## 2 Materials and Methods
### Tcn  
The convolutional neural network (CNN) has outstanding results for tasks in two dimensions, i.e., image processing. It is adopted in one dimension for the time series task. However, the main drawback of the basic architecture of CNN is that it is difficult to gain crucial information in long-term time series data. Thus, [PERSON] et al., proposed TCN as an  
Figure 1: The general architecture of the proposed method.  
enhancement of convolutional and recurrent networks due to being able to memorize sequence data with long relationships accurately [13]. TCN is capable of memorizing more historical data than a standard sequence of neural networks by widening the receptive field size [14]. A TCN model consists of residual blocks including dilated causal convolutional neural networks, residual connection, and others as shown in Figure 2.  
A residual block in the TCN model contains some basic layers, i.e., dilated causal convolutions, weight normalization, the rectified linear unit (ReLU) as activation function, and spatial dropout as regularization. Here, the causal convolution term refers to a convolution process obtaining output \(t\) by convolving elements from inputs at step \(t\) and the previous periods. Let us assume to predict output \(y_{t}\), we need elements \(x_{0},\ \dots,\ x_{t}\) from the previous layer as input.  
Afterward, dilated causal convolution is proposed to address the drawback of long-term dependency in causal convolutional neural networks. The dilated causal convolution exponentially enlarges the receptive fields to memorize the long historical information without adding other parameters [15]. The dilated causal convolution steps up the filter in input sequences and skips a certain point with a fixed stride. This convolution has tasks identical to the pooling concept but the input and output channel width in this convolution remains the same size. Figure 3 shows the dilated causal convolution process with dilation factors \(d\) of 1, 2, and 4 and filter size of \(k=3\). The blue and red colors refer to the original time series, while the yellow color denotes the padding. This dilated causal convolution can be formulated with Equation (1).  
\[F(x)=\sum_{l=0}^{k-1}h(l)f(x-d.l), \tag{1}\]  
where \(f(*)\) denotes the input of the dilated causal convolution and \(h(l)\) refers to the filter length \(k\). Additionally, \(d\) refers to the dilation factor, \(x\)-\(d.l\) denotes the past direction of element x, and \(F(*)\) is the output of this process.  
Figure 2: Residual block architecture in TCN model.  
Weight normalization is applied in the TCN model after the dilated causal convolution process to speed up and make the training process robust. ReLU activation function placed after weight normalization can enhance the sparsity of TCN and avoid strong dependency among parameters [16]. Afterward, the spatial dropout is added to avoid overfitting the network model. In addition, \(1\times 1\) convolution operation, as shown in Figure 2, is incorporated as the residual connection to adjust the channel width of the input and output layer which may have different sizes. Specifically, a residual connection is integrated to the output of TCN to prevent gradient attenuation. The input \(x\) is combined with the output of TCN formulated in Equation (2):  
\[O=Activation(x+F(x)). \tag{2}\]  
Subsequently, the output of the residual block from multiple layers is considered for the input of the next residual block.