Extreme-point Symmetric Mode Decomposition Method Research Articles

ESMD-WSST High-Frequency De-Noising Method for Bridge Dynamic Deflection Using GB-SAR

Ground-based synthetic aperture radar (GB-SAR), as a new non-contact measurement technique, has been widely applied to obtain the dynamic deflection of various bridges without corner reflectors. However, it will cause some high-frequency noise in the obtained dynamic deflection with the low signal-to-noise ratio. To solve this problem, this paper proposes an innovative high-frequency de-noising method combining the wavelet synchro-squeezing transform (WSST) method with the extreme point symmetric mode decomposition (ESMD) method. First, the ESMD method is applied to decompose the observed dynamic deflection signal into a series of intrinsic mode functions (IMFs), and the frequency boundary of the original signal autocorrelation is filtered by the mutual information entropy (MIE) for each IMF pair. Second, the high-frequency IMF components are fused into a high-frequency sub-signal. WSST is performed to remove the influence of noise to reconstruct a new sub-signal. Finally, the de-noised bridge dynamic deflection is reconstructed by the new sub-signal, the remaining IMF components, and the residual curve R. For the simulated signal with 5 dB noise, the signal-to-noise ratio (SNR) after noise reduction is increased to 11.13 dB, and the root-mean-square error (RMSE) is reduced to 0.30 mm. For the on-site experiment for the Wanning Bridge, the noise rejection ratio (NRR) is 5.48 dB, and ratio of the variance root (RVR) is 0.05 mm. The results indicate that the proposed ESMD-WSST method can retain more valid information and has a better noise reduction ability than the ESMD, WSST, and EMD-WSST methods.

Analysis of the Anomalous Environmental Response to the 2022 Tonga Volcanic Eruption Based on GNSS

On 15 January 2022, a violent eruption and tsunami of the Hunga Tonga-Hunga Ha’apai (HTHH) volcano in Tonga, South Pacific, caused widespread international concern. In order to detect the anomalous environmental response caused by the HTHH volcanic eruption based on GNSS ionospheric data, GNSS tropospheric data and GNSS coordinate time series, a new method combining the zenith non-hydrostatic delay difference method and the extreme-point symmetric mode decomposition (ESMD) method, was proposed to detect tropospheric anomalies. The moving interquartile range method and the ESMD method were introduced to detect ionospheric anomalous and coordinate time series anomalies, respectively. The results showed that 9–10 h before the eruption of the Tonga volcano and 11–12 h after the eruption of the Tonga volcano, obvious total electron content (TEC) anomalies occurred in the volcanic eruption center and its northeast and southeast, with the maximum abnormal value of 15 TECU. Significant tropospheric anomalies were observed on the day of the HTHH volcano eruption as well as 1–3 days and 16–17 days after the eruption, and the abnormal intensity was more than 10 times that of normal. The coordinate time series in direction E showed very significant anomalies at approximately 2:45 p.m. on 14 January, at approximately 4:30 a.m.–5:40 a.m. on 15 January, and at approximately 3:45 a.m. on 16 January, with anomalies reaching a maximum of 7–8 times daily. The abnormality in the direction north (N) is not obvious. Very prominent anomalies can be observed in the direction up (U) at approximately 4:30 a.m.–5:40 a.m., with the intensity of the anomalies exceeding the normal by more than 10 times. In this study, GNSS was successfully used to detect the anomalous environmental response during this HTHH volcano eruption.

A novel fault location method for hybrid lines based on traveling wave

In this paper, a novel fault traveling wave-based location method for hybrid lines is proposed. According to the refraction characteristics of fault traveling wave (TW) at junction nodes, the node ratio coefficient value Q is defined based on the amplitude of traveling wave. Then, by comparing the Q values of junction nodes and fault points, a fault section identification principle is proposed to determine the fault line section for hybrid lines. After identification of fault line section, a double-ended fault location method is revised via adding the location compensation factors to mitigate the effect of traveling wave velocity changes on fault location for multi-section hybrid lines. Additionally, considering the detection problems of arrival time for TW signals, the extreme-point symmetric mode decomposition (ESMD) method and Teager energy operator (TEO) are proposed. The ESMD method is applied to decompose the sampled fault traveling wave signals into several intrinsic modes to filter out the fundamental frequency and low frequency signals. The TEO is employed to track the characteristics of TW signals. Effectiveness of the proposed method is validated in PSCAD. Results show that the ESMD-TEO can detect TW signals effectively, and the proposed fault location method can determine the fault line section and locate fault positions correctly.

Temporal-spatial and non-stationarity characteristics of extreme precipitation in the Poyang Lake Basin, China

Extreme weather/climate events increased significantly because of global warming. Based on daily records from 16 meteorological stations in the Poyang Lake Basin (PLB) from 1959 to 2019, we comprehensively investigated the temporal-spatial and non-stationarity characteristics of extreme precipitation from three dimensions (intensity, frequency and duration) using PreWhite-ning Mann-Kendall (PWMK), extreme-point symmetric mode decomposition method (ESMD) and the generalized additive models for location, scale and shape (GAMLSS). The results showed that the intensity and frequency of extreme precipitation increased significantly in the PLB, while the duration of extreme precipitation decreased from 1959 to 2019. The extreme precipitation had features of high intensity, high frequency, and short duration in the PLB. There was a clear distinction between flood season and non-flood season for extreme precipitation. Extreme precipitation was concentrated in the northern and central PLB during the flood season and in the central PLB during the non-flood season. The increasing trend of extreme precipitation amount was 2.10 mm·a-1 in the Xinjiang basin, which had the largest increment over the PLB. In the flood season, the extreme precipitation had longer duration but weaker intensity and smaller range, contrasting with the status during the non-flood season. The intensity and frequency of extreme precipitation showed stationarity characteristics in the PLB. However, the duration of extreme precipitation showed non-stationarity characteristics. With the continuous increase of extreme precipitation amount, the risk of related disasters would increase.

Fault Diagnosis of Axial Piston Pump Based on Extreme-Point Symmetric Mode Decomposition and Random Forests

Aiming at fault diagnosis of axial piston pumps, a new fusion method based on the extreme-point symmetric mode decomposition method (ESMD) and random forests (RFs) was proposed. Firstly, the vibration signal of the axial piston pump was decomposed by ESMD to get several intrinsic mode functions (IMFs) and an adaptive global mean curve (AGMC) on the local side. Secondly, the total energy was selected as the data of feature extraction by analyzing the whole oscillation intensity of the signal. Thirdly, the data were preprocessed and the labels were set, and then, they were adopted as the training and testing set of machine learning samples. Lastly, the RFs model was created based on machine learning service (MLS) to diagnose the faults of the axial piston pump on the cloud. Using the test and verifying the data set for comparative testing, the fault diagnosis precision rates of the model are above 90.6%, the recall rates are more than 90.9%, the F1 score is higher than 90.7%, and the accuracy rate of this model reached 97.14%. A benchmark data simulation of mechanical transmission systems and an experimental data investigation of an axial piston pump are performed to manifest the superiority of the present method by comparing with classification and regression trees (CART) and support vector machine (SVM).

A denoising carbon price forecasting method based on the integration of kernel independent component analysis and least squares support vector regression

During the past few decades, accurately forecasting carbon price has become a significant research field and aroused concerns from both scholars and policymakers, which contributes the Organized Exchange to scientifically and rationally allocate a fixed-quantity of carbon emissions among prospective polluters. The conventional forecasting approaches, however, suffer from the poor prediction accuracy due to the nonlinearity and non-stationarity of the carbon price series. Meanwhile, monitoring and filtering the inherent noise in carbon price series, which are the main steps in the forecasting model, are perceived as the challenging tasks to work in. To address these obstacles, a denoising-hybridization procedure, which is a hybrid model of extreme-point symmetric mode decomposition (ESMD), kernel independent component analysis (KICA) and least squares support vector regression (LSSVR), is put forward for predicting the carbon price. Firstly, the carbon price is decomposed into several intrinsic mode functions (IMFs) via the ESMD method. Secondly, independent components (ICs), which reflect the internal formation mechanism, are separated out from the IMFs via KICA method. Further, the IC comprised of the noise is eliminated according to the results of noise monitoring. Finally, the LSSVR method is applied to the remaining ICs for achieving the forecasting results of carbon price, wherein the particle swarm optimization (PSO) algorithm is employed to synchronously optimize the hyper parameters in LSSVR. The empirical results on four carbon futures prices from European Union Emissions Trade System (EU ETS) demonstrate the effectiveness and robustness of the promoted denoising-hybridization procedure. Comparative experiments illustrate the superiority of the proposed method from the perspective of statistical performance criteria.

2D Extreme–Point Symmetric Mode Decomposition Method

This paper proposes a kind of 2D extension of Extreme–Point Symmetric Mode Decomposition Method. The 2D-sifting process is performed by 4 steps: 1 Extrema points detection (by neighboring window). 2 Delaunay triangulation is done over the grid of all extrema points. 3 Mid points of the triangulation are obtained. 4 Thin plate splines are used to interpolate the mid points. 5 Sifting processing is carried out. Finally, experiments are done with real images to show the efficiency of this 2D Extreme–Point Symmetric Mode Decomposition Method.

Spatiotemporal Characteristics of Drought in the Heihe River Basin Based on the Extreme-Point Symmetric Mode Decomposition Method

Assessment of spatiotemporal characteristics of drought under climate change is significant for drought mitigation. In this study, the standardized precipitation evapotranspiration index (SPEI) calculated at different timescales was adopted to describe the drought conditions in the Heihe River Basin (HRB) from 1961 to 2014. The period characteristics and spatiotemporal distribution of drought were analyzed by using the extreme-point symmetric mode decomposition (ESMD) and inverse distance weight interpolation methods. Four main results were obtained. (1) The SPEI series of the upper reaches of the HRB at different timescales showed an upward trend (not significant) during 1961–2014. In the middle and lower reaches, the SPEI series exhibited significant downward trends. (2) The annual SPEI series of the lower reaches was decomposed through the ESMD method and exhibited a fluctuating downward trend as a whole. The oscillation showed quasi-3.4-year and quasi-4.5-year periods in the interannual variation, while a quasi-13.5-year period occurred in the interdecadal variation. The interannual period plays a leading role in drought variation across the HRB. (3) The entire research period was divided into three subperiods by the Bernaola–Galvan segmentation algorithm: 1961–1966, 1967–1996, and 1997–2014. The spring drought frequency and autumn drought intensity arrived at their maxima in the lower reaches during 1997–2014, with values of 72.22% and 1.56, respectively. The high frequency and intensity areas of spring, summer, and autumn drought moved from the middle-upper reaches to the middle-lower reaches of the HRB during 1961–2014. (4) Compared to the wavelet transform, the ESMD method has self-adaptability for signal decomposition and is more accurate for drought period analysis. Extreme-point symmetric mode decomposition is a more efficient decomposition method for nonlinear and nonstationary time series and has important significance for revealing the complicated change features of climate systems.

Deformation Activity Analysis of a Ground Fissure Based on Instantaneous Total Energy.

This study proposes a novel instantaneous total energy method to perform an activity analysis of ground fissures deformation, which is calculated by integrating the extreme-point symmetric mode decomposition (ESMD) method and kinetic energy based on the time-series displacement acquired by shape acceleration array (SAA) sensors. The proposed method is tested on the Xiwang Road fissure in Beijing, China. First, to fully monitor the hanging wall and footwall of the monitored ground fissure, a 4 m-long SAA in the vertical direction and an 8 m-long SAA in the horizontal direction were embedded in a ground fissure to obtain an accurate time-series displacement with an accuracy of ±1.5 mm/32 m and a displacement acquisition frequency of once an hour. Second, to improve the accuracy of the activity analysis, the ESMD method and Spearman’s rho are applied to perform signal denoising of the original time-series displacement obtained by the SAA sensors. Finally, the instantaneous total energy is obtained to analyze the activity of the monitored ground fissure. The results demonstrate that the proposed method is more reliable to reflect the activity of a monitored ground fissure compared to the time-series displacement.

Damage Detection and Analysis of Urban Bridges Using Terrestrial Laser Scanning (TLS), Ground-Based Microwave Interferometry, and Permanent Scatterer Interferometry Synthetic Aperture Radar (PS-InSAR)

This paper presents a practical framework for urban bridge damage detection and analysis by using three key techniques: terrestrial laser scanning (TLS), ground-based microwave interferometry, and permanent scatterer interferometry synthetic aperture radar (PS-InSAR). The proposed framework was tested on the Beishatan Bridge in Beijing, China. Firstly, a Digital Surface Model (DSM) of the lower surface of the bridge was constructed based on the point cloud generated by using TLS to obtain the potential damage area. Secondly, the dynamic time-series displacement of the potential damage area was acquired by ground-based microwave interferometry, and the Extreme-Point Symmetric Mode Decomposition (ESMD) method was applied to detect damages by the use of signal decomposition and instantaneous frequency calculation. Lastly, the PS-InSAR technique was applied to obtain the surface deformation around Beishatan Bridge by using COSMO-SkyMed images with a ground resolution of 3 m × 3 m, and finally, we analyzed the causes of bridge damage. The experimental results showed that the proposed framework can effectively obtain the potential damage area of the bridge by the DSM from the point cloud by TLS and further judge whether the bridge was damaged by the ESMD method, based on the time-series displacement data. The results also showed that the subway shield construction may be the reason for damage to Beishatan Bridge.

Forecasting Models for Wind Power Using Extreme-Point Symmetric Mode Decomposition and Artificial Neural Networks

The randomness and volatility of wind power poses a serious threat to the stability, continuity, and adjustability of the power system when it is connected to the grid. Accurate short-term wind power prediction methods have important practical value for achieving high-precision prediction of wind farm power generation and safety and economic dispatch. Therefore, this paper proposes a novel combined model to improve the accuracy of short-term wind power prediction, which involves grey correlation degree analysis, ESMD (extreme-point symmetric mode decomposition), sample entropy (SampEn) theory, and a hybrid prediction model based on three prediction algorithms. The meteorological data at different times and altitudes is firstly selected as the influencing factors of wind power. Then, the wind power sub-series obtained by the ESMD method is reconstructed into three wind power characteristic components, namely PHC (high frequency component of wind power), PMC (medium frequency component of wind power), and PLC (low frequency component of wind power). Similarly, the wind speed sub-series obtained by the ESMD method is reconstructed into three wind speed characteristic components, called SHC (high frequency component of wind speed), SMC (medium frequency component of wind speed), and SLC (low frequency component of wind speed). Subsequently, the Bat-BP model, Adaboost-ENN model, and ENN (Elman neural network), which have high forecasting accuracy, are selected to predict PHC, PMC, and PLC, respectively. Finally, the prediction results of three characteristic components are aggregated into the final prediction values of the original wind power series. To evaluate the prediction performance of the proposed combined model, 15-min wind power and meteorological data from the wind farm in China are adopted as case studies. The prediction results show that the combined model shows better performance in short-term wind power prediction compared with other models.

An improved extreme-point symmetric mode decomposition method and its application to rolling bearing fault diagnosis

HHT (Hilbert-Huang Transform) which consist of EMD (Empirical Mode Decomposition) and HT (Hilbert Transform) now is the most widely used time-frequency analysis technique for rolling element bearing fault diagnosis, however, its fault characteristic information extraction accuracy is usually limited due to the problem of mode mixing in EMD. ESMD (Extreme-point symmetric mode decomposition) is a novel development of HHT which is promising to alleviate this limitation and it has been applied to some fields successfully, but its application for rolling bearing fault diagnosis has rarely been seen in the literature. In this paper, ESMD is applied to extract the bearing fault characteristics for rolling bearing fault detection, and the results proved that ESMD can have a better fault diagnose effect than EMD and HT. What’s more, for further improving bearing fault characteristic extraction accuracy of rolling bearing vibration signals, the sifting scheme is proposed for selecting the sensitive fault-related IMFs (intrinsic mode functions) generated by ESMD, in which a weighted kurtosis index is introduced for automatic selection and reconstruction of the fault-related IMFs, and then the original and reconstructed bearing fault vibration signal after performing Hilbert transform as the results to diagnose the incipient rolling bearing fault. ESMD combined with the proposed sifting scheme are applied to diagnose the simulated and experimental signals, and the results confirmed that the sifting scheme based ESMD is superior to the other conventional method in rolling bearings fault diagnosis.

Two-Level W-ESMD Denoising for Dynamic Deflection Measurement of Railway Bridges by Microwave Interferometry

Aiming to reduce the influences of noise for dynamic deflection measurements of railway bridges by microwave interferometry, this paper proposes a two-level wavelet-based extreme-point symmetric mode decomposition (ESMD) denoising algorithm on the basis of the characteristics of different noise frequency scales. First, the ESMD method is adopted to decompose the obtained dynamic deflection signal into a series of intrinsic mode functions (IMFs) from high to low frequency. Second, the first-level denoising is performed to eliminate the influences of high frequency noise by integrating the heursure threshold rule, soft thresholding, and different decomposition scales for high or low frequency IMFs, respectively. Third, the reconstructed first-level denoised signal is further decomposed into a series of IMFs, from high to low frequency, by the ESMD method. Last, the second-level denoising is performed to eliminate the influences of low frequency noise and some residual high frequency noise. For the low frequency noise, by using the heursure threshold rule, soft thresholding and a lower decomposition scale, and for some residual high frequency noise by using minimax threshold rule, hard thresholding and a moderate decomposition scale, the results on both simulated and real dynamic deflection signals show that the proposed two-level denoising algorithm is superior to wavelet threshold denoising and conventional empirical mode decomposition-based denoising techniques.

Dynamic Monitoring and Vibration Analysis of Ancient Bridges by Ground-Based Microwave Interferometry and the ESMD Method

In this paper, we propose to conduct a dynamic monitoring and vibration analysis of ancient bridges by means of ground-based microwave interferometry and the extreme-point symmetric mode decomposition (ESMD) method. Ground-based microwave interferometry, a novel non-contact technology with a high accuracy, is used to acquire dynamic time series displacements with environmental excitation factors and a transient load with a car, respectively. The ESMD method, a new alternative to the Hilbert-Huang transform (HHT), is adopted to conduct the instantaneous vibration analysis of Zhaozhou Bridge. Firstly, a series of intrinsic mode functions (IMFs) are obtained together with an optimal adaptive global mean (AGM) curve by using a mode symmetric about the maxima and minima points. Secondly, the instantaneous frequency of each IMF is obtained by the use of a direct interpolation algorithm, which can reconcile the conflict between the period and the frequency for the traditional time-frequency analysis methods. As a representative case, Zhaozhou Bridge, a well-known Chinese ancient bridge constructed more than 1400 years ago, is studied in detail. Four kinds of dynamic time series displacements—two of them acquired by considering only environmental excitation factors for the mid-span and 1/4-span points and the others obtained with the transient load of a car for the mid-span and 1/4-span points—are selected to pursue a comparison of the decomposed IMFs and the instantaneous frequencies to perform the instantaneous vibration analysis of Zhaozhou Bridge. By comparing the results obtained with HHT for the decomposed IMFs and the instantaneous frequencies, the results show that the proposed method has a powerful ability to evaluate the instantaneous dynamic response of ancient bridges.

Spatio‐temporal variations of nonlinear trends of precipitation over an arid region of northwest China according to the extreme‐point symmetric mode decomposition method

ABSTRACTClimate systems have both nonlinear and non‐stationary characteristics, and it is important to develop methods to reveal the nonlinear trends in these systems. Based on synthetic data with known signals and a precipitation anomaly time series from 74 meteorological stations in an arid region of northwest China (ARNC) from 1960 to 2015, the latest extreme‐point symmetric mode decomposition (ESMD) and ensemble empirical mode decomposition (EEMD) methods were employed to conduct multi‐scale mode decomposition. The results showed that both the ESMD and EEMD methods can separate synthetic signals and extract the nonlinear trends of precipitation in ARNC and can thus provide reliable decomposition results. This article employed the ESMD method to analyse the spatial and temporal variation features of nonlinear precipitation trends in ARNC from 1960 to 2015. The results indicated that over the past 50+ years, the overall precipitation in ARNC has exhibited an apparent nonlinear upwards trend. In addition, its changes have exhibited oscillation periods of 3, 6, and 11 years. The periods of the three components are significant (p < 0.05), and the variance contribution rate of the first intrinsic mode function (IMF1) is the largest, reaching 58%. Furthermore, there are obvious spatial differences in the nonlinear trends of average annual precipitation: northern Xinjiang has had a mainly rising trend; southern Xinjiang has had mainly rising and decreasing–rising trends; and the precipitation changes in the Hexi Corridor have been quite complicated. The ESMD method can reflect integrated changes in the nonlinear trends of precipitation over different time scales and has important practical significance and scientific value for revealing the complicated structural features of climate systems.

Correlation between hydrological drought, climatic factors, reservoir operation, and vegetation cover in the Xijiang Basin, South China

The Xijiang River is known as the Golden Watercourse because of its role in the development of the Pearl River Delta Regional Economic System in China, which was made possible by its abundant water resources. At present, the hydrological regime of the Xijiang River has now become complicated, the water shortages and successive droughts pose a threat to regional economic development. However, the complexity of hydroclimatological processes with emphasizes on drought has not been comprehended. In order to effectively predict and develop the adaptation strategies to cope with the water scarcity damage caused by hydrological droughts, it is essential to thoroughly analyze the relationship between hydrological droughts and pre/post-dependent hydroclimatological factors. To accomplish this, the extreme-point symmetric mode decomposition method (ESMD) was utilized to reveal the periodic variation in hydrological droughts that is characterized by the Standardized Drought Index (SDI). In addition, the cross-wavelet transform method was applied to investigate the correlation between large-scale climate indices and drought. The results showed that hydrological drought had the most significant response to spring ENSO (El Niño-Southern Oscillation), and the response lags in sub-basins were mostly 8–9months except that in Yujiang River were mainly 5 or 8months. Signal reservoir operation in the Yujiang River reduced drought severity by 52–95.8% from January to April over the 2003–2014 time period. Similarly, the cascade reservoir alleviated winter and spring droughts in the Hongshuihe River Basin. However, autumn drought was aggravated with severity increased by 41.9% in September and by 160.9% in October, so that the land surface models without considering human intervention must be used with caution in the hydrological simulation. The response lags of the VCI (Vegetation Condition Index) to hydrological drought were different in the sub-basins. The response lag for the Hongshuihe, Yujiang, and Liujiang River Basins were mostly 0–4months, 0–1months, and 2–3months, respectively, but there was no obvious regular change pattern in the Guijiang River Basin.

Acoustic Emission Monitoring and Failure Precursors of Sandstone Samples under Various Loading and Unloading Paths

To explore the failure precursors of hard rock, a series of triaxial loading and unloading experiments were carried out on sandstone sample using the acoustic emission systems. The extreme-point symmetric mode decomposition method (ESMD method) was used to denoise and reconstruct the AE data. The AE quiet period in Scheme I becomes much more obvious with the confining pressure increasing, which can be regarded as the precursor information of the sample failure under conventional triaxial compression. Unlike Scheme I, there are no obvious precursory characteristics before failure in Schemes II and III, and the count rate reaches the maximum at the peak point. When the stress ratio ranges from 0.8 to 1.0, the fractal values of acoustic emission can be used to investigate the failure precursors of samples at a lower confining pressure. When the time ratio is greater than 0.8 under higher confining pressures, the fractal values of sandstone samples under unloading paths are rapidly reduced, which can be used to predict rock failure at higher confining pressures.

The regional features of precipitation variation trends over Sichuan in China by the ESMD method

Based on a precipitation time series from 49 meteorological stations in Sichuan Province during the period from 1961 to 2011, the multi-scale characteristics of precipitation variability are analyzed using the extreme-point symmetric mode decomposition method (ESMD). Regional differences in variation trends and change-points were also preliminarily discussed. The results indicated that in the last 50+ years, the overall precipitation in Sichuan Province has exhibited a significant non-linear downward trend, and its changes have clearly exhibited an inter-annual scale (quasi-3 and quasi-8-year) and interdecadal scale (quasi-13-year). The variance contribution rates of each component demonstrated that the inter-annual change had a strong influence on the overall precipitation change in Sichuan Province, and the reconstructed inter-annual variation trend could describe the fluctuation state of the original precipitation during the study period. The reconstructed interdecadal variability revealed that the climate mode in Sichuan Province had divided into three distinct variation periods with 1973 and 1998 as the boundaries. Furthermore, there were regional differences in the non-linear changes and change-points of precipitation. In addition, in order to study the relations between the changing more or less of rising or decrease and meteorological station’s geographical position (latitude, longitude and elevation) i.e., the Cokriging interpolation technique is applied directly to precipitation variation trend components through ESMD decomposition. At the same time, the results also suggested that the ESMD method can effectively reveal variations in long-term precipitation sequences at different time scales and can be used for the complex diagnosis of non-linear and non-stationary signal changes.

EXTREME-POINT SYMMETRIC MODE DECOMPOSITION METHOD FOR DATA ANALYSIS

An extreme-point symmetric mode decomposition (ESMD) method is proposed to improve the Hilbert–Huang Transform (HHT) through the following prospects: (1) The sifting process is implemented by the aid of 1, 2, 3 or more inner interpolating curves, which classifies the methods into ESMD_I, ESMD_II, ESMD_III, and so on; (2) The last residual is defined as an optimal curve possessing a certain number of extreme points, instead of general trend with at most one extreme point, which allows the optimal sifting times and decompositions; (3) The extreme-point symmetry is applied instead of the envelop symmetry; (4) The data-based direct interpolating approach is developed to compute the instantaneous frequency and amplitude. One advantage of the ESMD method is to determine an optimal global mean curve in an adaptive way which is better than the common least-square method and running-mean approach; another one is to determine the instantaneous frequency and amplitude in a direct way which is better than the Hilbert-spectrum method. These will improve the adaptive analysis of the data from atmospheric and oceanic sciences, informatics, economics, ecology, medicine, seismology, and so on.