High-Dimensional Mapping Entropy Method and Its Application in the Fault Diagnosis of Reciprocating Compressors

Abstract

The effectiveness of feature extraction is a critical aspect of fault diagnosis for petrochemical machinery and equipment. Traditional entropy analysis methods are prone to disruption by noise, parameter sensitivity, and sudden entropy variations. This study establishes a high-dimensional mapping entropy (HDME) method characterized by robust noise resistance, addressing the issues of parameter sensitivity and inadequate noise suppression inherent in traditional feature extraction methodologies. A mapping theory of high-dimensional space based on kernel function pattern recognition is proposed, which reassembles the sample vector after phase space reconstruction of time series. The multi-scale high-dimensional mapping entropy (MHDME) and refined composite multi-scale high-dimensional mapping entropy (RCMHDME) algorithms are further studied based on the idea of refined composite multi-scale. Application to simulated signals shows that the suggested methods reduce parameter sensitivity and enhance entropy smoothness. The development of a methodology to identify faults through MHDME is proposed. This approach integrates signal preprocessing and intelligent preference techniques to achieve pattern recognition of reciprocating compressor bearings in various wear conditions. Moreover, the identification findings demonstrate that the suggested approach can effectively extract the characteristics of the signal and accurately distinguish the effects caused by different faults.

1. Introduction

Nowadays, reciprocating compressors are extensively utilized in the industrial compressible fluids sector, particularly in the petrochemical industry. It is imperative to conduct equipment monitoring, maintenance, and fault research which ensure the smooth operation of these compressors [1]. The vibration signals of reciprocating compressors exhibit strong non-stationary characteristics and multi-component coupling characteristics due to their complex structures and multiple excitations [2]. The inherent structural characteristics of reciprocating compressors result in vibration signals that encapsulate the vibrational attributes of various components, rendering the identification of fault characteristics challenging [3]. The crucial issue in fault diagnosis of mechanical equipment is the ability to accurately extract the features that reflect the state of the machine. In recent years, multiple methodologies for feature extraction in mechanical fault diagnosis have been devised by researchers [4]. These methodologies include analysis methods based on signal processing in the time domain, frequency domain, and time–frequency domain [5]. Feature extraction methods are informed by information theory, including information entropy [6] and mutual information [7]. Additionally, techniques based on sparsity measurement [8] and deep learning have gained prominence [9,10]. Nonetheless, sparsity analysis is often used to monitor the working trend and state changes of equipment, and the signals generated by reciprocating compressors may not have obvious sparsity, which makes it difficult to effectively use sparsity [11]. A significant amount of historical operating data is required to obtain features through deep learning [12]. In practical engineering applications, it may be difficult to obtain adequate operational data on reciprocating compressors. Hence, this article will not delve into the latter two methods.

Fault diagnosis is essentially a process of recognizing fault patterns and its success relies on effectively extracting and utilizing information from vibration signals. Empirical evidence indicates that the characteristic distributions and statistical parameters of reciprocating compressors in varying conditions are discernible, thereby offering a foundation for the development of fault classification models [13]. Traditional approaches may be inadequate for complex datasets as they frequently presuppose linearity and stationarity within the data. Yet, in practical engineering contexts, the data commonly present non-linear and non-stationary behaviors. This renders traditional time–frequency methods in signal processing limited in effectively extracting features from non-stationary vibration signals. The emergence of non-linear technology has led to the widespread use of analysis methods based on various information theories in the manufacture of mechanical equipment [14]. Therefore, entropy-based fault detection methods are increasingly garnering attention because of their ability to capture the intricacy and irregularity of signals.

Commonly utilized entropy techniques include approximate entropy (ApEn) [15], sample entropy (SE) [16], fuzzy entropy (FE) [17], permutation entropy (PE) [18], dispersion entropy (DE) [19], and others. The above methods are used to measure the complexity and irregularity of time series on a single time scale. In practical applications, Costa [20] considered the multi-scale characteristics of complex signal and introduced multi-scale sample entropy (MSE) to assess time series complexity and irregularity across different scales. Addressing the issue of substantial fluctuations in MSE entropy values, Wu [21] proposed refined composite multi-scale sample entropy (RCMSE). Among these methods, SE solves the issue of self-similarity that ApEn demonstrates and effectively measures the complexity and irregularity of time series. However, SE still has some restrictions, such as compromised estimation accuracy when dealing with time series data of high complexity, resulting in inadequate distinguishability [22]. Since sample entropy assesses the similarity between vectors by phase space reconstruction, while solely employing the plane Chebyshev distance to measure the distance of the time series, its outcome is vulnerable to noise interference and mutations. High-dimensional space mapping theory is extensively employed in the field of machine learning pattern recognition [23]. The vectors linearly indistinguishable in the original low-dimensional vector space are discerned through appropriate high-dimensional mapping, suppressing noise information, highlighting effective features, and achieving effective differentiation between vectors.

The original planar data space is transformed into a high-dimensional vector space using a high-dimensional mapping function φ(x), which captures the high-dimensional feature representation of the data [24]. This mapping function improves the separability of the sample data in the kernel space and increases the variability among the feature vectors. To accurately assess the similarities and differences between different sequences, the Euclidean distance calculation is employed to reorganize the vector samples in the high-dimensional kernel space. This approach effectively attenuates interfering factors, suppresses noise, and highlights the relevant information by reevaluating the vector distances. Therefore, combined with the sample entropy algorithm, a new high-dimensional mapping entropy (HDME) method is proposed. The HDME method proposed in this study focuses on the problems of noise interference and parameter sensitivity in sample entropy analysis methods. Compared with traditional entropy research, HDME can effectively suppress the noise and highlight the features after mapping in the proper high-dimensional vector kernel function space, thus improving the accuracy and stability of the fault features. The innovation of this paper is to introduce a new high-dimensional mapping method to improve the fault diagnosis effect. Meanwhile, combined with signal pre-processing and intelligent optimization technology, a set of fault feature extraction strategies is formed. Furthermore, this research achieves an accurate diagnosis of faults in reciprocating compressor bearings. The research motivation for this method comes from the in-depth analysis of the traditional sample entropy method. When the vector distance is measured in the planar phase space, the sample entropy is easily affected by the noise disturbance. In this paper, the theory of high-dimensional mapping is applied to the process of entropy feature extraction, and a new method, HDME, is formed. Another major innovation of this approach is that it combines the simplicity of linear processing with the power of non-linear mapping. In addition, this thesis develops multi-scale high-dimensional mapping entropy (MHDME) and refined composite multi-scale high-dimensional mapping entropy (RCMHDME). We examine the application of the proposed HDME under the presumption that it possesses strong noise resistance and differentiation capabilities.

The primary objectives of this research are outlined as follows: First, we propose a method aimed at enhancing the stability of signal complexity descriptions and increasing the accuracy of data separability. Second, we address the limitations inherent in single-scale complex datasets. Third, we formulate a novel fault feature extraction strategy by amalgamating signal preprocessing with an intelligent preference technique. Lastly, we apply the proposed fault diagnosis methodology to the bearings of reciprocating compressors.

In this study, we focus on the entropy calculation of high-dimensional vector spaces. In Section 2, we review the sample entropy approach to present the idea of high-dimensional mapping entropy. In Section 3, we explain the high-dimensional mapping theory and algorithms, analyse the simulated signals, and discuss the parameters. In Section 4, we propose multi-scale high-dimensional mapping entropy and apply simulated signal analysis. In addition, in Section 5, we construct a feature extraction method based on multi-scale high-dimensional mapping entropy and apply it to a case study of reciprocating compressor bearing failure. In Section 6, the conclusions are described.

2. Review of Sample Entropy (SE)

The sample entropy is obtained from Richman’s improved approximate entropy [15], which is obtained by the following algorithm [16]:

(1)

Reconstruct the time series $x\left(i\right)$ in phase space in m dimensions:

$$X_m\left(i\right)=\left\{x\left(i\right),\cdots ,x\left(i+m-1\right)\right\} (i=1,\cdots N-m+1)$$

(1)

(2)

Define the maximum in the absolute value as the Chebyshev distance between the two vectors $d\left[X_m\left(i\right),X_m\left(j\right)\right]$:

$$d\left[X_m\left(i\right),X_m\left(j\right)\right]=\underset{k=0,\cdots ,m-1}{max}\left[\left|x\left(i+k\right)-x\left(j+k\right)\right|\right] (i,j=1,\cdots N-m,i\ne j)$$

(2)

(3)

Preset the value of similarity tolerance $r$, count the number $B_i$ of all vectors whose Chebyshev distance $d\left[X_m\right(i),X_m(j\left)\right]$ is less than $r$, and ratio this number of vectors to the total number of distances N − m − 1 to obtain the ratio $B_i^m\left(r\right)$:

$$B_i^m\left(r\right)=\frac{1}{N-m-1}{B}_i (i=1,\cdots N-m)$$

(3)

(4)

Calculate $B_i^m\left(r\right)$ the average of $B^m\left(r\right)$:

$$B^m(r)=\frac{1}{N-m}{\sum }_{i=1}^{N-m}{B}_i^m(r)$$

(4)

(5)

Increase the reconstruction dimension to m + 1 and repeat the above steps, thus obtaining $B^{m+1}\left(r\right)$:

$$B^{m+1}\left(r\right)=\frac{1}{N-m}{\sum }_{i=1}^{N-m}{B}_i^{m+1}\left(r\right)$$

(5)

(6)

Finally, calculate the sample entropy of the time series:

$$SampEn(m,r)=\underset{N\to \infty }{lim}\left\{-\mathit{ln}\left[\frac{B^{m+1}\left(r\right)}{B^m\left(r\right)}\right]\right\}$$

(6)

Whereas, in practice, N is finite in value, so the sample entropy as:

$$SampEn(m,r,N)=-\mathit{ln}\left[\frac{B^{m+1}\left(r\right)}{B^m\left(r\right)}\right]$$

(7)

The sample entropy exhibits noise immunity and can suppress interference from amplitude noise below a threshold value of r. SE mitigates the issue of self-similarity by omitting self-matches from its counting scheme. During computation, Equation (3) quantifies the number of template vectors with distances less than the similarity tolerance r. The template vector would be excluded from this quantification to avoid self-comparison. This exclusion ensures non-overlapping when comparing two template sequences, thus solving the self-similarity problem that exists in ApEn [22]. When there are large oscillation points in the signal, this type of data constitutes a large inter-vector Chebyshev distance from neighbouring samples, which is also removed in the threshold check. While practical applications find that sample entropy is still disturbed by significant background noise, making it difficult to obtain comprehensive information on time series changes and reducing the accuracy of entropy calculation. Analysis of the sample entropy algorithm, where Equation (2) represents the plane metric Chebyshev distance, reflecting the characteristics of high-dimensional time series individually, indicates that plane space to deal with non-linear relationships is limited and will interfere with the time series characterization accuracy. The introduction of a high-dimensional vector space is proposed to reflect the characteristics of the data samples and achieve a non-linear transformation of the data samples from a low-dimensional to a high-dimensional space. Consequently, the direct solution is a valuable attempt from another dimension.

3. High-Dimensional Mapping Entropy

The main focuses of this paper include the discussion of high-dimensional mapping entropy, the construction of feature extraction methods and strategies, and the application of reciprocating compressor-bearing fault diagnosis. The specific research route is shown in Figure 1.

3.1. High-Dimensional Mapping Theory

The essence of high-dimensional mapping entropy theory can be attributed to the principle of high-dimensional space mapping and the idea of kernel function classification. The literature [11] proposes a Gaussian kernel-based clustering method, which adopts the Mercer kernel mapping idea to map samples to Hilbert space on the basis of retaining essential features and achieves kernel space sample separability by enhancing the variability of data features in the high-dimensional space. Moreover, the kernel function high-dimensional space mapping theory is also widely used in the field of support vector machine (SVM) pattern recognition. The mapping theory process is detailed in Figure 2.

Conducting a mapping of the sample data points into a high-dimensional space, aimed at identifying the optimal hyperplane for dividing the distinct data points. The hyperplane that implements the decision boundary for transforming a non-linear problem into a linear problem can be expressed as a linear equation [25]:

$${\omega }^T+\varphi \left(X\right)+b=0$$

(8)

$\mathrm{w}\mathrm{h}\mathrm{e}\mathrm{r}\mathrm{e} \varphi$: X $\to$ H is the mapping function, X is the input space, and H is the feature space (Hilbert space).

Traditional SE evaluates the distance between samples in a reconstructed phase space. HDME employs kernel mapping to project data samples into higher-dimensional feature spaces, enhancing the distinction between noise signals and characteristic signal data points. This approach aims to identify the optimal hyperplane for separating distinct data points, thereby converting the non-linear problem into a linear one. While preserving the original attributes of the sample, HDME generates a new dataset within this feature space that minimizes noise and outlier interference. Subsequently, it measures the similarities of different features within this newly defined feature space.

The critical points in high-dimensional mapping entropy are the choice of the high-dimensional mapping kernel function, the measurement of the distances in space, and the determination of the similarity tolerance. These aspects are essential for precisely calculating high-dimensional mapping entropy.

3.1.1. Kernel Function Selection

The primary function of kernel functions is to offer a non-linear mapping that transforms low-dimensional input data into representations in high-dimensional feature spaces. This mapping partially suppresses the original data in the new high-dimensional space, while highlighting another part of the information, resulting in better separability. There is no definitive way to choose the feature space H and the mapping function ϕ when determining the kernel function K(x, z). Multiple feature spaces may be chosen, and within the same feature space, diverse mapping rules may be selected. The choice of the kernel function relies on domain knowledge within practical applications, and it is necessary to verify the kernel function’s validity. Through the use of the mapping function ϕ, the kernel function K(x, z) is obtained through evaluating the internal product of ϕ(x) and ϕ(z) [25]. Revealing the underlying mapping function used in the classification process is our objective in this paper. The decision of the kernel function and the conditions it needs to satisfy are crucial considerations. There are several commonly used kernel functions, each with its own characteristics:

(1)

Linear kernel function: Linear kernel functions are straightforward and map input vectors directly into higher-dimensional spaces. They are suitable for effectively handle linearly separable problems and data that exhibit strong linear correlations;

(2)

Polynomial kernel function: Polynomial kernel functions introduce non-linearity by calculating polynomial combinations of input vectors. To some extent, non-linear relations can be handled. However, the application of the polynomial kernel function needs to be determined in advance, as using too high of an order can lead to overfitting problems;

(3)

Gaussian (RBF) kernel function: Gaussian kernel functions are one of the most commonly used and versatile kernel functions. They are based on the concept of a Gaussian distribution and maps input vectors to infinite-dimensional feature spaces. Gaussian kernel functions have excellent non-linear modelling capabilities and can adapt well to various data distributions.

The preference for Gaussian kernel functions stems from their robust non-linear modelling capabilities and adaptability. These functions generally exhibit superior performance across a multitude of machine-learning tasks. As a result, in this research paper, a mapping method for vector high-dimensional feature spaces are experimentally determined, and Gaussian radial basis kernel functions are adopted for classification using support vector machines (SVMs).

The kernel mapping allows the inner product in the transformed space to be transformed into some function in the original space for computation, thus avoiding the computation of $\varphi \left(x\right)$-specific mappings, and the inner product of the transformed high-dimensional feature space is computed using the kernel function. The kernel mapping enables the mapping of samples that are indivisible in the input space to samples that are divisible in the feature space. The steps for mapping to a high-dimensional kernel space are:

• Take any $N$ samples in the $d$ dimensional sample space $R^{d}S=\{x_1,x_2,\cdots ,x_N\}\subset R^d$
• Using non-linear mapping $\Phi :R^d\to F,x\to \varphi \left(x\right)$, map the samples in the original space $R^d$ to the feature space $F$ after the samples are $\left\{\varphi \right(x1),\varphi (x2),\cdots ,\varphi (xN\left)\right\}\subset F$.
  The inner product of two functions in the eigenspace is a binary function that is reproducible when its kernel matrix is a semi-positive definite matrix [26]:

$$\kappa (X_i,X_j)=\kappa (·,X_i{)}^T\kappa (·,X_j)$$

(9)

Thus, its inner product space is an endowment vector space which can be completed to obtain the Hilbert space, the regenerated kernel Hilbert space. It is obtained after constructing the kernel matrix for the kernel function as a necessary condition [27]:

$${\sum }_{i,j=1}^{m}G(X_i,X_j)=‖{\sum }_{i=1}^m\varphi (X_i{)}^2‖\ge 0$$

(10)

In practice, the existing and commonly used positive definite kernel functions are used—mainly the Gaussian radial basis kernel function. The application of the Gaussian radial basis kernel function [27] is:

$$K(X,Y)=\mathit{exp}\{-\gamma {‖X-Y‖}^2/{\sigma }^2\}$$

(11)

where $\gamma$ and $\sigma$ are kernel function parameters.

According to Equation (11), the kernel parameter is too large or too small, which will affect the spatial distribution of the mapping points and the distance between the points (degree of proximity). For this reason, the experimental Gaussian kernel function parameter $\gamma$ is taken as 1, and $\sigma \in \left[0.01,0.05\right]$.

The use of the Gaussian kernel function is founded opon the Gaussian distribution concept and projects the input vector into a feature space of high-dimensions. Notably, the Gaussian kernel function exhibits outstanding non-linear processing capacities and can adeptly adapt to diverse data distributions. In bearing signal fault mode recognition, the kernel function enables the identification and discrimination of various fault modes. The Gaussian kernel function can properly capture the non-linear features in the vibration signal by mapping the vibration signal to the high-dimensional feature space. This enables the pattern recognition algorithm to differentiate between normal working states and various fault modes, thus achieving precise fault diagnosis and monitoring.

3.1.2. Distance Measure of High-Dimensional Mapping Entropy

The high-dimensional mapping entropy is computed in a high-dimensional space, whereas the sample entropy distance is computed in a two-dimensional plane, and there are some differences between the high-dimensional and two-dimensional distances when metricizing the sample distances.

When using high-dimensional mapping entropy, a kernel-based non-linear function is employed to map the recombination vector samples to a high-dimensional kernel space by adopting the kernel method. In the high-dimensional space, the data has more dimensions and therefore the distances between data points have richer variations in more dimensions. In contrast, using only straight-line distances on a two-dimensional plane may not accurately reflect the relationships between data points. Commonly used distance metrics include Euclidean distance, Manhattan distance, Mahalanobis distance, and Chebyshev distance. The choice of a specific distance metric usually depends on the nature of the data and the needs of the application scenario:

(1)

The Euclidean distance, being the most commonly used distance measurement, quantifies the straight-line separation between two points. It effectively preserves the geometric characteristics of data in a high-dimensional space, making it an ideal choice for evaluating continuous features;

(2)

The Manhattan distance neglects the correlation between the variables, instead concentrating solely on the aggregation of discrepancies. This can result in inaccurate representation of true differences in the data;

(3)

The Mahalanobis distance can produce unreliable results when the covariance matrix of the data is computationally unstable or difficult to estimate. In addition, the computational complexity of the procedure increases significantly in high-dimensional datasets;

(4)

The Chebyshev distance measures the difference between vectors by calculating their maximum difference in each dimension. The distance calculation potentially overlooks correlation in other dimensions, thereby neglecting critical information across multiple dimensions.

In high-dimensional mapping entropy, we use the Euclidean distance to measure the distance between data points in a high-dimensional space. The Euclidean distance is superior in preserving the geometric structure of data in high-dimensional spaces and capturing the relationships between different dimensions. It can provide richer information including the magnitude and direction of the distance [28]. In high-dimensional spaces, discrepancies between sample points may exist across multiple dimensions. The Euclidean distance can more effectively capture these discrepancies and offer a more comprehensive representation of the data. Compared to other distance formulas, Euclidean distance calculations require only square and squaring operations, allowing for faster measurement of distances between high-dimensional sample points.

In conclusion, the use of Euclidean distance in high-dimensional mapping entropy can better capture the multi-dimensional relationship of data and provide richer information.

3.1.3. Determination of Similarity Tolerance for High-Dimensional Mapping Entropy

When selecting the similarity tolerance in a sample entropy plane space, it is important to strike a balance between preserving time series detail information and filtering out noise. A common approach is to choose a value for the similarity tolerance, denoted as r, based on the standard deviation (SD) of the original sequence. In the case of using Euclidean vector distances in high-dimensional mapping entropy, the selection of similarity tolerance involves considering the characteristic scale of the data or the specific requirements of the problem. The similarity tolerance can be determined based on the standard deviation of the primary sample, which reflects the fluctuation and complexity of the data. Choosing an appropriate similarity tolerance has an impact on the calculation of high-dimensional mapping entropy. A larger similarity tolerance may result in the loss of fine-grained information, while a smaller tolerance makes the calculation more sensitive to noise, potentially reducing the accuracy of entropy estimation.

The similarity tolerance 0.15 SD–0.25 SD is taken as the range of parameter r, which is the value adopted by the empirical mechanical system [22]. In this paper, when r = 0.15 SD, the empirical test shows that the algorithm can work normally. Therefore, it shows that r is set in the range of 0.15 SD–0.25 SD to obtain satisfactory results, which provides a valuable reference for researchers. This range can meet the requirements of reliability and practical applications.

3.2. High-Dimensional Mapping Entropy Algorithm

HDME combines kernel functions and high-dimensional mapping theory to improve SE and achieve an up-dimensional description of reconstructed information in space. The specific steps are as follows:

(1)

For a given N point time series $X=\left\{x_i\right\}$ with a pre-determined embedding dimension $m$, perform m-dimensional reorganization to obtain an $N-m+1$ vector:

$$X_m\left(i\right)=\left\{x\left(i\right),\cdots ,x\left(i+m-1\right)\right\} (i=1~N-m)$$

(12)

(2)

Map the recombinant vector samples to a high-dimensional kernel space using non-linear mapping based on the kernel method.

First, calculate the Euclidean distance of each vector from the rest of the and construct the distance matrix $d_k$:

$$d_{ij}^m=‖X_m\left(i\right)-X_m\left(j\right)‖ (1\le i,j\le N-m)$$

(13)

Second, convert the distance matrix into a kernel matrix:

$$D_k=\left[\begin{array}{ccc}{d}_{11}^m(X_1,X_1)& \cdots & d_{1N-m}^m(X_1,X_{N-m})\\ ⋮& \ddots & ⋮\\ d_{N-m1}^m(X_{N-m},X_1)& \cdots & d_{N-mN-m}^m(X_{N-m},X_{N-m})\end{array}\right]$$

(14)

Then, substitute the generated kernel matrix into different kernel functions. Sum the matrix to obtain the mapping points of corresponding vectors in space, denoted as:

$$K_{1\times N-m}={\sum }_{i=1}^{N-m}\mathit{exp}\{-\gamma {D_k}^2/{\sigma }^2\}$$

(15)

$$\mathrm{or}{K}_{1\times N-m}={\sum }_{i=1}^{N-m}(\gamma D_k+C{)}^d$$

(16)

Finally, match the initial m-dimensional reconstruction vector with the matrix $K_{1\times N-m}$ to obtain the high-dimensional spatial projection points;

(3)

To calculate the Chebyshev distance between two vectors $X_m\left(i\right)\prime$ and $X_m\left(j\right)$′ in a high-dimensional space $X_m$, we need to determine the absolute difference between the corresponding elements of the two vectors:

$$d\left[X_m{\left(i\right)}^{\prime },X_m{\left(j\right)}^{\prime }\right]=\underset{k=0,\cdots ,m}{max}\left|x\left(i+k\right)-x\left(j+k\right)\right| (i,j=\mathrm{1,2},\cdots ,N-m,i\ne j)$$

(17)

(4)

Pre-set the value of the similarity tolerance $r$ and count the $d\left[X_m\right(i)\prime ,X_m(j)\prime ]\le r$ of the number of distances, denoted as $B_i$, and compare this result with the total number of distances $N-m-1$ to obtain the ratio $B_i^m\left(r\right)$:

$$B_i^m\left(r\right)=\frac{1}{N-m-1}{B}_i (i=1,\cdots N-m)$$

(18)

(5)

Calculate the above $N-m$ of $B_i^m\left(r\right)$ of the above, denoted as $B^m\left(r\right)$:

$$B^m(r)=\frac{1}{N-m}{\sum }_{i=1}^{N-m}{B}_i^m(r)$$

(19)

(6)

Repeat the above steps 1–5 when increasing the reconstruction dimension to m + 1, and similarly obtain $B^{m+1}\left(r\right)$:

$$B^{m+1}(r)=\frac{1}{N-m}{\sum }_{i=1}^{N-m}{B}_i^{m+1}(r)$$

(20)

(7)

This results in the high-dimensional mapping entropy of this time series:

$$HDME\left(m,r,sigma,N\right)=\underset{N\to \mathrm{\infty }}{lim}\left[-\mathit{ln}\frac{B^{m+1}\left(r\right)}{B^m\left(r\right)}\right]$$

(21)

However, in practice N is finite in value, so the entropy of the high-dimensional mapping entropy is denoted as:

$$HDME\left(m,r,sigma,N\right)=\left[-\mathit{ln}\frac{B^{m+1}\left(r\right)}{B^m\left(r\right)}\right]$$

(22)

The process of feature extraction based on high-dimensional mapping entropy is shown in Figure 3.

3.3. Simulated Signal Analysis

To assess the performance of the suggested approach, a mixed signal is generated by superimposing a deterministic signal, a random signal, and a noise:

$$\mathit{mix}(N,p)=(1-p\left)X\right(i)+pY(i)$$

(23)

$$X_1=mix\left(3000,0.3\right)+wgn(1,3000,0.2)$$

(24)

$$X_2=mix\left(3000,0.5\right)+wgn(1,3000,0.5)$$

(25)

where $X\left(i\right)=\sqrt{3}\mathit{sin}(\pi /4)$,$Y\left(i\right)$ is the random signal $Y\left(i\right)\in [-\sqrt{5},\sqrt{5}]$, N the length of the signal, and the ratio p $\in \left[\mathrm{0,1}\right]$. The sample entropy and high-dimensional mapping entropy of the signals $X_1$ and $X_2$ are denoted as SampEn1, SampEn2, HDME1, and HDME2, respectively, and the entropy of the mixed signals with different similarity tolerances r are shown in Figure 4.

As depicted in Figure 4, both the SE and HDME entropy of the mixed signals exhibit better similarity in general. As the similarity tolerance r changes, SampEn1 and SampEn2 show larger fluctuations. SampEn1 and SampEn2 are also more sensitive to changes. The sensitivity to these variations could present issues during the analysis of complex signals. Notable fluctuations in sample entropy may result in misleading interpretations of the dynamic behavior of signals and affect the reliability of diagnostic conclusions. In contrast, both the HDME1 and HDME2 curves are more uniform, with better performance in parameter consistency and smoothness. This indicates that HDME captures the dynamic characteristics of signals with greater stability and less susceptibility to parameter selection, which is advantageous for practical applications. It is important to note that the signal ratio has a low signal-to-noise ratio, resulting in an irregular signal and a high level of uncertainty. Consequently, the entropy of the signal surpasses that of the majority of theoretical thresholds, signifying a higher degree of complexity in the signal, in line with the principles of entropy theory. A low signal-to-noise ratio increases the difficulty of signal processing as it necessitates differentiation between fault characteristics in the signal and background noise. Under these conditions, HDME retains its efficacy in revealing the intrinsic complexity of the signal, thereby validating its effectiveness as a metric for signal complexity assessment.

3.4. Parameter Discussion

The feature information is conducted by using the proposed high-dimensional mapping entropy algorithm. Since varying parameter values can impact entropy, the study of parameters for the HDME value requires the application of such parameters. HDME involves the following parameters: the embedding dimension m, the similarity tolerance r, and the time delay λ. An experimental analysis was carried out using a section of normal state rolling bearing data from Case Western Reserve University [29]:

(1)

Embedded dimension m: This parameter defines the number of points in time to consider when reconstructing the phase space. Bearing fault signals often contain multiple dynamic behaviors, so m is needed to capture these complexities. Increasing m provides more information for sequence reconstruction but at the cost of reduced computational efficiency. Selecting a small value of m may result in insufficient data length and potential loss of important information. As per the literature, in this paper it is set as m = 2;

(2)

Similarity tolerance r is utilised to assess the level of similarity between two vectors. An appropriate r can help distinguish between normal and faulty dynamics. In scenarios where r is small, the entropy findings are prone to noise interference, and the statistical effect is insufficient. Conversely, if r is large, the detailed information of the time series can be forfeited. Generally, r is taken as 0.1 SD–0.25 SD; in this paper, it is taken as r = 0.15 SD;

(3)

Time delay $\lambda$: The degree to which a time series is sampled is determined by $\lambda$, which affects the MHDME measure of signal complexity. As shown in Figure 5, if the λ becomes too large, it will lead to the loss of characteristic features in the time series and distortion of the sampled data. The time delay needs to be selected according to the sampling frequency of the signal and the fault characteristics of interest to ensure that the fault characteristics are effectively captured. It is generally taken as $\lambda$ = 1;

(4)

Data length N: The data length represents the operating state of the bearing. The data length needs to be sufficient to cover the entire cycle of bearing failure development. Comparing the entropy of mixed signals with different lengths N is shown in Figure 6. The analysis indicates that the multi-scale high-dimensional mapping entropy is consistent with different values of N, and the signal length requirement is low;

(5)

Scale factor $\tau$: This determines the granularity of the signal complexity analysis. For bearing fault signals, different scales may highlight different fault characteristics. From N = $10^m~{30}^m$, m = 2, so N $\ge$ $100$, the scale factor $\tau \ge \mathrm{}$ has a small effect on the entropy; in this paper, it is set as $\tau$ = 20;

(6)

The value of the kernel parameter sigma affects the distance between reconstructed vectors, and if it is too large or too small it will lead to inaccuracy when calculating the spacing. The determined kernel parameters should be able to distinguish between signal fluctuations under normal operating conditions and signal changes caused by failures. In this paper, we take the Gaussian kernel parameter as $\sigma$ = 0.05 and the polynomial kernel values as d = 2 and $\gamma$ = 1.

4. Multi-Scale High-Dimensional Mapping Entropy (MHDME)

4.1. Refined Composite Multi-Scale Studies

4.1.1. Multi-Scale High-Dimensional Mapping Entropy

Combined with the idea of multi-scaling to determine the original sequence coarse-graining law, matching forms the multi-scale high-dimensional mapping entropy (MHDME) algorithm as follows:

(1)

Pre-set the original sequence with r, m, and N to create a new coarse-grained sequence:

$$y_j^{\tau }=\frac{1}{\tau }{\sum }_{i=\left(j-1\right)\tau +1}^{j\tau }{x}_i(1\le j\le \frac{N}{\tau })$$

(26)

where $\tau$ represents the scale factor. When τ = 1, $y_i\left(1\right)$ is the primary sequence. For the non-zero positive integer τ, the primary sequence is divided into τ coarse grain sequences $y_j^{\tau }$, each with length $\frac{N}{\tau }$;

(2)

Calculate the high-dimensional mapping entropy for each coarse-grained sequence to obtain the variation of the multi-scale high-dimensional mapping entropy with increasing scale factor:

$$MHDME\left(x,m,r,\tau ,sigma,N\right)=HDME({y_j}^{\tau },m,r,sigma)$$

(27)

The MHDME reflects the regularity and complexity of time series at various scales. With the rising scale factor, the length of the coarse-grained sequence declines, indicating that the time series structure contains more information under the smaller scale factor.

4.1.2. Refined Composite Multi-Scale High-Dimensional Mapping Entropy (RCMHDME)

The RCMHDME algorithm improves the accuracy of entropy estimation by summing all coarse-grained sequences under the scale factor τ and then calculating the high-dimensional mapping entropy, and the probability of undefined entropy appearing in MHDME is effectively reduced. The steps are as follows:

(1)

The Equation (26) is used to preprocess the original time series $\{x_1,x_2,\cdots ,x_n\}$ and obtain $\tau$ a coarse-grained time series of ${y_k}^{\tau }=\{y_{k,1}^{\left(\tau \right)},y_{k,2}^{\left(\tau \right)},\cdots ,y_{k,p}^{\left(\tau \right)}\}$:

$$y_k^{\tau }=\frac{1}{\tau }{\sum }_{i=(j-1)\tau +k}^{j\tau +k-1}{x}_i (1\le j\le \frac{N}{\tau },1\le k\le \tau )$$

(28)

(2)

Given a statistically determined scale factor $\tau$, under which all coarse-grained time series satisfy the conditional vector pair $N_{k,\tau }^m$ and $N_{k,\tau }^{m+1}$.

(3)

Calculate the mean ${\bar{N}}_{k,\tau }^m$, ${\bar{N}}_{k,\tau }^{m+1}$ of individual $N_{k,\tau }^m$ and $N_{k,\tau }^{m+1}$$(1\le k\le \tau )$, respectively. To log the ratio of ${\bar{N}}_{k,\tau }^m$ to ${\bar{N}}_{k,\tau }^{m+1}$, define RCMHDME:

$$CMHDME\left(x,\tau ,m,r,sigma\right)=-\mathit{ln}\frac{{\bar{N}}_{k,\tau }^{m+1}}{{\bar{N}}_{k,\tau }^m}$$

(29)

Collating the above equation yields:

$$RCMHDME\left(x,\tau ,m,r,sigma\right)=-\mathit{ln}\frac{{\sum }_{k=1}^{\tau }{N}_{k,\tau }^{m+1}}{{\sum }_{k=1}^{\tau }{N}_{k,\tau }^m}$$

(30)

Only when both $N_{k,\tau }^m$ and $N_{k,\tau }^{m+1}$ are 0, the entropy presents undefined results, and compared with the multi-scale coarse-grained process, the refined composite process is more suitable for the analysis of short-time series, the computational accuracy is improved, and the obtained entropy results are better in terms of stability and consistency.

4.2. Simulation Comparative Analysis

To investigate the law of variation and measure its robustness, white noise is used as an example. The time-domain waveform diagram and spectrum diagram of white noise with data length N = 3000 and noise intensity of 0.3 are shown in Figure 7. What can be concluded from Figure 7 is that the white noise remains constant in energy and complexity in each frequency band.

The multi-scale high-dimensional mapping entropy (MHDME) and fine composite multi-scale high-dimensional mapping entropy (RCMHDME) with multi-scale sample entropy (MSE) and refined composite multi-scale sample entropy (RCMSE) methods were used for complexity feature extraction analysis of Gaussian white noise signals, respectively. The parameters were set as follows: dimension m = 2, similarity tolerance r = 0.15 SD, time delay $\lambda$ = 1, and data length N = 3000. Entropy was computed for under 50 datasets, followed by plotting the respective error bar graphs. The outcomes are presented in Figure 8.

Compared with MSE and RCMSE, the MHDME and RCMHDME methods proposed in this paper have smaller error bars. Consequently, the accuracies of MHDME and RCMHDME are superior, yielding measurements that more closely approximate true values with reduced deviation. Results derived from the MHDME and RCMHDME methods demonstrate high consistency, exhibit resistance to random error, and present improved stability. Notably, the RCMHDME method displays minimal entropy fluctuation at larger scales, signifying that the algorithm’s performance remains more uniform and consistent across various time scales. The entropy of the original multi-scale algorithm produces slight fluctuations at some scales, and the refined composite multi-scale entropy has smaller errors and better smoothness at a wide range of scales. Therefore, it has been proven that the high-dimensional mapping entropy algorithm proposed in this paper is more accurate and smoother.

5. Application for Reciprocating Compressor

5.1. Feature Extraction Method

In this section, multi-scale high-dimensional mapping entropy is applied to feature extraction in reciprocating compressors. Firstly, an adaptive decomposition method suitable for reciprocating compressor signal characteristics is selected to remove interference and background noise components. Secondly, the modal components abundant in faulty information are selected by the correlation coefficient for signal reconstruction. Then, the multi-scale high-dimensional mapping entropy of the reconstructed signals is calculated to construct a state feature set. Furthermore, the important feature elements of the set are intelligently selected to form feature vectors. Finally, the feature vectors are input into the pattern recognizer for classification and recognition to achieve the diagnosis of different types of faults. The details are as follows:

(1)

Signal adaptive decomposition processing

There exist emerging and effective research methods for applying the combination of adaptive decomposition methods and entropy for fault feature extraction. Generally used tools include EMD [30], LMD [31], and VMD [32]. EMD and LMD decomposition have constraints in suppressing modal aliasing, and the adaptability to compressor signals is restricted. VMD utilizes the features of data at various scales and possesses efficient processing capabilities alongside strong noise resistance. For comprehensive consideration, the VMD method is used to preprocess the signal in the experiments of this paper;

(2)

Characteristic component screening and reconstructing signals

Reciprocating compressor vibration signals are preprocessed by VMD to form a series of band-limited intrinsic mode functions (BLIMFs). Correlation analysis was performed according to the correlation coefficient method to calculate the value of the coefficient between the modal components and the original signal. The components with high correlation in the decomposition results are selected as BLIMF components that contain the main state information. The fault information primarily exists in the first several components of the decomposition, and the characteristics of the reciprocating compressor, in particular, are concentrated in the high-frequency band;

(3)

Computing MHDME to construct the set of feature matrices

For filtering the main components related to the signal, signal reconstruction is performed to solve the entropy of the multi-scale high-dimensional mapping. The set of feature matrices obtained for each state is fused to form the set of state feature matrices as:

$$T_k=\left[\begin{array}{ccccc}{t}_{k11}& \cdots & t_{k1\tau }& \cdots & t_{k1\alpha }\\ ⋮& & & & \\ t_{kp1}& & t_{kp\tau }& & ⋮\\ ⋮& & & & \\ t_{k\beta 1}& & \cdots & & t_{k\beta \alpha }\end{array}\right]$$

(31)

where $T_k(k=1,\cdots ,\omega )$ is the $k$ set of feature matrices of the state, $t_{kp\tau }$ denotes the entropy when the scale factor $\tau (1\le \tau \le \alpha )$ of the $p(1\le p\le \beta )$ group of samples is in the state $k$, $\beta$ is the number of groups in the corresponding state, and $\alpha$ is the maximum scale factor;

(4)

Intelligent optimization

When the feature vectors are directly input into the classifier, the dimension of the feature space increases, resulting in the redundancy of information, which reduces the performance of the classifier. Therefore, an intelligent selection algorithm is used to select and rank the importance of the elements $t_{kpr}$ of the feature vector matrix set $T_k$. By extracting the representative state information, a feature vector with good separability is formed to improve the computational efficiency and classification accuracy.

Max-relevance and min-redundancy (mRMR) [33], Fisher score (FS) and Laplacian Score (LS) [34] are commonly used preference learning techniques. The mRMR is not suitable for high-dimensional datasets based on specific metrics [35]. The FS ranks features by estimating the discriminative ability of each feature vector for different classes without considering the redundancy among the features. Meanwhile, the LS can effectively preserve the holistic geometrical structure information that is embedded in the state feature set, facilitating state feature extraction and identification in complex devices. Overall, the LS-based feature intelligent preference strategy is used to rank the importance of the features and select the first few orders of importance to construct the state feature vector.

(5)

Support vector machine pattern recognition

Common classification models include logistic regression (LR), decision tree classifiers, and support vector machines (SVMs). Logistic regression is suitable for dealing with classification problems that are close to linearly divisible, and when the feature space is large, the performance of logistic regression is weak. Decision tree classifiers are more suitable for discrete features. One of the most widely used classification algorithms in machine learning is the SVM. It is able to handle large feature spaces and interactions between non-linear features without relying on the data as a whole.

Therefore, the set of state feature vectors is fed into the SVM for training to obtain a trained model for various bearing faults. This allows differentiation and identification of different states of reciprocating compressors.

The process of fault diagnosis is shown in Figure 9.

5.2. Reciprocating Compressor Bearing Clearance Failure Experimental Study

The reciprocating compressor is one of the most commonly used power tools for compressing natural gas for long-distance transportation. This type of compressor is highly efficient and reliable, making it adaptable to different operational conditions and environmental situations [36,37]. The reciprocating compressor structure is complex, with multiple parts, mainly including body parts, power transmission mechanisms, working chamber, and auxiliary accessories. The connecting rods of connecting rod members in the reciprocating compressor drive train are connected to the crankshaft and the crosshead pins through rotary joints. In practical constructions, plain bearings are usually used as heavily loaded joints. Over time, plain bearings experience an increase in clearance due to friction and wear [38]. Excessive clearance produces large impact and friction-damping problems under reciprocating inertia forces, which seriously affects the performance of the equipment. Therefore, a set of processing methods suitable for reciprocating compressor-bearing fault signals is proposed, which realizes feature extraction and pattern recognition of different faults through adaptive decomposition preprocessing and the fusion of the entropy analysis method and the feature preference strategy, and has relatively high value for engineering applications. The bearing clearance vibration signal of the reciprocating compressor 2D12-70 is selected as the data source for this paper. The main technical parameters of the 2D12-70 reciprocating compressor are shown in

Table 1. Main technical parameters of the 2D12-70 reciprocating compressor.

Parameter Type	Parameter Valve
Shaft power (kW)	500
Exhaust volume (m3/min)	70
Primary exhaust pressure (MPa)	0.2746~0.2942
Secondary exhaust pressure (MPa)	1.2749
Piston stroke (mm)	240
Crank speed (rpm)	496

[39,40].

A study was conducted on the failure of a double-acting reciprocating compressor of type 2D12. In the actual test, sensor A was set above the crankcase to record the vibration signal. Figure 10 illustrates the three-dimensional and planar positioning of sensor A. The data acquisition analyzer used in this study was a portable 16-channel device, specifically the UT9001 model. The sensor employed was an ICP accelerometer with a sensitivity of 10 mv/unit. The frequency of the reciprocating compressor was determined to be 8.27 Hz, derived from a rotational speed of 496 r/min. The sampling time was set at 4 s, the sampling frequency at 50 kHz, and continuous sampling was conducted 100 times. Data were collected for three different states: normal condition bushing with a 0.1 mm gap, moderate wear state with a 0.2 mm gap, and severe wear state with a 0.3 mm gap [41]. Bushings in normal and different wear states are shown in Figure 11.

The fault simulation experiments were conducted on a type 2D12 opposed double-acting reciprocating compressor, selecting technical parameters reflective of actual operational conditions. In the wear diagnosis of reciprocating compressor bearings, high-dimensional mapping entropy (HDME) should accurately reflect fault characteristics under varying wear conditions and possess strong discriminatory power regarding the severity of faults. This paper discusses the applicability of HDME, focusing primarily on the capture of characteristic features at different stages of bearing wear within an engineering production context and quantifying states of normal, moderate, and severe wear. A normal bearing at 0.1 mm clearance, a moderately worn bearing at 0.2 mm clearance, and a severely worn bearing at 0.3 mm clearance were chosen. These gaps represent typical wear gaps encountered in engineering experimental production operations. Consequently, this selection ensures the signal complexity inherent in fault scenarios is realistically captured.

5.3. Application for Reciprocating Compressor Fault Diagnosis

MHDME serves as a non-linear analytical tool for time series data, designed to quantify shifts in a system’s dynamic complexity. In diagnosing faults in reciprocating compressor bearings, MHDME is capable of detecting variations in signal patterns attributable to changes in bearing condition.

MHDME effectively captures specific fault characteristics associated with reciprocating compressor bearings by analyzing vibration acceleration signal data. Its utility lies in the detection and quantification of diverse patterns and dynamics within the vibration signals that signify bearing failures, which typically present as irregularities in the signal.

When the sample data is projected into a high-dimensional space applying MHDME, the prominence of fault features that may be confused by noise is enhanced. MHDME quantifies signal complexity and uncertainty through phase space reconstruction of the vibration signal, transforming it into a multi-dimensional representation, and computing the information entropy of the point set within this space. This methodology not only exposes variations in signal patterns resulting from alterations in bearing conditions but also delineates the temporal evolution characteristics of the signal. Meanwhile, the high-dimensional mapping entropy method is integrated with time series analysis in the process of phase space reconstruction of bearing signals. This integration can reflect the dynamic behaviour of bearing faults more comprehensively, thus improving the accuracy and reliability of fault diagnosis.

5.3.1. Vibration Signal Preprocessing of Different States

In the experiment, experimental data for different bushing wears of reciprocating compressors were collected, including normal state signal, moderate wear state, and heavy wear state. The spatial location of sensor measurement point A is visually depicted in Figure 10. According to the speed and sampling frequency of the reciprocating compressor, one working cycle is 0.12 s, and the data of two cycles are taken in each operating state for analysis. The vibration signals of the compressor bearings under normal and fault conditions are shown in Figure 12.

Several modal components are obtained in each state using the VMD method. The decomposition scale of the original sampled signal is preset to be $K=7$ and the bandwidth parameter $\alpha =2000$. The BLIMF1-BLIMF7 components obtained from the decomposition of the different states are shown in Figure 13. The amount of VMD decomposition is humanly determined. If K is too small, incomplete decomposition and modal aliasing may occur. Whereas, if K is too large, over-decomposition may occur, resulting in loss of modal information. To address this problem, in this paper we screen the valid modal components of the original signal information by the correlation degree between the BLIMF components.

5.3.2. Fault Principal Component Selection in Different States

The value of the correlation degree between the seventh-order modal components and the original signal is calculated in

Table 2. The correlation coefficient of the BLIMF component and the original signal.

Fault Type	Value of the Correlation Coefficient between the Modal Component and the Original Signal
	BLIMF1	BLIMF2	BLIMF3	BLIMF4	BLIMF5	BLIMF6	BLIMF7
Normal state	0.3683	0.5439	0.5462	0.5069	0.3954	0.1945	0.0700
Moderate wear condition	0.2081	0.2714	0.3429	0.5265	0.6930	0.5243	0.1172
Heavy wear condition	0.3224	0.3279	0.5874	0.6996	0.3805	0.2582	0.0460

. From the table, it is known that under the normal state of the bearing, over-decomposition occurs when $K\ge 5$, the fifth-order modal component is distorted, and the correlation degree with the original signal decreases significantly. Therefore, the number of modal components of each state is unified, and the first five-order modal components are taken for signal reconstruction.

The three different state signals were subjected to VMD decomposition to derive the seventh-order BILMFs shown sequentially in Figure 14. The amplitude of the normal state signal after decomposition is the lowest and shows periodic changes. The signal after decomposition of the moderate and moderately worn state has a higher impact and higher amplitude. Reciprocating compressor vibration signals form a series of BLIMF after VMD. The information contained in each component is inconsistent, and the decomposition results are filtered for noise to highlight the main state information. As shown in Table 2, the initial five orders of decomposition signal exhibit a high correlation coefficient value with the original signal. When k = 6, the normal state and heavy wear state signal components have a low correlation with the original signal and cannot truly represent the original signal state information. Therefore, the “clean vectors” of different states are constructed after the reconstruction of BLIMF in the first five orders. The amplitudes of the three states after reconstruction are lower than the original signal values, discarding the low-frequency baseline oscillations and low-frequency noise, and highlighting the fault information in the high-frequency band.

5.3.3. Calculate the Different Entropies in the Three States

The signal samples of different states are preprocessed using the VMD method, and the components containing the main feature information are preferred for signal reconstruction. The MHDME algorithm was used to analyse and process each group of reconstructed signals, and the entropy under 20 scales for each state was calculated as state features. At the same time, the same process was used to calculate the MSE and MPE of the pre-processed signals by VMD, and the pre-selected scales were set at $K=5$ in VMD. To assess the stability of various algorithms, the values were calculated for three distinct working conditions. Parameter selection for entropy calculation was as follows: $m=2$, $\tau =20$, $r=0.15\times SD$ (SD is the standard deviation), time delay t $=$ $1$ in MHDME, $m=6$, $\tau =20$, $t=1$ in MPE, $m=2$, $\tau =20$, $r=0.15\times SD$, t $=$ $1$ in MSE. The alterations in entropy for distinct bearing conditions are depicted in Figure 15a–c.

It can be seen from Figure 15a that, within 20 scales of the MHDME, there is no crossover of entropy between different states, and the separability and stationarity between fault types are preferable. The entropy information of different compressor fault states on a large scale not only shows the necessity of multi-scale analysis but also shows the complexity of vibration signals. In the normal state, the unit operation has randomness and irregularity, so the signal chaos degree is larger, and the entropy amplitude is larger. When the bearing of a reciprocating compressor has a failure with different degrees of wear, the distinction between the states is obvious. The entropy of the bearing under a normal state is greater than that under other fault states, indicating that the signal under a normal state is relatively random and complex, and when the fault occurs, the signal has a certain regularity and the signal complexity is reduced.

According to the analysis in Figure 15b,c, it can be seen that the entropy curves of MPE and MSE after pretreatment have different degrees of crossover under different scale ranges. When the scale factor τ = 1, only comparing the MPE in different states is insufficient to achieve accurate prediction and diagnostic outcomes. MPE exhibits some degree of crossover at a scale factor 2 ≤ τ ≤ 5, and the separability of different states is poor at larger scales. In addition, MSE fluctuates more on 20 scales, and the crossover of different states is serious at scales 2 ≤ τ ≤ 7. At the same time, the normal state and moderate wear faults appear to be intermixed at larger scales. The experimental data of MPE and MSE show that the two methods can achieve pattern differentiation, but the stability and separability are unfavorable, resulting in lower recognition accuracy. MHDME exhibits a narrower error margin compared to MPE and MSE, which demonstrate broader error margins. This indicates that the MHDME algorithm yields more stable results, with enhanced consistency between measurements, thereby providing more accurate information. In conclusion, the MHDME, MPE and MSE of the three working conditions are compared and analyzed. The stability of both MPE and MSE is unsatisfactory at most scales, while the MHDME algorithm exhibits stability and discernibility. The MHDME algorithm can accurately depict the defect features exhibited by the vibration signal of the reciprocating compressor bearing.

5.3.4. Intelligent Preference of Features for Different States

Due to the substantial volume of data, information redundancy exists within the feature vectors. If the high-dimensional matrix set is directly input into SVM for pattern recognition, the recognition accuracy and computational efficiency will be reduced, and the divisibility between states will become worse. Therefore, the Laplacian score (LS) algorithm is selected to intelligently rank the importance of extracted state features, select features with better separability to form feature vectors, and then input them into SVM, which not only improves the diagnostic accuracy but also shortens the calculation time. The results of LS sorting are as follows:

$$\begin{gathered} L{S}_{12}<L{S}_{13}<L{S}_{14}<L{S}_{10}<L{S}_{11}<L{S}_9<L{S}_{15}<L{S}_{16}<L{S}_{17}<L{S}_8<L{S}_{18}< \\ L{S}_{19}<L{S}_5<L{S}_6<L{S}_{20}<L{S}_7<L{S}_2<L{S}_4<L{S}_3<L{S}_1 \end{gathered}$$

(32)

where the LS subscript represents the scale factor.

The smaller the Laplace score, the more important the features at this scale are, and the more prominent the state information is. The diagnosis result is affected by the feature vector dimension. If the feature number is insufficient, it is limited to completely distinguish and reflect the fault types. Conversely, if there are too many features, there may be cross-redundancy of information, leading to decreased accuracy and efficiency of recognition. To prove this point, feature vectors composed of values of different dimensions sorted by LS are selected and input into SVM respectively. After running SVM 10 times, the results of training time and recognition rate are shown in

Table 3. The output of SVM when the characteristic dimension changes.

Measurable Indicator	SVM (X = 2)	SVM (X = 3)	SVM (X = 4)	SVM (X = 5)	SVM (X = 6)	SVM (X = 7)	SVM (X = 8)
Average training time (s)	4.02	4.27	3.81	3.80	3.76	3.75	3.72
Average fault recognition rate (%)	80.81	92.24	96.15	97.32	99.68	99.53	99.11

.

Table 3 shows the selected SVM intelligent recognition classifiers combined. When the number of features covering a larger amount of information is reduced, the average training time is decreased, and recognition accuracy gradually declines. Nevertheless, average training time increases with the number of features. The results in

Table 4. Comparison of the results of 20 SVM runs of the three methods.

Troubleshooting Methods	Normal State Recognition Rate (%)	Moderate Wear Condition Recognition Rate (%)	Heavy Wear Condition Recognition Rate (%)	Average Recognition Rate (%)
VMD-MHDME	100 $\pm$ 0.21	98 $\pm$ 1.86	97 $\pm$ 0.73	98.33 $\pm$ 0.93
VMD-MPE	95 $\pm$ 1.86	96 $\pm$ 2.55	88 $\pm$ $2.09$	93.00 $\pm$ 2.17
VMD-MSE	86 $\pm$ 1.29	90 $\pm$ $2$.56	75 $\pm$ 3.72	83.67 $\pm$ 2.52

prove the feasibility and reasonableness of selecting the first six state features in this paper. Therefore, the optimal features under the first six scale factors are combined with a feature vector τ = [9,10,11,12,13,14], which is input into SVM as a new feature vector for pattern classification recognition.

5.3.5. Support Vector Machine Pattern Recognition Results

The proposed MHDME method via VMD preprocessing, MPE method via VMD preprocessing, and MSE method via VMD preprocessing are used to input feature vectors into SVM for classification and recognition. The results of the examination are shown in Table 4. Through comparative analysis, the fault feature extraction capability of the three methods is MHDME > MPE > MSE. This indicates that the approach enables deep and accurate acquisition of feature information, which highlights its higher feature extraction capability and fault recognition rate.

As depicted in Table 4, during the normal and moderate wear stages of reciprocating compressor bearings, all three methods exhibit the ability to identify and diagnose faults. Among these methods, VMD–MHDME demonstrates a higher level of identification accuracy compared to the others. However, in the heavy wear state, the VMD–MSE and VMD–MPE methods exhibit a high misclassification rate, which is unfavorable for accurate fault diagnosis. Conversely, the feature extraction method of VMD–MHDME demonstrates the highest recognition accuracy in a single state. In the results of 20 SVM runs, VMD–MHDME exhibits the best average state feature recognition rate with a small standard deviation and maintains a stable fault recognition rate in multi-fault diagnosis. The experimental results demonstrate that VMD–MHDME can accurately and stably extract state feature information, highlighting its high feature extraction capability and fault recognition rate.

6. Conclusions

In this paper, a high-dimensional mapping entropy (HDME) algorithm is proposed, which involves the characteristics of vibration signals. The application of a combination algorithm provides a new way for equipment fault diagnosis:

(1)

HDME, grounded in the noise reduction theory of high-dimensional space mapping, aims to effectively mitigate the issue of traditional entropy’s abrupt changes. It integrates sample entropy vector distance measurement steps to achieve efficient vector differentiation in high-dimensional spaces. When applied to simulated signals, the method exhibits a smooth transition with similarity tolerance and demonstrates strong performance in terms of parameter consistency and sensitivity. By introducing the scale factor, the MHDME is proposed, and the RCMHDME is proposed by integrating the refined composite multi-scale criteria. Simulation results indicate that RCMHDME exhibits low sensitivity to parameters, smaller entropy error bars, and improved stability;

(2)

The state feature extraction strategy for MHDME is constructed to achieve the identification and diagnosis of different fault state types. The VMD method is used for preprocessing to quantitatively describe the state signals in terms of MHDME and construct the feature set. The LS intelligent algorithm is used to perform feature preference, form feature vectors, and combine with SVM to achieve complete fault feature extraction and diagnosis;

(3)

The MHDME method has been applied to extract features from vibration signals of reciprocating compressors experiencing three bearing clearance fault states. The results show that the single fault diagnosis recognition rate and average recognition rate using the MHDME method are the highest among the three methods, and the variance value is the smallest. Therefore, this study confirms the proposed method’s authenticity and reliability in detecting bearing clearance faults within reciprocating compressors.

The HDME has the advantage of measuring the distance directly after reconstructing the planar phase space with traditional entropy. In this way, we increase the dimensionality of the sample space and have better hyperplanes in the higher dimensional space to distinguish these data. In this way, the difference features of different samples can be better extracted, and the stability and accuracy of entropy can be improved. The method introduced in this study demonstrates the susceptibility of analogue signals to single-parameter modifications. However, the assessment of multi-parameter interactions and their significance requires further comprehensive exploration. Future research should address the following needs to improve the understanding of the HDME algorithm and increase its application effectiveness: A comprehensive sensitivity analysis including multi-parameter interactions, introduction of further data features and comparisons of fault diagnosis techniques for evaluation of the noise immunity and robustness of HDME, and testing of the generalization ability of HDME using different data sets from other domains.

Overall, the effectiveness and feasibility of the proposed method are verified by comparing three cases of bearing signal analysis and fault diagnosis of reciprocating compressors. This study has potential value in predicting early wear or failure and fault diagnosis.