suppressed the shortcomings of poor stability caused by the length of the original signals in the multiscale permutation entropy model. Validity and accuracy are considered in the numerical experiments, and then compared with the computational efficiency of the multiscale permutation entropy method. Secondly, the entropy values of the multiscale permutation entropy/compound multiscale permutation entropy under different scales are regarded as the input of the particle swarm optimization–support vector machine models for fulfilling the fault identification, the classification accuracy is used to verify the effectiveness of the multiscale permutation entropy/compound multiscale permutation entropy with particle swarm optimization–support vector machine. Finally, the experimental results show that the classification accuracy of the compound multiscale permutation entropy model is higher than that of the multiscale permutation entropy.

Multiscale Permutation Entropy (MPE) analysis is a powerful ordinal tool in the measurement of information content of time series. MPE refinements, such as Composite MPE (cMPE) and Refined Composite MPE (rcMPE), greatly increase the precision of the entropy estimation by modifying the original method. Nonetheless, these techniques have only been proposed as algorithms, and are yet to be described from the theoretical perspective. Therefore, the purpose of this article is two-fold. First, we develop the statistical theory behind cMPE and rcMPE. Second, we propose an alternative method, Refined Composite Downsampling Permutation Entropy (rcDPE) to further increase the entropy estimation’s precision. Although cMPE and rcMPE outperform MPE when applied on uncorrelated noise, the results are higher than our predictions due to inherent redundancies found in the composite algorithms. The rcDPE method, on the other hand, not only conforms to our theoretical predictions, but also greatly improves over the other methods, showing the smallest bias and variance. By using MPE, rcMPE and rcDPE to classify faults in bearing vibration signals, rcDPE outperforms the multiscaling methods, enhancing the difference between faulty and non-faulty signals, provided we apply a proper anti-aliasing low-pass filter at each time scale.

An improved permutation entropy (PE) algorithm named coded permutation entropy (CPE) is proposed in this paper to optimize the problems existing in PE based on the secondary partitioning. The principle of CPE algorithm is given, and the performance of it for dynamical change detection is analyzed using synthetic signal, logistic map and Lorenz map. The detection ability of CPE algorithm in different signal-to-noise ratios (SNR) is studied and the algorithm complexity is discussed. The results show that CPE can accurately capture minor feature information and amplify the detection results of dynamical changes compared with PE, weighted permutation entropy (WPE) and amplitude-aware permutation entropy (AAPE), but it has less robustness to noise and requires a higher computation cost than the others. Finally, we use the new algorithm to analyze the rolling bearing fault signals. The application of actual signals illustrates that CPE performs better in detecting abnormal pulse of the rolling bearing when the embedded dimension is small. From all the analyses in this paper, we find that CPE has a better performance for dynamical change detection compared with the other three algorithms when there is a larger repetition rate of permutation pattern in the position sequences.

Warships play an important role in the modern sea battlefield. Research on the line spectrum features of warship radio noise signals is helpful to realize the classification and recognition of different types of warships, and provides critical information for sea battlefield. In this paper, we proposed a novel linear spectrum frequency feature extraction technique for warship radio noise based on complete ensemble empirical mode decomposition with adaptive noise (CEEMDAN), duffing chaotic oscillator (DCO), and weighted-permutation entropy (W-PE). The proposed linear spectrum frequency feature extraction technique, named CEEMDAN-DCO-W-PE has the following advantages in comparison with other linear spectrum frequency feature extraction techniques; (i) as an adaptive data-driven algorithm, CEEMDAN has more accurate and more reliable decomposition performance than empirical mode decomposition (EMD) and ensemble EMD (EEMD), and there is no need for presetting parameters, such as decomposition level and basis function; (ii) DCO can detect the linear spectrum of narrow band periodical warship signals by way of utilizing its properties of sensitivity for weak periodical signals and the immunity for noise; and (iii) W-PE is used in underwater acoustic signal feature extraction for the first time, and compared with traditional permutation entropy (PE), W-PE increases amplitude information to some extent. Firstly, warship radio noise signals are decomposed into some intrinsic mode functions (IMFs) from high frequency to low frequency by CEEMDAN. Then, DCO is used to detect linear spectrum of low-frequency IMFs. Finally, we can determine the linear spectrum frequency of low-frequency IMFs using W-PE. The experimental results show that the proposed technique can accurately extract the line spectrum frequency of the simulation signals, and has a higher classification and recognition rate than the traditional techniques for real warship radio noise signals.

The recognition of abnormal dynamic pressure wave signal is key to pipeline leakage detection. Permutation entropy algorithm can effectively amplify slight changes of time series data and has advantages of easily computing and good performance in change detection. The weighted-permutation entropy which is based on the permutation entropy considers the amplitude information. In this study, the weighted-permutation entropy is utilized to detect the pipeline leakage. To prove the effectiveness of the method, typical experiments are performed on the test platform. Results show that the weighted-permutation entropy method can successfully detect the pipeline leakage.

Entropy quantification approaches have gained considerable attention in engineering applications. However, certain limitations persist, including the strong dependence on parameter selection, limited discriminating power, and low robustness to noise. To alleviate these issues, this paper introduces two novel algorithms for time series analysis: the ensemble improved permutation entropy (EIPE) and multiscale EIPE (MEIPE). Our approaches employ a new symbolization process that considers both permutation relations and amplitude information. Additionally, the ensemble technique is utilized to reduce the dependence on parameter selection. We performed a comprehensive evaluation of the proposed methods using various synthetic and experimental signals. The results illustrate that EIPE is capable of distinguishing white, pink, and brown noise with a smaller number of samples compared to traditional entropy algorithms. Furthermore, EIPE displays the potential to discriminate between regular and non-regular dynamics. Notably, when compared to permutation entropy, weighted permutation entropy, and dispersion entropy, EIPE exhibits superior robustness against noise. In practical applications, such as RR interval data classification, bearing fault diagnosis, marine vessel identification, and electroencephalographic (EEG) signal classification, the proposed methods demonstrate better discriminating power compared to conventional entropy measures. These promising findings validate the effectiveness and potential of the algorithms proposed in this paper.