## 3 Methodology
### CNN-Based Airport Delay Level Prediction Model
#### 3.2.2 CNN-Based Feature Extraction and Rank Prediction  
This model predicts airport delay levels based on convective weather and air traffic conditions. It utilizes a multi-channel weather-traffic image (MTWSI) generated by combining relevant data. The output corresponds to delay categories created through FCM clustering analysis.  
During the construction of feature maps, the entire image is scanned by a processing unit, and the state of this unit is recorded at the corresponding position in the feature map. This process is equivalent to a convolution operation followed by a bias addition and a sigmoid activation function, represented as follows:  
\[y^{(d)}=\sigma\Big{(}Wy^{(d-1)}+b\Big{)} \tag{1}\]  
where \(d\) denotes the depth of the convolutional layer, \(W\) is the weight matrix, and \(b\) is the bias term. In fully connected neural networks, the weight matrix is dense, meaning that each input is connected to every neuron via distinct weights. However, in convolutional  
Figure 7: Distribution of delay features in different clusters.  
neural networks (CNNs), due to weight sharing, the weight matrix \(W\) is sparse. Its structure can be described as follows:  
\[\begin{bmatrix}w&\dots&0\\ \vdots&\ddots&\vdots\\ 0&\dots&w\end{bmatrix} \tag{2}\]  
Here, \(w\) is a matrix with dimensions matching the receptive field of the unit. The use of sparse weight matrices reduces the number of trainable parameters, thereby enhancing the model's generalization capability. Multiplying the input by the weight matrix \(W\) is equivalent to convolving the input with a learnable filter \(w\).  
If the input to the \((d-1)\)-th layer has a spatial size of N \(\times\) N, and the receptive field of the units in layer d is m \(\times\) m, then the resulting feature map will have dimensions (N \(-\) m + 1) \(\times\) (N \(-\) m + 1). Specifically, the value at position \((i,j)\) in the feature map is calculated as follows:  
\[y_{ij}^{(d)}=\sigma\Big{(}x_{ij}^{(d)}+b\Big{)} \tag{3}\]  
\[x_{ij}^{(d)}=\sum_{a=0}^{m-1}\sum_{b=0}^{m-1}w_{ab}y_{(i+a)(j+b)}^{(d-1)} \tag{4}\]  
where the bias \(b\) is a scalar. By applying Equations (3) and (4) across all input positions \((i,j)\), the feature map for the corresponding plane is generated.  
The CNN model contains two convolution layers and three dense layers. The first convolutional layer employs six \(3\times 3\) kernels, and the second uses sixteen \(3\times 3\) kernels to extract features. A ReLU activation function follows each dense layer (with 120 and 84 nodes, respectively), and the final layer consists of 2 nodes corresponding to the two classes--delay and no delay--with probabilities computed via a SoftMax function. The probability of each category is calculated using a SoftMax activation function. The model has a learning rate of 0.001 and a batch size of 128. The data distribution is learned from training-set samples by iteratively optimizing the network model parameters. Usually, the optimization direction is determined by the objective function, which consists of an error term (J) and a regularity term (R) [28]:  
\[\theta=argminL(X,Y)=argmin(J,\lambda R) \tag{5}\]  
\[J=-\frac{1}{m}\sum_{i=1}^{N}f\big{(}y(i),y^{\prime}(i)\big{)} \tag{6}\]  
In the equation, \(X\) and \(Y\) are the inputs and outputs of the model. The loss function and regularization term are denoted by \(J\) and \(R\), respectively. \(\theta\) denotes the parameters of the deep neural network, and \(\lambda\) is the weight of the regularization term. In this paper, the cross-entropy is used as the loss function and is determined by \(f(\cdot)\)where \(y(i)\) and \(y^{\prime}(i)\) are the real and actual labels of the first \(i\) training sample. The use of a shedding layer effectively reduces the occurrence of overfitting, and the Adam optimizer is applied to improve the performance of the gradient descent algorithm. Figure 8 demonstrates the CNN Architecture.