# Fault Diagnosis of Vehicle Gerboxes Based on Adaptive Wavelet Threshold and LT-PCA-NGO-SVM
## 2 Principle of the Method
### Dimensionality Reduction of Fault Feature Construction
#### 2.2.2 Log-Transformed Principal Component Analysis Dimensionality Reduction  
Principal component analysis (PCA) is a widely used technique for dimensionality reduction [27]. Its core idea involves mapping high-dimensional data to a low-dimensional space through a linear transformation while retaining as much variance in the data as possible. Dimensionality reduction is achieved by finding the main components of the data and projecting them into a new coordinate system. PCA effectively mitigates information loss by downscaling the data using the feature matrix [28]. However, outliers with large variance values can impact the data's variance during downscaling, influencing the direction of each principal component of PCA. Simply removing outliers may result in the loss of valuablesample information. To address this, we propose a new dimensionality reduction method in this section--logarithmic transformation principal component analysis (LT-PCA). To stabilize fluctuations in the dataset and maintain as constant a variance as possible, logarithmic transformation is chosen for data processing. This transformation compresses the data, reduces fluctuations, and eliminates large differences in orders of magnitude. We define the sample as \(Z_{t}\), the sample mean \(E(Z_{t})=\mu_{t}\), and set the distribution standard deviation to be linearly correlated with the mean; then, \(\sqrt{Var(Z_{t})}=\mu_{t}\times\sigma\), where \(\sigma\) is the correlation coefficient. When \(x\) is sufficiently small, \(\log(1+x)\approx x\), leading to the following formula:  
\[\log(Z_{t})\approx\log(\mu_{t})+\frac{Z_{t}-\mu_{t}}{\mu_{t}} \tag{22}\]  
Furthermore, \(E\left(\frac{Z_{t}-\mu_{t}}{\mu_{t}}\right)=0\), \(E(\log(Z_{t}))\approx\log(\mu_{t})\), as demonstrated in the following derivation process:  
\[ \begin{array}{ll}Var[\log(Z_{t})]&=E[\log(Z_{t})-E(\log(Z_{t}))]^{2}=E[\log( Z_{t})-\log(\mu_{t})]^{2}\\ &\approx E\left(\frac{Z_{t}-\mu_{t}}{\mu_{t}}\right)^{2}=\frac{1}{(\mu_{t})^{ 2}}E(Z_{t}-E(Z_{t}))^{2}\\ &=\frac{1}{(\mu_{t})^{2}}Var(Z_{t})\approx\sigma^{2}\end{array}  \tag{23}\]  
To summarize, \(E(\log(Z_{t}))\approx\log(\mu_{t})\), \(Var[\log(Z_{t})]\approx\sigma^{2}\). Logarithmic transformations can effectively convert a model prone to significant fluctuations with increasing independent variables into a more stable model. Assuming \(D\) is a dataset with \(m\) samples and \(n\) variables, the transformation matrix is defined by the formula:  
\[P=\log_{n}D \tag{24}\]  
In the formula, \(P\) represents the logarithmic transformation form of \(D\). Compared with \(D\), \(P\) offers greater stability and is used to replace \(D\) in PCA modeling. Taking the logarithm of sample data has the effect of decreasing the difference between large and small values on a logarithmic scale, thereby reducing data dispersion. When the variance of the data is associated with the mean, the logarithmic transformation stabilizes the data, creating a more homogeneous relationship between variance and mean. This proves beneficial for specific data analysis and modeling tasks, aiding in the reduction of heteroskedasticity and facilitating easier data processing and analysis. It is essential to note that the logarithmic transformation may not be suitable for all datasets, especially those containing negative numbers or zeros. The differences between the LT-PCA method and the PCA method are exemplified by the coordinate images below. From the left panel of Figure 7, it is evident that outliers with large difference values significantly affect the principal component direction. Although these outliers can be addressed during dimensionality reduction, their noticeable deviation from the main data direction contradicts the requirement for effective data dimensionality reduction. However, with LT-PCA, the first principal component direction can encapsulate most of the data information, markedly mitigating the impact of outlier points.