## 2 Automatic POD Diagnosis Method
### Regular Convolutional Auto-Encoder  
The auto-encoder (AE) serves as a data dimensionality reduction technique [23]. Traditionally, the framework of AE consists of two components, the encoder and decoder. The encoder is mainly responsible for segmenting the input image and extracting valid features, and the decoder is used to reconstruct input data with feature information [24]. The number of nodes in the hidden layer is much lower than the input layer so that the encoder can provide effective representation for the input data.  
Given an input amplitude time-frequency data set \(\mathbf{x}=\{x_{1},x_{2},\ldots,x_{l}\}\) with shape size \(m\times n\) for each element, where \(x_{l}\) denotes the \(l\)th sample of \(\mathbf{x}\). Note that the \(\mathbf{x}\) is the two-dimensional matrix set, which represents the time-frequency matrix caused by the DRST. The \(m\) and \(n\) represent the height and width of the time-frequency matrix, namely, the \(m\) and \(n\) are 256 and 512, respectively. In the auto-encoder, the data can be viewed as multiple two-dimensional images stacked on top of each other. Each image can be viewed as a feature mapping. The output mapping \(\mathbf{z}_{i}^{r}\) of the encoder in the \(r\)th layer can be expressed as  
\[\mathbf{z}_{i}^{r}=f(\mathbf{W}*\mathbf{x}+\mathbf{b}) \tag{9}\]  
where \(\mathbf{W}\) and \(\mathbf{b}\) are the \(d\times d\times h\) dimensional convolutional kernel (also called convolution filter) weight and offset vector, respectively. The \(h\) is the depth of the convolution layer, \(i=0,\,1,\,\ldots,\,h\). The \(d\) is the size of the convolution filter. The sign \(*\) is the convolution operation, \(f(\cdot)\) is the activate function. We choose the rectified linear unit as the activate function to improve the convergence speed [25]. The size of \(\mathbf{z}_{i}^{r}\) becomes \(m\times n\times h\) which the zero is padding into the blank space.  
Following the convolutional layer in the encoder, a max-pooling layer, which plays a role of the downsampling operation, is applied to reduce the input dimension [20]. The output feature \(\mathbf{z}_{pi}^{r}\) of the max-pooling layer is defined as \(\mathbf{z}_{pi}^{r}=\text{max}\,\left(\mathbf{z}_{i}(p\times p\times h)_{s}\right)\), where \(p\times p\times h\) is the size of the pooling layer with the stride length \(s\). It indicates that the maximum value per \(p\times p\) size in \(\mathbf{z}_{i}\) is the output of max-pooling. The size of pooled feature vectors \(\mathbf{z}_{pi}^{r}\) is determined with the formula \(\lfloor(m-p)/s\rfloor\times\lfloor(n-p)/s\rfloor\times h\), where \(\lfloor\cdot\rfloor\) denotes rounding down. The target features \(\mathbf{z}_{pi}^{r}\) extraction by the AE are fed to the classifier.  
To reconstruct the signal, the decoder constructs a symmetric AE structure by upsampling. Similarly, the hidden layer representation \(\mathbf{z}_{pi}^{r}\) is then reversely mapped to the same input size. This inverse mapping process is called decoder, the reconstructed output \(\mathbf{x}^{\prime}\) of the nonlinearity form is  
\[\mathbf{x}^{\prime}=f\Big{(}\mathbf{W}^{\prime}*\mathbf{z}_{pi}^{\prime}+\mathbf{ b}^{\prime}\Big{)} \tag{10}\]  
where \(\mathbf{W}^{\prime}\) and \(\mathbf{b}^{\prime}\) are the weight and offset vector, respectively. The dimension of \(\mathbf{x}^{\prime}\) is expanded in up-sampling by interpolation so that the \(\mathbf{W}^{\prime}\) and \(\mathbf{b}^{\prime}\) have the same dimension corresponding to the encoder.  
The regular AE can be used to compress complex PQDs to obtain representative features. However, it is difficult for AE to maintain effectiveness in shallow layers when the real-time diagnosis of PQDs is taken into account.