Research on an Adaptive Variational Mode Decomposition with Double Thresholds for Feature Extraction

The variational mode decomposition (VMD) method for signal decomposition is severely affected by the number of components of the VMD method. In order to determine the decomposition modal number, K, in the VMD method, a new center frequency method of the multi-threshold is proposed in this paper. Then, an improved VMD (MTCFVMD) algorithm based on the center frequency method of the multi-threshold is obtained to decompose the vibration signal into a series of intrinsic modal functions (IMFs). The Hilbert transformation is used to calculate the envelope signal of each IMF component, and the maximum frequency value of the power spectral density is obtained in order to effectively and accurately extract the fault characteristic frequency and realize the fault diagnosis. The rolling element vibration data of the motor bearing is used to test the effectiveness of proposed methods. The experiment results show that the center frequency method of the multi-threshold can effectively determine the number, K, of decomposed modes. The proposed fault diagnosis method based on MTCFVMD and Hilbert transformation can effectively and accurately extract the fault characteristic frequency, rotation frequency, and frequency doubling, and can obtain higher diagnostic accuracy.

Variational mode decomposition has been widely applied to machinery fault diagnosis during these years. However, it remains difficult to set proper hyperparameters for the variational mode decomposition, including number of decomposed modes, initial center frequencies, and balance parameter. Moreover, the low efficiency of the existing variational mode decomposition methods hinders their applications to practical diagnostic task. This article proposes an adaptive and efficient variational mode decomposition method after thoroughly investigating its convergence property characteristic. A convergent tendency phenomenon is discovered and is explained mathematically for the first time. Motivated by the convergent tendency phenomenon, the proposed method rapidly and adaptively determines the number and the optimal initial center frequencies of signal latent modes with the guidance of the convergent tendencies of the initial center frequencies changing from small to large. In the proposed method, the number of decomposed modes and initial center frequencies are not hyperparameters that require to be set in advance any more, but are parameters learned from the analyzed signals. The determined parameters enable efficient extraction of the main latent modes contained in the analyzed signals. Therefore, the proposed variational mode decomposition method represents a major improvement in parameter adaption and decomposition efficiency over the existing variational mode decomposition methods. In the application for bearing fault diagnosis, the faulty modes are selected adaptively and the corresponding balance parameters are further optimized efficiently. Two experimental cases validate the proposed method and its superiority over the existing variational mode decomposition methods and the classical fast spectral kurtosis in bearing fault diagnosis.

This paper proposes a novel method to improve the variational mode decomposition (VMD) method and to automatically acquire the sensitive intrinsic mode function (IMF). First, since fault signals are impulsive and periodic, a weighted autocorrelative function maximum (AFM) indicator is constructed based on the Gini index and autocorrelation function to serve as the optimization objective function. The mode number K and the penalty parameter α of VMD are automatically obtained through an optimal parameter searching process underpinned by the improved particle swarm optimization (PSO) algorithm with a variety of inertia weights. This improvement solves one of the major drawbacks of the conventional VMD method, that is, the need to manually set parameters. Then, an optimal IMF automatic selecting process is performed for single-failure faults and compound faults, according to the principles of the maximum weighted AFM indicator and maximum spectrum peak ratio (SPR), respectively. The sensitive IMFs are then subjected to an envelope demodulation analysis to obtain the fault characteristic frequency. The results of simulations and experiments show that the proposed method can effectively identify fault characteristics early, especially compound faults, demonstrating great potential for real-world applications.

Machines learning models have recently been proposed for predicting rivers water temperature (Tw) using only air temperature (Ta). The proposed models relied on a nonlinear relationship between the Tw and Ta and they have proven to be robust modelling tools. The main motivation for this study was to evaluate how the variational mode decomposition (VMD) contributed to the improvement of machines learning performances for river Tw modelling. Measured data collected at five stations located in Poland from 1987 to 2014 were acquired and used for the analysis. Six machines learning models were used and compared namely, K-nearest neighbor's regression (KNNR), least square support vector machine (LSSVM), generalized regression neural network (GRNN), cascade correlation artificial neural networks (CCNN), relevance vector machine (RVM), and locally weighted polynomials regression (LWPR). The six models were developed according to three scenarios. First, the models were calibrated using only the Ta as input and obtained results show that the models were able to predict consistently water temperature, showing a high determination coefficient (R2) and Nash-Sutcliffe efficiency (NSE) with values near or above 0.910 and 0.915, respectively, and in overall the six models worked equally without clear superiority of one above another. Second, the air temperature was combined with the periodicity (i.e., day, month and year number) as input variable and a significant improvement was achieved. Both models show their ability to accurately predict river Tw with an overall accuracy of 0.956 for R2 and 0.955 for NSE values, but the LSSVM2 have some advantages such as a small errors metrics, and high fitting capabilities and it slightly surpasses the others models. Thirdly, air temperature was decomposed into several intrinsic mode functions (IMF) using the VMD method and the performances of the models were evaluated. The VMD parameters appeared to cause much influence on the prediction accuracy, exhibiting an improvement of about 40.50% and 39.12% in terms of RMSE and MAE between the first and the third scenarios, however, some models, i.e., GRNN and KNNR have not benefited from the VMD. This research has demonstrated the high capability of the VMD algorithm as a preprocessing approach in improving the accuracies of the machine learning models for river water temperature prediction.

This paper proposes a study on gearbox fault feature extraction of wind turbine under variable speed condition using improved adaptive variational mode decomposition (VMD). Frequent changes in wind speed and critical noise interference make the vibration signal exhibit non-stationary characteristics. Although computational order tracking can transform non-stationary signals in the time domain into stationary angular signals, and then extract fault features from order spectrum obtained by FFT, the obtained order spectrum is liable to be polluted by noise or to appear order aliasing. To avoid order aliasing and eliminate noise interference, the original non-stationary signal is firstly processed using the improved adaptive VMD method called adaptive differential evolution – VMD (ADE-VMD). ADE-VMD can not only utilise the advantages of traditional VMD but also adaptively select narrow-band intrinsic mode function (NBIMF) to construct the reconstructed signal with less noise and without order aliasing. In the experiment, we compared the ADE-VMD method with other VMD methods such as GA-VMD, PSO-VMD and DE-VMD, and the results showed that ADE-VMD has excellent adaptive processing ability, and its convergence and optimisation speed are more remarkable. ADE-VMD can effectively filter the noise inference and avoid the order aliasing, so it is well suitable for fault feature extraction under variable speed.

Recently, magnetocardiography (MCG) has attracted increasing attention as a non-invasive and non-contact technique for detecting electrocardioelectric functions. However, the severe background noise makes it difficult to extract information. Variational Mode Decomposition (VMD), which is an entirely non-recursive model, is used to decompose the non-stationary signal into the intrinsic mode functions (IMFs). Traditional VMD algorithms cannot control the bandwidth of each IMF, whose quadratic penalty lacks adaptivity. As a result, baseline drift noise is still present or medical information is lost. In this paper, to overcome the unadaptable quadratic penalty problem, an improved VMD model via correlation coefficient and new update formulas are proposed to decompose MCG signals. To improve the denoising precision, this algorithm is combined with the interval threshold algorithm. First, the correlation coefficient is calculated, to determine quadratic penalty, in order to extract the first IMF made up of baseline drift. Then, the new update formulas derived from the variance that describes the noise level are used, to perform decomposition on the rest signal. Finally, the Interval thresholding algorithm is performed on each IMF. Theoretical analysis and experimental results show that this algorithm can effectively improve the output signal-to-noise ratio and has superior performance.

The research about online monitoring and leakage automatic location of water distribution networks (WDN) has a wide range of applications that include water resource protection, monitoring, and allocation. Variational mode decomposition (VMD) and cross-correlation (CC) based leakage location is a popular and effective method in WDN. However, the value of K intrinsic mode functions (IMFs) based on VMD decomposition needs to be determined artificially, which affects the separation effect of signal frequency band characteristics directly. Hence, this work proposes an adaptive method to determine the parameter K of leakage vibration signal’s IMFs, which will be applied to automatic leakage location in WDN. Firstly, the number of saddle points in the frequency domain envelope of the sampled signal in different step sizes is calculated. The parameter K is determined according to the curvature change of the number of saddle points and the sampled signal. Finally, the selective IMFs are reconstituted into a new signal, which can determine a leak position using CC based time-delay estimation (TDE). To verify the effectiveness of the proposed algorithm, the different methods based on EMD and Fast ICA are compared. The experimental results demonstrate that the proposed parameter K value adaptive VMD (KVA-VMD) decomposition method is more suitable for leakage location in WDN.

Aiming at the problems of difficulty in extracting vibration signal features and low recognition accuracy of bearing fault, a new fault diagnosis method based on ISSA-VMD and IMSE was proposed. This method leverages an improved sparrow search algorithm (ISSA) to optimize the variational mode decomposition (VMD) method and employs an improved multiscale sample entropy (IMSE) for fault feature extraction. Initially, the ISSA algorithm optimizes two critical parameters of the VMD method: the number of modes K and the penalty factor α, to obtain the optimal parameter combination [K, α]. The optimized VMD method is then used to decompose the bearing vibration signals for signal reconstruction analysis. Subsequently, the IMSE entropy algorithm is applied to the reconstructed signals to extract fault features, resulting in the required fault feature vector set. Finally, this extracted fault feature vector set is input into a multi-kernel extreme learning machine model for fault classification and diagnosis. The results show that the identification accuracy of the fault diagnosis of reciprocating compressor plain bearings and automobile rolling bearings is significantly improved, and the method has better fault feature extraction effect.

An optimized VMD method and its applications in bearing fault diagnosis

The variational mode decomposition (VMD) method has been widely applied in the field of mechanical fault diagnosis as an excellent non-recursive signal processing tool. The performance of VMD depends on its inherent prior parameters. Searching for the key parameters of VMD using intelligent optimization algorithms poses challenges for the internal essence and fitness function selection of intelligent optimization algorithm. Moreover, the computational complexity of optimization is high. Meanwhile, such methods are not competitive in evaluating orthogonality between intrinsic mode functions and the reconstruction error of the signal as a joint indictor for the termination of decomposition. Therefore, this paper proposes a new auto-regulative sparse variational mode decomposition method (ASparse–VMD) to achieve accurate feature extraction. The regularization term of the VMD handles sparsification by constructing an L2-norm with a damping coefficient ε, and mode number K is set adaptively in a recursive manner to ensure appropriateness. The penalty parameter α is dynamically selected according to the number of modes and sampling frequency. The update step τ of the VMD algorithm is set using the signal-to-noise ratio to ensure the singleness and orthogonality of the modal components and suppress mode aliasing. The experimental results of the simulation signal and measured signal demonstrate the effectiveness of the proposed strategies for improving the inherent defects of VMD. Extensive comparisons with state-of-the-art methods show that the proposed algorithm is more effective and practical for hybrid feature extraction in mechanical faults.

The bearing fault diagnosis is of vital significance in maintaining the safety of rotation machine. Among various fault detection techniques, the diagnosis based on vibration signal is widely applied in monitoring the condition of rotation machine. Variational mode decomposition (VMD) is a novel signal analysis method, which can decompose a multi-component signal into a certain number of band-limited intrinsic mode functions (BLIMFs) nonrecursively. VMD could overcome some problems such as mode mixing, the inference of noise, the determination of wavelet base, which exist in empirical mode decomposition, ensemble empirical mode decomposition, wavelet transform, respectively. However, the empirical selection of the parameters for VMD would affect the result of the decomposition. This paper presents an adaptive VMD method with parameter optimization for detecting the localized faults of rolling bearing. Kurtosis, sensitive to transient impulsive components, is employed as optimization index to evaluate the performance of the VMD. Two parameters in the VMD, namely the number of decomposition modes and data-fidelity constraint, are optimized synchronously based on the kurtosis index through artificial fish swarm algorithm. Executing VMD with the acquired parameters, the optimal BLIMF is obtained. The spectrum analysis of the optimal BLIMF could identify the characteristic frequency caused by the localized crack effectually. The validity of the proposed method is proved by means of a cyclic transient impulse response signal and two experiments with practical vibration signals of rolling bearings. Compared to several existing methods, the proposed method demonstrates reinforced results.

Amyotrophic lateral sclerosis (ALS), also known as motor neuron disease, is a neurological disease that occurs as a result of damage to the nerves in the brain and restriction of muscle movements. Electroencephalography (EEG) is the most common method used in brain imaging to study neurological disorders. Diagnosis of neurological disorders such as ALS, Parkinson's, attention deficit hyperactivity disorder is important in biomedical studies. In recent years, deep learning (DL) models have been started to be applied in the literature for the diagnosis of these diseases. In this study, event-related potentials (ERPs) were obtained from EEG signals obtained as a result of visual stimuli from ALS patients and healthy controls. As a new method, variational mode decomposition (VMD) is applied to the produced ERP signals and the signals are decomposed into subbands. In addition, empirical mode decomposition (EMD), one of the popular decomposition methods in the literature, was also analyzed, and ERP signals were divided into subbands and compared with the VMD method. Subband signals were classified in two stages with the one-dimensional convolutional neural network (1D CNN) model, which is one of the DL techniques proposed in the study. Accuracy, sensitivity, specificity, and F1-Score measurements were obtained using 5- and 10-fold cross-validation to evaluate classifier performance. In the first stage of classification, only VMD and EMD subband signals were used and 92.95% classification accuracy was obtained by the VMD method. In the second stage, VMD, EMD subband signals, and original ERP signals were all classified together with the VMD+ERP model achieving the maximum classification accuracy rate of 90.42%. It is thought that the results of the study will contribute to the diagnosis of similar neurological disorders such as ALS, attention studies based on visual stimuli, and the development of brain-computer interface (BCI) systems using the method applied to the proposed ERP signals.

In this study, a new approach is proposed to detection of Parkinson’s disease by using Electroencephalography (EEG) signals and three different subband signals generated using Variational Mode Decomposition (VMD) method. In the proposed method, EEG signals and subband signals are applied separately to the generated 1D CNN model and the classification results are compared. The classification results showed that the VMD-sub band signals obtained from EEG signals were successful in diagnosing Parkinson’s. The highest classifier accuracy was obtained from second VMD subband data by 98.10%.

Ferromagnetic materials are extensively utilized in industrial settings where the early detection and repair of defects is paramount for ensuring industrial safety. During the enhanced magnetic memory detection of micro-defects, many interference signals appear in the detection signal, which makes it difficult to accurately extract the characteristics of the micro-defect signals, significantly affecting detection effectiveness. When improved complete ensemble empirical mode decomposition with adaptive noise (ICEEMDAN) is employed independently for signal denoising, the noise and feature signals of the transition components are retained or removed. When variational mode decomposition (VMD) is employed independently for signal denoising, the denoising effect is restricted because of the difficulty in determining the penalty factor α and the number of decomposition layers m. To solve these problems, a denoising method for enhanced magnetic memory detection signals based on ICEEMDAN and VMD, called ICEEMDAN–VMD, is proposed in this paper. First, a comprehensive index (CI) combining information entropy (IE) and the correlation coefficient R is proposed, then the signal components obtained by performing decomposition with the ICEEMDAN method are divided into noise-dominant components, transition components, and useful signal components based on the CI. Subsequently, VMD is employed to perform secondary decomposition on the transition components obtained from the ICEEMDAN method and calculate the correlation coefficients. Ultimately, the optimal VMD components and useful signal components obtained by the ICEEMDAN method are selected for signal reconstruction to obtain a denoised signal. To validate the effectiveness of the proposed method, the denoising effects of the ICEEMDAN–VMD, ICEEMDAN, and VMD methods were compared based on the signal-to-noise ratio (SNR) and fuzzy entropy (FE). The comparison indicated that the ICEEMDAN–VMD denoising method significantly enhanced the denoising effect, and the SNRs of the components of the magnetic field signal could be increased by up to 69.426%. The SNR of each gradient component of the magnetic field signal could be improved by up to ten times, and the FEs of the signal components and their corresponding gradient components could be reduced by 24.198%–81.011%, respectively.

According to the non-stationary characteristic of rotating machinery vibration signals of a rotor system with a loose pedestal fault, variational mode decomposition was applied in the pedestal looseness fault diagnosis for such a rotor system. Variational mode decomposition is used to decompose the rotor vibration signal into several stable components. This can achieve the separation of the pedestal looseness fault signal from the background signals, and extract the fault characteristic of a vibration signal from a rotor system with pedestal looseness. Experimental data from a rotor system with pedestal looseness were used to verify the proposed method. The results showed that the stable components of the rotor vibration signal obtained by variational mode decomposition have obvious amplitude modulation characteristics. The components which contain fault information were analyzed by envelope demodulation, which can extract the pedestal looseness fault features of a rotor vibration signal. Therefore, the variational mode decomposition method can be effectively applied to the pedestal looseness fault diagnosis of such a rotor system.