# An Applicable Predictive Maintenance Framework for the Absence of Run-to-Failure Data
## 2 Background  
### Autoencoder (AE)  
Autoencoder is one of the artificial neural networks used to learn data by unsupervised learning [21]. Unlike other neural networks, the basic purpose of an autoencoder is to learn the representation of the data. In other words, the autoencoder is trained to construct a neural network whose input and output are identical, learning the reconstruction of the input data while training the networks [13]. To enable this, the autoencoder consists of three layers: an input layer, a hidden layer, and an output layer. The basic structure of an autoencoder is shown in Figure 1 below. Autoencoder can be divided into two parts: the encoder and the decoder. The input layer and the hidden layer are called the encoder, and the hidden layer and the output layer are called the decoder. In this case, the autoencoder can be expressed as below. Given an input dataset \(\mathbf{x}=\left\{x_{1},\ x_{2},\ \ldots,\ x_{p}\right\}\in\mathbb{R}^{1 \times p}\), the encoder transforms the input data to the hidden representation \(\mathbf{h}=\left\{h_{1},\ h_{2},\ \ldots,\ h_{n}\right\}\in\mathbb{R}^{1 \times n}\), the expression of an encoding process can be described as in (1)  
\[\mathbf{h}=f_{activation}\left(\mathbf{W}_{\textit{sh}}\mathbf{x}+\mathbf{b}_ {\textit{sh}}\right) \tag{1}\]  
where \(\mathbf{W}_{\textit{sh}}\) and \(\mathbf{b}_{\textit{sh}}\) are the weight matrix and bias vector of the neural network, and \(f_{activation}\) is the activation function which is the nonlinear mapping function. Subsequently, the decoder takes the hidden representation from the encoder as the input and reconstructs the original input data \(\mathbf{x}\). The decoder maps the hidden representation  
Figure 1: The basic structure of an autoencoder. to the original input data in the same way as the encoder maps the original input to the hidden representation. The expression of the decoding process can be defined as follows:  
\[\mathbf{z}=g_{activation}(\mathbf{W}_{hx^{\prime}}\mathbf{h}+\mathbf{b}_{hx^{\prime}}) \tag{2}\]  
where \(\mathbf{W}_{hx^{\prime}}\) and \(\mathbf{b}_{hx^{\prime}}\) is the weight and bias of the network, \(g_{activation}\) is the nonlinear mapping function and \(\mathbf{z}=\left\{z_{1},\ z_{2},\ \dots,\ z_{p}\right\}\in\mathbb{R}^{1 \times p}\) is called the reconstructed data. The difference between original input data and reconstructed data is a reconstruction error as in (3). \[\mathbf{reconstruction\ error}\ =||\mathbf{x}-\mathbf{z}|| \tag{3}\]  
The autoencoder learns to minimize this reconstruction error using the backpropagation algorithm. In general, the mean square error (MSE) or the mean absolute error (MAE) is chosen as the loss function of the autoencoder. In this paper, the mean absolute error is used as the loss function because the minimizing process using the mean square error can be extremely affected by the noise or outliers in the input data [13]. In this paper, we use standardized data as the input data, not normalized data. Unlike deep learning, especially the convolutional neural networks (CNNs), equipment sensor data often fall outside the normal range. The standardization formula is expressed in (4). \[x_{standardized}=\frac{x-\mu}{\sigma} \tag{4}\]  
where \(\mu\) is the mean of given data, \(\sigma\) is the standard deviation of given data. In addition, the nonlinearity of the autoencoder depends not only on the number of nodes in the hidden layer but also on the activation functions \(f_{activation}\) and \(g_{activation}\). In this paper, \(f_{activation}\) uses tanh, one of the most commonly used functions, and \(g_{activation}\) uses the linear function to reconstruct the input data. Since standardized data are used as the input data, the output may theoretically have a range of real numbers. Therefore, if using a function such as sigmoid or relu as \(g_{activation}\), it would be difficult to restore the input data, and it is better to use the linear function as \(g_{activation}\).