## 3 Damage Identification Paradigm
### Formation of Damage Features Using AANNs
#### 3.1.2 Damage Feature Extraction  
NLPCA was previously introduced in the field of structural damage detection, where it was mainly used to deal with environmental effects [52; 53]. In this article, it helps build damage features.  
Considering that the WPNEs vector \(e_{p}\) is a column vector, a matrix \(E\) is constructed from the WPNEs vectors \(e_{p}\) of \(m\) samples as follows:  
\[E=\left[\begin{array}{ccccccc}e_{1,1}&e_{1,2}&\cdots&e_{1,p}&\cdots&e_{1,m}\\ e_{2,1}&e_{2,2}&\cdots&e_{2,p}&\cdots&e_{2,m}\\ \vdots&\vdots&\ddots&\vdots&\ddots&\vdots\\ e_{64,1}&e_{64,2}&\cdots&e_{64,p}&\cdots&e_{64,m}\end{array}\right]=\left( \begin{array}{ccccccc}e_{1}&e_{2}&\cdots&e_{p}&\cdots&e_{m}\end{array}\right) \tag{11}\]  
Extraction of damage features from WPNEs is tackled by the NLPCA strategy that is implemented by the AANNs, with the structure depicted in Figure 2. The number of neurons at the input layer or output layer (\(M\)) fits the dimensions of the WPNEs. According to [49], the number of neurons of either the mapping layer or the remapping layer \(N\) should be chosen such that:  
\[2N<M \tag{12}\]  
The bottleneck layer will produce NPCs that represent the extracted damage features. Therefore, the number of NPCs (\(r\)) equals the number of neurons in the bottleneck layer. \(r\) is properly chosen to ensure that the extracted damage features contain the sufficient damage information stored in the WPNEs. The activation functions in the mapping and remapping layers are the sigmoid functions \(\sigma\) and \(\lambda\), respectively, whereas the activation functions at the bottleneck and output layers are the pure line functions \(\beta\) and \(\delta\), respectively, as indicated in the circle of the corresponding layer in Figure 2. The cost function, measured by the mean squared error (\(MSE\)), directs the training of the AANNs, with the \(MSE\) defined by [50]:  
\[MSE=\sum_{p=1}^{m}\sum_{i=1}^{64}\bigl{(}e_{i,p}-\hat{e}_{i,p}\bigr{)}^{2} \tag{13}\]  
where \(m\) is the number of samples. \(e_{i,p}\) and \(\hat{e}_{i,p}\) are the target values and prediction values of the AANNs, respectively.  
When the inputs and the target output are both set to WPNEs, the AANNs are forced to build a mapping from the inputs to the outputs. This mapping can produce a converted low-dimensional vector of WPNEs, i.e., the NPCs of WPNEs. In a well-trained AANN, the matrix of NPCs \(C_{r\times m}\) with \(r\ll 64\) is generated from the bottleneck layer. The \(C\) matrix is formed:  
\[C=\left[\begin{array}{ccccccc}c_{1,1}&c_{1,2}&\cdots&c_{1,p}&\cdots&c_{1,m} \\ c_{2,1}&c_{2,2}&\cdots&c_{2,p}&\cdots&c_{2,m}\\ \vdots&\vdots&\ddots&\vdots&\ddots&\vdots\\ c_{r,1}&c_{r,2}&\cdots&c_{r,p}&\cdots&c_{r,m}\end{array}\right]=\left(\begin{array} []{ccccccc}c_{1}&c_{2}&\cdots&c_{p}&\cdots&c_{m}\end{array}\right) \tag{14}\]  
where \(c_{p}\) is a vector of NPCs for a damage sample. The NPCs carry almost all the damage information of WPNEs but have lower dimensionality than the WPNEs. This peculiarity enables NPCs to be valid damage features for intelligent damage pattern recognition.