## 3 Materials and Methods
### Model Construction
#### 3.2.4 Dilated Causal Convolution  
Dilated Causal Convolution is a technique that combines causal convolution and dilated convolution. In causal convolution, it is ensure that the model's predictions depend only on the current or past input data, without using future information. This is crucial for time-series forecasting tasks, as future data cannot be'seen' when predicting current values. Dilated convolution, on the other hand, introduces spacing between the convolutional kernel elements, allowing the model to handle longer time spans of data without increasing computational cost. The combination of these two techniques enables the model to effectively capture long-term temporal dependencies, thereby improving forecasting accuracy for time series. Dilated convolutions address this limitation by introducing a dilation factor \(d\), which determines the spacing between input values used in the convolution [37]. The formula for a dilated convolution becomes:\[(y_{t})=\sum_{i=0}^{k-1}w_{i}\cdot x_{t-d\cdot i} \tag{19}\]  
where \(d\) is the dilation factor. For example, with \(d=2\), the convolution skips every other input value, effectively doubling the receptive field. This allows the model to cover a larger context without increasing the number of parameters. The receptive field of a dilated convolution is given by:  
\[\text{Receptive Field}=(k-1)\cdot d+1 \tag{20}\]  
This formula shows how the dilation factor \(d\) exponentially increases the receptive field, enabling the model to capture long-range dependencies efficiently. In time-series modeling, causality is a critical requirement. A causal convolution ensures that the output \(y_{t}\) at time \(t\) only depends on the inputs \(x_{t},x_{t-1},\ldots\), and not on any future inputs \(x_{t+1},x_{t+2},\ldots\). This is achieved by masking future inputs during the convolution operation. Combining causality with dilation, the formula for a dilated causal convolution becomes:  
\[(y_{t})=\sum_{i=0}^{k-1}w_{i}\cdot x_{t-d\cdot i},\quad\text{where }t-d\cdot i\geq 0 \tag{21}\]  
This masking ensures that the model adheres to the causality constraint, making it suitable for autoregressive tasks. To capture increasingly larger contexts, multiple layers of dilated causal convolutions can be stacked. At each layer, the dilation factor is typically increased exponentially (e.g., \(d=1,2,4,\ldots\)), which further expands the receptive field without increasing the computational cost significantly. The effective receptive field for \(L\) layers is:  
\[\text{Effective Receptive Field}=\sum_{l=0}^{L-1}(k-1)\cdot d_{l}+1 \tag{22}\]  
where \(d_{l}\) is the dilation factor at layer \(l\). Dilated causal convolution combines the strengths of dilated convolution and causal convolution [25], ensuring that the model effectively captures long-term dependencies while maintaining temporal order. By expanding the receptive field of the convolution kernel, dilated convolution excels at extracting key information over extended time spans, while mitigating the impact of short-term fluctuations on long-term trends. Figure 5 illustrates the operation of dilated causal convolution, where the kernel size is set to \(K=2\) and the dilation factor \(d\) is configured as \((1,2,4)\). When \(d=1\), the kernel slides sequentially, processing adjacent input points. The causality constraint ensures that the convolution output at each time step depends solely on the current and previous inputs. When \(d=2\), the convolution operation skips one input point, leaving a gap of one point between the inputs corresponding to each convolution output. This expands the receptive field of the neuron, allowing the convolution to capture information over a longer time span without increasing computational cost. When \(d=4\), the convolution operation skips three input points, further enlarging the receptive field and enabling the model to capture a broader range of input data.