Time Shift Multiscale Ensemble Fuzzy Dispersion Entropy and Its Application in Bearing Fault Diagnosis

Abstract

Accurate detection of surface defects such as wear, cracks, and flaws in metallic components is critical for equipment reliability and longevity, representing a core challenge in surface integrity engineering. To solve the information loss, low estimation accuracy and poor noise immunity associated with Multiscale Dispersion Entropy (MDE) are utilized to address the sensitivity to parameter selection and overfitting susceptibility of the Least Squares Twin Support Vector Machines (LSTSVM). A brand new fault diagnosis method which combined Time Shift Multiscale Ensemble Fuzzy Dispersion Entropy (TSMEFuDE) with binary tree LSTSVM (BT LSTSVM) was proposed. Firstly, a time shift method based on Higuchi Fractal Dimension was introduced to TSMEFuDE, resolving the continuity loss between coarse-grained levels. Second, four mapping techniques, linear, NCDF, tansig and logsig, are introduced. This synergetic combination of each advantage results in the improvement of entropy output stability. Furthermore, triangular and trapezoidal membership functions are incorporated into dispersion patterns and abolished in the round function, therefore enhancing the boundaries between the classes after signal mapping to discrete classes. Lastly, the proposed BT LSTSVM algorithm decomposes the multi-classification problem to a binary classification problem, which promotes the robustness of the algorithm. Simulation experiments maintain that TSMEFuDE has stronger adaptability, higher stability, and better noise resistance. In the fault diagnosis experiment, when compared to the Multiscale Fuzzy Dispersion Entropy (MFuDE) combined with the BT TSVM method, the TSMEFuDE combined with BT LSTSVM method improved the accuracy of bearing fault diagnosis by 5.65% and 2.82%.

1. Introduction

As the core “joint” of the industry equipment, the performance of the bearing directly determines the reliability of the rotating machine. The timely detection of surface defects—such as pitting, cracks, and wear in metallic components—is critical for preventing catastrophic machinery failures, minimizing downtime, and ensuring operational safety. Fault diagnosis techniques are essential for identifying these defects early. According to the data, bearings account for 84% of the faults in wind power transmission systems, with main bearings contributing 30% individually [1]. In the aero engine area, the maintenance of the main bearing costs over 60% of the total cost [2]. Complex working conditions frequently lead to metal fatigue and wear [3]. Traditional predictive models have struggled to provide effective warnings. Once a failure occurs, it can trigger a chain reaction of equipment downtime, financial loss, and even safety accidents. In the field of bearing fault diagnosis, the academic community has proposed various fault diagnosis techniques. Acoustic emission detection captures early micro-defects by making use of the high-frequency stress wave signals [4]. Although its sensitivity has significantly improved, it still needs a complex noise reduction process. The fault diagnosis techniques based on oil analysis focus on analyzing wear particles, pollutants and other components in oil, which can assess the degree of wear and the type of fault in the equipment [5]. However, both of these detection methods have certain limitations in bearing fault diagnosis. Mechanical equipment generates vibration signals during work, and fault diagnosis based on vibration signals has been attracting growing attention from the academy and industry.

Vibration signal-based fault feature extraction encompasses time domain, frequency domain, time–frequency analysis, and multiscale entropy methods. While time and frequency domain approaches offer intuitive primary feature extraction [6], they exhibit limitations for non-stationary signals. Time–frequency techniques (e.g., Wavelet Transform) address this constraint [7]. Multiscale entropy methods—including Multiscale Sample Entropy (MSE), Multiscale Permutation Entropy (MPE), and Multiscale Dispersion Entropy (MDE) [8]—demonstrate enhanced operational adaptability and feature robustness by quantifying multi-scale complexity and uncertainty. However, three persistent limitations remain: ① Inadequate representation of nonlinear dynamic features. ② Loss of continuous state information during coarse-graining. ③ Sensitivity to noise interference and short-sequence data.

These constraints have motivated recent innovations in entropy-based feature extraction. Scholars have developed various multiscale entropy methods, with classical approaches exhibiting distinct characteristics: Multiscale Sample Entropy (MSE) [9] overcomes single-scale constraints of Sample Entropy (SE) [10] through multiscale frameworks, enhancing signal analysis accuracy and robustness. However, it suffers from: (1) sensitivity to time series length; (2) limited handling of abnormal data; (3) inability to identify adjacent amplitude features. Multiscale Attention Entropy (MAE) evaluates multi-scale complexity but experiences entropy value fluctuations at high scale factors due to shortened coarse-grained sequences, impairing feature stability [11]. Multiscale Dispersion Entropy (MDE) [12] integrates discrete entropy with Multiscale analysis to improve fault detection and noise resilience, yet loses critical high-frequency details at large scales. Multiscale Slope Entropy (MSlopEn) [13] neglects segment data correlations during coarse-graining, causing statistical information loss. Multiscale Permutation Entropy (MPE) [14] offers algorithmic simplicity and noise robustness but reduces statistical reliability as scale factors increase. Multiscale Fuzzy Entropy (MFE) [15] captures multiscale features effectively but exhibits parameter sensitivity and results instability.

To overcome the limitations of the previous methods, researchers have proposed many improved multiscale entropy methods in recent years. The Time Shift Entropy method has been introduced into these approaches [16], leading to the development of several important variants: Time shift Multiscale Dispersion Entropy (TSMDE), Time shift Multiscale Permutation Entropy (TSMPE) [17], Time shift Multiscale Slope Entropy (TSMSlopEn) [18], Time Shift Multiscale Fuzzy Entropy (TSMFuE) [19], Time shift Multiscale Increment Entropy (TSMIncrE) [20], and Time shift Multiscale Range Entropy (TSMRE) [21]. These methods use time shift strategies to analyze signal complexity patterns across different scales. When processing information from a large amount of data and non-stationary signals, the TSMDE approach performs very well. Though Kaixuan Shao’s TSMDE captures signal complexity better than similar methods, it still struggles with noise interference [22].

To address the issue of poor noise resistance, recent research has focused on multimodal fusion. Mostafa Rostaghi et al. [23] proposed the Fuzzy Dispersion Entropy (FuDE) by introducing the fuzzy membership function, which reduces the information loss that occurs when the signal is mapped to the dispersion mode. Experimental results demonstrate that FuDE has stronger resistance to noise. Therefore, Yuxing Li et al. [24] proposed the Multiscale Fuzzy Dispersion Entropy (MFuDE), a method that combines the previously mentioned Multiscale entropy approach and FuDE and demonstrates its stronger capabilities in feature extraction. Hamed Azami et al. [25] proposed Ensemble Entropy (EE), which combines multiple algorithms or parameters and significantly improves the robustness and noise resistance of signal analysis. Furthermore, it may capture new characteristics missed by conventional approaches. However, the EE method has limitations, such as high computational complexity and the calculation results depend on the design of the ensemble strategy.

In summary, the current entropy methods still face four bottleneck issues: weak noise resistance, loss of coarse-grained information, insufficient dynamic representation, and low stability [26]. Although the latest improvement strategies can partially alleviate the problems, there comes new flaws: MFuDE enhances noise resistance through fuzzy membership functions, but stability decreases at high-frequency scales, and the settings depend on previous experience. EE mixes multiple algorithms to improve robustness, but the computational complexity is high and interpretability is poor. The TSM improves the retention of coarse-grained information but does not significantly enhance noise resistance and is easily affected by dynamic signal interference. A single method is difficult to address all four major bottleneck issues simultaneously, requiring collaborative combination.

Pattern recognition plays a crucial role in fault diagnosis. Commonly used pattern recognition methods include statistical methods [27], supervised learning, and convolutional neural networks. Of these, the Support Vector Machine (SVM) is the most frequently utilized in supervised learning [28]. Support Vector Machine (SVM) enhances generalization ability by maximizing the classification margin, making it suitable for high-dimensional data and small sample circumstances. Kernel functions can address nonlinear problems and resist overfitting. However, the standard SVM is essentially a binary classification model, while fault diagnosis in practical engineering often involves multi-condition recognition. To address this, various improvements have been proposed in the academic circle to extend SVM for solving multi-class problems and further enhance its performance. OVO SVM reduces inter-class interference through a one-vs-one strategy, making it suitable for multi-class problems. This method offers higher classification accuracy and computational efficiency when handling complex boundaries, making it especially suitable for fault diagnosis in high-dimensional feature spaces. Yang Liu [29] applied the OVO SVM method for text sentiment classification and achieve a higher test accuracy. The OvR SVM proposed by Xiaobin Xu et al. [30] has achieved high prediction accuracy in rotating machinery fault diagnosis. The model is simple to train, requires fewer samples, and has high computational efficiency, making it especially suitable for applications to a small number of categories and significant inter-class differences. Unlike traditional SVM, which finds a hyperplane to maximize the internal between two classes of samples, the Twin Support Vector Machine (TSVM) does classification by constructing two non-parallel hyperplanes. Zhiwen Liu et al. [31] significantly reduced computational complexity by applying TSVM to mechanical fault diagnosis. LSTSVM was proposed by M.A. Ganaie et al. [32], which replaces quadratic programming optimization by solving a system of linear equations. Although LSTSVM significantly improves computational efficiency compared to TSVM, both are limited to binary classification tasks and require one-vs-rest (OvR) or hierarchical extension strategies to achieve multi-fault pattern identification.

To address the above issues, a fault diagnosis method based on Time Shift Multiscale Ensemble Fuzzy Dispersion Entropy (TSMEFuDE) and BTLSTSVM is proposed. The following is a summary of the innovations and the main contributions of this work:

A Time Shift Multiscale coarse-grained method (TSM) is introduced into Ensemble Fuzzy Dispersion Entropy, which enhances the coarse-grained sequences. This method can analyze the complexity of signals and the phase distribution of signals at different time scales, solving the problem of continuous sequence point loss encountered by traditional coarse-grained methods.

A new single scale entropy ensemble fuzzy dispersion entropy (EFuDE) is proposed, enhancing the noise resistance and stability of the entropy feature. The specific innovations are as follows: ① Based on the diversity of the signal amplitude, four mapping methods (linear, NCDF, tansig, and logsig) are used to map the original signal into multiple discrete categories. This approach fully takes advantage of each mapping method, effectively reducing sensitivity to noise and outliers, and shows more stability, especially when handling short-duration signals. ② EFuDE excludes the lossy round classification function, instead employing trapezoidal and triangular membership functions to maximize entropy and strengthen inter-category boundaries after signal mapping. ③ Ensemble entropy is introduced to handle more complex synthetic data, which includes different types of noise and mixing processes. It utilizes its superior ability to distinguish between different types of noise, thereby improving the signal feature extraction and noise resistance capabilities.

TSMEFuDE significantly enhances the stability of feature extraction, noise robustness, and adaptability to signal length, while reducing parameter sensitivity, by introducing a time-shifted multiscale strategy to address information loss and integrating various mapping functions and membership functions to address stability and noise issues. This provides more reliable feature inputs for subsequent fault diagnosis.

By innovatively introducing the binary tree structure (BT) into LSTSVM, the accuracy and flexibility of fault diagnosis for complex systems are significantly improved.

The organization of this paper is as follows: In Section 2, the theory of Time shift Multiscale Ensemble Fuzzy Dispersion Entropy (TSMEFuDE) is elaborated in detail, with a quantitative analysis of the impact of parameters such as embedding dimension and time shift step size on TSMEFuDE. Stability, noise resistance, and multi-class fault feature separability of TSMEFuDE are systematically verified. In Section 3, the theory of BT-LSTSVM is fully derived. Section 4 presents the fault diagnosis method based on TSMEFuDE and BT-LSTSVM, along with an algorithm flowchart. Section 5 conducts fault diagnosis experiments on triaxial bearings and aeroengine bearings, comparing and analyzing the feature extraction performance and classifier recognition accuracy. The experimental conclusions are summarized in Section 6, which also covers the value of engineering applications.

2. Time Shift Multiscale Ensemble Fuzzy Dispersion Entropy

2.1. Time Shift Multiscale Ensemble Fuzzy Dispersion Entropy

(1)

An innovative method has been designed to introduce the time shift multi-scale analysis of Higuchi Fractal Dimension (HFD) into EFuDE. With the help of HFD, this stable numerical analysis method may accurately manage different kinds of time series data, such as stationary, non-stationary, deterministic, random, and stationary signals. This method can effectively alleviate the problem of correlated information being lost in continuous sequences during the coarse-graining process. For the original signal $X=\left\{x_1,x_2,\cdots ,x_L\right\}$, coarse-graining is performed according to Equation (1), resulting in the sequence $y_t^s$ at different scales s [33].

$$y_t^s=\left(x_s,x_{s+t},x_{s+2t},\cdots ,x_{s+⌊{\displaystyle \frac{N-s}{t}}⌋t}\right)$$

(1)

In the formula, ${\displaystyle \frac{N-s}{t}}$ is the rounded integer value, where s and t represent the initial time point and time interval, respectively, and $\left(1\le t\le s\right)$ the starting point of the time series. The time-shifted coarse-graining process for a scale factor s = 3 is shown in Figure 1.

(2)

To retain as much information as possible contained in the signal and enhance the stability of the entropy value, we apply the averaging output strategy shown in Equation (2) to the coarse-grained sequence at each scale, calculating the average complexity across all scales. The computation steps of EFuDE are as follows:

$$TSMEFuDE\left(X,m,c,s,d\right)={\displaystyle \frac{1}{s}}\sum _{t=1}^s EFuE\left(y_t^s,m,c,d\right)$$

(2)

Step 1: Fully utilize the advantages of different mapping methods. Each sequence applies four mapping methods—linear, logsig, tansig, and NCDF—corresponding to Formulas (3)–(6) [25]. The $\tau$ coarse-grained sequences are then mapped into c discrete classes.

$${\eta }_j=\left({\displaystyle \frac{x-\mathit{min}\left(x_s\right)}{\mathit{max}\left(x_s\right)-\mathit{min}\left(x_s\right)}}\cdot \left(1-ϵ\right)+ϵ\right)\cdot nc+0.5$$

(3)

$${\xi }_j={\displaystyle \frac{1}{1+e^{-\frac{x_s-\mu }{\sigma }}}}$$

(4)

$${\kappa }_j={\displaystyle \frac{2}{1+e^{-2\frac{x_s-\mu }{\sigma }}}}-1$$

(5)

$${\lambda }_j={\displaystyle \frac{1}{\sigma \sqrt{2\pi }}}\underset{-\infty }{\overset{x_s}{\int }} e^{{\displaystyle \frac{-(t-\mu {)}^2}{2{\sigma }^2}}}dt$$

(6)

In the formula, $x_s$ represents the coarse-grained sequence, and $ϵ$ in Formula (3) is a very small constant used to avoid values which equal to 0 or 1. $nc$ is the grouping length. In Formulas (4)–(6), $\sigma$ and $\mu$ represent the standard deviation and the mean of the sequence, respectively.

The recognition accuracy of the BT LSTSVM algorithm is the highest among all methods, except for MPE. It is clear from this that, in comparison to other models, the BT LSTSVM model has a definite advantage in terms of fitting ability.

Although TSMEFuDE does not obtain an absolute advantage over all prediction algorithms, it can be seen that, in most circumstances, its prediction accuracy is not very different from the optimal one.

Except for the four basic prediction models, such as BPNN, where TSMEFuDE performs poorly, the accuracy achieved by TSMEFuDE is quite remarkable, fully demonstrating that the features extracted by this method have better distinguishability.

The degradation process of mechanical parts lasts a long time, and the categories and boundaries of fault states are fuzzy. Therefore, it is difficult for traditional features to characterize the degradation trend of parts. DE can measure the complexity and chaotic characteristics of the signal, but its performance is not very good in tracking the state of check valve and bearing.

Because of the use of normal cumulative distribution function (NCDF) mapping, the trend in vibration data is not completely taken into consideration in the DE feature, and because the description of the vibration data distribution characteristics by DE is not sufficiently accurate. In addition, DE is easily disturbed by small fluctuations and noise, and the reliability of tracking is poor. Therefore, a sliding dispersion entropy (SDE) based on sliding window down-sampling and TANSIG mapping is proposed. Assuming that the vibration signal of the mechanical part at the k-th $(k=1,2,\cdots ,K)$ time point has been collected, the SDE feature of the vibration signal at the current time point can be expressed as $SD{E}_k$, and its calculation procedures are as follows.

Among them, the following are the advantages of these four algorithms:

①

Linear mapping: Easy to implement and low-cost, specializing in linear relationship modeling, dimensionality reduction and data standardization.

②

Logsig function mapping: Specifically suitable for probabilistic prediction and binary classification tasks, the output is limited to a range of 0 to 1, but gradient vanishing may be faced in deep networks.

③

Tansig function mapping: The mapping output is to the interval −1 to 1, which is suitable for processing complex signals with positive and negative values, but it is also possible to encounter gradient vanishing in deep network training.

④

NCDF mapping: Optimize data uniformity by converting to normal distribution, which significantly improves model stability and prediction accuracy, being especially suitable for data standardization.

The comprehensive use of linear, logsig, Tansig and NCDF mapping methods can make full use of their advantages in data preprocessing, such as maintaining the original scale of data, processing nonlinear relationships and normalizing distribution. This multi-strategy approach not only optimizes the stability of model training, but also significantly improves the prediction accuracy and the adaptability of the model to complex data.

Step 2: Calculate the affiliation. Since ${\mathit{y}}_p^{\prime }$ is mapped to c discrete classes, the ambiguity of the boundaries between classes and the change in signal amplitude caused by noise may affect the classification results. This work departs from the conventional round classification function to solve this problem. By using two kinds of fuzzy membership functions MK, we can deal with each category more accurately, so as to enhance the class boundary after the signal is mapped to the classification sequence. In the calculation process, class 1 and class c apply trapezoidal membership functions (see Equations (7) and (8)), while other categories (K ≠ 1 and K ≠ c) use triangular membership functions (see Equation (9)).

In this paper, class 1 and class c use trapezoidal membership functions, while other classes (K ≠ 1 and K ≠ c) use triangular membership functions. The specific fuzzy membership functions are as follows:

$${\mu }_{M_1}\left(\alpha \right)=\left\{\begin{array}{ccc}0& & \alpha >2\\ 2-\alpha & & 1\le \alpha \le 2\\ 1& & \alpha <2\end{array}\right.$$

(7)

$${\mu }_{Mk}\left(\alpha \right)=\left\{\begin{array}{ccc}0& & \alpha <k+1\\ k+1-\alpha & & k\le \alpha \le k+1\\ \alpha -k+1& & k-1\le \alpha \le k\\ 0& & \alpha <k-1\end{array}\right.$$

(8)

$${\mu }_{Mc}\left(\alpha \right)=\left\{\begin{array}{ccc}1& & \alpha >c\\ \alpha -c+1& & c-1\le \alpha \le c\\ 0& & \alpha <c-1\end{array}\right.$$

(9)

where Equation (8) k = 2 to c − 1.

Step 3: Build pattern vectors. For a given embedding dimension m, delay d, and fuzzy set number nc, all possible pattern vectors need to be constructed, and there are a total of nc^m possible pattern vectors. Each ${\eta }^c$, ${\xi }^c$, ${\kappa }^c$, ${\lambda }^c$ and ${\omega }^c$ is mapped to the corresponding scatter mode ${\pi }_{v_0{v}_1\dots v_{m-1}}$. The calculation method is shown in Equation (10), where $j=\mathrm{1,2},\dots ,N-(m+1)d.$

$$z_j^{m,c}=\left\{z_j^c,z_{j+\left(1\right)d}^c,\dots ,z_{j+\left(m-1\right)d}^c\right\}$$

(10)

Step 4: Probability calculation of the scattering fuzzy pattern. Each pattern vector is mapped to a different scatter pattern ${\pi }_{v_0{v}_1\dots v_{m-1}}$ based on its membership degree. Each sample point $z_{j+\left(m-1\right)d}^c$ of each pattern vector has a corresponding membership degree according to its category $v_i$. Only when the membership degrees of all sample points are true (that is, non-zero), the membership degree of the whole pattern is true. We use t-norm for judgment, such as Equation (11). Where i is the index of the number of time delay steps starting from the current sample point j and looking back.

$${\mu }_{\pi v0v1\dots vm-1}\left(z_j^{m,c}\right)=\prod _{i=0}^{m-1}{\mu }_{M_{vi}}\left(z_{j+\left(i\right)d}^c\right)$$

(11)

Step 5: Calculate the probability of fuzzy dispersion mode. Determine the probability of each mode according to Equation (12). $p\left({\pi }_{v_0,v_1,\dots ,v_{m-1}}\right)$ reflects the frequency of the pattern in the whole time series.

$$p\left({\pi }_{v_0,v_1,\dots ,v_{m-1}}\right)={\displaystyle \frac{{\sum }_{j=1}^{N-\left(m-1\right)d} {\mu }_{{\pi }_{v_0{v}_1\dots v_{m-1}}}\left({\mathit{z}}_j^{m,c}\right)}{N-\left(m-1\right)d}}$$

(12)

Step 6: Calculate the set fuzzy dispersion entropy. Finally, based on the information entropy theory and combined with the fuzzy dispersion entropy calculation results under four different fuzzy mapping methods.

$$\begin{array}{c}EFuDE\left(X,m,c,tau\right)=\\ {\displaystyle \sum _{\pi =1}^{c^m}} p\left({\pi }_{v_0,v_1,\dots ,v_{m-1}}\right)·ln p\left({\pi }_{v_0,v_1,\dots ,v_{m-1}}\right)\end{array}$$

(13)

(3)

Iterative scale S, i.e., S = S + 1, repeat the calculation process of steps (1) and (2) until S reaches the preset value. The entire calculation process is shown in Figure 2, where the red box mainly describes some important updates and compares with the EDE to show the improvements made by TSMEFuDE. We completed the entire TSMEFuDE calculation.

2.2. The Influence of Parameters on TSMEFuDE

Sequence length, embedding dimension m, time delay t, and number of classes c are key parameters that influence the entropy-based features of TSMEFuDE. In order to further study the specific impact of these core parameters on the TSMEFuDE characteristics of the time series, a series of numerical simulations was conducted, based on white noise (WGN) and 1/f noise. Each noise-type dataset contains 3000 randomly selected data points, with a sampling frequency set at 5000 Hz. Figure 3 displays the time domain diagram and spectrum. Notably, the amplitude of WGN shows a random distribution in both time and frequency domains. In contrast to 1/f noise, WGN demonstrates greater variability and unpredictability.

To verify the superior performance of TSMEFuDE under different parameter settings, we conducted a comparative analysis. TSMEFuDE was compared with Composite Multiscale Fuzzy Dispersion Entropy (CMFuDE), Refined Composite Multiscale Fuzzy Dispersion Entropy (TSMDE), Reverse-Cumulative Multiscale Fuzzy Dispersion Entropy (RCMEFuDE), and Time-Shifted Multiscale Ensemble Fuzzy Dispersion Entropy (TSMEFuDE). All experiments were performed on a 12th Gen Intel ^® Core ™i7-12700 processor configured at 2.30 GHz and supported by MATLAB 2023b software.

2.2.1. The Impact of Series Length N on TSMEFuDE

Increasing the signal sequence length N can better reveal the intrinsic complexity of the sequence and reduce the noise and boundary effects. Nevertheless, this results in reduced computational efficiency. To explore the relationship between computational efficiency and accuracy, we conducted experiments using WGN and 1/f noise with sequence lengths of 2000, 4000, 6000, 8000, and 10,000 respectively. The entropy or complexity values extracted by each method are shown in Figure 4a. Except for Figure 4c–g, which all have a Y-axis scale of 0.4, Figure 4c uses a Y-axis scale of 2, and Figure 4g uses a Y-axis scale of 0.8. The following conclusions can be drawn from the analysis results:

①

By comparing Figure 4a MFuDE with Figure 4b TSMFuDE, as well as Figure 4e MEFuDE with Figure 4h TSMEFuDE, it is observed that the curves become smoother. This indicates that the introduction of the time-shifted analysis method significantly enhances the stability of the entropy values.

②

A comparison between Figure 4c,d and Figure 4g,h shows that the introduction of Ensemble Entropy helps to smooth the curves. From the figure, it is obvious that the entropy value becomes more stable, leading to a smoother of the curve.

③

The TSMEFuDE method we proposed combines the benefits of the previously mentioned approaches, as illustrated in Figure 4h, the entropy curves for the two types of noise almost overlap, indicating that TSMEFuDE exhibits the best stability and noise resistance.

To evaluate the stability of TSMEFuDE under different sequence lengths, the Coefficient of Variation (CV) was adopted as the evaluation metric. The calculation formula of CV is $CV=(\sigma /\mu )\times 100\mathrm{\%}$, where σ and μ, respectively, represent the standard deviation and mean value of TSMEFuDE at the same scale. Analysis of the experimental results for WGN and 1/f noise under five different sequence lengths is presented in Figure 5a,b. The figure clearly shows that TSMEFuDE consistently yields the lowest CV values across all tested sequence lengths. This provides strong evidence of TSMEFuDE outstanding stability under different values of N.

2.2.2. The Impact of Embedding Dimension m, Time Delay t and Class Number c on TSMEFuDE

The embedding dimension m, time delay t, and class number c are the most fundamental and commonly used parameters in dispersion entropy analysis. These characteristics have been incorporated into the proposed TSMEFuDE algorithm. Therefore, we constructed a WGN with a length of 3000 and 1/f noise to analyze the influence of m, t, and c on TSMEFuDE. Setting the embedding dimension m too low may result in an inability to effectively capture the nonlinear features of the time series [34]. Conversely, an excessively huge m, will lead to lower computing efficiency. For a detailed analysis of the effect of embedding dimension m on TSMEFuDE, the time delay t and class number c were fixed at 1 and 5, respectively. WGN and 1/f noise sequences of length 3000 were constructed, and the entropy values were taken from a computed form ranging from 2 to 5. To evaluate the impact of m on computation time, error bar charts were used to illustrate the average, maximum, and minimum computation times corresponding to each value of m, as shown in Figure 6a–c. Based on these results, we can draw the following conclusions: ① With the increase in M, entropy will increase, but the fluctuation of entropy curve will decrease. ② The time required to calculate entropy grows as parameter m increases. ③ Regardless of how m varies, the TSMEFuDE entropy curve always has the smoothest value.

During the process of phase space reconstruction, the separation between neighboring data points that were used to create phase space vectors is determined by the time delay t. In order to record the dynamic activity of time series data, this parameter is necessary. This study examines the impact of time delay t on TSMEFuDE. For this purpose, we fixed the embedding dimension m at 2 and parameter c at 5, then computed TSMEFuDE values for two sets of time series with distinct noise levels across t ranging from 1 to 5. As shown in Figure 6d–f, the results indicate that, under two noise conditions, the TSMEFuDE entropy curves corresponding to different t values are closely clustered. This demonstrates the high robustness of the method. Meanwhile, we also observed that the computation efficiency is higher when the time delay t is set to 3 or 4.

Class number c determines how many discrete intervals the continuous time series data are divided into, which affects the precision of the sequence classification. Selecting the appropriate number of classes c is very important for effectively capturing the complexity of time series, and the inappropriate number of classes may lead to the loss or oversimplification of information. In this study, we set the embedding dimension m to 2 and the time delay t to 1 to explore the effect of different classes c on entropy. As shown in Figure 6g–i, we observe that the TSMEFuDE method has a higher entropy than the other two methods, suggesting that the method may be more effective in revealing the complexity of time series. However, we also note that as the number c increases, the computation time also increases significantly, although the entropy increases accordingly.

2.3. Validity Verification Experiment of TSMEFuDE

2.3.1. Validity Verification Experiment of Time Shift

To validate the superiority of TSMEFuDE, we conducted a comparative analysis between the TS (Time Shift) coarse-graining strategy and the M (Multiscale) coarse-graining strategy. By constructing five distinct entropy methods and applying two coarse-graining strategies, respectively, we can compare the relative advantages of the TS (Time Shift) and M (Multiscale) approaches. These entropy methods include permutation entropy (PE), slope entropy (SlopEn), dispersion entropy (DE), fuzzy dispersion entropy (FuDE), and ensemble fuzzy dispersion entropy (EFuDE). The influence of different coarse-grained methods on error bar, coefficient of variation and computing time is analyzed by constructing 30 groups of WGN and 1/f noises with a length of 3000. As shown in the figure, the stability and calculation duration of each entropy method under different scale factors are shown by observing and summarizing the characteristics of the diagram, the following conclusions can be drawn: ① By comparing Figure 7a and Figure 7c, it can be seen that when white noise and 1/f noise are processed by different methods, the entropy value changes little under different scale factors, showing good robustness. This shows that these methods have strong adaptability under different types of noise. It should be noted that the curve presented by MPE method in WGN and 1/f noise is a straight line, which cannot effectively express the complexity characteristics of WGN and 1/f noise. However, the curve of MFuDE under 1/f noise decreases with the increase in scale factor, which does not conform to the curve characteristic diagram of 1/f noise. Therefore, MPE and MFuDE methods will not be considered in future discussions. ② By comparing Figure 7b with Figure 7d, it can be seen that the proposed TSMEFuDE method has the smallest CV. Under two groups of noise, CV values under different scale factors show that the stability of TSMEFuDE is the highest. Figure 7c,f shows the maximum calculation time and average calculation time of various entropy methods under 30 groups of signals. It can be shown from the figure that the M-based method is much better than the TS based method in computing time. However, the main disadvantage of the M-based method is that its stability is poorer than the TS method. In addition, the introduction of set entropy also significantly increases the calculation time, making MEFuDE and TSMEFuDE the longest in the same kind of methods. To sum up, although TSMEFuDE takes a long time to calculate, its excellent accuracy and stability make it a compromise solution. Although other methods are fast in calculation, they have no obvious advantages in stability and volatility, so TSMEFuDE is considered to be a better choice.

2.3.2. Verification of Amplitude Frequency Characterization Ability and Noise Resistance Performance

In order to verify the characterization ability of the proposed TSMEFuDE for amplitude and frequency, we constructed five groups of AM FM signals with length of 2000, as shown in Equation (14), and the amplitude and frequency change at equal intervals. The entropy value curve is shown in Figure 8. The following conclusions can be drawn:

$$\left\{\begin{array}{c}x1=[1+0.1\mathit{cos}(30\pi t)]\mathit{cos}(20\pi t+30\pi t^2)\\ x2=[1+0.3\mathit{cos}(30\pi t)]\mathit{cos}(40\pi t+30\pi t^2)\\ x3=[1+0.5\mathit{cos}(30\pi t)]\mathit{cos}(60\pi t+30\pi t^2)\\ x4=[1+0.7\mathit{cos}(30\pi t)]\mathit{cos}(80\pi t+30\pi t^2)\\ x5=[1+0.9\mathit{cos}(30\pi t)]\mathit{cos}(100\pi t+30\pi t^2)\end{array}\right.$$

(14)

Under all scale factors, the five entropy curves obtained by (f) TSMEFuDE have higher coincidence degree, and the resulting curves are smoother, which shows that the noise robustness of the proposed method is the best.

For the verification experiment of anti-noise performance of TSMEFuDE model, we constructed a set of AM-FM signals with a length of 2000, as shown in Equation (15). Then, we added white noise with signal to noise ratios (SNR) of 5 dB, 10 dB, 15 dB, 20 dB and 25dB, respectively, to obtain five groups of signals, namely ‘Pure signal’, ‘SNR = 15’, ‘SNR = 20’, ‘SNR = 25’, ‘SNR = 30’. The entropy values under these five conditions are shown in the figure. The analysis results show that ① The five entropy curves obtained by TSMEFuDE method almost coincide under all scale factors, indicating that the method has excellent noise robustness. ② It can be seen from Figure 8a,d that the introduction of time-shifted multi-scale analysis method significantly improves the noise robustness. ③ From the comparison between Figure 8d and Figure 8e, it can be concluded that the introduction of set entropy improves noise robustness.

$$\begin{array}{c}x\left(t\right)=\left[\begin{array}{c}2+0.1cos\left(8\pi t\right)\left)cos\right(150\pi t+2\cos \left(5\pi t\right))\end{array}\right]\\ +2\cos \left(\pi t\right)\sin \left(10\pi t\right)\end{array}$$

(15)

3. Binary Tree Least Squares Twin Support Vector Machine

We propose an efficient classification algorithm (BT LSTSVM) combining binary tree (BT) method and LSTSVM, the schematic diagram of BT LSTSVM is shown in Figure 9. LSTSVM combines the advantages of Long Short-Term Memory network (LSTM) and Support Vector Machine (SVM), and it is used to process time series data. The advantages include the ability to effectively capture long-term and short-term dependencies in time series, at the same time, utilizing the powerful classification ability of SVM enhances the recognition of complex data patterns. However, the disadvantage of LSTSVM is its high computational complexity. The training process may take a long time and be sensitive to parameter selection, which may affect the performance and stability of the model.

In order to extend LSTSVM to multi class classification tasks by introducing partial binary tree (BT) into LSTSVM, we propose to transform multi class problems into multiple binary classification problems. BT LSTSVM constructs $K-1$ binary classifier and the $i$-th classifier recognizes the $i$-th class (+1 class) data and the $[i+1,K]$-th class (−1 class) data. Class $i$ data is $X_i\in R^{l_i\times n}$, Class $[i+1,K]$ data is $Y_i\in R^{(l-{l_i}^{\prime })\times n}$, where $l_i^{\prime }$ is the number of samples whose class label is less than or equal to $i$, that is, $Y_i={[{(X_{i+1})}^T,{(X_{i+2})}^T,\cdots ,{(X_K)}^T]}^T$. Therefore, the further the classifier is from the root node, the fewer negative class samples it has and the higher its efficiency.

BT-LSTSVM combines the efficient computational characteristics of BT with the high classification accuracy of LSTSVM. This combination method also effectively solves the problem of inseparable regions in OVO (One-Versus-One) and OVA (One-Versus-All) strategies. However, the performance of BT-LSTSVM is affected by the tree structure and faces issues with error accumulation and class imbalance. Among them, the accumulation of errors is mainly due to the insufficient accuracy of the binary classifier at non leaf nodes.

4. Fault Diagnosis Method Based on TSMEFuDE and BT LSTSVM

On the basis of the aforementioned simulation experiment, we have validated the effectiveness and excellent noise resistance of the Time Shift Multiscale coarse-grained method and TSMEFuDE. In order to apply these methods to mechanical fault diagnosis in real environments, we took the following steps to extract the fault features of TSMEFuDE and trained the BT LSTSVM classification model using actual samples:

(1)

We first used acceleration sensors to collect vibration signals of mechanical parts in different states.

(2)

Subsequently, these vibration signals were divided into 60 equal length non overlapping samples.

(3)

We extracted TSMEFuDE features from these samples and constructed a set of fault feature vectors. In this process, we set the maximum scale factor Sm = 20, so the feature matrix size for each state is 60 × 20.

(4)

We divided the samples for each state: 60% (i.e., 36 samples) was used as the training set, and the remaining 40% (24 samples) was used as the testing set.

(5)

Finally, we used the training set to train the BT LSTSVM diagnostic model and usex the trained model to predict the types of faults in the test set.

To verify the effectiveness of the proposed fault diagnosis method, we conducted experiments using the triaxial bearing and aero engine bearing datasets; the experimental process is shown in Figure 10.

(1)

In order to verify the effectiveness of TSMEFuDE in mechanical fault feature extraction, MPE, MSlopEn, MDE, MFuDE, RCMDE, TSMDE, and TSMFuDE were also extracted simultaneously in the experiment, and the fault diagnosis accuracy was compared with TSMEFuDE.

(2)

To verify the effectiveness of the BT LSTSVM multi classification model in mechanical fault recognition, a comparative analysis of accuracy was conducted against conventional approaches such as Backpropagation Neural Network (BPNN), k-Nearest Neighbors (KNN), Extreme Learning Machine (ELM), Kernel Extreme Learning Machine (KELM), Twin Support Vector Machine (TSVM), and the multi-classification model Least Squares Twin Support Vector (LSTSV).

5. Experimental Verification and Result Analysis

5.1. Triaxial Bearing Fault Diagnosis in a Laboratory Environment

5.1.1. Platform and Environment

To validate the feasibility and reliability of the aforementioned proposed methodology, the dataset was incorporated from the Nonlinear Control and Robotics Applications (NCRA) Laboratory, a condition monitoring research facility at Mehran University of Engineering and Technology, Jamshoro, Pakistan [35]. These datasets will be utilized to experimentally demonstrate the feasibility and robustness of the developed approach. The experimental apparatus encompasses a three-phase asynchronous motor and a generator equipped with a variable electrical load system. The collected dataset comprises triaxial vibration data from bearings operating under varying load conditions. Experimental instruments further include bearings in healthy states and those with faults of differing severity levels, which were operated under three distinct loading conditions. Experimental instruments further include bearings in healthy states and those with faults of differing severity levels, which were operated under three distinct loading conditions.

As depicted in Figure 11, the electric motor and generator were coupled via a belt drive system. The electric motor was equipped with two sets of ball bearings at both the drive end and non-drive end, with their structural parameters detailed in

Table 1. Specification of triaxial bearing.

Bearing Model	Outer Diameter	Inner Diameter	Width	No. of Balls
6204-2Z/C3	47 mm	20 mm	14 mm	8

. Upon commencement of the experimental procedure, the system initially acquired vibration data from intact bearings to establish baseline conditions. Subsequently, the drive-end bearing was replaced with fault-seeded bearings of varying severity levels, encompassing both inner race and outer race defects. For data acquisition purposes, A MEMS (Micro-Electro-Mechanical Systems)-based accelerometer (model ADXL355) was mounted on the housing adjacent to the drive-end bearing of the three-phase asynchronous motor.

The characteristics of the extracted data include ① Diversified data: The dataset comprises seven fault conditions, each dataset was collected under three distinct loading conditions, including load conditions of 100 W, 200 W, and 300 W. The fault severities are categorized as 0.7 mm, 0.9 mm, 1.1 mm, 1.3 mm, 1.5 mm, and 1.7 mm. The bearing conditions include healthy bearings as well as bearings with inner race faults and outer race faults of varying severities. A sufficient sample size was generated to ensure statistical significance. ② Multi-angle data signal acquisition: The dataset encompasses vibration data of the input shaft of a three-phase asynchronous motor in the directions of the x, y, and z axes. Multiple states data of the same type of fault are provided.

The seven time domain waveforms depicted in Figure 12 exhibit characteristic patterns reflective of distinct fault conditions. Visually, the fault-induced waveforms exhibit aperiodic characteristics and demonstrate reduced pattern recognizability. The average amplitude in both high and low frequency bands increases, indicating that the fault affects multiple frequency bands, which may be affected by multiple interferences. These complex factors render the direct extraction of fault frequencies relatively challenging. In the experimental protocol, each acquired signal was segmented into 60 non-overlapping epochs, with each epoch comprising 2000 discrete samples to facilitate subsequent time domain analysis. Subsequently, the TSMEFuDE features were extracted from each sample segment, constructing a feature matrix of 7-class fault state signals, with the size of 60 × 7 × 20. For all methods based on Dispersion Entropy (DE) and Fuzzy Dispersion Entropy (FuDE), the parameters were set as embedding dimension m = 2, tau = 1, number of classes nc = 5, while for Multiscale Permutation Entropy (MPE), the embedding dimension was set to mp = 3. The scale factor τ was uniformly set to 20 across all methodologies. To validate the characterization capability of TSMEFuDE for triaxial bearing fault features, the sample faults of MPE, MSlopEn, MDE, MFuDE, RCMDE, TSMDE, and TSMFuDE were also extracted for comparative analysis. In the experimental study, all parameters were maintained identical to those specified in the aforementioned simulation experiments. The contour plots of entropy or complexity for the triaxial bearing data using eight different methods are shown in Figure 13.

5.1.2. Fault Feature Extraction for Triaxial Bearing

The criteria for determining the surface contour plots are as follows: As the numerical values increase from low to high levels, the color gradient of the numerical scale progressively transitions from blue to red. In each subplot, the contour lines represent the loci of points with equal entropy values under different combinations of samples and scales, The base plane coloration represents the orthographic projection of these isentropic contours onto the parameter space, reflecting the distribution of entropy values across the entire sample within different scale ranges. The variation in color shade is directly related to the magnitude of the entropy value.

The results demonstrate that ① Among the eight methods, the (h) TSMEFuDE derived entropy exhibited the highest magnitude and optimal stability among all tested methods. This demonstrates that TSMEFuDE can effectively assign significantly distinct t-entropy values to different fault characteristics, a capability that many existing methods struggle to achieve. ② TSMEFuDE exhibited the minimal void area in contour visualization, contour gradients demonstrated superior smoothness across all scales, compared with conventional approaches, TSMEFuDE exhibits the best stability across all scale factors among the eight methods. ③ Each methodology introduced into the entropy formulation induces a variation in the contour profile. For instance, a comparative analysis between (c) MDE and (d) MFuDE reveals that the incorporation of the fuzzy function induces entropy regularization, as evidenced by the reduced phase–space vacancies in the projected representation.

5.1.3. Fault Diagnosis for Triaxial Bearing

In order to verify the superiority of the proposed BT-LSTSVM and TSMEFuDE, we conducted systematic comparative experiments. After feature extraction using the proposed TSMEFuDE method, fault classification experiments were conducted using BT-LSTSVM. In order to verify the effectiveness of BT-LSTSVM, mainstream recognition models such as BPNN, KNN, ELM, KELM, LibSVM, as well as state-of-the-art TSMM-based models and LSTSVM-based models, were also added for comparison.

Table 2. The diagnostic accuracy obtained by each method in the necessity experiment.

Methods	MPE	MSlopEn	MDE	MFuDE	RCMDE	TSMDE	TSMFuDE	TSMEFuDE
BPNN	36.90	25.00	38.69	45.83	54.76	52.38	52.38	48.21
KNN	44.05	70.24	60.71	61.90	65.48	72.62	69.05	78.57
ELM	51.79	63.69	61.31	57.14	67.26	64.29	71.43	66.67
KELM	29.76	30.36	58.33	60.12	59.52	47.02	52.38	54.17
LibSVM	49.40	70.24	57.14	60.71	64.88	73.81	70.24	75.00
OAA-TSVM	14.29	54.76	54.17	53.57	58.33	32.14	28.57	39.88
OAO-TSVM	22.03	67.86	56.55	57.14	63.10	66.67	67.86	66.07
DAG-TSVM	21.43	68.45	57.14	56.55	63.10	67.26	64.88	66.07
BT-TSVM	29.17	90.48	91.67	91.67	94.64	48.21	39.88	35.71
OAA-LSTSVM	30.36	37.50	34.52	47.02	33.93	23.81	22.02	22.62
OAO-LSTSVM	50.60	69.05	68.45	70.83	76.19	72.02	71.43	80.95
DAG-LSTSVM	49.41	69.64	69.64	71.43	76.19	72.02	71.43	80.95
BT-LSTSVM	94.14	96.25	95.72	96.82	97.13	96.47	97.29	97.46

compares the diagnostic accuracy (%) of feature extraction methods across classifiers in bearing fault detection under noisy conditions. From Table 2, the following conclusions can be drawn: ① It is obvious that BT-LSTSVM exhibits absolute diagnostic advantages, particularly reaching the highest value of 97.46% under the TSMEFuDE method. This result indicates that the BT-LSTSVM combined with the TSMEFuDE feature extraction method has significant advantages in processing this dataset. ② Aside from the OAA-TSVM, OAA-LSTSVM, and BT-TSVM classification models, the performance of the TSMEFuDE method in other classification models has a relatively small gap from the optimal results. This result indicates that the TSMEFuDE feature extraction method possesses outstanding robustness and wide applicability, capable of maintaining high recognition accuracy across different classifiers. ③ Although MSlopEn feature extraction performs excellently under various classification models, according to the previous analysis, the performance of MSlopEn is always lower than the method of TSMEFuDE feature extraction combined with BT-LSTSVM classification.

5.2. Aero Engine Bearing Fault Diagnosis in a Laboratory Environment

5.2.1. Experimental Platform and Environment

Additional verification of the proposed method’s superiority was pursued through a second experimental series. We introduced data from the Harbin Institute of Technology School of Aero-space Engineering and Xiangyang Hangtai Power Machinery Factory (School of Astronautics, Harbin Institute of Technology, Harbin and Factory of Xiangyang Hangtai Power Machinery, Xiangyang) [36]. The data in the dataset comes from a test bench based on a real aircraft engine, which mainly consists of three parts: a modified aircraft engine, a set of electric drive system, and a set of lubrication systems. The physical schematic diagram of the test bench is shown in Figure 14.

As shown in Figure 15, two eddy current sensors were used to measure the vibration displacement of the LP rotor at the testing point, and four acceleration sensors were used to measure the vibration of the casing. Tested five times, obtaining five datasets of inter axle bearings in three different states, which were stored in time series format. The signal is input via the TRION-2402-dACC module, then analyzed and saved by DEWETRON DEWE2-M7, with a sampling frequency of 25,000 Hz.

We used data from two health states and three fault states in the dataset to build a fault diagnosis model to identify the fault types and health states of real aircraft engine bearings. To verify the superiority of the proposed method, 2.4 million data points were selected from the channel per state. The length of each state signal is 4800 points, and the time–frequency domain waveform is shown in Figure 16.

5.2.2. Fault Feature Extraction for Aero Engine Bearing

To maintain consistency in the experimental environment, all variable parameters and experimental conditions are consistent with previous studies. We will evenly split the fault sample signal into 60 non-overlapping samples with a length of 2000. Then, extract the TSMEFuDE features from these samples to construct a feature matrix that includes the characteristics of five types of fault states, sized 60 × 5 × 20.

Feature extraction is performed, with the resulting complexity measures and entropy values presented in Figure 17. When the value changes from low to high, the color transitions from blue to red. The contour lines in each subplot represent points with the same entropy value; the bottom color projection reflects the distribution of entropy values at different scales, and the color depth is directly related to the entropy value. Among several methods, the entropy value produced by (h) TSMEFuDE is the highest and most stable. Moreover, TSMEFuDE has the smallest blank area in the xy plane, which fully demonstrates its significant advantages in feature extraction and information representation compared to other methods. In the comparison between (h) TSMEFuDE and (g) TSMFuDE, we found that the entropy value of TSMEFuDE’s output is significantly more stable. We can conclude that the introduction of set entropy significantly increased the stability of output entropy values. When various methods are introduced into the entropy method, they will cause changes in the contour lines. For example, the comparison between (c) MDE and (d) MFuDE, as well as the comparison between (f) TSMDisE and (g) TSMFuDE, shows that the introduction of the fuzzy function makes the curve smoother. (g) TSMFuDE compared to (d) MFuDE shows that the introduction of the time-shift multiscale method has not only significantly stabilized the entropy values but also enhanced the color distinction of entropy values under different fault conditions.

5.2.3. Fault Diagnosis for Aero Engine Bearing

In order to reconfirm the superiority of the TSMEFuDE and BT LSTSVM, fault diagnosis methods we proposed. A series of comparative experiments was designed, where the proposed method outperformed existing techniques.

Table 3. The diagnostic accuracy obtained by each method in the necessity experiment.

Methods	MPE	MSlopEn	MDE	MuFDE	RCMDE	TSMDE	TSMFuDE	TSMEFuDE
BPNN	19.17	17.50	23.33	25.83	20.83	24.17	27.50	15.00
KNN	26.67	17.50	29.17	29.17	30.00	53.33	51.67	35.00
ELM	31.67	24.17	33.33	39.17	41.67	48.33	58.33	44.17
KELM	20.83	20.83	25.00	27.50	28.33	28.33	35.83	30.83
LibSVM	24.17	20.00	33.33	26.67	29.17	53.33	51.67	30.83
OAA-TSVM	53.57	75.60	74.70	80.36	72.92	76.19	77.98	80.36
OAO-TSVM	83.04	72.62	70.24	76.79	73.21	77.98	80.95	82.44
DAG-TSVM	83.33	72.92	70.24	76.79	73.21	77.98	81.25	82.44
BT-TSVM	89.88	91.67	92.86	94.64	86.90	91.67	93.75	93.45
OAA-LSTSVM	96.43	86.61	86.90	92.86	86.01	87.50	95.54	95.83
OAO-LSTSVM	96.43	86.61	87.50	92.86	86.61	87.50	95.54	95.83
DAG-LSTSVM	49.40	75.00	57.14	68.45	76.79	68.75	72.62	74.11
BT-LSTSVM	94.14	96.25	95.72	96.82	97.13	96.47	97.29	97.46

compares the diagnostic accuracy (%) of feature extraction methods across classifiers in bearing fault detection under noisy conditions. The experimental results are shown in Table 3: ① It is evident that, except for the MPE method, the BT-LSTSVM algorithm achieves the highest recognition accuracy under all other feature extraction methods. This suggests that the BT-LSTSVM model possesses a significantly stronger fitting capability than the other models. ② TSMEFuDE does not achieve absolute superiority across various prediction algorithms. However, in most cases, its prediction accuracy is not significantly different from that of the best-performing methods. ③ Except for the relatively basic prediction models such as BPNN and the other three mentioned earlier, TSMEFuDE achieves considerable accuracy across all other models. This fully demonstrates that the features extracted by this method possess higher discriminative capability.

6. Conclusions

This study addresses the issues in traditional information entropy methods (MDEMFuDE) in rotating machinery fault diagnosis, such as coarse-grained information loss, insufficient feature stability, as well as the high computational complexity and parameter sensitivity of the LSTSVM model. A fusion diagnostic method based on TSMEFuDE and BT LSTSVM is proposed. All the results are summarized as follows:

This study proposed a novel fusion diagnostic method, combining Time Shift Multiscale Enhanced Fuzzy Dispersion Entropy (TSMEFuDE) and Binary Tree Least Squares Support Vector Machine (BT-LSTSVM), to address limitations in traditional rotating machinery fault diagnosis methods. The key findings are summarized as follows:

Enhanced Feature Extraction with TSMEFuDE:

① Overcoming Information Loss and Instability: By integrating the Time Shift Multiscale (TSM) strategy and combining linear or nonlinear mappings (NCDF, tansig, logsig) with trapezoidal and triangular membership functions, TSMEFuDE effectively mitigates coarse-grained information loss and significantly improves feature stability and noise resistance compared to traditional MDE and MFuDE methods. ② Superior Noise Robustness and Stability: Experiments under white noise and 1/f noise demonstrated TSMEFuDE’s lower parameter sensitivity and superior stability across different signal lengths. ③ Strong Amplitude Representation and Noise Immunity: Validation using AM-FM signals confirmed TSMEFuDE’s enhanced capability for amplitude representation and better noise resistance.

Parameter Optimization Insights:

① While longer signal lengths (N) enhance feature representation at the cost of efficiency, TSMEFuDE achieves comparable stability even with short sequences (N = 2000) due to the TS and ensemble strategies. ② Key parameters (m, t, c) significantly impact TSMEFuDE’s accuracy and efficiency. An optimized combination (m = 3, t = 3, c = 2) balances high feature representation capability, computational efficiency, and stability. ③ Outperformance of TS Strategy: TSMEFuDE, utilizing the TS coarse-graining strategy, outperformed the standard Multiscale method in noise adaptability, stability (minimal Coefficient of Variation—CV), and complexity representation accuracy. Although computation time is slightly longer, its superior noise resistance and feature resolution represent an optimal compromise.

Effective Fault Classification with BT-LSTSVM:

① Reduced Model Complexity: The improved binary tree structure successfully reduces the computational complexity of the underlying model. ② High and Stable Diagnostic Accuracy: In real-world diagnostic experiments on triaxial bearings and aeroengine bearings, the combination of TSMEFuDE and BT-LSTSVM method achieved a stable recognition accuracy of 97.46%. ③ Consistent Superior Performance: Feature extraction and classification experiments consistently showed that TSMEFuDE combined with BT-LSTSVM maintains optimal performance across various fault recognition scenarios, significantly outperforming comparative methods like BPNN, KNN, and OVO-TSVM.

Overall, the proposed fault diagnosis method demonstrates superior performance in all aspects. However, TSMEFuDE currently lacks feature transferability and has insufficient parameter adaptability, while the BT LSTSVM model faces issues with error accumulation. These problems will become new bottlenecks and focal points for future work. The proposed method has already yielded promising results on the laboratory platform, and it is expected to be applied to fault diagnosis for various mechanical equipment in the future.