Empirical Mode Decomposition Approach Research Articles

Optimal averages for nonlinear signal decompositions—Another alternative for empirical mode decomposition

The empirical mode decomposition (EMD) is an algorithm pioneered by Huang et al. as an alternative technique to the traditional Fourier and wavelet methods for analyzing nonlinear and non-stationary signals. It aims at decomposing a signal, via an iterative sifting procedure, into several intrinsic mode functions (IMFs), and each of the IMF has better behaved instantaneous frequency analysis. This paper presents an alternative approach for EMD. The main idea is to replace the average of upper and lower envelopes in the sifting procedure of EMD by a local average obtained by variational optimization framework. Therefore, an IMF can be produced by simply subtracting the average from the signal without iteration. Our numerical examples illustrate that the resulting decomposition is convergent and robust against noise.

Quantile-Based Empirical Mode Decomposition: An Efficient Way to Decompose Noisy Signals

The main goal of this paper is to propose a new approach of empirical mode decomposition (EMD) that analyzes noisy signals efficiently. The EMD has been widely used to decompose nonlinear and nonstationary signals into some components according to intrinsic frequency called intrinsic mode functions. However, the conventional EMD may not be efficient in decomposing signals that are contaminated by noninformative noises or outliers. This paper presents a new EMD procedure that analyzes noisy signals effectively and is robust to outliers with holding the merits of the conventional EMD. The key ingredient of the proposed method is to apply a quantile smoothing method to a noisy signal itself instead of interpolating local extrema of the signal when constructing its mean envelope. Through simulation studies and texture image analysis, it is demonstrated that the proposed method produces substantially effective results.

Application of Multivariate EMD to Improve Quality VLF-EM Data: Synthetic and Fields Data

A method for enhancing VLF˗EM data based on Multivariate Empirical Mode Decomposition (EMD) was presented. The noise assisted multivariate empirical mode decomposition (NA-MEMD) approach to simultaneously decompose bivariate data.The NA-MEMD is applied to enhance bivariate VLF˗EM data. The method was also tested on a synthetic and two fields VLF-EM data sets. The results indicate that the filtered VLF˗EM data based on the NA-MEMD results better data and easier to interpret or further analyzed. In addition, the 2D resistivity profile result estimated from the inversion of filtered VLF˗EM data is appropriate to geological condition.

Nonlinear and adaptive undecimated hierarchical multiresolution analysis for real valued discrete time signals via empirical mode decomposition approach

Hierarchical multiresolution analysis is an important tool for the analysis of signals. Since this multiresolution representation provides a pyramid like framework for representing signals, it can extract signal information effectively via levels by levels. On the other hand, a signal can be nonlinearly and adaptively represented as a sum of intrinsic mode functions (IMFs) via the empirical mode decomposition (EMD) algorithm. Nevertheless, as the IMFs are obtained only when the EMD algorithm converges, no further iterative sifting process will be performed directly when the EMD algorithm is applied to an IMF. As a result, the same IMF will be resulted and further level decompositions of the IMFs cannot be obtained directly by the EMD algorithm. In other words, the hierarchical multiresolution analysis cannot be performed via the EMD algorithm directly. This paper is to address this issue by performing a nonlinear and adaptive hierarchical multiresolution analysis based on the EMD algorithm via a frequency domain approach. In the beginning, an IMF is expressed in the frequency domain by applying discrete Fourier transform (DFT) to it. Next, zeros are inserted to the DFT sequence and a conjugate symmetric zero padded DFT sequence is obtained. Then, inverse discrete Fourier transform (IDFT) is applied to the zero padded DFT sequence and a new signal expressed in the time domain is obtained. Actually, the next level IMFs can be obtained by applying the EMD algorithm to this signal. However, the lengths of these next level IMFs are increased. To reduce these lengths, first DFT is applied to each next level IMF. Second, the DFT coefficients of each next level IMF at the positions where the zeros are inserted before are removed. Finally, by applying IDFT to the shorten DFT sequence of each next level IMF, the final set of next level IMFs are obtained. It is shown in this paper that the original IMF can be perfectly reconstructed. Moreover, computer numerical simulation results show that our proposed method can reach a component with less number of levels of decomposition compared to that of the conventional linear and nonadaptive wavelets and filter bank approaches. Also, as no filter is involved in our proposed method, there is no spectral leakage in various levels of decomposition introduced by our proposed method. Whereas there could be some significant leakage components in the various levels of decomposition introduced by the wavelets and filter bank approaches.

THE MULTI-FREQUENCY CORRELATION BETWEEN EUA AND sCER FUTURES PRICES: EVIDENCE FROM THE EMD APPROACH

Currently European Union Allowances (EUA) and secondary Certified Emission Reduction (sCER) have become two dominant carbon trading assets for investors and their linkage attracts much attention from academia and practitioners in recent years. Under this circumstance, we use the empirical mode decomposition (EMD) approach to decompose the two carbon futures contract prices and discuss their correlation from the multi-frequency perspective. The empirical results indicate that, first, the EUA and sCER futures price movements can be divided into those triggered by the long-term, medium-term and short-term market impacts. Second, the price movements in the EUA and sCER futures markets are primarily caused by the long-term impact, while the short-term impact can only explain a small fraction. Finally, the long-term (short-term) effect on EUA prices is statistically uncorrelated with the short-term (long-term) effect of sCER prices, and there is a medium or strong lead-and-lag correlation between the EUA and sCER price components with the same time scales. These results may provide some important insights of price forecast and arbitraging activities for carbon futures market investors, analysts and regulators.

A novel EMD selecting thresholding method based on multiple iteration for denoising LIDAR signal

Empirical mode decomposition (EMD) approach has been believed to be potentially useful for processing the nonlinear and non-stationary LIDAR signals. To shed further light on its performance, we proposed the EMD selecting thresholding method based on multiple iteration, which essentially acts as a development of EMD interval thresholding (EMD-IT), and randomly alters the samples of noisy parts of all the corrupted intrinsic mode functions to generate a better effect of iteration. Simulations on both synthetic signals and LIDAR signals from real world support this method.

Sea level rise around Korea: Analysis of tide gauge station data with the ensemble empirical mode decomposition method

This study estimates the trend and acceleration of sea level rise (SLR) at the tide gauge stations around Korea as now the data from the stations are recorded for approximately 50 years, which allow us to estimate the acceleration of SLR. We use not only traditional polynomial regression analysis but also the ensemble empirical mode decomposition (EEMD) approach, whose strength is the elimination of long-term variations in sea level data, to process this relatively short-term data. The relative sea level data from the five tide gauge stations around Korea, including Mokpo, Jeju, Busan, Ulsan and Mukho are used in this study. With Jeju showing the highest trend of SLR, all stations exhibit rising trend of sea level regardless of the methods used in this study. Positive accelerations of SLR are found in four out of five stations with the highest acceleration at Mukho. We also compare the results based on the EEMD approach and those based on the regression analysis and find generally consistent results between two. While the trends show the slight differences at the five stations with no tendency toward over- or under-estimation, the accelerations are slightly lower with the EEMD approach than with the regression analysis at all four stations with positive accelerations.

Empirical mode decomposition of long-term polar motion observations

We use the Empirical Mode Decomposition (EMD) method to study the decadal variations in polar motion and its long-term trend since year 1900. The existence of the so-called “Markowitz wobble”, a multidecadal fluctuation of the mean pole of rotation whose nature has long been debated since its discovery in 1960, is confirmed. In the EMD approach, the Markowitz wobble naturally arises as an empirical oscillatory term in polar motion, showing significant amplitude variations and a period of approximately 3 decades. The path of the time-averaged, non-cyclic component of polar motion matches the results of previous investigations based on classical spectral methods. However, our analysis also reveals previously unnoticed steep variations (change points) in the rate and the direction of secular polar motion.

Multi-scale analysis of streamflow using the Hilbert-Huang Transform

Purpose– The purpose of this paper is to simulate and analyze accurately the multi-scale characteristics, variation periods and trends of the annual streamflow series in the Haihe River Basin (HRB) using the Hilbert-Huang Transform (HHT).Design/methodology/approach– The Empirical Mode Decomposition (EMD) approach is adopted to decompose the original signal into intrinsic mode functions (IMFs) in multi-scales. The Hilbert spectrum is applied to each IMF component and the localized time-frequency-energy distribution. The monotonic residues obtained by EMD can be treated as the trend of the original sequence.Findings– The authors apply HHT to 14 hydrological stations in the HRB. The annual streamflow series are decomposed into four IMFs and a residual component, which exhibits the multi-scale characteristics. After the Hilbert transform, the instantaneous frequency, center frequency and mean period of the IMFs are obtained. Common multi-scale periods of the 14 series exist, e.g. 3.3a, 4∼7a, 8∼10a, 11-14a, 24∼25a and 43∼45a. The residues indicate that the annual streamflow series has exhibited a decreasing trend over the past 50 years.Research limitations/implications– The HHT method is still in its early stages of application in hydrology and needs to be further tested.Practical implications– It is helpful for the study of the complex features of streamflow.Social implications– This paper will contribute to the sustainable utilization of water resources.Originality/value– This study represents the first use of the HHT method to analyze the multi-scale characteristics of the streamflow series in the HRB. This paper provides an important theoretical support for water resources management.

Rotor Local Rubbing Fault Feature Analysis Based on Teager-Huang Transform

An analysis method based on Teager-Huang transform for rotor local rubbing fault diagnosis is introduced. Firstly, the original vibration signal is decomposed into some Intrinsic Mode Function (IMF) components by using Empirical Mode Decomposition (EMD) approach, secondly, Teager energy operator is applied to estimate the instantaneous amplitude and instantaneous frequency of each IMF component, so the time-frequency distribution of the signal is obtained. The rotor local rubbing fault is simulated on a rotor test rig. The analysis results show that this method compared with Hilbert-Huang transform (HHT) can track the occurrence of rotor local rubbing fault better and can extract the characteristics of rotor local rubbing fault effectively. It provides a reliable method for timely and accurate diagnostic to the rotor local rubbing fault.

Speech Enhancement with EMD and Hurst-Based Mode Selection

This paper presents a speech enhancement technique for signals corrupted by nonstationary acoustic noises. The proposed approach applies the empirical mode decomposition (EMD) to the noisy speech signal and obtains a set of intrinsic mode functions (IMF). The main contribution of the proposed procedure is the adoption of the Hurst exponent in the selection of IMFs to reconstruct the speech. This EMD and Hurst-based (EMDH) approach is evaluated in speech enhancement experiments considering environmental acoustic noises with different indices of nonstationarity. The results show that the EMDH improves the segmental signal-to-noise ratio and an overall quality composite measure, encompassing the perceptual evaluation of speech quality (PESQ). Moreover, the short-time objective intelligibility (STOI) measure reinforces the superior performance of EMDH. Finally, the EMDH is also examined in a speaker identification task in noisy conditions. The proposed technique leads to the highest speaker identification rates when compared to the baseline speech enhancement algorithms and also to a multicondition training procedure.

HHT–SVM: An online method for detecting profile injection attacks in collaborative recommender systems

Collaborative recommender systems are known to be particularly vulnerable to profile injection attacks, in which malicious users insert fake profiles into the rating database in order to bias the systems’ output. To reduce this risk, a number of methods have been proposed to detect such attacks. However, almost all of them operate in batch mode, i.e., they require examining and processing the entire rating database to detect the attacks. With this problem in mind, we propose an online method (called HHT–SVM) to detect profile injection attacks by combining Hilbert–Huang transform (HHT) and support vector machine (SVM), which can operate incrementally. The underpinning idea of HHT–SVM is the feature extraction method based on an individual user profile. In this paper, we first construct rating series for each user profile based on the novelty and popularity of items. Then, by introducing HHT we use the empirical mode decomposition (EMD) approach to decompose each rating series and extract Hilbert spectrum based features to characterize the profile injection attacks. Finally, we exploit SVM to detect profile injection attacks based on the proposed features. We conduct experiments on the MovieLens 1M dataset and compare the performance of HHT–SVM with PCA-VarSelect and Batch-SVM to demonstrate the effectiveness of the proposed approach.

Fast, simultaneous and robust VLF-EM data denoising and reconstruction via multivariate empirical mode decomposition

The measurement of Very Low Frequency Electromagnetic (VLF-EM) is important in many different applications, i.e, environmental, archeological, geotechnical studies, etc. In recent years, improving and enhancing VLF-EM data containing complex numbers (bivariate) was presented by several authors in order to produce reliable models, generally using univariate empirical mode decomposition (EMD). Applying univariate EMD separately on each data is problematic. This results in a different number of misaligned Intrinsic Mode Functions (IMFs) which can complicate the selection of some IMFs for denoising process. Thus, a filtering method based on the multivariate empirical mode decomposition (MEMD) approach to decompose simultaneously bivariate data is proposed. In this paper we address two issues by employing the recently introduced noise assisted MEMD (N-A MEMD) for improving bivariate VLF-EM data. Firstly, the N-A MEMD to decompose bivariate measurement of the VLF-EM data into IMFs and a residue is defined as VLF-EM signal or unwanted noise. Secondly, the proposed method is used to enhance VLF-EM data and to reject unwanted noise. Finally, the proposed method is applied to a synthetic data with two added sinusoids. To demonstrate the robustness of the N-A MEMD method, the method was tested on added-noise synthetic data sets and the results were compared to the Ensemble EMD (EEMD) and Bivariate EMD (BEMD). The N-A MEMD gave more robust and accurate results than the EEMD and BEMD methods and the method required less CPU time to obtain the IMFs compared to EEMD. The method was also tested on several field data sets. The results indicate that the filtered VLF-EM data based on the N-A MEMD make the data easier to interpret and to be analyzed further. In addition, the 2D resistivity profile estimated from the inversion of filtered VLF-EM data results was appropriate to the geological condition.

A Single-Channel ICA-R Method for Speech Signal Denoising combining EMD and Wavelet

According to the problem of speech signal denoising, we propose a novel method in this paper, which combines empirical mode decomposition (EMD), wavelet threshold denoising and independent component analysis with reference (ICA-R). Because there is only one mixed recording, it is a single-channel independent component analysis (SCICA) problem in fact, which is hard to solve by traditional ICA methods. EMD is exploited to expand the single-channel received signal into several intrinsic mode functions (IMFs) in advance, therefore traditional ICA of multi-dimension becomes applicable. First, the received signal is segmented to reduce the processing delay. Secondly, wavelet thresholding is applied to the noise-dominated IMFs. Finally, fast ICA-R is introduced to extract the object speech component from the processed IMFs, whose reference signal is constructed by assembling the high-order IMFs. The simulations are carried out under different noise levels and the performance of the proposed method is compared with EMD, wavelet thresholding, EMD-wavelet and EMD-ICA approaches. Simulation results indicate that the proposed method exhibit superior denoising performance especially when signal-to-noise ratio is low, with a half shorter running time.

Hilbert-Huang processing for single-exposure two-dimensional grating interferometry

Single-shot crossed-type fringe pattern processing and analysis method called Hilbert-Huang grating interferometry (HHGI) is proposed. It consist of three main procedures: (1) crossed pattern is resolved into two fringe families using novel orthogonal empirical mode decomposition approach, (2) separated fringe sets are filtered using modified automatic selective reconstruction aided by enhanced fast empirical mode decomposition and mutual information detrending, and (3) Hilbert spiral transform is employed for fringe phase demodulation. Numerical and experimental studies corroborate the validity, versatility and robustness of the proposed HHGI technique. It can be successfully applied to multiplicative and additive type crossed patterns with sinusoidal and binary orthogonal component structures. Efficient adaptive filtering enables successful fast processing and analysis of complex and defected patterns.

Joint Empirical Mode Decomposition and Sparse Binary Programming for Underlying Trend Extraction

This paper presents a novel methodology for extracting the underlying trends of signals via a joint empirical mode decomposition (EMD) and sparse binary programming approach. The EMD is applied to the signals and the corresponding intrinsic mode functions (IMFs) are obtained. The underlying trends of the signals are obtained by the sums of the IMFs where these IMFs are either selected or discarded. The total number of the selected IMFs is minimized subject to a specification on the maximum absolute differences between the denoised signals (signals obtained by discarding the first IMFs) and the underlying trends. Since the total number of the selected IMFs is minimized, the obtained solutions are sparse and only few IMFs are selected. The selected IMFs correspond to the components of the underlying trend of the signals. On the other hand, the L <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">∞</sub> norm specification guarantees that the maximum absolute differences between the underlying trends and the denoised signals are bounded by an acceptable level. This forces the underlying trends to follow the global changes of the signals. As the IMFs are either selected or discarded, the coefficients are either zero or one. This problem is actually a sparse binary programming problem with an L <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">0</sub> norm objective function subject to an L <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">∞</sub> norm constraint. Nevertheless, the problem is nonconvex, nonsmooth, and NP hard. It requires an exhaustive search for solving the problem. However, the required computational effort is too heavy to be implemented practically. To address these difficulties, we approximate the L <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">0</sub> norm objective function by the L <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</sub> norm objective function, and the solution of the sparse binary programming problem is obtained by applying the zero and one quantization to the solution of the corresponding continuous-valued L <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</sub> norm optimization problem. Since the isometry condition is satisfied and the number of the IMFs is small for most of practical signals, this approximation is valid and verified via our experiments conducted on practical data. As the L <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</sub> norm optimization problem can be reformulated as a linear programming problem and many efficient algorithms such as simplex or interior point methods can be applied for solving the linear programming problem, our proposed method can be implemented in real time. Also, unlike previously reported techniques that require precursor models or parameter specifications, our proposed adaptive method does not make any assumption on the characteristics of the original signals. Hence, it can be applied to extract the underlying trends of more general signals. The results show that our proposed method outperforms existing EMD, classical lowpass filtering and the wavelet methods in terms of the efficacy.

Multifractal description of wind power fluctuations using arbitrary order Hilbert spectral analysis

The objectives are to study and model the aggregate wind power fluctuations dynamics in the multifractal framework. We present here the analysis of aggregate power output sampled at 1 Hz during three years. We decompose the data into several Intrinsic Mode Functions (IMFs) using Empirical Mode Decomposition (EMD). We use a new approach, arbitrary order Hilbert spectral analysis, a combination of the EMD approach with Hilbert spectral analysis (or Hilbert–Huang Transform) and the classical structure-function analysis to extract the scaling exponents or multifractal spectrum ζ(q): this function provides a full characterization of a process at all intensities and all scales. The application of both methods, i.e. structure-function and arbitrary-order Hilbert spectral analyses, gives similar results indicating that the aggregate power output from a wind farm, possesses intermittent and multifractal properties. In order to check this result, we generate stochastic simulations of a Multifractal Random Walk (MRW) using a log-normal stochastic equation. We show that the simulation results are fully compatible with the experimental results.

A Flattest Constrained Envelope Approach for Empirical Mode Decomposition

Empirical mode decomposition (EMD) is an adaptive method for nonlinear, non-stationary signal analysis. However, the upper and lower envelopes fitted by cubic spline interpolation (CSI) may often occur overshoots. In this paper, a new envelope fitting method based on the flattest constrained interpolation is proposed. The proposed method effectively integrates the difference between extremes into the cost function, and applies a chaos particle swarm optimization method to optimize the derivatives of the interpolation nodes. The proposed method was tested on three different types of data: ascertain signal, random signals and real electrocardiogram signals. The experimental results show that: (1) The proposed flattest envelope effectively solves the overshoots caused by CSI method and the artificial bends caused by piecewise parabola interpolation (PPI) method. (2) The index of orthogonality of the intrinsic mode functions (IMFs) based on the proposed method is 0.04054, 0.02222±0.01468 and 0.04013±0.03953 for the ascertain signal, random signals and electrocardiogram signals, respectively, which is lower than the CSI method and the PPI method, and means the IMFs are more orthogonal. (3) The index of energy conversation of the IMFs based on the proposed method is 0.96193, 0.93501±0.03290 and 0.93041±0.00429 for the ascertain signal, random signals and electrocardiogram signals, respectively, which is closer to 1 than the other two methods and indicates the total energy deviation amongst the components is smaller. (4) The comparisons of the Hilbert spectrums show that the proposed method overcomes the mode mixing problems very well, and make the instantaneous frequency more physically meaningful.

ANALYSIS OF SIGNAL TREND BY A PSEUDO-EMD METHOD WITH APPLICATIONS TO WEATHER AND SPEECH DATA

Empirical mode decomposition (EMD) has been established as a valuable tool in determining nonlinear signal trend. EMD decomposes a one-dimensional (1D) signal into hierarchical components known as intrinsic mode functions (IMFs) and a residue, based on the local properties of the signal. The first IMF depicts the highest local oscillations, while the residue depicts the trend of a signal/data. In each iteration of the EMD process, interpolation is applied to some local maxima and minima points to form upper and lower envelopes, respectively. But, the application of interpolation methods causes huge computation time and other artifacts in the decomposition, which limits the use of EMD for many real life signals. This paper proposes an effective method that replaces the interpolation step by direct envelope estimation using order statistics filters, which results in decreased computation time, following a similar EMD approach that has been recently proposed for two-dimensional data or image analysis. The modified EMD of this paper called pseudo EMD (P-EMD) method is particularly useful in determining, analyzing, and/or modifying the trend of various signals to obtain and/or produce some desired results/outcomes. Several synthetic and real-life signals such as speech signal and sea level pressure and temperature are tested to verify the effectiveness of the P-EMD. From the results, P-EMD has been found as a superior alternative for trend analysis of signal/data, since it results in more accurate trend compared to the other interpolation based EMD methods such as classical EMD (CEMD) and a modified EMD (MEMD), and also facilitates faster computation.

Segmentation of Pathological Heart Sound Signal Using Empirical Mode Decomposition

The Phonocardiogram (PCG) is the graphicalrecording of acoustic energy produced by the mechanicalactivity of various cardiac. Due to the complicated mechanismsinvolved in the generation of in the PCG signal, it is consideredas multicomponent non stationary signal. Empirical modedecomposition (EMD) allows decomposing an observedmulticomponent signal into a set of monocomponent signals,called Intrinsic Mode Functions (IMFs). The goal of this paperis to segment some pathological HS signals into the murmursrelated to cardiac diseases. EMD approach allows toautomatically selecting the most appropriate IMFscharacterizing the murmur using the noise only model. Real-lifesignals are used in the various cases such as Early AorticStenosis (EAS), Late Aortic Stenosis (LAS), MitralRegurgitation (MR) and Aortic Regurgitation (AR) to validate,and demonstrate the effectiveness of the proposed method.