## 3 High-Dimensional Mapping Entropy
### 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 \(\lambda\). 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 \(\lambda\) 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 \(N=10^{m}\sim 30^{m}\), \(m=2\), so \(N\geq 100\), the scale factor \(\tau\geq\) 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 \(\gamma=1\).  
Figure 5: Comparison of entropy for different time delays \(\lambda\).