# Temporal Convolutional Network for Carbon Tax Projection: A Data-Driven Approach
## 3 Data and Methodology
### Temporal Convolutional Network  
Our choice of the TCN model for carbon tax price projection was informed by recent advancements in time series forecasting techniques. While traditional methods have been widely used [20], there is a growing body of literature exploring the application of machine learning techniques to environmental and energy-related forecasting problems [21; 22]. TCNs represent a class of deep learning models specifically designed for sequence modeling tasks. They offer several advantages over traditional recurrent architectures: Parallelism: TCNs allow for parallel processing of inputs, leading to faster training and inference times. This is particularly beneficial when dealing with large datasets or when real-timepredictions are required. Flexible receptive field: Through dilated convolutions, TCNs can efficiently capture long-range dependencies. This allows the model to consider both recent and distant past information when making predictions. Stable gradients: The convolutional structure mitigates the vanishing gradient problem common in recurrent networks, allowing for more effective training of deep architectures. Constant memory usage: Unlike recurrent neural networks, a TCN's memory usage does not grow with the length of the input sequence, making it more suitable for processing long time series.  
Our TCN model architecture consists of the following components: Input layer: accepts a sequence of historical carbon prices and related features. The input shape is (batch_size, sequence_length, n_features), where the sequence length is 5 years of monthly data (60 time steps) and n_features is the number of input variables.  
Dilated causal convolutional layers: a stack of 1D convolutional layers with increasing dilation rates. We use 4 layers with dilation rates of 1, 2, 4, and 8, allowing the model to capture dependencies over a wide range of time scales. Each layer uses 64 filters with a kernel size of 3.  
Residual connections: to facilitate gradient flow and enable deeper network training. Each convolutional layer is wrapped in a residual block, which adds the input to the layer's output. This helps in training very deep networks by allowing gradients to flow directly through the network.  
Normalization and activation: After each convolutional layer, we apply layer normalization followed by a ReLU activation function. This helps in stabilizing the training process and introduces non-linearity into the model.  
Dropout: To prevent overfitting, we apply dropout with a rate of 0.2 after each convolutional layer.  
Output layer: a dense layer producing the projected carbon tax price for the next time step.  
The mathematical formulation of the dilated causal convolution used in our TCN model can be expressed as  
\[F(s)=(x*_{d}f)(s)=\sum_{i=0}^{k-1}f(i)\cdot x_{s-d\cdot i} \tag{2}\]  
where \(x\) is the input sequence, \(f\) is the filter, \(k\) is the filter size, \(d\) is the dilation factor, and \(s\) is the current time step.  
The model was trained to minimize the mean squared error between predicted and actual carbon tax prices. We used the Adam optimizer with a learning rate of 0.001 and trained for 100 epochs with early stopping based on validation loss. The learning rate was reduced by a factor of 0.5 if the validation loss did not improve for 10 consecutive epochs.