# DDTree: A Hybrid Deep Learning Model for Real-Time Waterway Depth Prediction and Smart Navigation
## 3 Domain Analysis and Definition of Conceptual Model
### Deep Neural Network  
Prediction of waterway depth requires learning through a DNN as a nonlinear input-output model. Figure 4 shows the structure diagram of a DNN, in which the number of hidden layers is seven.  
For the prediction of waterway depth, the DNN model uses as the inputs of different AIS attribute data \(x=(x_{1},x_{2},\dots,x_{n})\) and produces the output y as prediction of depth; \(f_{1},f_{2},\dots,f_{n}\) are the activation functions corresponding to each layer in the deep neural network; \(\hat{y}(x)\) is the output corresponding to each input data. Therefore, for layer \(l\) of the network, it can be expressed as Equation (4):  
\[f_{w_{l},b_{l}}(x)=f(\sum_{j=1}^{N_{l}}w_{lj}x_{j}+b_{l})=f\Big{(}w_{l}^{T}x_{ l}+b_{l}\Big{)} \tag{4}\]where \(N_{l}\) is the number of nodes on the first layer of the network, \(w_{l}\in R^{N_{l}\times N_{l-1}}\) and \(b\in R\), are the weight and bias respectively in the network, and \(f_{w,b}\) is the network activation function of each layer.  
The optimal attribute is selected by the preceding decision tree model to provide the appropriate input attribute parameters for the DNN. Then, we need to set the initial parameters of the DNN. By iteratively adjusting the weights \(w_{i}\) and offsets \(\theta_{i}\) of the DNN, the prediction accuracy is maximized.  
For the prediction of the waterway depth of the unknown waterway, DNN aims to predict the waterway depth information of the unknown waterway depth based on the waterway depth data of the known area. The prediction results \(y(x)\) can be expressed as Equation (5):  
\[y(x)=\mathbf{x}_{a+h,b+h}^{m}=\left(\begin{array}{c}x_{a+h,b+h}^{1}\\ \ldots\\ x_{a+h,b+h}^{m}\end{array}\right), \tag{5}\]  
where the waterway depth is \(x_{a,b}^{m}\), and the latitude and longitude of the geographical position of the point is (\(a\), \(b\)).  
For the entire region of the prediction of depth data, the prediction is given by Equation (6):  
\[\mathbf{x}^{m}=\left(\begin{array}{c}x_{a-h,b-h}^{1}\cdots,x_{a+h,b+h}^{1}\\ \ldots\ldots\\ x_{a-h,b-h}^{m}\cdots,x_{a+h,b+h}^{m}\end{array}\right) \tag{6}\]  
where \(n\) represents points corresponding to different geographic locations, and (\(a\), \(b\)) represents the corresponding longitude and latitude.  
After preprocessing and analyzing the AIS data, the most important factor for the prediction capability of DNN is to discover the optimal hyperparameter values to achieve the best prediction performance of the entire network. We use the stochastic gradient descent [40] for the network optimization. The range of the network parameters can be specified as Equation (7):  
\[\begin{split} f\in\{tanh(x),sigmoid(x),max(x,0)\}\,\ \lambda\in\{10^{-4},10^{-2}\},\ n\in\{1,15\},N_{l}\in\{1,200\}^{n}\\ x_{a+h,b+h}^{m}=(f_{1}\cdot f_{2}\cdot f_{n})x^{m},\ f_{l}=f\left(w_{l}^{T} x_{l}+b_{l}\right),\end{split} \tag{7}\]  
Figure 4: The structure of a deep neural network.  
where \(\lambda\) is the rate of learning, \(f\) is the activation function, \(N_{I}\) is the number of iterations, \(n\) is the number of network layers, \(x_{a+h,b+h}^{m}\) is the prediction of depth model structure, and \(f_{I}\) is the expression of each neural network.