## 3 The Proposed PCA-LSTM Model
### PCA Based on Mechanism  
The traditional PCA was introduced in Section 2.1. Due to the purely data-driven process, historical data is analyzed in PCA without any prior knowledge, which may bring about that some redundancies may be considered despite the causal relationship. Therefore, human experience will be introduced to interfere with the variable selection procedure ahead of PCA, based on the known mechanism. The monitoring system always contains a broad range of sensors data related to the target object. Some of the data is on control variables, while some of the data is just on display variables that visualize the parameters. Assuming the sensor system obtains an initial dataset:  
\[\mathbf{X}=(\mathbf{x}_{1},\mathbf{x}_{2},\ldots,\mathbf{x}_{i-m},\mathbf{x}_{i-m+1},\ldots,\mathbf{x }_{i},\ldots,\mathbf{x}_{k}) \tag{11}\]  
where \(\mathbf{x}_{i}\) represents the \(i\)-th sensor equipped in the system. \[\mathbf{x}_{i}=[\mathbf{x}_{i1},\mathbf{x}_{i2},\ldots,\mathbf{x}_{ij},\ldots,\mathbf{x}_{in}]^{T} \tag{12}\]  
where \(\mathbf{x}_{ij}\) represents the \(j\)-th data obtained by \(i\)-th sensor. When studying the working mechanism of the target, variables causal relationship will be analyzed and some of the redundant variables will be deleted, as well as some meaningless display parameters. This creates a new sample set:  
\[\mathbf{X}^{\prime}=(\mathbf{x}_{1},\mathbf{x}_{2},\ldots,\mathbf{x}_{i-m},\mathbf{x}_{i},\ldots, \mathbf{x}_{k}) \tag{13}\]  
PCA based on human experience method obtains a hyperplanar representation of all samples through recent reconstruction, realizing the dimension reduction from \(k\) to \(k^{\prime}\) with the least loss. The samples are centralized firstly as:  
\[\sum_{i}\mathbf{x}_{i}=0 \tag{14}\]  
Then a new coordinate system can be obtained after projection transformation:  
\[\mathbf{W}=(\mathbf{w}_{1},\mathbf{w}_{2},\ldots\mathbf{w}_{i-m},\mathbf{w}_{i},\ldots,\mathbf{w}_{k}) \tag{15}\]  
where \(\mathbf{w}_{i}\) is the standard orthogonal basis vector. \[\left\|\mathbf{w}_{i}\right\|_{2}=1 \tag{16}\]  
\[\mathbf{w}_{i}{}^{T}\mathbf{w}_{j}=0,(i\
eq j) \tag{17}\]  
If a portion of the coordinate is abandoned, namely the dimension is reduced from \(k\) to \(k^{\prime}\) (\(k^{\prime}<k\)), the projection of samples \(\mathbf{x}_{i}\) in the low-dimensional coordinate system will be:  
\[z_{ij}=(z_{i1},z_{i2},\ldots,z_{in^{\prime}}) \tag{18}\]\[z_{ij}={\mathbf{w}_{j}}^{T}{\mathbf{x}_{i}} \tag{19}\]  
where \(z_{ij}\) is the \(j\)-th coordinate of \({\mathbf{x}_{i}}\) in low-dimensional space; and \({\mathbf{x}_{i}}\) can be reconstructed as:  
\[\hat{{\mathbf{x}}_{i}}=\sum\limits_{j=1}^{k^{\prime}}z_{ij}{\mathbf{w}_{j}} \tag{20}\]  
For the whole training dataset, the distance between original samples \({\mathbf{x}_{i}}\) and the reconstructed samples \(\hat{{\mathbf{x}}_{i}}\) can then be determined as:  
\[ \begin{split}&\sum\limits_{i=1}^{k}\left\|\sum\limits_{j=1}^{n^{ \prime}}z_{ij}{\mathbf{w}_{j}}-{\mathbf{x}_{i}}\right\|_{2}^{2}=\sum\limits_{i=1}^{k}{ \mathbf{z}_{i}}^{T}{\mathbf{z}_{i}}-2\sum\limits_{i=1}^{k}{\mathbf{z}_{i}}^{T}W^{T}{\mathbf{x }_{i}}+const\\ &\propto-tr({\mathbf{W}^{T}(\sum\limits_{i=1}^{k}x_{i}{\mathbf{x}_{i}}^ {T})}{\mathbf{W}})\end{split}  \tag{21}\]  
where \(const\) is the constant item, and \({\mathbf{W}}\) can be obtained by Equation (15). Because \(\sum\limits_{i=1}^{k}{\mathbf{x}_{i}}{\mathbf{x}_{i}}^{T}\) is a covariance matrix, the target distance function can be minimized as:  
\[\left\{ \begin{array}{l}Min_{W}~{}-tr\big{(}{\mathbf{W}^{T}{\mathbf{X}^{\prime}}{ \mathbf{X}^{\prime}}{\mathbf{W}}}\big{)}\\ s.t.~{}{\mathbf{W}^{T}}{\mathbf{W}}={\mathbf{I}}\end{array} \right. \tag{22}\]  
where \({\mathbf{I}}\) is the identity matrix.