Weighted Permutation Entropy Research Articles

A Method for Single-Phase Ground Fault Section Location in Distribution Networks Based on Improved Empirical Wavelet Transform and Graph Isomorphic Networks

When single-phase ground faults occur in distribution systems, the fault characteristics of zero-sequence current signals are not prominent. They are quickly submerged in noise, leading to difficulties in fault section location. This paper proposes a method for fault section location in distribution networks based on improved empirical wavelet transform (IEWT) and GINs to address this issue. Firstly, based on kurtosis, EWT is optimized using the N-point search method to decompose the zero-sequence current signal into modal components. Noise is filtered out through weighted permutation entropy (WPE), and signal reconstruction is performed to obtain the denoised zero-sequence current signal. Subsequently, GINs are employed for graph classification tasks. According to the topology of the distribution network, the corresponding graph is constructed as the input to the GIN. The denoised zero-sequence current signal is the node input for the GIN. The GIN autonomously explores the features of each graph structure to achieve fault section location. The experimental results demonstrate that this method has strong noise resistance, with a fault section location accuracy of up to 99.95%, effectively completing fault section location in distribution networks.

EI-ISOA-VMD: Adaptive denoising and detrending method for nuclear circulating water pump impeller

As a key component of the nuclear circulating water pump, it is difficult to extract effective information from low-quality impeller signal because of the interference noise and non-stationary trend, such as water flow impact, vibrations from unrelated components, and sensor noise. Traditional signal denoising methods, particularly those utilizing variational mode decomposition, often require manual parameter tuning, resulting in unsatisfactory outcome. To address these issues, this paper proposes a signal denoising and detrending method based on the evaluation index driven the improved seagull algorithm optimized for variational mode decomposition (EI-ISOA-VMD). During the optimization process, sinusoidal chaotic mapping is utilized for initializing the seagull population, while a tanh function is employed to design nonlinearly decreasing inertia weights. Furthermore, the Levy flight mechanism enhances the randomness of position updates and optimizes seagull individuals beyond the boundary range in order to form the improved seagull optimization algorithm. The denoising, detrending process, and evaluation index proposed in this paper are utilized as the fitness function for the improved seagull optimization algorithm to optimize key parameters of variational mode decomposition. During the denoising and detrending process, the trend component is initially selected using the mean ratio method. The useful, low noise and high noise intrinsic mode functions are screened by correlation coefficient and weighted permutation entropy. Subsequently, the low noise components are denoised by wavelet thresholding method, and the high noise components are discarded to finally reconstruct the signal. The experimental results demonstrate that the proposed method effectively eliminates noise and trend components in low-quality signal while preserving more valuable information.

Detecting stochasticity in population time series using a non‐parametric test of intrinsic predictability

Abstract Many ecological systems dominated by stochastic dynamics can produce complex time series that inherently limit forecast accuracy. The ‘intrinsic predictability’ of these systems can be approximated by a time series complexity metric called weighted permutation entropy (WPE). While WPE is a useful metric to gauge forecast performance prior to model building, it is sensitive to noise and may be biased depending on the length of the time series. Here, we introduce a simple randomized permutation test (rWPE) to assess whether a time series is intrinsically more predictable than white noise. We apply rWPE to both simulated and empirical data to assess its performance and usefulness. To do this, we simulate population dynamics under various scenarios, including a linear trend, chaotic, periodic and equilibrium dynamics. We further test this approach with observed abundance time series for 932 species across four orders of animals from the Global Population Dynamics Database. Finally, using Adélie (Pygoscelis adeliae) and emperor penguin (Aptenodytes forsteri) time series as case studies, we demonstrate the application of rWPE to multiple populations for a single species. We show that rWPE can determine whether a system is significantly more predictable than white noise, even with time series as short as 10 years that show an apparent trend under biologically realistic stochasticity levels. Additionally, rWPE has statistical power close to 100% when time series are at least 30 time steps long and show chaotic or periodic dynamics. Power decreases to ~10% under equilibrium dynamics, irrespective of time series length. Among four classes of animal taxa, mammals have the highest relative frequency (28%) of time series that are both longer than 30 time steps and indistinguishable from white noise in terms of complexity, followed by insects (16%), birds (16%) and bony fishes (11%). rWPE is a straightforward and useful method widely applicable to any time series, including short ones. By informing forecasters of the inherent limitations to a system's predictability, it can guide a modeller's expectations for forecast performance.

Neuroimaging markers of aberrant brain activity and treatment response in schizophrenia patients based on brain complexity

The complexity of brain activity reflects its ability to process information, adapt to environmental changes, and transition between states. However, it remains unclear how schizophrenia (SZ) affects brain activity complexity, particularly its dynamic changes. This study aimed to investigate the abnormal patterns of brain activity complexity in SZ, their relationship with cognitive deficits, and the impact of antipsychotic medication. Forty-four drug-naive first-episode (DNFE) SZ patients and thirty demographically matched healthy controls (HC) were included. Functional MRI-based sliding window analysis was utilized for the first time to calculate weighted permutation entropy to characterize complex patterns of brain activity in SZ patients before and after 12 weeks of risperidone treatment. Results revealed reduced complexity in the caudate, putamen, and pallidum at baseline in SZ patients compared to HC, with reduced complexity in the left caudate positively correlated with Continuous Performance Test (CPT) and Category Fluency Test scores. After treatment, the complexity of the left caudate increased. Regions with abnormal complexity showed decreased functional connectivity, with complexity positively correlated with connectivity strength. We observed that the dynamic complexity of the brain exhibited the characteristic of spontaneous, recurring “complexity drop”, potentially reflecting transient state transitions in the resting brain. Compared to HC, patients exhibited reduced scope, intensity, and duration of complexity drop, all of which improved after treatment. Reduced duration was negatively correlated with CPT scores and positively with clinical symptoms. The results suggest that abnormalities in brain activity complexity and its dynamic changes may underlie cognitive deficits and clinical symptoms in SZ patients. Antipsychotic treatment partially restores these abnormalities, highlighting their potential as indicators of treatment efficacy and biomarkers for personalized therapy.

Refined composite multivariate multiscale weighted permutation entropy and multicluster feature selection-based fault detection of gearbox

Refined composite multivariate multiscale weighted permutation entropy and multicluster feature selection-based fault detection of gearbox

Generalized weighted permutation entropy analysis of SPEI index in Serbia as a proxy of corn production

AbstractIn this work we analyse yearly standardized precipitation evapotranspiration index for accumulation timescale of 6 months (SPEI‐6) for August which has been identified as a strong proxy for corn production in Serbia. By applying recently proposed method generalized weighted permutation entropy (GWPE), which provides the information about predictability of large/small fluctuations in nonstationary time series, on SPEI index calculated from high‐resolution gridded dataset for the period 1951–2022, we found that the small fluctuation sequences of consecutive 4‐year SPEI values are the most predictable (indicated by lowest entropy values). The order of large fluctuation sequences is less predictable, while the order of sequences considering all magnitudes of fluctuations are the least predictable (indicated by highest entropy values). We also analysed 4‐month SPEI‐6 sequences for May, June, July and August (during the growing season of corn) along the years under study and found that they display lower GWPE values and thus are more predictable than yearly SPEI‐6 August sequences (small fluctuations displaying lower entropy values than large fluctuations). Regarding spatial distributions, in both cases SPEI‐6 sequences show similar pattern for large fluctuations: higher predictability (lower entropy values) in northern and eastern part of Serbia and lower predictability (higher entropy values) in southern and western part. For small fluctuations spatial distribution of GWPE values indicates that yearly SPEI‐6 August series are more predictable in the western part of the country while May to August monthly SPEI‐6 sequences are more predictable in the eastern part.

Bearing fault diagnosis of induction machines using VMD-DWT and composite multiscale weighted permutation entropy

PurposeBearings play a critical role in the reliable operation of induction machines, and their failure can lead to significant operational challenges and downtime. Detecting and diagnosing these defects is imperative to ensure the longevity of induction machines and preventing costly downtime. The purpose of this paper is to develop a novel approach for diagnosis of bearing faults in induction machine.Design/methodology/approachTo identify the different fault states of the bearing with accurately and efficiently in this paper, the original bearing vibration signal is first decomposed into several intrinsic mode functions (IMFs) using variational mode decomposition (VMD). The IMFs that contain more noise information are selected using the Pearson correlation coefficient. Subsequently, discrete wavelet transform (DWT) is used to filter the noisy IMFs. Second, the composite multiscale weighted permutation entropy (CMWPE) of each component is calculated to form the features vector. Finally, the features vector is reduced using the locality-sensitive discriminant analysis algorithm, to be fed into the support vector machine model for training and classification.FindingsThe obtained results showed the ability of the VMD_DWT algorithm to reduce the noise of raw vibration signals. It also demonstrated that the proposed method can effectively extract different fault features from vibration signals.Originality/valueThis study suggested a new VMD_DWT method to reduce the noise of the bearing vibration signal. The proposed approach for bearing fault diagnosis of induction machine based on VMD-DWT and CMWPE is highly effective. Its effectiveness has been verified using experimental data.

Generalized weighted permutation entropy analysis of satellite hot-pixel time series in Brazilian biomes

In this work we analyze hot-pixel satellite data, proxy for natural and anthropogenic vegetation and forest fires, using recently proposed method Generalized weighted permutation entropy (GWPE). The current analysis of data from 2002 to 2022 for six Brazilian biomes sheds new light on differences among the biomes in terms of ordinal pattern entropy, for both small and large fluctuations. It is found that for all biomes large fluctuations in fire temporal series show higher complexity level than small fluctuations, small fluctuations being more predictable as indicated by lower entropy values. Moreover, for fire sensitive biomes Amazon and Atlantic Forest the three dimensional complexity entropy curves in the complexity-entropy-scale causality box (CESCB) are rather similar (indicating similar dynamics for both small and large fluctuations), as well as for fire dependent biomes Pantanal and Pampa, while fire independent Caatinga biome shows different, distinct behavior. Finally, the similarity in lower parts of the CESCB curves (corresponding to small fluctuations) between Amazon and Cerrado reflects the changes of fire dynamics due to human activities.

Epileptic Seizure Detection in EEG Signal Using Optimized Convolutional Neural Network with Selected Feature Set

This article implements novel epileptic seizure detection by EEG signal that includes various phases: Feature extraction, Feature selection, and Classification. Originally, the input EEG signal is subjected to the preprocessing phase, and from the preprocessed signal, certain features are extracted in which the features like TDFs, FDFs, and TFDFs are extracted. Here, the Time Domain Features (TDFs) include Non-linear energy (NE), Permutation Entropy (PE), and Weighted Permutation Entropy (WPE); the Frequency Domain Features (FDFs) consists of Intensity Weighted Mean Frequency (IWMF); and the Time-Frequency Domain Features (TFDFs) consists of the “Weighted Multi-scale Rényi Permutation Entropy (WMRPE)”. Moreover, the extracted features are subjected to the feature selection phase. An improved chi-square model is proposed for selecting the appropriate features. Then, the chosen features are subjected to the classification process. Here, the classification is performed using an Optimized Convolutional Neural Network (CNN). To make the detection more precise, the CNN weights are tuned optimally through a Weighting Factor Based Shark Smell Optimization (IWFSSO) model. The MCC of the implemented CNN+IWFSSO method achieved a better value for a learning percentage 90%; nevertheless, the compared existing models accomplish a smaller value for dataset 2.

Ensemble Improved Permutation Entropy: A New Approach for Time Series Analysis.

Entropy quantification approaches have gained considerable attention in engineering applications. However, certain limitations persist, including the strong dependence on parameter selection, limited discriminating power, and low robustness to noise. To alleviate these issues, this paper introduces two novel algorithms for time series analysis: the ensemble improved permutation entropy (EIPE) and multiscale EIPE (MEIPE). Our approaches employ a new symbolization process that considers both permutation relations and amplitude information. Additionally, the ensemble technique is utilized to reduce the dependence on parameter selection. We performed a comprehensive evaluation of the proposed methods using various synthetic and experimental signals. The results illustrate that EIPE is capable of distinguishing white, pink, and brown noise with a smaller number of samples compared to traditional entropy algorithms. Furthermore, EIPE displays the potential to discriminate between regular and non-regular dynamics. Notably, when compared to permutation entropy, weighted permutation entropy, and dispersion entropy, EIPE exhibits superior robustness against noise. In practical applications, such as RR interval data classification, bearing fault diagnosis, marine vessel identification, and electroencephalographic (EEG) signal classification, the proposed methods demonstrate better discriminating power compared to conventional entropy measures. These promising findings validate the effectiveness and potential of the algorithms proposed in this paper.

Denatured recognition of biological tissue based on multi-scale phase weighted-permutation entropy during HIFU treatment

In the course of high-intensity focused ultrasound (HIFU) treatment, the capacity to precisely recognize biological tissue that has been denatured is crucial to ensuring the security and availability of HIFU treatment. Multi-scale permutation entropy (MPE) and its variant multi-scale weighted-permutation entropy (MWPE), as common methods to measure the complexity of nonlinear time series, are often used to recognize denatured biological tissue during HIFU treatment. In order to improve the inevitable disadvantages of MPE and MWPE in some cases, a new complexity method called multi-scale phase weighted-permutation entropy (MPWPE) is put forward. The proposed MPWPE improves MPE and MWPE by adding phase information through the Hilbert transform. The simulated analysis result indicates that the MPWPE can detect more dynamic changes in the synthetic signal compared with MPE and MWPE. Finally, based on the key MPWPE feature extraction algorithm, a novel intelligent biological tissue denatured recognition technology combined with the classifier is proposed. The actual HIFU echo signals of biological tissues are employed to verify the effectiveness of the proposed method. The results show that compared with MPE and MWPE, the MPWPE features can distinguish non-denatured and denatured tissues at multi scales with better performance and achieve higher recognition accuracy.

Spatio-temporal transferability of environmentally-dependent population models: Insights from the intrinsic predictabilities of Adélie penguin abundance time series

Ecological predictions are necessary for testing whether processes hypothesized to regulate species population dynamics are generalizable across time and space. In order to demonstrate generalizability, model predictions should be transferable in one or more dimensions, where transferability is the successful prediction of responses outside of the model data bounds. While much is known as to what makes spatially-oriented models transferable, there is no general consensus as to the spatio-temporal transferability of ecological time series models. Here, we examine whether the intrinsic predictability of a time series, as measured by its complexity, could limit such transferability using an exceptional long-term dataset of Adélie penguin breeding abundance time series collected at 24 colonies around Antarctica. For each colony, we select a suite of environmental variables from the Community Earth System Model, version 2 to predict population growth rates, before assessing how well these environmentally-dependent population models transfer temporally and how reliably temporal signals replicate through space. We show that weighted permutation entropy (WPE), a model-free measure of intrinsic predictability recently introduced to ecology, varies spatially across Adélie penguin populations, perhaps in response to stochastic environmental events. We demonstrate that WPE can strongly limit temporal predictive performance, although this relationship could be weakened if intrinsic predictability is not constant over time. Lastly, we show that WPE can also limit spatial forecast horizon, which we define as the decay in spatial predictive performance with respect to the physical distance between focal colony and predicted colony. Irrespective of intrinsic predictability, spatial forecast horizons for all Adélie penguin breeding colonies included in this study are surprisingly short and our population models often have similar temporal and spatial predictive performance compared to null models based on long-term average growth rates. For cases where time series are complex, as measured by WPE, and the transferability of biologically-motivated mechanistic models are poor, we advise that null models should instead be used for prediction. These models are likely better at capturing more generalizable relationships between average growth rates and long-term environmental conditions. Lastly, we recommend that WPE can provide valuable insights when evaluating model performance, designing sampling or monitoring programs, or assessing the appropriateness of preexisting datasets for making conservation management decisions in response to environmental change.

A mechanical part fault diagnosis method based on improved multiscale weighted permutation entropy and multiclass LSTSVM

Multi-scale permutation entropy (MPE) has poor stability, accuracy and noise immunity, recognition accuracy of twin SVM is low, so a diagnosis method based on improved multi-scale weighted permutation entropy (IMWPE) and least squares TSVM (LSTSVM) is proposed. The innovations include that the improved variational mode decomposition (IVMD) based on energy and mutual information is used to suppress noise, and novel composite coarse-grained and permutation pattern weighting methods are introduced to construct IMWPE, after which the noise immunity, stability and accuracy of features have been improved. LSTSVM is extended to a multi-class LSTSVM through one vs one (OVO) and binary tree (BT) strategies, the performance of the proposed methodology is significantly improved. The results show that signal-to-noise ratio of signal is enhanced, the stability of IMWPE is improved, the diagnosis accuracies for bearing and check valve are increased by 17.084 % and 12.498 %, which provides a reference for fault diagnosis under noise.

Online Chatter Identification for Thin-Walled Parts Machining Based on Improved Multisensor Signal Fusion and Multiscale Entropy

Thin-walled workpieces frequently appear to chatter during milling, causing the creation of vibration patterns on the surface of the workpiece, which affects the machining quality and workpiece performance. Identification of online chatter is important for thin-walled parts. The vibration signal associated with chatter changes dynamically throughout the milling process due to changes in tool position, removal of workpiece material, and transmission-related vibration signal attenuation characteristics, making reliable chatter identification difficult. This research proposes a unique online chatter identification approach for the machining of thin-walled parts based on improved multi-sensor signal fusion and multi-scale entropy. In order to acquire the fused vibration signals, the vibration signals produced during the machining of thin-walled parts are first acquired by sensors at various test sites and adaptively given weight coefficients by the improved Hausdorff distance and distance factor. Second, using the multi-scale sample entropy (MSSE) and multi-scale weighted permutation entropy (MSWPE), the eigenvalues of the fused vibration signals are obtained. Finally, the trained Logistic regression (LR) classification model is used to determine the vibration status of thin-walled parts. The analysis’s results demonstrated that the technique can properly identify the vibration state of thin-walled parts, and that the fusion signal can reflect the vibration state at the tool contact site more accurately.

SVMWPE-HLLE Based Fault Diagnosis Approach for Rolling Bearing

Multi-scale permutation entropy (MPE) is an analytical method describing the complexity of time series, which has been applied to the fault diagnosis of rolling bearings. To solve the problems of MPE in coarse-grained process and permutation entropy calculation, multi-scale weighted permutation entropy based on sliding variance (SVMWPE) was proposed in this paper. By analyzing WGN signal and 1/f noise signal, the parameter selections for SVMWPE were studied, and the stability and superiority were investigated by comparing SVMWPE with MPE, MWPE, and SVMPE. The high-dimensional matrix obtained by MPE feature extraction to pattern recognition was solved by introducing Hessian local linear embedding (HLLE) dimension reduction method, and the feature extraction method based on SVMWPE-HLLE was proposed. The clustering effect was studied by comparing SVMWPE-HLLE with SVMWPE-LLE through the analysis of three simulation signals. Fault diagnosis method for rolling bearing was proposed by combining SVMWPE-HLLE with extreme learning machine (ELM), which was applied to two experimental cases of rolling bearings for analysis. The experimental results showed that the proposed method can realize intelligent diagnosis of different fault types and degrees of rolling bearing, and the fault recognition rate of the proposed method was higher than other methods.

Quantifying inherent predictability and spatial synchrony in the aphid vector Myzus persicae: field-scale patterns of abundance and regional forecasting error in the UK.

Sugar beet is threatened by virus yellows, a disease complex vectored by aphids that reduces sugar content. We present an analysis of Myzus persicae population dynamics with and without neonicotinoid seed treatment. We use 6 years' yellow water trap and field-collected aphid data and two decades of 12.2m suction-trap aphid migration data. We investigate both spatial synchrony and forecasting error to understand the structure and spatial scale of field counts and why forecasting aphid migrants lacks accuracy. Our aim is to derive statistical parameters to inform regionwide pest management strategies. Spatial synchrony, indicating the coincident change in counts across the region over time, is rarely present and is best described as stochastic. Uniquely, early season field populations in 2019 did show spatial synchrony to 90 km compared to the overall average weekly correlation length of 23 km. However, 70% of the time series were spatially heterogenous, indicating patchy between-field dynamics. Field counts lacked the same seasonal trend and did not peak in the same week. Forecasts tended to under-predict mid-season log10 counts. A strongly negative correlation between forecasting error and the proportion of zeros was shown. Field populations are unpredictable and stochastic, regardless of neonicotinoid seed treatment. This outcome presents a problem for decision-support that cannot usefully provide a single regionwide solution. Weighted permutation entropy inferred that M. persicae 12.2m suction-trap time series had moderate to low intrinsic predictability. Early warning using a migration model tended to predict counts at lower levels than observed. © 2022 The Authors. Pest Management Science published by John Wiley & Sons Ltd on behalf of Society of Chemical Industry.

Instability of Low-Velocity and High Water-Cut Oil–Water Two-Phase Flow Based on Microwave Signals

Accurately understanding the nonlinear and complex dynamics underlying low-velocity and high water-cut oil–water two-phase flow is a significant but challenging problem to the study of flow pattern boundary transformation and flow heat transfer. In this article, we propose a nonlinear analysis method to characterize and distinguish patterns from low-velocity and high water-cut oil–water two-phase flow based on the fractional entropy algorithm that have found great usage in complex science. First, we obtain the variable measurement and images of the oil–water two-phase flow by a microwave sensor and a high-speed camera, respectively. Second, we analyze the complexity of flow structure using recurrence algorithms and adaptive optimal kernel time–frequency representation. Finally, we propose a dynamic nonlinear analysis framework to characterize flow instability based on multiscale fractional weighted permutation entropy. The results show that the entropy rate and its mean value are sensitive to the flow structure. These interesting and significant findings suggest that the multiscale fractional weighted permutation entropy can potentially be a powerful tool for uncovering the underlying dynamics leading to the formation of two-phase flow.

Distribution Network Fault-Line Selection Method Based on MICEEMDAN–Recurrence Plot–Yolov5

Distribution system fault signals contain severe noise components. In order to solve the problem of distribution network fault-line selection, a fault-line selection method based on modifying the Improved Complete Ensemble Empirical Mode Decomposition Adaptive Noise (MICEEMDAN) algorithm, Recurrence Plot, and Yolov5 network is proposed. First, ICEEMDAN is optimized using multi-scale weighted permutation entropy (MWPE). MICEEMDAN can decompose an electrical signal into a series of intrinsic mode functions (IMFs). Recurrence Plot transformation of all IMFs, obtained from decomposition and stitching from top to bottom, realizes the conversion of 1D time series to 2D images. Then, the recurrence maps obtained from all lines in the distribution network are stitched to obtain the distribution network recurrence map, realizing the mining of the fault-signal features of the whole distribution network. Finally, the Yolov5 network is used to mine the fault features of the recurrence map of the distribution network autonomously to realize the fault-line selection. The experiments show that the method has a good noise immunity and 99.98% fault-selection accuracy, which can effectively complete the distribution network fault selection.

Exploring the impact of flow values on multiscale complexity quantification of airport flight flow fluctuations

Being able to characterize the dynamic characteristics of the air traffic system and embody the dynamic evolution thereof, the complexity of air traffic flows has attracted the broad attention of scholars and researchers. To fully consider amplitude information, we adopt the improved multiscale weighted permutation entropy (IMWPE) and propose an improved multivariate multiscale weighted permutation entropy (IMMWPE) to explore the complexity of flight flows of the top ten busy airports in China from a univariate and a multivariate perspective, respectively. In addition, we propose an improved multiscale weight contingency to investigate the inhomogeneity between fluctuation pattern distributions in departure and arrival flight flows. At the advantages of the multiple coarse-graining procedures, we obtain more reliable and stable entropy and contingency results, considering amplitude information. Comparisons between the weighted and unweighted results show that the weighted methods are as capable as the unweighted in exactly characterizing the dynamic characteristics of flight flow fluctuations, i.e., temporal scale dependency, spike distributions, differences between univariate and multivariate perspectives. Results also show that the weight has a stronger impact on the complexity of univariate time series than the multivariate. From the univariate perspective, the complexity of departure and arrival flight flows of ZLXY and ZGGG is more sensitive to the amplitude information. From the multivariate perspective, ZUCK could be a typical representative of the complexity of the total flight flows of the ten airports.

Fault Diagnosis Method Based on AUPLMD and RTSMWPE for a Reciprocating Compressor Valve.

In order to effectively extract the key feature information hidden in the original vibration signal, this paper proposes a fault feature extraction method combining adaptive uniform phase local mean decomposition (AUPLMD) and refined time-shift multiscale weighted permutation entropy (RTSMWPE). The proposed method focuses on two aspects: solving the serious modal aliasing problem of local mean decomposition (LMD) and the dependence of permutation entropy on the length of the original time series. First, by adding a sine wave with a uniform phase as a masking signal, adaptively selecting the amplitude of the added sine wave, the optimal decomposition result is screened by the orthogonality and the signal is reconstructed based on the kurtosis value to remove the signal noise. Secondly, in the RTSMWPE method, the fault feature extraction is realized by considering the signal amplitude information and replacing the traditional coarse-grained multi-scale method with a time-shifted multi-scale method. Finally, the proposed method is applied to the analysis of the experimental data of the reciprocating compressor valve; the analysis results demonstrate the effectiveness of the proposed method.