Multiscale Entropy Research Articles

A Noise Reduction Method for Signal Reconstruction and Error Compensation of a Maglev Gyroscope Under Persistent External Interference

To eliminate the noise interference caused by continuous external environmental disturbances on the rotor signals of a maglev gyroscope, this study proposes a noise reduction method that integrates an adaptive particle swarm optimization variational modal decomposition algorithm with a strategy for error compensation of the trend term in reconstructed signals, significantly improving the azimuth measurement accuracy of the gyroscope torque sensor. The optimal parameters for the variational modal decomposition algorithm were determined using the adaptive particle swarm optimization algorithm, allowing for the accurate decomposition of noisy rotor signals. Additionally, using multi-scale permutation entropy as a criterion for discriminant, the signal components were filtered and summed to obtain the denoised reconstructed signal. Furthermore, an empirical mode decomposition algorithm was employed to extract the trend term of the reconstructed signal, which was then used to compensate for the errors in the reconstructed signal, achieving significant noise reduction. On-site experiments were conducted on the high-precision GNSS baseline of the Xianyang Yuan Tunnel in the second phase of the project to divert water from the Han River to the Wei River, where this method was applied to process and analyze multiple sets of rotor signals. The experimental results show that this method effectively suppresses continuous external environmental interference, reducing the average standard deviation of the compensated signals by 46.10% and the average measurement error of the north azimuth by 45.63%. Its noise reduction performance surpasses that of the other four algorithms.

A Feature Extraction Method of Ship Underwater Noise Using Enhanced Peak Cross-Correlation Empirical Mode Decomposition Method and Multi-Scale Permutation Entropy

A feature extraction method based on the combination of improved empirical modal decomposition (IEMD) and multi-scale permutation entropy (MPE) is proposed to address the problem of inaccurate recognition and classification of ship noise signals under complex environmental conditions. In order to eliminate the end effects, this paper proposes an extended model based on the principle of peak cross-correlation for improved empirical modal decomposition (EMD). In this paper, the IEMD method is used to decompose three ship underwater noise signals to extract the MPE features of the highest order intrinsic modal function (IMF) of energy. The results show that the IEMD-MPE method performs well in extracting the feature information of the signals and has a strong discriminative ability. Compared with the IEMD-aligned entropy (IEMD-PE) method, which describes the signals only at a single scale, the IEMD-MPE method achieves an improvement in the minimum difference distance ranging from 101.36% to 212.98%. In addition, two sets of highly similar ship propulsion noise signals were applied to validate the IEMD-MPE method, and the minimum differences of the experimental results were 0.0814 and 0.0057 entropy units, which verified the validity and generality of the method. This study provides theoretical support for the development of ship target recognition technology for propulsion.

Modified multiscale Renyi distribution entropy for short-term heart rate variability analysis.

Multiscale sample entropy (MSE) is a prevalent complexity metric to characterize a time series and has been extensively applied to the physiological signal analysis. However, for a short-term time series, the likelihood of identifying comparable subsequences decreases, leading to higher variability in the Sample Entropy (SampEn) calculation. Additionally, as the scale factor increases in the MSE calculation, the coarse-graining process further shortens the time series. Consequently, each newly generated time series at a larger scale consists of fewer data points, potentially resulting in unreliable or undefined entropy values, particularly at higher scales. To overcome the shortcoming, a modified multiscale Renyi distribution entropy (MMRDis) was proposed in our present work. The MMRDis method uses a moving-averaging procedure to acquire a family of time series, each of which quantify the dynamic behaviors of the short-term time series over the multiple temporal scales. Then, MMRDis is constructed for the original and the coarse-grained time series. The MMRDis method demonstrated superior computational stability on simulated Gaussian white and 1/f noise time series, effectively avoiding undefined measurements in short-term time series. Analysis of short-term heart rate variability (HRV) signals from healthy elderly individuals, healthy young people, and subjects with congestive heart failure and atrial fibrillation revealed that MMRDis complexity measurement values decreased with aging and disease. Additionally, MMRDis exhibited better distinction capability for short-term HRV physiological/pathological signals compared to several recently proposed complexity metrics. MMRDis was a promising measurement for screening cardiovascular condition within a short time.

A novel epilepsy detection approach using intrinsic multiscale entropy analysis and DSEAM-enhanced 1D-ResNets

Abstract Epilepsy, a prevalent neurological disorder, typically requires a complex diagnostic process involving medical history inquiry, physical examination, head computed tomography, and electroencephalogram (EEG) visual examination. Among the existing epilepsy automated detection algorithms, machine learning methods require manual feature selection. Most of deep learning algorithms for automatic detection have high complexity and computational complexity. To address this issue, this study proposes a high-precision, robust, and low computational epilepsy automatic detection algorithm based on EEG signal processing. The study utilizes ensemble empirical mode decomposition to preprocess the original EEG signal, breaking it down into intrinsic mode functions (IMFs) across various frequency bands. These IMFs contain information about epilepsy occurrence within the signal at different frequency bands. To enhance computational efficiency and reduce data dimension, the refined composite multiscale dispersion entropy of each IMF is further computed at different scales, referred to as intrinsic multiscale entropy (IME) analysis. IME analysis consolidates epilepsy occurrence information from EEG signals across different frequency bands and scales, linking entropy values to generate feature vectors. Drawing inspiration from successful deep residual networks and Squeeze-and-Excitation networks, the study introduces a double Squeeze-and-Excitation attention module enhanced one-dimensional residual network to classify one-dimensional feature vectors. The proposed method was tested on epilepsy dataset from University of Bonn, and the results demonstrated superior classification performance. In the experiment, the distinction between normal and epileptic EEG signals achieved 100% accuracy rate, while distinguishing between normal, epileptic interval, and epileptic EEG signals achieved accuracy rate of 99.41%.

Generalized Gaussian Distribution Improved Permutation Entropy: A New Measure for Complex Time Series Analysis.

To enhance the performance of entropy algorithms in analyzing complex time series, generalized Gaussian distribution improved permutation entropy (GGDIPE) and its multiscale variant (MGGDIPE) are proposed in this paper. First, the generalized Gaussian distribution cumulative distribution function is employed for data normalization to enhance the algorithm's applicability across time series with diverse distributions. The algorithm further processes the normalized data using improved permutation entropy, which maintains both the absolute magnitude and temporal correlations of the signals, overcoming the equal value issue found in traditional permutation entropy (PE). Simulation results indicate that GGDIPE is less sensitive to parameter variations, exhibits strong noise resistance, accurately reveals the dynamic behavior of chaotic systems, and operates significantly faster than PE. Real-world data analysis shows that MGGDIPE provides markedly better separability for RR interval signals, EEG signals, bearing fault signals, and underwater acoustic signals compared to multiscale PE (MPE) and multiscale dispersion entropy (MDE). Notably, in underwater target recognition tasks, MGGDIPE achieves a classification accuracy of 97.5% across four types of acoustic signals, substantially surpassing the performance of MDE (70.5%) and MPE (62.5%). Thus, the proposed method demonstrates exceptional capability in processing complex time series.

Gearbox Fault Diagnosis Based on Adaptive Variational Mode Decomposition-Stationary Wavelet Transform and Ensemble Refined Composite Multiscale Fluctuation Dispersion Entropy.

Existing gearbox fault diagnosis methods are prone to noise interference and cannot extract comprehensive fault signals, leading to misdiagnosis or missed diagnosis. This paper proposes a method for gearbox fault diagnosis based on adaptive variational mode decomposition-stationary wavelet transform (AVMD-SWT) and ensemble refined composite multiscale fluctuation dispersion entropy (ERCMFDE). Initially, the kurtosis coefficient and autocorrelation coefficient are presented, and the Intrinsic Mode Functions are denoised through the application of AVMD-SWT. Secondly, the coarse-grained processing method of composite multiscale fluctuation dispersion entropy is extended to encompass three additional approaches: first-order central moment, second-order central moment, and third-order central moment. This enables the comprehensive extraction of feature information from the time series, thereby facilitating the formation of an initial hybrid feature set. Subsequently, recursive feature elimination (RFE) is employed for feature selection. Ultimately, the outcomes of the faults diagnoses are derived through the utilization of a Support Vector Machine with a Sparrow Search Algorithm (SSA-SVM), with the actual faults data collection and analysis conducted on an experimental platform for gearbox fault diagnosis. The experiments demonstrate that the method can accurately identify gearbox faults and achieve a high diagnostic accuracy of 98.78%.

Switch monitoring algorithm for 220 kV terminal substation startup process based on multi time scale graph model

The switch monitoring in the startup process of 220 kV terminal substation is a dynamic and changeable process, which has obvious characteristics of diversified scales in time, so as to solve the problem of switch monitoring under multivariable and multi-scale interference. The switch monitoring framework for the startup process of 220KV terminal substation is built. The process layer uses the graphic configuration software and combines the internal and external models of the substation to build the objective function for the generation of the substation graphic model. The particle swarm optimization algorithm is used to solve it to generate the optimal substation graphic model. Through the online monitoring device, the signals such as sensors, normally open and normally closed contacts are collected in real time, and the status of the switchgear is obtained through processing, the reduced half trapezoidal cloud model multivariable multi-scale sample entropy similarity tolerance criterion is used for softening treatment, and the multivariable multi-scale cloud sample entropy is determined to achieve the extraction of multiple time scale switch fault feature vectors, which are used as the input of the SE-DSCNN fault diagnosis model. Combined with the substation diagram, the switch fault identification during the startup process of the substation is realized, and the fault switch position is located. The experimental results show that the algorithm can accurately generate the substation model, which includes all the equipment in the substation, accurately describes the connection relationship and operation status of the equipment, and has high accuracy, integrity and aesthetics; The algorithm can effectively extract and analyze the MMCES entropy characteristics of different types of switch faults; The algorithm can realize the switch monitoring during the startup of substation C, and determine the fault switch and fault location.

Resting-State Electroencephalogram Complexity Is Associated With Oral Ketamine Treatment Response: A Bayesian Analysis of Lempel-Ziv Complexity and Multiscale Entropy.

Subanesthetic doses of ketamine are a promising novel treatment for suicidality; however, the evidence for predictive biomarkers is sparse. Recently, measures of complexity, including Lempel-Ziv complexity (LZC) and multiscale entropy (MSE), have been implicated in ketamine's therapeutic action. We evaluated electroencephalogram (EEG)-derived LZC and MSE differences between responders and nonresponders to oral ketamine treatment. A total of 31 participants received six single, weekly (titrated) doses of oral racemic ketamine (0.5-3mg/kg) and underwent EEG scans at baseline (Week 0), post-treatment (Week 6), and follow-up (Week 10). Resting-state (eyes closed and open) recordings were processed in EEGLAB, and complexity metrics were extracted using the Neurokit2 package. Participants were designated responders or nonresponders by clinical response (Beck Suicide Scale [BSS] score reduction of ≥50% from baseline to the respective timepoint or score ≤6) and then compared in terms of complexity across resting-state conditions and time. Employing a Bayesian mixed effects model, we found strong evidence that LZC was higher in the eyes-open compared to eyes-closed condition, as were MSE scales 1-3. At a global level, responders displayed elevated eyes-open baseline complexity compared to nonresponders, with these values decreasing from baseline to post-treatment (Week 6) in responders only. Exploratory analyses revealed that the elevated baseline eyes-open LZC in responders was spatially localized to the left frontal lobe (F1, AF3, FC1, and F3). EEG-complexity metrics may be sensitive biomarkers for evaluating and predicting oral ketamine treatment response, with the left prefrontal cortex bein a possible treatment response region.

Complexity of electrodermal activity to mental stress is changed during adolescent age-period.

Complexity characterizes behaviour of all physiological systems whose components interact in multiple ways usually quantified by entropy techniques. However, complexity analysis regarding electrodermal activity (EDA)-related sympathetic cholinergic nervous system is rare. Thus, we aimed to study EDA dynamics complexity changes from aspect of various embedding dimensions (m) and timescales (τ) (sample entropy (SampEn) with m ∈ <2,7>, and multiscale entropy (MSE) in τ ∈ <1,20>) in association with traditionally used EDA indices (skin conductance level (SCL) and non-specific skin conductance responses (NS.SCRs)) to mental stress (mental arithmetic test - MAT) in healthy participants at critical adolescent age. The cohort (total group) consisted of 60 adolescents (17.5 ± 0.5 yrs) divided into three groups: Group-1: early (13.1 ± 0.3 yrs), Group-2: middle (16.6 ± 0.2 yrs) and Group-3: late (22.9 ± 0.1 yrs) adolescence. SampEn (m > 2) and MSE (for all τ) were significantly higher during MAT than baseline in total group and Group-2 (p < 0.05). Index MSE for all τ was significantly higher during MAT than baseline in total group, and Group-2; for τ ∈ <2,13> in Group-1 (p < 0.05). Additionally, while SCL was significantly higher during MAT than baseline in all groups, NS.SCRs was lower during stress only in Group-3 (p < 0.05). In conclusion, this study revealed distinct EDA complexity characteristics in individual examined groups indicating importance of complexity evaluation in stress-related sympathetic regulatory mechanisms within individual adolescent age ranges.

Combined ICEEMDAN and CMPE noise reduction algorithm for dynamic deformation data

Abstract To effectively segregate complex noise components within Global Navigation Satellite System (GNSS) deformation data, a noise reduction algorithm is proposed, which integrates the improved complete ensemble empirical mode decomposition with adaptive noise (ICEEMDAN) and composite multiscale permutation entropy (CMPE). The algorithm initially employs the ICEEMDAN algorithm to decompose the observed data into a series of intrinsic modal functions (IMFs), and then develops a CMPE-based effective coefficient sieving method to accurately classify the IMF that contains effective information components, thus obtaining the GNSS deformation data after noise separation. The processing outcomes of both simulated and empirically measured data reveal that the adopted approach surpasses a solitary filtering algorithm in achieving optimal noise mitigation. Notably, the application of this method leads to a diminution of the root mean square error (RMSE) metric associated with the sequence of elevation coordinates, resulting in a reduction to 2.837 mm post-denoising and an enhancement of 32.9% in overall precision.

Implication of heart rhythm complexity in predicting long-term outcomes in pulmonary hypertension

Pulmonary hypertension (PH) is a serious disease, however simple tools to predict outcomes are lacking. In our previous investigation, we found that heart rate variability (HRV) and heart rhythm complexity (HRC) were associated with the detection and severity of PH, however their association with PH mortality remains unclear.The aim of this study was to investigate these metrics as a tool for determining long-term outcomes in PH patients. We enrolled 74 Asian PH patients with WHO PH group 1 or 4 at a single hospital in Taiwan between March 2012 and June 2018. After a median follow-up duration of 58 months (to January 2023), 22 patients had died. The patients who died had a significantly lower lean body mass index (BMI), impaired renal function, higher N-terminal pro B-type natriuretic peptide (NT-proBNP) level, lower very low-frequency (VLF), lower short-term detrended fluctuation analysis α1 (DFAα1), and lower multiscale entropy scale 5 value. In multivariable analysis, BMI, VLF and multiscale entropy scale 5 were significantly associated with survival. The best cut-off VLF and scale 5 values were 115.13 and 0.738, respectively. We then categorized the study population into three groups: both elevated VLF/scale 5 (group 1), either depressed VLF or depressed scale 5 (group 2), and both depressed VLF/scale 5 (group 3). The results showed that group 1 had the best outcomes, whereas group 3 had the worst survival (P < 0.001).Combining HRV and HRC metrics appears to be a good non-invasive tool to predict the long-term outcomes of patients with PH.

A time scale measurement method for dynamic temporal networks

Selecting an appropriate time scale is paramount for analysing dynamic temporal networks. This paper presents a systematic and data-driven framework for selecting suitable time scales for analysing these networks. The concept of multi-scale entropy is initially introduced to describe the complexity and stochasticity of time series and to determine a suitable range of time scales. The optimal time scale is defined and calculated, and a comprehensive assessment is conducted regarding node characteristics, edge characteristics, and the overall structure. Finally, the effectiveness of the proposed approach is verified by identifying pivotal nodes and using Susceptible-Infected-Recovered (SIR) dynamics propagation modelling, utilising three genuine traffic datasets as case studies. In addition, results from the empirical study on the Email-Eu-core network indicate that the proposed approach applies to social networks. This method enhances the scientific rigour, efficiency and practicality of dynamic temporal network analysis while providing novel conceptual frameworks and analytical tools for the field.

Metacontrol Regulates Creative Thinking: An EEG Complexity Analysis Based on Multiscale Entropy.

Previous studies have shown that creative thinking is associated with metacontrol, but its neural basis is unknown. The present study explored the neural basis of both by assessing EEG complexity through multiscale entropy. Subjects were engaged in a metacontrol task and an Alternative Uses Task, grouped according to task performance, and the EEG was analysed by multiscale entropy. The results showed that EEG complexity was significantly higher in the high-metacontrol and high-creativity groups than in the low-metacontrol and low-creativity groups, respectively, at high time scales. The metacontrol adaptability score and multipurpose task score were significantly and positively correlated with the EEG complexity at multiple electrode sites. It suggests that metacontrol and creativity are dependent on the activation of long-duration neural networks.

Denoising diffusion probabilistic model enhanced tool condition monitoring method under imbalanced conditions

Abstract In recent years, tool condition monitoring (TCM) based on deep learning has been widely considered and achieved remarkable success. However, these methods typically require relatively large training samples to produce significant results, which are both imbalanced and rather troublesome to obtain in the practical application of TCM. To address this issue, a novel TCM method combined with multiscale permutation entropy (MPE), denoising diffusion probabilistic model (DDPM) and a residual network (ResNet) is proposed under conditions of sample imbalance. First, the one-dimensional sensing signal data is converted to a grayscale recurrence plot (RP) by minimizing the MPE of the signals in each channel. Second, combine and splice these grayscale RPs from different channels in each sample into color RPs. After that, the generated RP images using DDPM are added to the imbalanced dataset to augment the data to achieve a balanced state of the dataset. Finally, the balanced mixed data set of real and fake samples is input into a ResNet for recognition and monitoring tool conditions. TCM experiments are conducted to verify the performance of the proposed method with imbalanced dataset, and the results of experimental investigation demonstrate that the accuracy of the proposed method improved by 2%–18.8% compared to that of the other four sample augmentation methods using ResNet18 when the imbalance rate is 1:200.

Complexity and synchronization of carbon and new energy markets based on multiscale entropy

AbstractThis study provides a comprehensive analysis of the dynamic evolution of complexity and synchrony between the carbon market and the new energy market from 2019 to 2023, employing multiscale sample entropy and cross‐sample entropy methods. The study shows that, both in the long term and short term, the complexity of the new energy market consistently exceeds that of the carbon market. Additionally, the carbon and new energy markets exhibit higher synchronization on smaller scales. As the time scale increases, interactions between the carbon and new energy markets become more complex and diverse. According to these findings, it is recommended that policymakers improve the transparency of the new energy market, optimize the operational mechanisms of the carbon market, and ensure that carbon reduction strategies and new energy policies are mutually coordinated. This holds significant importance for both carbon reduction targets and the development of new energy.

Real-time EEG-based detection of driving fatigue using a novel semi-dry electrode with self-replenishment of conductive fluid

A novel semi-dry electrode that can realize self-replenishment of conductive liquid is proposed in this study. Driving fatigue is detected by extracting the refined composite multiscale fluctuation dispersion entropy (RCMFDE) features in electroencephalogram (EEG) signals collected by this electrode. The results show that the new semi-dry electrode can automatically complete the conductive fluid supplement according to its own humidity conditions, which not only notably improves the effective working time, but also significantly reduces the skin impedance. By comparing with the classical entropy algorithms, the computational speed and the stability of the RCMFDE method are Substantially enhanced.

A New Algorithm for Predicting Dam Deformation Using Grey Wolf-Optimized Variational Mode Long Short-Term Neural Network

To solve the problems of difficult-to-model parameter selections, useful-signal extraction and improper-signal decomposition in nonlinear, non-stationary dam displacement time series prediction methods, we propose a new predictive model for grey wolf optimization and variational mode decomposition and long short-term memory (GVLSTM). Firstly, we used the grey wolf optimization (GWO) algorithm to optimize the parameters of variable mode decomposition (VMD), obtaining the optimal parameter combination. Secondly, we used multiscale permutation entropy (MPE) as a standard to select signal screening, determining and recon-structing the effective modal components. Finally, the long short-term memory neural network (LSTM) was used to learn the dam deformation characteristics. The result shows that the GVLSTM model can effectively reduce the estimation deviation of the prediction model. Compared with VMDGRU and VMDANN, the average RMSE and MAE value of each station is increased by 19.11%~28.58% and 27.66%~29.63%, respectively. We used determination (R2) coefficient to judge the performance of the prediction model, and the value of R2 was 0.95~0.97, indicating that our method has good performance in predicting dam deformation. The proposed method has outstanding advantages of high accuracy, reliability, and stability for dam deformation prediction.

Bearing fault diagnosis based on novel hierarchical multiscale dispersion entropy in corresponding color block images

Abstract The rolling bearing is a critical component of mechanical equipment, and its failure can lead to serious consequences. In order to effectively extract fault features of rolling bearings and improve fault diagnosis performance, a fault diagnosis framework based on hierarchical multiscale dispersion entropy (HMDE) and improved histogram of oriented gradient (HOG) is proposed by combining entropy method with image recognition method. Firstly, the original vibration signal is subjected to moving average filtering to eliminate sudden noise and outliers. Then, HMDE is used for the extraction of fault features. HMDE can evaluate the complexity of the signal at different levels and scales, thereby extracting more comprehensive information. Based on HMDE, entropy color block (ECB) images are generated and the improved HOG of the images are extracted. Finally, K-nearest neighbor (KNN) is used to classify the improved HOG features, completing the recognition of different working states of rolling bearings. The validity and robustness of the proposed fault diagnosis framework are proved by the verification experiments on the public bearing datasets of Case Western Reserve University and Southeast University.

Method and Application of Spillway Radial Gate Vibration Signal Denoising on Multiverse Optimization Algorithm-Optimized Variational Mode Decomposition Combined with Wavelet Threshold Denoising

To address the noise issue in the measured vibration signals of spillway radial gate discharge, this paper utilizes the Multiverse Optimization Algorithm (MVO) to optimize the number of decomposition modes (K) and the penalty factor (α) in Variational Mode Decomposition (VMD). This approach ensures improved efficiency of VMD decomposition while maintaining accuracy. Subsequently, the obtained Intrinsic Mode Functions (IMFs) from VMD decomposition are classified based on Multi-scale Permutation Entropy (MPE). IMFs are divided into pure components and noisy components; the noisy components are processed with Wavelet Threshold Denoising (WTD), while the pure components are overlaid and reconstructed to obtain the denoised vibration signal of the gate. Comprehensive comparisons involving artificial signal simulations, gate flow-induced vibration model tests, and numerical simulations lead to the following conclusions: compared to other algorithms, the proposed combined denoising method (MVO-VMD-MPE-WTD) achieves the highest signal-to-noise ratio (SNR) in both the frequency and time domains for artificial signals, while yielding the lowest mean square error (MSE). In the gate flow-induced vibration model tests, the method significantly reduces noise in the vibration signals and effectively preserves characteristic information. The error in preserving characteristic information across model tests and numerical simulations is kept below 1%. Furthermore, compared to other optimization algorithms, the MVO demonstrates higher computational efficiency. The parameter-optimized combined denoising method proposed in this study provides insights into denoising measured vibration signals of hydraulic spillway radial gates and other drainage structures, and it opens possibilities for exploring more efficient optimization algorithms for achieving online monitoring in the future.

Dynamically adjusted normalized multi-scale symbolic dynamic entropy for fault diagnosis of rotating machinery in strong noise

Dynamically adjusted normalized multi-scale symbolic dynamic entropy for fault diagnosis of rotating machinery in strong noise