## 2 Proposed Method
### Network Details
#### 2.2.2 Feature-Extraction Module  
Based on the input of decomposed components from different measurement points in the pre-defined time window, this module would make use of the neural network to learn features in each component. In particular, a convolution neural network is applied. The CNN is exclusively applied in the 2D image feature extraction [34; 35; 36], and also is useful for 1D signals like sound waves [37], electrocardiogram [38] and accelerations [23; 25]. The input and output of the CNN are both the feature maps with \(I\in\mathbb{R}^{F_{in}\times D_{in}}\) and \(O\in\mathbb{R}^{F_{out}\times D_{out}}\), respectively. In the proposed method, the input of the feature extraction module would be the decomposed components in the dimension of \((9\times N,M)\), i.e., 9 channels of decomposed components. The \((9\times N,M)\) decomposed components in the previous module can be regarded as a special feature map. \(O^{d_{out}}\in\mathbb{R}^{F_{out}},1\leqslant d_{out}\leqslant D_{out}\) is a slice vector in \(O\), and \(I^{d_{in}}\in\mathbb{R}^{F_{in}},1\leqslant d_{in}\leqslant D_{in}\) is a slice vector in \(I\). In the CNN network, the convolutional kernel \(Q\in\mathbb{R}^{W\times D_{in}\times D_{out}}\) is defined, with \(w^{d_{in}d_{out}}\in\mathbb{R}^{L}\) being the one-dimensional convolutional kernel, \(1\leqslant d_{in}\leqslant D_{in},1\leqslant d_{out}\leqslant D_{out}\). \(L\) is the length of the one-dimensional convolutional kernel. The convolutional kernels applied in this network would slide along only one direction. \(O^{d_{out}}\) is calculated by  
\[O^{d_{out}}=a\left(z^{d_{out}}\right)=a\left(\sum_{d^{in}=1}^{D_{in}}w^{d_{in} d_{out}}\otimes I^{d_{in}}+b^{d_{out}}\right), \tag{2}\]  
where \(b^{d_{out}}\) is a scalar bias, \(\otimes\) the operator of cross-correlation, \(a(\cdot)\) a activation function. The value of each element in the kernel, i.e., the weight in the kernel, and the bias are learnable. In order to increase the receptive field of the convolution kernel without increasing the number of parameters or introducing additional computational costs, \(D_{dil}-1\) zero values can be inserted in the conventional kernel \(w^{d_{in}d_{out}}\), where \(D_{dil}\) is the dilation rate. As shown in Figure 1, in this module, there are 3 conventional layers in total, namely Conv #1, Conv #2 and Conv #3. The length of the kernel (kernel size) in all three layers are set to 8. The output dimension \(D_{out}\) for the three layers, which is also the filter numbers, are set to 32. In order to increase the receptive field for deeper layers, \(D_{dil}\) of the three layers is set to 1, 2 and 4, for Conv #1, Conv #2 and Conv #3, respectively. The stride in all 3 layers is set to 1. Table 1 give the details of the configurations of the three layers. Figure 2: An example of a signal in time domain, together with the corresponding 1 low frequency component and 8 high frequency components. The signal in time domain is a Gaussian white noise, which is shown in red in the first row. The low frequency component \(a_{8}\) is shown in blue curve in the second row. The high frequency components, namely \(d_{1}\) to \(d_{8}\), are shown in the green curves. All 9 components are input to the following feature extraction module in the network. In order to improve the training efficiency and performance of the network, in each conventional layer, the network is regularized with batch normalization (BN) and weight normalization (WN). The BN is applied to \(z^{d_{out}}\) in Equation (2), i.e., \(\sum_{d^{in}=1}^{D_{in}}w^{d_{in}d_{out}}\otimes I^{d_{in}}+b^{d_{out}}\). Suppose that the mean and variance of \(z^{d_{out}}\) for a small batch of training data is \(\mu\) and \(\sigma^{2}\), the BN is defined as  
\[BN(z^{d_{out}})=\frac{z^{d_{out}}-\mu}{\sqrt{\sigma^{2}+\epsilon}}, \tag{3}\]  
where \(\epsilon\) is a pre-defined small constant. The BN would the convolved output to a standard normal distribution with a mean of 0 and a variance of 1.