# Surface Roughness Prediction of Bearing Ring Precision Grinding Based on Feature Extraction
## 3 Grinding Surface Roughness Feature Extraction and Fusion
### Entropy Characteristics
#### 3.2.1 Information Entropy  
Information entropy is a concept that describes the amount of information, and its definition involves the concept of self-information. Directly calculating the information entropy of the time-domain signal only reflects the uncertainty in time, ignoring the information in space. Therefore, the signal is usually preprocessed using methods such as power spectrum analysis and singular value decomposition to generate power spectrum entropy and singular spectrum entropy. (1) Power spectrum entropy  
Power spectrum entropy is an index used to measure the uncertainty of a signal via power spectrum division. Therefore, power spectrum entropy provides a quantitative description of the complexity of the energy distribution of a signal in the frequency-domain [32]. To solve the signal power spectrum and power spectrum density, the following equation is used:  
\[s(k)=\frac{1}{N}|y(k)|^{2} \tag{8}\] \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Frequency-Domain** & **Calculation Formulas** & **Frequency-Domain Features** & **Calculation Formulas** \\ \hline Sample mean of spectral & \(x_{12}=\frac{\sum_{k-1}^{K}s(k)}{K}\) & Change in the position of the main frequency band & \(x_{19}=\sqrt{\frac{\sum_{k-1}^{K}f_{k}^{K}s(k)}{\sum_{k-1}^{K}f_{k}^{K}s(k)}}\) \\ Sample variance of spectral & \(x_{13}=\frac{\sum_{k-1}^{K}(s(k)-x_{12})^{2}}{(K-1)}\) & Concentration of frequency & \(x_{20}=\frac{\sum_{k-1}^{K}f_{k}^{K}s(k)}{\sqrt{\sum_{k-1}^{K}s(k)}\sum_{k-1}^ {K}f_{k}^{K}s(k)}\) \\ Skewness of spectral & \(x_{14}=\frac{\sum_{k-1}^{K}(s(k)-x_{12})^{3}}{K\left(\sqrt{x_{13}}\right)^{3}}\) & Coefficient of variation & \(x_{21}=\frac{x_{17}}{x_{16}}\) \\ Kurtosis of spectral & \(x_{15}=\frac{\sum_{k-1}^{K}(s(k)-x_{12})^{4}}{Kx_{13}^{2}}\) & Skewness of frequency & \(x_{22}=\frac{\sum_{k-1}^{K}(f_{k}-x_{16})^{3}s(k)}{Kx_{17}^{3}}\) \\ Mean frequency & \(x_{16}=\frac{\sum_{k-1}^{K}f_{k}^{K}(s)}{\sum_{k-1}^{K}s(k)}\) & Kurtosis of frequency & \(x_{23}=\frac{\sum_{k-1}^{K}(f_{k}-x_{16})^{4}s(k)}{Kx_{17}^{3}}\) \\ Variance of frequency & \(x_{17}=\sqrt{\frac{\sum_{k-1}^{K}(f_{k}-x_{16})^{2}s(k)}{K}}\) & Standardized mean of spectral amplitude & \(x_{24}=\frac{\sum_{k-1}^{K}(f_{k}-x_{16})^{\frac{1}{2}s(k)}}{K\sqrt{x_{17}}}\) \\ Root Mean Square (RMS) & \(x_{18}=\sqrt{\frac{\sum_{k-1}^{K}f_{k}^{K}s(k)}{\sum_{k-1}^{K}s(k)}}\) & & \\ \hline \hline \end{tabular} Note: \(s(k)\) denotes the spectrum of signal sequence \(x(n)\), \(k=1,2\), \(3\), \(\ldots\), \(K\); \(K\) denotes the number of spectral lines.  
\end{table} 
Table 2: Frequency-domain features. \begin{table}
\begin{tabular}{c c} \hline \hline
**Entropy Feature** & **Calculation Formulas** \\ \hline Power spectrum entropy & Obtained from Equation (10) \\ Singular spectrum entropy & Obtained from Equation (13) \\ Permutation entropy & Obtained from Equation (17) \\ Sample entropy & Obtained from Equation (21) \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Features of entropy-domain. where \(k\) denotes frequency point, \(k=1,2,3,\ldots,N-1;y(k)\) denotes the application of a discrete Fourier transform to the signal. \[p_{i}=\frac{s(k)}{\sum\limits_{k=0}^{N-1}s(k)} \tag{9}\]  
Here, \(p_{i}\) denotes the power spectral density corresponding to the frequency of \(i\). Standardized power spectral density is used to calculate power spectrum entropy:  
\[H_{\text{{\it{\it{\it{\it{\it{\it{\it{\it{\it{\it{\it{\it{\it{\it{\it{\it{\it{ \it{ { }}}}}}}}}}}}}}}= \sum\limits_{i=1}^{N}p_{i}\log_{2}p_{i} \tag{10}\]  
where N denotes the total frequency count. (2) Singular spectrum entropy  
The singular spectrum entropy method is an advanced method for signal analysis. It combines singular spectrum analysis and information entropy. By calculating the singular spectrum of the signal and further obtaining its information entropy, the complex state features of the time series can be quantitatively described [33].