## 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. [leftmargin=*]
2. 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 \begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{**Troubleshooting Methods**} & **Normal State Recognition Rate (\%)** & **Moderate Wear Condition Recognition Rate (\%)** & **Heavy Wear Recognition Rate (\%)** \\ \hline 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 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Comparison of the results of 20 SVM runs of the three methods. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multirow{2}{*}{**Measurable Indicator**} & **SVM** & **SVM** & **SVM** & **SVM** & **SVM** & **SVM** \\  & **(X = 2)** & **(X = 3)** & **(X = 4)** & **(X = 5)** & **(X = 6)** & **(X = 7)** & **(X = 8)** \\ \hline 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 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: The output of SVM when the characteristic dimension changes. 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.