## 2 Fundamental Theories
### Nonlinear Principal Component Analysis  
Nonlinear principal component analysis (NLPCA) is deemed to be a nonlinear generalization of principal component analysis (PCA) [48], generally used for data compression and dimension reduction, and can be realized by the following procedure.  
Let \(X\) be an original data set of high dimension. \(X\) is projected by a nonlinear function vector \(U\) into a new data set \(T\) of low dimension:  
\[T=U(X) \tag{6}\]  
\(T\) can be projected back into the original space by another nonlinear function vector \(V\), giving \(\hat{X}\)  
\[\hat{X}=V(T) \tag{7}\]  
\(\hat{X}\) is the reconstruction of \(X\) with the residual error \(\widetilde{X}\)  
\[\widetilde{X}=X-\hat{X} \tag{8}\]  
The functions \(U\) and \(V\) are properly chosen to minimize the residual error. With appropriate \(U\) and \(V\), Equations (6) and (7) can be used to project a high-dimensional data set \(X\) to a low-dimensional data set \(T\) with little information loss, which is the essence of NLPCA.  
Compared to the PCA, the NLPCA has two distinctive features: (i) it has advantages over the standard PCA in removing not only linear but also nonlinear correlations in variables; (ii) the nonlinear principal components (NPCs) from NLPCA are parallel in importance rather than being in decreasing order like those of the PCA.  
Recent studies have reported that ANNs can implement some component analysis methods such as sensitivity analysis [49]. Likewise, they can be used to conduct NLPCA. Auto-associative neural networks (AANNs) have been demonstrated as a sophisticated strategy to carry out NLPCA [50]. The technological key lies in constructing an auto-associative learning framework by placing the high-dimensional data set \(X\) at both the input and target output layers. As shown in Figure 2, AANNs consist of five layers: the input layer, the mapping layer, the bottleneck layer (low-dimensional space), the remapping layer, and the output layer. After being well trained by controlling the resident error defined by Equation (8), AANNs perform Equation (6) by projecting \(X\) of high dimension from the input layer to \(T\) of low dimension in the bottleneck layer via the mapping layer; meanwhile, AANNs perform Equation (7) by projecting \(T\) from the bottleneck layer to \(\hat{X}\) in the output layer via the remapping layer.  
Figure 2: Auto-associative neural networks (AANNs) for damage feature extraction from wavelet packet node energies (WPNEs). (**A**) mapping layer; (**B**) bottleneck layer; (**C**) remapping layer.  
Alternatively, AANNs can be seen as a combination of two individual networks, NN1 and NN2, with symmetrical structures. NN1 consists of the input layer, the mapping layer, and the bottleneck layer, whereas NN2 consists of the bottleneck layer, the remapping layer, and the output layer. The two networks integrate by sharing the bottleneck layer. It has been illustrated that for a neural network, one hidden layer with a nonlinear activation function is sufficient for modeling arbitrary nonlinear functions, so both NN1 and NN2 are able to model a nonlinear function if their activation functions in the hidden layer are nonlinear [50]. To ensure that the AANNs are capable for NLPCA, they should have nonlinear activation functions for the mapping and remapping layers.