# Machinery Fault Signal Detection with Deep One-Class Classification
## 3 Fault Signal Detection with Deep Support Vector Data Description  
In this study, we propose to utilize the deep SVDD method for the fault signal detection framework. Hence, this section presents a more detailed description of the deep SVDD method. The deep SVDD [12] builds a hypersphere enclosing as many target signals as possible in the latent feature space generated by deep neural network structures. In the deep SVDD, the input time-series signal \(\mathcal{X}\subseteq\mathbb{R}^{d}\) (\(d\) denotes the dimension of time-series signal (i.e., the number of time points of the raw signal)) is mapped into the latent feature space \(\mathcal{T}\subseteq\mathbb{R}^{p}\) (p is the length of latent feature vector) as \(\phi(\cdot;\mathcal{W})\ :\ \mathcal{X}\rightarrow\mathcal{T}\), the neural network structures whose set of weights are \(\mathcal{W}=\{\mathcal{W}^{1},\ldots,\mathcal{W}^{L}\}\). The \(\mathcal{W}^{\ell}\) denotes the weights of \(\ell\)-th hidden layer \(\ell\in\{1,\ldots,L\}\) (\(L\) represents the number of total hidden layers). In the deep SVDD, the latent feature space is analogous to the feature mapping function used in the traditional SVDD model, and the features obtained from training the deep SVDD model comprise the latent feature space. Thus, the deep SVDD attempts to generate better latent feature vectors where the optimal hypersphere can be constructed. The overall training procedure on the deep SVDD is illustrated in Figure 1.  
As shown in Figure 1, the deep SVDD is trained by both pre-training and fine-tuning procedures. In order to more appropriately find the optimal latent feature space, deep SVDD first learns a stacked AE network as a pre-training procedure. The stacked AE network is composed of both encoding and decoding parts. The encoding part attempts to summarize the input time-series signal into smaller latent feature vectors, whereas the decoding part is used to reconstruct the input signals from the encoded latent feature vectors. The encoding part of the stacked AE is used as the initial network structure of the deep SVDD, and it helps the latent feature space can accommodate the intrinsic properties of the input time-series signal. By doing so, the initial weights of the neural network structure for the deep SVDD can be used to construct the hypersphere with a more accurate fault detection performance. However, the stacked AE might not be localized characteristics of the time-series signal in that the hidden layers of the stacked AE are fully connected to each other. Hence, in this study, we propose to employ the one-dimensional CAE network instead of stacked AE because the one-dimensional CAE (1D-CAE) structures can successfully handle the time-series signal data owing to the convolutional layer and pooling layers [51; 52]. In the 1D-CAE, the fully connected layers of the stacked AE are substituted as one-dimensional convolutional layers and pooling layers. The convolutional layer helps to consider localized properties within adjacent time points of input signals by using a convolutional kernel  
Figure 1: Graphical illustration of fault signal detection process using the deep SVDD. filter, and these convolutional kernel filters share weights across all regions of the input time-series signals. The output feature map of the \(j\)-th channel in the \(\ell\)-th convolutional layer, \(f_{j}^{\ell}\), is calculated as follows:  
\[f_{j}^{\ell}=\sigma\left(\sum_{m=1}^{C^{\ell-1}}f_{m}^{\ell-1}*\Psi_{m,\;c}^{ \ell}+b_{c}^{\ell}\right),\;\;c=1,\;2,\;\ldots,\;C^{\ell},\;\;\ell=1,2\ldots,L, \tag{3}\]  
where \(f_{m}^{\ell-1}\) is the output feature map of the m-th channel in the (\(\ell\)-1)-th layer transformed by the kernel filter \(\Psi_{m,c}^{j}\) of the \(c\)-th channel in the \(\ell\)-th layer. \(C^{\ell}\) is the number of convolutional channels in \(\ell\) layer and \(b_{c}^{\ell}\) is the bias of \(c\)-th channel in the \(\ell\)-th layer. In addition, * is the convolutional operator, and \(\sigma(\cdot)\) denotes the nonlinear activation function.