Permutation Entropy Research Articles

Tool Wear State Identification Based on the IWOA-VMD Feature Selection Method

Complex, thin-walled components are the most important load-bearing structures in aircraft equipment. Monitoring the wear status of milling cutters is critical for enhancing the precision and efficiency of thin-walled item machining. The cutting force signals of milling cutters are non-stationary and non-linear, making it difficult to detect wear stages. In response to this issue, a system for monitoring milling cutter wear has been presented, which is based on parameterized Variational Mode Decomposition (VMD) Multiscale Permutation Entropy. Initially, an updated whale optimization technique is used, with the joint correlation coefficient serving as the fitness value for determining the VMD parameters. The improved VMD technique is then used to break down the original signal into a series of intrinsic mode functions, and the Multiscale Permutation Entropy of each effective mode is determined to generate a feature vector. Finally, a 1D Convolutional Neural Network (1D CNN) is employed as the input model for state monitoring using the feature vector. The experimental findings show that the suggested technique can efficiently extract characteristics indicating the wear condition of milling cutters, allowing for the precise monitoring of milling cutter wear states. The recognition rate is as high as 98.4375%, which is superior to those of comparable approaches.

Variable contribution analysis in multivariate process monitoring using permutation entropy

Multivariate statistical process monitoring is widely used in industrial processes for performance monitoring and fault detection. Once a fault has been detected, fault diagnoses must be performed to identify the variables that are responsible for the fault or the deviation from normal operation. In this paper, we present a new data-driven method for variable contribution analysis to a fault or abnormal behaviour in a process. The method is based on permutation entropy (PE), which measures the order of fluctuations within a time series. The PE method for variable contribution analysis is independent of fault detection statistics, and is used to evaluate which variables experienced the greatest change in their expected behaviour over time, immediately after the fault has been identified. We show how PE can be used to identify and interpret the variables responsible for, or affected by faults in an industrial process. The well-known Tennessee Eastman Process (TEP) is used to illustrate the application of PE for variable contribution analysis. The results show that PE is an efficient analysis tool for specifying variable contributions to faults.

Multichannel multiscale increment entropy and its application in roller bearing fault diagnosis

As a new index to measure the complexity of time series, increment entropy, which takes into account the fluctuation directions and amplitude of time series, has better performance than the traditional entropy analysis methods such as sample entropy and permutation entropy. However, the increment entropy value of time series at single scale cannot completely reflect the dynamic change of time series. In this paper, the multichannel multiscale increment entropy (MMIE) is proposed by introducing the coarse-graining and multichannel analysis tools of time series to explore the complexity of multichannel time series over multiple different scales. MMIE considers the dynamic relationships between multichannel data and the relevant cross-channel variations that are often overlooked in single-channel analysis, thus achieving a full utilization of state information. Finally, an MMIE-based fault diagnosis method is proposed for roller bearing. The analysis results of the simulation signals and the measured data of roller bearings indicate that MMIE performs better than MMDE, MMPE, and MMSE approaches in time costing and fault identification rates.

Laser ultrasonic spatially resolved acoustic spectroscopy for grain size study based on Improved Variational Mode Decomposition (IVMD)

Laser Ultrasonic Spatially Resolved Acoustic Spectroscopy (SRAS) is a technique used for detecting the grain size of metal surfaces. It involves exciting Surface Acoustic Waves (SAW) using a grated light source and employs optical deflection techniques to detect ultrasound. The differences in sound velocity between different grains are utilized for grain size characterization. However, this technique is susceptible to the influence of sample surface roughness, resulting in issues like low signal-to-noise ratio and waveform distortion. To address these challenges, a study was conducted on grain size characterization using Laser Ultrasonic SRAS based on Improved Variational Mode Decomposition (IVMD). This involved optimizing Variational Mode Decomposition (VMD) by introducing Permutation Entropy (PE), Weighted Spectral Kurtosis (WSK), and Dynamic Time Warping (DTW). Adaptive decomposition and selection of simulated surface wave signals were employed to achieve signal recovery. A comparison with Empirical Mode Decomposition (EMD) demonstrated that the proposed algorithm had higher noise robustness. The Laser Ultrasonic SRAS detection system was used to characterize the grain size of Ti-6Al-4V casting alloys, and the data were processed using the proposed algorithm. The results were consistent with optical microscopy images, further confirming the reliability and effectiveness of the improved algorithm. The research findings indicate that Laser Ultrasonic Spatially Resolved Acoustic Spectroscopy based on IVMD can effectively reduce the impact of noise on detection signals, thereby enhancing the detectability of grain size measurement.

Research on Sea State Signal Recognition Based on Beluga Whale Optimization-Slope Entropy and One Dimensional-Convolutional Neural Network.

This study introduces a novel nonlinear dynamic analysis method, known as beluga whale optimization-slope entropy (BWO-SlEn), to address the challenge of recognizing sea state signals (SSSs) in complex marine environments. A method of underwater acoustic signal recognition based on BWO-SlEn and one-dimensional convolutional neural network (1D-CNN) is proposed. Firstly, particle swarm optimization-slope entropy (PSO-SlEn), BWO-SlEn, and Harris hawk optimization-slope entropy (HHO-SlEn) were used for feature extraction of noise signal and SSS. After 1D-CNN classification, BWO-SlEn were found to have the best recognition effect. Secondly, fuzzy entropy (FE), sample entropy (SE), permutation entropy (PE), and dispersion entropy (DE) were used to extract the signal features. After 1D-CNN classification, BWO-SlEn and 1D-CNN were found to have the highest recognition rate compared with them. Finally, compared with the other six recognition methods, the recognition rates of BWO-SlEn and 1D-CNN for the noise signal and SSS are at least 6% and 4.75% higher, respectively. Therefore, the BWO-SlEn and 1D-CNN recognition methods proposed in this paper are more effective in the application of SSS recognition.

Noise reduction analysis of deformation data based on CEEMD-PE-SVD modeling

In the field of high slope deformation monitoring, the deformation data obtained are often characterized by high volatility, strong nonlinearity, and noise content, due to the influence of factors such as the surrounding environment, human operation, and the complexity of the project. To overcome these problems, this paper proposes deformation data based on the coupling of complementary ensemble empirical modal decomposition (CEEMD), permutation entropy (PE), and singular value decomposition (SVD). Firstly, CEEMD decomposition is performed on the deformation data. Secondly, a randomness test is performed on the obtained modal components, and if the entropy value of the component arrangement entropy is greater than 0.5, SVD decomposition will be used for noise reduction. Finally, the components that meet the conditions are reconstructed to realize the noise reduction of the deformation data. The results show that the algorithm model has an excellent noise elimination effect, which can reflect the internal detail information of deformation data and provide a well-established method for future research on the processing of deformation monitoring data.

Dispersion complexity-entropy curves: An effective method to characterize the structures of nonlinear time series.

The complexity-entropy curve (CEC) is a valuable tool for characterizing the structure of time series and finds broad application across various research fields. Despite its widespread usage, the original permutation complexity-entropy curve (PCEC), which is founded on permutation entropy (PE), exhibits a notable limitation: its inability to take the means and amplitudes of time series into considerations. This oversight can lead to inaccuracies in differentiating time series. In this paper, drawing inspiration from dispersion entropy (DE), we propose the dispersion complexity-entropy curve (DCEC) to enhance the capability of CEC in uncovering the concealed structures within nonlinear time series. Our approach initiates with simulated data including the logistic map, color noises, and various chaotic systems. The outcomes of our simulated experiments consistently showcase the effectiveness of DCEC in distinguishing nonlinear time series with diverse characteristics. Furthermore, we extend the application of DCEC to real-world data, thereby asserting its practical utility. A novel approach is proposed, wherein DCEC-based feature extraction is combined with multivariate support vector machine for the diagnosis of various types of bearing faults. This combination achieved a high accuracy rate in our experiments. Additionally, we employ DCEC to assess stock indices from different countries and periods, thereby facilitating an analysis of the complexity inherent in financial markets. Our findings reveal significant insights into the dynamic regularities and distinct structures of these indices, offering a novel perspective for analyzing financial time series. Collectively, these applications underscore the potential of DCEC as an effective tool for the nonlinear time series analysis.

A New Method for Denoising Underwater Acoustic Signals Based on EEMD, Correlation Coefficient, Permutation Entropy, and Wavelet Threshold Denoising

A New Method for Denoising Underwater Acoustic Signals Based on EEMD, Correlation Coefficient, Permutation Entropy, and Wavelet Threshold Denoising

Permutation Entropy Does Not Track the Electroencephalogram-Related Manifestations of Paradoxical Excitation During Propofol-Induced Loss of Responsiveness: Results From a Prospective Observational Cohort Study.

During the anesthetic-induced loss of responsiveness (LOR), a "paradoxical excitation" with activation of β-frequencies in the electroencephalogram (EEG) can be observed. Thus, spectral parameters-as widely used in commercial anesthesia monitoring devices-may mistakenly indicate that patients are awake when they are actually losing responsiveness. Nonlinear time-domain parameters such as permutation entropy (PeEn) may analyze additional EEG information and appropriately reflect the change in cognitive state during the transition. Determining which parameters correctly track the level of anesthesia is essential for designing monitoring algorithms but may also give valuable insight regarding the signal characteristics during state transitions. EEG data from 60 patients who underwent general anesthesia were extracted and analyzed around LOR. We derived the following information from the power spectrum: (i) spectral band power, (ii) the spectral edge frequency as well as 2 parameters known to be incorporated in monitoring systems, (iii) beta ratio, and (iv) spectral entropy. We also calculated (v) PeEn as a time-domain parameter. We used Friedman's test and Bonferroni correction to track how the parameters change over time and the area under the receiver operating curve to separate the power spectra between time points. Within our patient collective, we observed a "paradoxical excitation" around the time of LOR as indicated by increasing beta-band power. Spectral edge frequency and spectral entropy values increased from 19.78 [10.25-34.18] Hz to 25.39 [22.46-30.27] Hz (P = .0122) and from 0.61 [0.54-0.75] to 0.77 [0.64-0.81] (P < .0001), respectively, before LOR, indicating a (paradoxically) higher level of high-frequency activity. PeEn and beta ratio values decrease from 0.78 [0.77-0.82] to 0.76 [0.73-0.81] (P < .0001) and from -0.74 [-1.14 to -0.09] to -2.58 [-2.83 to -1.77] (P < .0001), respectively, better reflecting the state transition into anesthesia. PeEn and beta ratio seem suitable parameters to monitor the state transition during anesthesia induction. The decreasing PeEn values suggest a reduction of signal complexity and information content, which may very well describe the clinical situation at LOR. The beta ratio mainly focuses on the loss of power in the gamma-band. PeEn, in particular, may present a single parameter capable of tracking the LOR transition without being affected by paradoxical excitation.

Multiscale permutation entropy gray image coding method and its application in bearing fault diagnosis

Bearing fault diagnosis is of great significance to the normal operation of machinery, and its performance and life span directly affect the operational efficiency and safety of the whole equipment. For existing image coding methods which detecting the bearing fault with a large number of training samples and complex neural networks to achieve the desired detection performance, the Multiscale Permutation Entropy Gray Image Coding (MPEGIC) method is proposed. In order to fully extract the feature information of the time series signal, this paper uses the Multiscale Permutation Entropy (MPE) method to construct a new image coding method by calculating the alignment information of the time series to reflect the complexity and randomness of the time series at different scales, and dividing the obtained feature matrix and mapping it to the gray-scale image domain. And it is experimentally verified by Case Western Reserve University (CWRU) bearing dataset and self-made rotor experimental platform bearing dataset. The results show that the method in this paper effectively reduces the number of training samples and the number of model parameters, and maintains a better detection performance even in a strong noise background.

A permutation entropy analysis of Bitcoin volatility

Cryptocurrencies are widely regarded as volatile and less predictable assets by financial participants. The behaviour and dynamics of Bitcoin’s daily volatility, obtained by fitting GARCH models, are investigated for a period of 8 years using permutation entropy which is represented by the variable H for calculations. The best fitting GARCH models selected are the FIGARCH(1,0.7,1) and SGARCH(1,1) models based on maximum likelihood estimation, Akaike Information Criterion and Bayesian Information Criterion. Simulated volatilities are also obtained from the best fitting GARCH models using their respective parameters, to confirm how well the models fit. The results obtained show that the H values of Bitcoin are generally low and that the dynamics of Bitcoin’s volatility is quite predictable, as Bitcoin’s volatility is most likely to decline over time than increase or have an alternating movement. Also, the simulated volatilities show good agreement with the real-world volatility, confirming the models as good fits.

Entropy of Difference: A New Tool for Measuring Complexity

We propose a new tool for estimating the complexity of a time series: the entropy of difference (ED). The method is based solely on the sign of the difference between neighboring values in a time series. This makes it possible to describe the signal as efficiently as prior proposed parameters, such as permutation entropy (PE) or modified permutation entropy (mPE). Firstly, this method reduces the size of the sample that is necessary to estimate the parameter value, and secondly it enables the use of the Kullback–Leibler divergence to estimate the “distance” between the time series data and random signals.

A hybrid model with combined feature selection based on optimized VMD and improved multi-objective coati optimization algorithm for short-term wind power prediction

With the continuous global increase in installed wind power capacity and subsequent surge in power generation, the contradiction between the safe operation of the grid and the efficient consumption of new energy after large-scale grid connection has become increasingly prominent. This paper presents a hybrid model prediction method to further improve the stability and accuracy of wind power prediction. Firstly, variational modal decomposition (VMD) optimized by the coati optimization algorithm (COA) is employed to decompose original wind power, mitigating the non-stationary characteristics of the power sequence. Subsequently, permutation entropy (PE) is used to recombine the decomposed components, and the combined feature selection method is achieved by integrating the Spearman correlation coefficient (SCC) and the autocorrelation function (ACF). Then, the multivariate combined model is constructed, and the improved multi-objective coati optimization algorithm (IMOCOA) determines the weight coefficients of each model to enhance the performance of the hybrid model. Finally, research and analysis are conducted from multiple scenarios and time scales using historical operating data from a wind farm in Xinjiang. The experimental results show that the proposed prediction model effectively improves the accuracy and stability of the wind power prediction compared with other popular prediction models.

A new parameter-free entropy based on fragment oscillation and its application in fault diagnosis

For finite time series, it is difficult to provide a convincing determination scheme for specifying relevant parameters of entropy. This article proposes a new entropy, σkEn, based on the amplitude fluctuations of time series segments. It uses the optimal variance estimation under the principle of minimizing AMSE(σˆ2) to adaptively select the window width, and then realizes the no-argument calculation process of volatility entropy through kernel density estimation. By testing with artificial data and applying to real-world data, it has been verified anti-noise and stability properties of σkEn. In a background noise environment with an intensity of 20%, σkEn distinguish various chaotic systems and random signals efficiently. In addition, this manuscript also proposes the plane of (σkEn,NkEn) and (σ̄,σkEn) , which also successfully identifies 6 types of chaotic systems, and 3 random noise signals. Furthermore, (<σ̄>,<σkEn>) curve exhibits different dynamical behavior patterns for these chaotic systems. Finally, we employ (σ̄,σkEn) as a feature and take 8 usual machine learning models, such as Linear Discriminant, Gaussian Naive Bayes and Subspace Discriminant, etc, to perform anomaly detection and fault classification tests on 6 open datasets. Given the appropriate classifiers, our method can achieve a classification accuracy of 100% for four tasks, and an average accuracy of over 96% for the remaining two. This is significantly superior than the accuracy obtained by using Permutation Entropy as a feature.

Fetal QRS extraction from single-channel abdominal ECG using adaptive improved permutation entropy.

Fetal electrocardiogram (fECG) monitoring is crucial for assessing fetal condition during pregnancy. However, current fECG extraction algorithms are not suitable for wearable devices due to their high computational cost and multi-channel signal requirement. The paper introduces a novel and efficient algorithm called Adaptive Improved Permutation Entropy (AIPE), which can extract fetal QRS from a single-channel abdominal ECG (aECG). The proposed algorithm is robust and computationally efficient, making it a reliable and effective solution for wearable devices. To evaluate the performance of the proposed algorithm, we utilized our clinical data obtained from a pilot study with 10 subjects, each recording lasting 20min. Additionally, data from the PhysioNet 2013 Challenge bank with labeled QRS complex annotations were simulated. The proposed methodology demonstrates an average positive predictive value ([Formula: see text]) of 91.0227%, sensitivity (Se) of 90.4726%, and F1 score of 90.6525% from the PhysioNet 2013 Challenge bank, outperforming other methods. The results suggest that AIPE could enable continuous home-based monitoring of unborn babies, even when mothers are not engaging in any hard physical activities.

Precise Lightning Strike Detection in Overhead Lines Using KL-VMD and PE-SGMD Innovations

When overhead lines are impacted by lightning, the traveling wave of the fault contains a wealth of fault information. The accurate extraction of feature quantities from transient components and their classification are fundamental to the identification of lightning faults. The extraction process may involve modal aliasing, optimal wavelet base issues, and inconsistencies between the lightning strike distance and the fault point. These factors have the potential to impact the effectiveness of recognition. This paper presents a method for identifying lightning strike faults by utilizing Kullback–Leibler (KL) divergence enhanced Variational Mode Decomposition (VMD) and Symmetric Geometry Mode Decomposition (SGMD) improved with Permutation Entropy (PE) to address the aforementioned issues. A model of a 220 kV overhead line is constructed using real faults to replicate scenarios of winding strike, counterstrike, and short circuit. The three-phase voltage is chosen and then subjected to Karenbaren decoupling in order to transform it into zero mode, line mode 1, and line mode 2. The zero-mode voltage is decomposed using KL-VMD and PE-SGMD methods, and the lightning identification criteria are developed based on various transient energy ratios. The research findings demonstrate that the criteria effectively differentiate between winding strike, counterstrike, and short-circuit faults, thus confirming the accuracy and efficacy of the lightning fault identification criteria utilizing KL-VMD and PE-SGMD.

Scale based entropy measures and deep learning methods for analyzing the dynamical characteristics of cardiorespiratory control system in COVID-19 subjects during and after recovery

COVID-19, known as Coronavirus Disease 2019 primarily targets the respiratory system and can impact the cardiovascular system, leading to a range of cardiorespiratory complications. The current forefront in analyzing the dynamical characteristics of physiological systems and aiding clinical decision-making involves the integration of entropy-based complexity techniques with artificial intelligence. Entropy-based measures offer promising prospects for identifying disturbances in cardiorespiratory control system (CRCS) among COVID-19 patients by assessing the oxygen saturation variability (OSV) signals. In this investigation, we employ scale-based entropy (SBE) methods, including multiscale entropy (MSE), multiscale permutation entropy (MPE), and multiscale fuzzy entropy (MFE), to characterize the dynamical characteristics of OSV signals. These measurements serve as features for the application of traditional machine learning (ML) and deep learning (DL) approaches in the context of classifying OSV signals from COVID-19 patients during their illness and subsequent recovery. We use the Beurer PO-80 pulse oximeter which non-invasively acquired OSV and pulse rate data from COVID-19 infected patients during the active infection phase and after a two-month recovery period. The dataset comprises of 88 recordings collected from 44 subjects(26 men and 18 women), both during their COVID-19 illness and two months post-recovery. Prior to analysis, data preprocessing is performed to remove artifacts and outliers. The application of SBE measures to OSV signals unveils a reduction in signal complexity during the course of COVID-19. Leveraging these SBE measures as feature sets, we employ two DL techniques, namely the radial basis function network (RBFN) and RBFN with dynamic delay algorithm (RBFNDDA), for the classification of OSV data collected during and after COVID-19 recovery. To evaluate the classification performance, we employ standard metrics such as sensitivity, specificity, false positive rate (FPR), and the area under the receiver operator characteristic curve (AUC). Among the three scale-based entropy measures, MFE outperformed MSE and MPE by achieving the highest classification performance using RBFN with 13 best features having sensitivity (0.84), FPR (0.30), specificity (0.70) and AUC (0.77). The outcomes of our study demonstrate that SBE measures combined with DL methods offer a valuable approach for categorizing OSV signals obtained during and after COVID-19, ultimately aiding in the detection of CRCS dysfunction.

Generalized weighted permutation entropy analysis of satellite hot-pixel time series in Brazilian biomes

In this work we analyze hot-pixel satellite data, proxy for natural and anthropogenic vegetation and forest fires, using recently proposed method Generalized weighted permutation entropy (GWPE). The current analysis of data from 2002 to 2022 for six Brazilian biomes sheds new light on differences among the biomes in terms of ordinal pattern entropy, for both small and large fluctuations. It is found that for all biomes large fluctuations in fire temporal series show higher complexity level than small fluctuations, small fluctuations being more predictable as indicated by lower entropy values. Moreover, for fire sensitive biomes Amazon and Atlantic Forest the three dimensional complexity entropy curves in the complexity-entropy-scale causality box (CESCB) are rather similar (indicating similar dynamics for both small and large fluctuations), as well as for fire dependent biomes Pantanal and Pampa, while fire independent Caatinga biome shows different, distinct behavior. Finally, the similarity in lower parts of the CESCB curves (corresponding to small fluctuations) between Amazon and Cerrado reflects the changes of fire dynamics due to human activities.

Research on a Fault Diagnosis Method for Crankshafts Based on Improved Multi-Scale Permutation Entropy

As the crucial part of a transmission assembly, the monitoring of the status of the crankshaft is essential for the normal working of a reciprocating machinery system. In consideration of the interaction between crankshaft system components, the fault vibration feature is typically non-stationary and nonlinear, and the single-scale feature extraction method cannot adequately assess the fault features, therefore a novel impact feature extraction method based on genetic algorithms to optimize multi-scale permutation entropy is proposed. Compared with other traditional feature extraction methods, the proposed method illustrates good robustness and high adaptability in the signal processing of crankshaft vibrations. Firstly, the improved complete ensemble empirical mode decomposition with adaptive noise (ICEEMDAN) method is developed on the signal to obtain several intrinsic mode function (IMF) components, and the IMF components with a large kurtosis are selected for array reorganization. Then, the parameters of multi-scale permutation entropy (MPE) are optimized based on genetic algorithm (GA), the multi-scale permutation entropy is calculated and the feature vector set is constructed. The feature vector set is input into the support vector machine (SVM) and optimized by a particle swarm optimization (PSO) model for training and final pattern recognition, where the Variational Mode Decomposition(VMD)-GA-MPE with a PSO-SVM recognition model and the ICEEMDAN-MPE with PSO-SVM recognition model without GA optimization are constructed for a comparison with the proposed method. The research result illustrates that the proposed method, which inputs the genetic algorithm optimized multi-scale permutation entropy extracted from the ICEEMDAN decomposition into the PSO-SVM, performs well in impact feature extraction and the pattern recognition of crankshaft vibrations.

A novel process control method for remote calibration based on improved multiscale permutation entropy and dynamic Hotelling’s T2 control chart

Remote calibration (RC) is a new promising technology for electric power instrument calibration. However, due to the inevitable impact of external environmental changes and internal insulation aging during the transportation, installation, and measurement processes involved in RC, the metrological performance of relevant instruments may deteriorate. Therefore, quality control of electric power instruments during RC is of great significance. In this paper, a novel process control method for RC is proposed. First, from the physical characteristic perspective, an improved multiscale permutation entropy (IMPE) algorithm is designed to detect the complexity change point of the instrument system. Second, from the statistical characteristic perspective, a dynamic multivariable Hotelling’s T2 control chart (DMHTCC) is developed to detect the outliers within a time series measurement signal. Finally, a fusion scheme of IMPE and DMHTCC is presented to promote the validity and reliability of process control. The effectiveness of the proposed approach and its superiority over some traditional process control techniques is demonstrated through both simulative and experimental case studies.