# Machinery Fault Signal Detection with Deep One-Class Classification
## 3 Fault Signal Detection with Deep Support Vector Data Description
Hence, the output feature map having \(C^{\ell}\) number of convolutional channels is computed. In general, each of the convolutional layers is followed by a pooling layer, which integrates adjacent values within the feature map into one value. Among various pooling manners, max-pooling, which picks the maximum value within a local region of the input feature map, is the most widely used. By employing the pooling layer, the size of the feature map and computational burden can be reduced. Both convolutional and max-pooling layer pairs (referred to as a convolutional block) can be repeated multiple times in the encoding part. At the end of the encoding part, the feature maps obtained by multiple convolutional blocks are flattened into a one-dimensional vector, and the flattened vector is finally mapped to a smaller latent features vector (dimension of latent feature vector (i.e., the number of latent features) is denoted as \(q\)) by the fully connected layer. Then, in the decoding part, the input time-series signals are reconstructed from the latent variables of input time-series signals. The decoding part of the 1D-CAE comprises symmetrically arranged layers to the encoding part. In other words, the \(q\)-dimensional latent features vector is connected to a fully connected layer and reshaped into a one-dimensional feature map, and they are inversely reconstructed by transposed convolutional layers and up-sampling layers, which substitute the convolutional layers and max-pooling layers, respectively. The transposed convolutional layer expands the input feature map, conversely to the convolutional layer in the encoding part. The input feature map of the decoding part (i.e., the latent features vector) is transformed by a transposed convolutional layer as follows:  
\[\widetilde{f}_{m}^{\ell}=\sigma\left(\sum_{m=1}^{C^{\ell-1}}\widetilde{f}_{m }^{\ell-1}\;\;@\;\;\widetilde{W}_{m,c}^{\ell}+\widetilde{b}_{c}^{\ell}\right),\;\;c=1,\;2,\;3,\;\ldots,\;C^{\ell},\;\;\ell=1,2\ldots,L, \tag{4}\]  
where \(@\;\;\) denotes the transposed convolutional operator which enlarges the input feature map, and \(\widetilde{f}_{m}^{\ell-1}\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; \;\;\;\;\;\;\;where \(\mathcal{X}_{i}\) and \(\widetilde{\mathcal{X}}_{i}\) are the \(i\)-th input time-series signal and its reconstructed outcomes, respectively, and \(n\) is the number of total input time-series signals. By the encoding part of the 1D-CAE, the initial latent feature vectors can retain the intrinsic properties of input time-series signals. Hence, the weight set in the encoding part, denoted as \(\mathcal{W}\), obtained from the 1D-CAE training procedure are used as initial weights of the deep support vector data description model \(\phi(\cdot;\mathcal{W})\). Although the latent feature vector of the encoding part of the 1D-CAE neural network structure can accommodate intrinsic properties of training target time-series signals, it might not be optimized to detect anomalous signals. Thus, the deep neural network structure should be improved for fault signal detection tasks. To this end, the weight set W obtained from pre-training is updated to be optimized for the fault signal detection task. The deep SVDD performs subsequent fine-tuning procedures to minimize a loss function formulated for the fault signal detection task. The deep SVDD models can be divided into soft-boundary deep SVDD and one-class deep SVDD, according to how the loss function is defined. 1.