# Research on an ID-PCA Early Fault Detection Method for Rolling Bearings
## 2 Basic Theory
### Early Fault Feature Extraction  
Due to the limitations of the principal component analysis method, in the process of extracting the principal components, some fault features may be discarded, resulting in the failure of the principal component analysis to accurately detect the weak early fault information [29]. The depth decomposition theorem establishes a depth principal component model by means of linear projection. This model further decomposes the collected vibration signals to obtain a multi-data processing layer. Through the depth decomposition, the weak fault features in the vibration signal of the rolling bearing can be accurately detected, and the early fault detection of the rolling bearing can be realized. Let the data set \(X=[x_{1},x_{2}\ldots,x_{n}]^{T}\in R^{n+m}\). \(n\) is sample number, \(m\) is observed variable. After normalization, we can obtain:  
\[\overline{X}=\left( \begin{array}{cccc}\overline{x}_{1}(1)&\overline{x}_{2}( 1)&\ldots&\overline{x}_{n}(1)\\ \overline{x}_{1}(2)&\overline{x}_{2}(2)&\ldots&\overline{x}_{n}(2)\\ \ldots&\ldots&\ldots&\ldots\\ \overline{x}_{1}(m)&\overline{x}_{2}(m)&\ldots&\overline{x}_{n}(m)\\ \end{array} \right) \tag{1}\]  
The pivot space and residual space of dataset \(X\) are obtained by the PCA method:  
\[X=X_{11}+X_{12} \tag{2}\]  
where \(X_{11}\) and \(X_{12}\) represent the first-order sub-principal component space and sub-residual space of the original matrix \(X\), respectively. It can be expressed as:  
\[ \begin{array}{l}X_{11}=P_{11}P_{11}^{T}X\\ X_{12}=(I-P_{11}P_{11}^{T})X\\ \end{array}  \tag{3}\]  
Let \(P_{11}\) be the principal component vector of \(X\), and \(I\) is the identity matrix. In order to obtain more information about the second-order principal component space and residual space, \(X_{11}\) and \(X_{12}\) can be further decomposed:  
\[ \begin{array}{l}X_{11}=X_{21}+X_{22}\\ X_{12}=X_{23}+X_{24}\\ \end{array}  \tag{4}\] \begin{table}
\begin{tabular}{c c c c} \hline
**Time-Domain Index** & **Expression** & **Time-Domain Index** & **Expression** \\ \hline Mean index \(\overline{x}\) & \(\overline{x}=(\sum_{i=1}^{n}x(i))/\text{n}\) & Root mean square index \(x_{rms}\) & \(x_{rms}=((\sum_{i=1}^{n}x_{i}^{2})/\text{n})^{1/2}\) \\ Peak index \(x_{p}\) & \(x_{p}=\max(x(i))\) & Root index \(x_{r}\) & \(xr=((\sum_{i=1}^{n}|x|)^{1/2}/\text{n})^{1/2}\) \\ Mean square index \(x_{c}\) & \(x_{c}\)\(=(\sum_{i=1}^{n}x_{i}^{2})/\text{n}\) & Standard deviation index \(x_{std}\) & \(x_{std}\)\(=((\sum_{i=1}^{n}(x_{i}-\overline{x})/(\text{n}-1))^{1/2}\) \\ \hline \end{tabular}
\end{table} 
Table 1: Dimensioned time-domain index. \begin{table}
\begin{tabular}{c c c c} \hline
**Time-Domain Index** & **Expression** & **Time-Domain Index** & **Expression** \\ \hline Shape factor index \(x_{SF}\) & \(x_{SF}=x_{rms}/\overline{x}\) & Inpulse factor index \(x_{IF}\) & \(x_{SF}=x_{p}/x_{rms}\) \\ Crest factor index \(x_{CF}\) & \(x_{CF}=x_{p}/\overline{x}\) & Margin factor index \(x_{CF}\) & \(x_{SF}=x_{p}/x_{r}\) \\ \hline \end{tabular}
\end{table} 
Table 2: Dimensionless time-domain index. Let \(P_{21}\) and \(P_{22}\) be the principal component vector and residual vector of \(X_{11}\) respectively; \(P_{23}\) and \(P_{24}\) are the principal component vector and residual vector of \(X_{12}\) respectively, then the original data \(X\) can be expressed as:  
\[X=X_{21}+X_{22}+X_{23}+X_{24} \tag{5}\]  
where  
\[ \begin{array}{c}X_{21}=P_{21}P_{21}^{T}X_{11}\\ X_{22}=(I-P_{21}P_{21}^{T})X_{11}\\ X_{23}=P_{23}P_{23}^{T}X_{12}\\ X_{24}=(I-P_{23}P_{23}^{T})X_{12}\end{array}  \tag{6}\]  
By analogy, the original dynamic kernel matrix \(X\) can be expressed as the sum of \(2^{i}\) subspaces, where \(i\) represents the order. The subspace \(X_{ij}\) can be expressed as:  
\[X_{i,j}=\left\{