## 5 Application for Reciprocating Compressor  
### 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;  
Figure 8: Comparison of entropy of white noise under different algorithms. (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:  
\[T_{k}= \begin{bmatrix}t_{k11}&\cdots&t_{k1\tau}&\cdots&t_{k1\alpha}\\ \vdots&&&&\\ t_{kp1}&&t_{kp\tau}&&\vdots\\ \vdots&&&&\\ t_{k\beta 1}&&\cdots&&t_{k\beta a}\end{bmatrix}  \tag{31}\]  
where \(T_{k}(k=1,\cdots,\omega)\) is the \(k\) set of feature matrices of the state, \(t_{kpt}\) denotes the entropy when the scale factor \(\tau(1\leq\tau\leq\alpha)\) of the \(p(1\leq p\leq\beta)\) 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].