EMD-based Methods Research Articles

Integrating fast iterative filtering and ensemble neural network structure with attention mechanism for carbon price forecasting

Accurate carbon price forecasts are crucial for policymakers and enterprises to understand the dynamics of carbon price fluctuations, enabling them to formulate informed policies and investment strategies. However, due to the non-linear and non-stationary nature of carbon price, traditional models often struggle to achieve high prediction accuracy. To address this challenge, this study proposes a novel integrated prediction framework designed to enhance forecast accuracy. First, the carbon price series is decomposed into a series of smoother subsequences using fast iterative filtering (FIF). Subsequently, an integrated prediction model, AM-TCN-LSTM, is constructed, incorporating the attention mechanism (AM), temporal convolutional networks (TCN), and long short-term memory (LSTM) neural networks. The attention mechanism adaptively captures complex features from multiple factors, while the TCN-LSTM efficiently extracts temporal features from the sequences. Finally, the results from each subsequence are aggregated to generate the final prediction. Five carbon markets in china: Guangdong, Hubei, Shenzhen, Beijing, and Shanghai were selected to verify the validity of the proposed model. Various comparative models and evaluation metrics were employed to assess performance. The results demonstrate that: (1) the TCN-LSTM model achieves higher prediction accuracy compared to single models. (2) FIF is a more effective decomposition method with superior performance compared to EMD-based methods. (3) The proposed model exhibits the highest predictive capability, with MAE values of 0.0964, 0.1403, 1.9476, 2.0848, and 0.5029 for the five carbon markets, significantly outperforming comparison models. (4) The attention mechanism effectively captures the influence of multiple factors on carbon price, particularly within the short-term components.

EMD-based time-frequency analysis methods of audio signals

Using appropriate signal processing tools to analyze time series data accurately is essential for correctly interpreting the underlying processes. Commonly employed methods include kernel-based transforms that utilize base functions and modifications to depict time series data. This paper refers to the analysis of audio data using two such transforms: the Fourier transform and the wavelet transform, both based on assumptions regarding the signal's linearity and stationarity. However, in audio engineering, these assumptions often do not hold as the statistical characteristics of most audio signals vary over time, making them unsuitable for treatment as outputs from a Linear Time-Invariant (LTI) system. Consequently, more recent methods have shifted towards breaking down signals into various modes in an adaptive, data-specific manner, potentially offering benefits over traditional kernel-based methods. Techniques like empirical mode decomposition and Holo-Hilbert Spectral Analysis are examples of this. The effectiveness of these methods was tested through simulations using speech signals for both kernel-based and adaptive decomposition methods, demonstrating that these adaptive methods are effective for analyzing audio data that is both nonstationary and an output of the nonlinear system.

Speech emotion recognition based on improved masking EMD and convolutional recurrent neural network.

Speech emotion recognition (SER) is the key to human-computer emotion interaction. However, the nonlinear characteristics of speech emotion are variable, complex, and subtly changing. Therefore, accurate recognition of emotions from speech remains a challenge. Empirical mode decomposition (EMD), as an effective decomposition method for nonlinear non-stationary signals, has been successfully used to analyze emotional speech signals. However, the mode mixing problem of EMD affects the performance of EMD-based methods for SER. Various improved methods for EMD have been proposed to alleviate the mode mixing problem. These improved methods still suffer from the problems of mode mixing, residual noise, and long computation time, and their main parameters cannot be set adaptively. To overcome these problems, we propose a novel SER framework, named IMEMD-CRNN, based on the combination of an improved version of the masking signal-based EMD (IMEMD) and convolutional recurrent neural network (CRNN). First, IMEMD is proposed to decompose speech. IMEMD is a novel disturbance-assisted EMD method and can determine the parameters of masking signals to the nature of signals. Second, we extract the 43-dimensional time-frequency features that can characterize the emotion from the intrinsic mode functions (IMFs) obtained by IMEMD. Finally, we input these features into a CRNN network to recognize emotions. In the CRNN, 2D convolutional neural networks (CNN) layers are used to capture nonlinear local temporal and frequency information of the emotional speech. Bidirectional gated recurrent units (BiGRU) layers are used to learn the temporal context information further. Experiments on the publicly available TESS dataset and Emo-DB dataset demonstrate the effectiveness of our proposed IMEMD-CRNN framework. The TESS dataset consists of 2,800 utterances containing seven emotions recorded by two native English speakers. The Emo-DB dataset consists of 535 utterances containing seven emotions recorded by ten native German speakers. The proposed IMEMD-CRNN framework achieves a state-of-the-art overall accuracy of 100% for the TESS dataset over seven emotions and 93.54% for the Emo-DB dataset over seven emotions. The IMEMD alleviates the mode mixing and obtains IMFs with less noise and more physical meaning with significantly improved efficiency. Our IMEMD-CRNN framework significantly improves the performance of emotion recognition.

Pattern Recognition of sEMG Recordings on Arm Muscles among Different Training Motions

The paper develops a prediction model based on the surface electromyography (sEMG) signals to recognize the five different arm-related motions. We collect 100 groups of the three-channels signals on Biceps, Triceps, Brachioradialis as the raw data set. Then we extract four features from the data to describe the characteristics of different motions in the data set. In the pre-processing stage, we compare three types of EMD-based filtration methods to denoise signals and choose the most efficient one, then use principle component analysis to reduce the dimension of the data. We use three different methods to classify and predict the processed data for a higher accuracy. The contributions are following: This project successfully automated the pattern recognition of arm-related motions. Moreover, the project generated a valuable data set; Compared performances of EMD, EEMD and CEEMDAN in sEMG signals; Found that FCNN is the most accurate algorithm in this project. Future works include obtain more data from more volunteers to expand the data set.

Separation of Heart and Lung-related Signals in Electrical Impedance Tomography Using Empirical Mode Decomposition.

Electrical impedance tomography (EIT) can be used for continuous monitoring of pulmonary ventilation. However, no proper method has been developed for the separation of pulmonary ventilation and perfusion signals and the measurement of the associated ventilation/ perfusion (V/Q) ratio. Previously, various methods have been used to extract these components; however, these have not been able to effectively separate and validate cardiac- and pulmonary- related images. This study aims at validating and developing a novel method to separate cardiac- and pulmonary- related components based on the EIT simulation field of view and to simultaneously reconstruct the individual images instantly. Our approach combines the advantages of the principal component analysis (PCA) and processes that originally measure EIT data instead of handling a series of EIT images, thus introducing the empirical mode decomposition (EMD). The PCA template functions for cardiacrelated imaging and intrinsic mode functions (IMFs) of EMD for lung-related imaging are then adapted to input signals. The proposed method enables the separation of cardiac- and lung-related components by adjusting the proportion of the key components related to lung imaging, which are the fourth component (PC4) and the first component (IMF1) in PCA- and EMD-based methods, respectively. The preliminary results on the application of the method to real human EIT data revealed the consistently better performance and optimal computation compared with previous methods. This study proposes a novel method for applying EIT to evaluate the best time of V/Q matching on the cardiovascular and respiratory systems; this aspect can be investigated in future research.

Biomedical Signal Denoising Via Permutating, Thresholding and Averaging Noise Components Obtained from Hierarchical Multiresolution Analysis-Based Empirical Mode Decomposition

Biomedical signals are usually contaminated with interfering noise, which may result in misdiagnosis of diseases. Additive white Gaussian noise (AWGN) is a common interfering noise, and much work has been proposed to suppress AWGN. Recently, hierarchical multiresolution analysis-based empirical mode decomposition (EMD) denoising method is proposed and shows potential performance. In order to further improve performance of hierarchical multiresolution analysis-based EMD denoising, this paper combines hierarchical multiresolution analysis-based EMD, thresholding operation and averaging operation together. In particular, EMD is applied to the first intrinsic mode function (IMF) in the first level of decomposition to obtain IMFs in the second level of decomposition. The first IMF in the second level of decomposition is chosen as the noise component. For each realization, this noise component is segmented into various pieces, and these segments are permutated. By summing up this permutated IMF to the rest of IMFs in both the first level of decomposition and the second level of decomposition, new realization of the noisy signal is obtained. Next, for original signal and each realization of newly generated noisy signal, EMD is performed again. IMFs in the first level of decomposition are obtained. Then, consecutive mean squared errors-based criterion is used to classify IMFs in the first level of decomposition into the information group of IMFs or the noise group of IMFs. Next, EMD is applied to IMFs in the noise group in the first level of decomposition and IMFs in the second level of decomposition are obtained. Detrended fluctuation analysis is used to classify IMFs in the second level of decomposition into the information group of IMFs or the noise group of IMFs. After that, thresholding is applied to IMFs in the noise group in the second level of the decomposition to obtain denoised signal. Finally, the above procedures are repeated, and several realizations of denoised signals are obtained. Then, denoised signal obtained by applying thresholding to each realization is averaged together to obtain final denoised signal. The extensive numerical simulations are conducted and the results show that our proposed method outperforms existing EMD-based denoising methods.

Error Compensation for Optical Encoder via Local-Sinusoidal-Assisted Empirical Mode Decomposition With an Optimization Scheme

The measurement precision of the optical encoder is influenced by coupling interferences from the system and the environment. To further improve the measurement precision of the optical encoder, in this article, a nonlinear error compensation method is proposed to extract the optimal underlying trend of the measurement error, which is based on the empirical mode decomposition (EMD) with a local-sinusoidal-assisted (LSA) scheme and an adaptive intrinsic mode function (IMF) selection scheme. To solve the inherent problem of mode-mixing phenomena existing in the decomposed IMFs from the measurement error, an LSA scheme is proposed to adaptively change the extreme distribution of the measurement error, which is based on an abnormal segment discrimination rule. An adaptive IMF selection scheme is proposed to construct the optimal underlying trend of the measurement error, which is first formulated as an optimization problem. Next, a significant factor is defined to select the eligible IMFs with more contributions to the underlying trend. Comparison experiments indicate that the eligible IMFs can be adaptively selected, and the proposed compensation method achieves an excellent compensation performance with the root-mean-square error of 0.380 <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\mu$</tex-math></inline-formula> m in the 95% confidence interval of <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$-$</tex-math></inline-formula> 0.017–0.016 <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\mu$</tex-math></inline-formula> m, which is superior to the previous EMD-based compensation methods.

A Fault Diagnosis Method Based on EEMD and Statistical Distance Analysis

Serious vibration or wear with large friction usually appear when faults occur, which leads to more serious faults such as the destruction of the oil film, bringing great damages to both the society and economic sector. Therefore, the accurate diagnosis of a fault in the early stage is important for the safety operation of machinery. To effectively extract the fault features for diagnosis, EMD-based methods are widely used. However, these methods spend lots of efforts diagnosing faults and require plenty of professional knowledge of diagnosis. Although many intelligent classifiers can be used to automatically diagnose faults such as wear, a broken tooth and imbalance, the combing EMD-based method, the scarcity of samplings with labels hinder the application of these methods to engineering. It is because the model of the intelligent classifier must be constructed based on sufficient samplings with a label. To solve this problem, we propose a novel fault diagnosis method, which is performed based on the EEMD and statistical distance analysis. In this method, the EEMD is used to decompose one original signal into several IMFs and then the probability density distribution of each IMF is calculated. To diagnose the fault of the machinery, the Euclidean distance between the signal acquired under an unknown fault with the other referenced signals acquired previously under various fault types is calculated. At last, the fault of the signal is the same with the referenced signal when the distance is the smallest. To verify the effectiveness of our proposed method, a dataset of bearings with different faults, and a dataset of 2009 Prognostics and Health Management (PHM) data challenge, including gear, bearing and shaft faults are used. The result shows that the proposed method can not only automatically diagnose faults effectively, but also fewer samplings with a label are used compared with the intelligent methods.

Denoising method of natural gas pipeline leakage signal based on empirical mode decomposition and improved Bhattacharyya distance

Considering that the noise in the pipeline seriously affects the accuracy of leakage detection, this paper proposes a new denoising method (EMD-BD-WT) that combines Empirical Mode Decomposition (EMD), the improved Bhattacharyya distance (BD) and wavelet transform (WT). BD is first designed to measure the distance between the probability contribution function (PCF) and the normalization function (NF) to select the effective modes. Then WT is used to process the non-effective modes. Finally, the selected effective modes and the non-effective modes after wavelet processing are used to construct the filtered signal. The theoretical analysis, simulation experiments and filtering results of the actual natural gas pipeline leakage signal show that compared with other EMD-based filtering methods, the proposed method has superior performance.

BP-AdaBoost Algorithm Based on Variational Mode Decomposition Optimized by Envelope Entropy for Diagnosing the Working Conditions of a Slideway Seedling-Picking Mechanism

Highlights A method of monitoring the working conditions of a slideway seedling-picking mechanism based on variational mode decomposition (VMD), envelope entropy, and energy entropy is proposed. Based on the criterion of envelope entropy minimization, the combination of the decomposition layer number and penalty factor in VMD is optimized to yield a satisfactory decomposition effect of the analyzed vibration signal. The BP-AdaBoost algorithm is used to improve the working condition classification performance for the slideway seedling-picking mechanism. The working-condition identification effect with the proposed method are compared with those through EMD-based, LMD-based, and EEMD-based methods. Abstract . The slideway seedling-picking mechanism is a type of rotating machinery. This study proposes a novel method of identifying the working conditions of slideway seedling-picking mechanisms for early fault diagnosis by utilizing a back-propagation adaptive boosting (BP-AdaBoost) algorithm based on variational mode decomposition (VMD) optimized by the envelope entropy. The experimental results demonstrate that the proposed method can effectively verify the four working conditions (normal state, slideway failure, cam failure, and spring failure). The overall recognition accuracy reaches 90.0% under the optimal combination of the decomposition layer number K and penalty factor a in VMD determined through the envelope entropy minimization criterion. Classification comparisons with empirical mode decomposition (EMD), local mean decomposition (LMD) and ensemble empirical mode decomposition (EEMD) integrated into the BP-AdaBoost algorithm indicate that the overall recognition accuracy of the proposed method is 18.1%, 16.9%, and 15.6% higher than the accuracies of the three conventional methods, respectively. Compared with the K-means, support vector machine (SVM) algorithms, BP-AdaBoost algorithm demonstrates a more dependable capability for identifying the working conditions. This study provides a useful reference for monitoring the working conditions of slideway seedling-picking mechanisms. Keywords: BP-AdaBoost algorithm, Energy entropy, Envelope entropy, Slideway seedling-picking mechanism, Variational mode decomposition, Working conditions.

Fault Diagnosis of Composite Features Rolling Bearing Based on Variational Mode Decomposition

In order to extract the vibration signal feature of rolling bearing with non-steady feature and improve the fault diagnosis rate accurately and stably, a variational mode decomposition (VMD) feature extraction method is proposed. Particle swarm optimization (PSO) is used to optimize the parameters of support vector machine to construct a fault diagnosis model to achieve fault diagnosis of rolling bearings. Firstly, change the modal decomposition of the known fault signal under the same load to get the modal function, and the modal function is further extracted by the singular value decomposition. The time domain, frequency domain feature and modal feature of the original signal are extracted. Constructing hybrid features to achieve efficient fault feature extraction. Optimizing SVM parameters through PSO algorithm to construct fault diagnosis models to achieve efficient fault diagnosis. Finally, by comparing with EMD-based feature extraction methods in the same load, the method shows better classification performance and the overall fault recognition rate remains above 99.17%, which verifies the reliability and effectiveness of the method.

Fiber optic gyro noise reduction based on hybrid CEEMDAN-LWT method

Abstract The noise involved in drift signal of fiber optic gyroscope (FOG) mostly comes from electronic component, detection circuit and variable working environment. Gaussian white noise and fractional noise submerged in FOG output is difficult to be eliminated by conventional methods because of the non-stationary characteristics. The complete ensemble empirical mode decomposition with adaptive noise (CEEMDAN) method is a novel nonlinear and non-stationary signal processing method, which is exploited in FOG denoising. Lifting wavelet transform (LWT) technology is employed to combine with CEEMDAN method to expedite the computing efficiency and improve filtering accuracy, so that a hybrid CEEMDAN-LWT-based model is achieved. Comparison analysis with other filtering methods based on empirical mode decomposition (EMD) and its improved version is done. And ensemble empirical mode decomposition (EEMD) method combined with LWT considered as EEMD-LWT method is also applied as compared. Experimental analysis results show that new hybrid method outperforms other EMD-based filtering methods. The new method requires only 11.3% sifting iterations of the EEMD-LWT method. Meanwhile, the rate white noise, bias instability and quantization noise buried in FOG output signal decreases from 0 . 0031 ° / h , 0 . 0352 ° / h and 0 . 5412 ° to 0 . 0005 ° / h , 0 . 0056 ° / h and 0 . 0231 ° , respectively. Furthermore, de-trended fluctuation analysis (DFA) algorithm is employed to evaluate the effectiveness of hybrid method for the FOG signal filtering.

Fully Adaptive Denoising of ECG Signals Using Empirical Mode Decomposition with the Modified Indirect Subtraction and the Adaptive Window Techniques

Electrocardiogram (ECG) is one of the major methods for the diagnosis of heart malfunctions. ECG signals are susceptible to both high-frequency and low-frequency noises such as electromyography (EMG), power line interference (PLI) and baseline wander noises, respectively. These noises deteriorate the quality of ECG signals and challenge the proper identification of heart illnesses. In this article, we report an improved fully adaptive method for canceling high-frequency noises using the empirical mode decomposition (EMD) with the modified indirect subtraction and adaptive window techniques. As high-frequency noises are approximately of zero mean and lower-order intrinsic mode functions (IMFs) contain high-frequency noises, a statistical test is performed to determine whether a particular combination of IMFs has zero mean. First, to remove the PLI noise, the modified indirect subtraction technique is used. The sum of IMFs which are dominated by the noise is passed to a Butterworth band-pass filter, and the resultant filtered signal is subtracted directly from the noisy ECG signal. Then, the EMG noise is removed by an improved adaptive window-based noise reduction technique. By exploiting this technique, in contrast to common EMD-based methods, the duration of the QRS complex is computed regarding the location of the peak of the R wave and the widest possible QRS complex which makes the proposed technique applicable to all types of ECG signals. The quantitative results of simulations performed on several records from the MIT–BIH arrhythmia database prove the better performance of the proposed method than the compared methods at different noise levels.

Noise suppression in 2D and 3D seismic data with data-driven sifting algorithms

We have developed an empirical-mode decomposition (EMD) algorithm for effective suppression of random and coherent noise in 2D and 3D seismic amplitude data. Unlike other EMD-based methods for seismic data processing, our approach does not involve the time direction in the computation of the signal envelopes needed for the iterative sifting process. Instead, we apply the sifting algorithm spatially in the inline-crossline plane. At each time slice, we calculate the upper and lower signal envelopes by means of a filter whose length is adapted dynamically at each sifting iteration according to the spatial distribution of the extrema. The denoising of a 3D volume is achieved by removing the most oscillating modes of each time slice from the noisy data. We determine the performance of the algorithm by using three public-domain poststack field data sets: one 2D line of the well-known Alaska 2D data set, available from the US Geological Survey; a subset of the Penobscot 3D volume acquired offshore by the Nova Scotia Department of Energy, Canada; and a subset of the Stratton 3D land data from South Texas, available from the Bureau of Economic Geology at the University of Texas at Austin. The results indicate that random and coherent noise, such as footprint signatures, can be mitigated satisfactorily, enhancing the reflectors with negligible signal leakage in most cases. Our method, called empirical-mode filtering (EMF), yields improved results compared to other 2D and 3D techniques, such as [Formula: see text] EMD filter, [Formula: see text] deconvolution, and [Formula: see text]-[Formula: see text]-[Formula: see text] adaptive prediction filtering. EMF exploits the flexibility of EMD on seismic data and is presented as an efficient and easy-to-apply alternative for denoising seismic data with mild to moderate structural complexity.

The application of EMD-based methods for diagnosis of winding faults in a transformer using transient and steady state currents

The application of signal processing techniques is a fundamental step for fault diagnostic methodologies. The application of empirical mode decomposition (EMD)-based methods such as classic EMD, ensemble EMD (EEMD), and complete EEMD (CEEMD) is presented in this work for the analysis of inrush current signals. This analysis leads to the detection of short-circuited turns in transformers. Results show that CEEMD provides the best performance, as it readily extracts the information related to the fault, requiring of acceptable computational resources. Actual inrush current signals of a transformer with short-circuited turns are also considered. The number of short-circuited turns ranges from 5 to 40. Useful indices, such as the Shannon Entropy, Energy, and root-mean-square value, are obtained from the information provided by the CEEMD approach. These indices are analyzed for both the transient state and the steady state of the current signals, providing the proper quantification of the fault severity.

Quasi-bivariate variational mode decomposition as a tool of scale analysis in wall-bounded turbulence

The identification and separation of multi-scale coherent structures is a critical task for the study of scale interaction in wall-bounded turbulence. Here, we propose a quasi-bivariate variational mode decomposition (QB-VMD) method to extract structures with various scales from instantaneous two-dimensional (2D) velocity field which has only one primary dimension. This method is developed from the one-dimensional VMD algorithm proposed by Dragomiretskiy and Zosso (IEEE Trans Signal Process 62:531–544, 2014) to cope with a quasi-2D scenario. It poses the feature of length-scale bandwidth constraint along the decomposed dimension, together with the central frequency re-balancing along the non-decomposed dimension. The feasibility of this method is tested on both a synthetic flow field and a turbulent boundary layer at moderate Reynolds number ( $$Re_{\tau }$$ = 3458) measured by 2D particle image velocimetry (PIV). Some other popular scale separation tools, including pseudo-bi-dimensional empirical mode decomposition (PB-EMD), bi-dimensional EMD (B-EMD) and proper orthogonal decomposition (POD), are also tested for comparison. Among all these methods, QB-VMD shows advantages in both scale characterization and energy recovery. More importantly, the mode mixing problem, which degrades the performance of EMD-based methods, is avoided or minimized in QB-VMD. Finally, QB-VMD analysis of the wall-parallel plane in the log layer (at $$y/\delta$$ = 0.12) of the studied turbulent boundary layer shows the coexistence of large- or very large-scale motions (LSMs or VLSMs) and inner-scaled structures, which can be fully decomposed in both physical and spectral domains.

Variational Mode Extraction: A New Efficient Method to Derive Respiratory Signals from ECG.

ECG-derived respiratory (EDR) signal is an effective and inexpensive method to monitor the respiration. Previous studies have shown that the empirical mode decomposition (EMD) techniques can satisfactorily extract the EDR signal, however, their performances are degraded at the presence of noise. On the other hand, variational mode decomposition (VMD) performs good robustness against noise. In applications such as EDR extraction, where a specific mode is in interest, VMD imposes unnecessary computational cost. In this paper, we consider the extraction of EDR as a problem of obtaining a specific mode of a signal and suggest a new method named as variational mode extraction (VME). The method is established on the similar basis as VMD, with a new criterion: The residual signal after extracting the specific mode should have no or less energy at the center frequency of the mode. In this regard, VME is capable of solving the EDR problem by considering the EDR signal as a mode with approximate center frequency of zero. For verification, the respiratory rate signal is detected from EDR signal extracted by VME and compared it with those obtained by VMD, EMD-based methods, and bandpass filtering. The results confirm that the new method can extract the EDR signal with a better accuracy, while performing much lower computational cost and higher convergence rate.

Ensemble EMD‐based signal denoising using modified interval thresholding

Empirical mode decomposition (EMD) is extensively realised in its potential of non-parametric signal denoising. Ensemble EMD (EEMD) is an improved self-adapting signal decomposition approach that can produce signal components with no frequency aliasing. In this study, the interval thresholding and iteration operation of EMD-based denoising techniques are applied to the EEMD and found not entirely feasible in the EEMD case. A modified interval thresholding is proposed, which can be adjustable for the intrinsic mode functions from EEMD. By taking advantage of the characteristics of EEMD, the internal and external iterations are compared and properly adopted in the EEMD-based denoising strategy. As a result, the EEMD-based denoising methods are proposed by combining the modified interval thresholding and the iterations. The denoising results on synthetic and real-life signals indicate that the presented methods exhibit better performance comparing with EMD-based methods, especially for signals with low signal-to-noise ratio. Based on the time complexities of the proposed methods, the acceptable sampling frequencies of the methods in real-time denoising are given.

Variational mode decomposition denoising combined with the Hausdorff distance.

Variational mode decomposition (VMD) is a recently introduced adaptive signal decomposition algorithm with a solid theoretical foundation and good noise robustness compared with empirical mode decomposition (EMD). However, there is still a problem with this algorithm associated with the selection of relevant modes. To solve this problem, this paper proposes a novel signal-filtering method that combines VMD with Hausdorff distance (HD) in the VMD-HD method. A noisy signal is first decomposed into a given number K of band-limited intrinsic mode functions by VMD. The probability density function is then estimated for each mode. The aim of this method is to reconstruct the signal using the relevant modes, which are selected on the basis of noticeable similarities between the probability density function of the input signal and that of each mode. Various similarity measures are investigated and compared, and the HD is shown to offer the best performance. The results of filtering of simulation signals illustrate the validity of the proposed method when compared with EMD-based methods under comprehensive quantitative evaluation criteria. As a specific example, the proposed method is successfully used for filtering the pipeline leakage signal as evaluated by the de-trended fluctuation analysis algorithm.

A joint framework for multivariate signal denoising using multivariate empirical mode decomposition

In this paper, a novel multivariate denoising scheme using multivariate empirical mode decomposition (MEMD) is proposed. Unlike previous EMD-based denoising methods, the proposed scheme can align common frequency modes across multiple channels of a multivariate data, thus, facilitating direct multichannel data denoising. The key idea in this work is to extend our earlier MEMD based denoising method for univariate signal in Hao et al. (2016) [19] to the multivariate data. The MEMD modes (known as intrinsic mode functions) for separating noise components are first adaptively selected on the basis of a similarity measure between the probability density function (pdf) of the input multivariate signal and that of each mode by Frobenius norm. The selected modes are then denoised further by a local interval thesholding procedure followed by reconstruction of the thresholded IMFs. The resulting method operates directly in multidimensional space where input signal resides, owing to MEMD, and also benefits from its mode-alignment property. Furthermore, subspace projection is introduced within the framework of the proposed method to exploit the inter-channel dependence among IMFs with the same index, enabling the diversity reception of the signal. Performance of the proposed method against standard multiscale denoising schemes is demonstrated on both synthetic and real world data.