## 3 Proposed Approach
### Temporal Convolutional Network for IFGD Process Identification  
Convolution neural networks (CNNs) were first introduced to deal with problems in the computer vision field. More recently, in [26], a kind of CNN known as TCN was specially designed to deal with timeseries data. It was shown that TCNs outperform recurrent networks in terms of processing speed of timeseries with long-term dependencies, which is due to the parallel computation property of convolution. With most of the proposals over the year, TCNs is established as a timeseries model and used for forecasting purposes (see, e.g., [27; 28]). Here, TCNs are introduced to identify an IFGD process, and the structure of TCNs used in our work is first described. In TCNs, 1D causal convolution rather than standard convolution is employed, which suggests the output of convolution operation at time \(t\) is solely on the basis past values. To model the IFGD process with long-term dependencies, it is of critical importance to expand the scope of local receptive field. However, for a standard causal convolution, there are two ways of expanding the coverage of model input, namely increasing the depth of layer or kernel size. Evidently, both approaches would impose a heavy computational burden for the network training. A simpler alternative is to introduce dilation convolution; let the input and output of an SISO process be denoted as \(u\) and \(y\), respectively, then the output of a dilated convolution layer (denoted by \(o_{t}\)) is written as,  
\[o_{t}=\sum_{i=0}^{k-1}a_{i}y_{t-d\cdot i}+\sum_{i=0}^{k-1}b_{i}u_{t-d\cdot i} \tag{6}\]  
with \(d\) being the dilation factor, \(k\) being the size of filter and \(a_{i}/b_{i}(i=0,1,\cdots,k-1)\) being the kernel weight. An illustrative comparison among conventional convolution, causal convolution and dilated causal convolution is presented in Figure 3 below, wherein the layer number corresponding to three types of convolution operations remains the same, yet values of relevant hyperparameters are slightly different. As shown above, with the same kernel size, the receptive field of causal CNN and dilated causal CNN are three and five, respectively. This implies that the introduction of dilation operation is an effective way to expand the receptive field, and \(d\) always increases exponentially with the depth of network. For causality-related CNN structures, the length of each output in hidden layer is made to be identical using zero padding. Another way to increase the receptive field is to deepen TCNs, however, this brings problems related to training deep neural network. Fortunately, the residual learning framework, proposed in [29], is effective in handling such problems, and the central idea of residual structure is to add the input of a layer directly to some layer output, which in the TCNs process modeling context can be formulated as  
\[Y_{T}=\gamma(X_{T},W_{r})+W_{s}X_{T} \tag{7}\]  
where, the input multivariate signal \(X_{T}\triangleq\{x_{1T},\cdots,x_{nT}\}\) and the component \(x_{iT}(i=1,2,\cdots,n)\) is one channel in \(X_{T}\). \(Y_{T}\) denotes a signal of appropriate dimension, \(\gamma(\cdot)\) is known as the residual function which is of great flexibility and may consist of more than  
Figure 3: Comparison of different convolution structure. From left to right: conventional CNN with kernel size 3, causal CNN with kernel size 2 and dilated causal CNN with kernel size 2, dilation size 2.  
one layer, and \(W_{r}\) represents all parameters therein. Moreover, the weight matrix \(W_{s}\) is introduced in the shortcut connection to keep the dimension of \(\gamma(\cdot)\) consistent with that of shortcut's output. Overall, the TCN model used in our work is established by cascading several residue blocks, which takes the form of (6) and \(\gamma(\cdot)\) is realized by dilated causal convolution layer followed by a weight normalization layer, a ReLU nonlinearity and a dropout layer.