# Vertical Force Monitoring of Racing Tires: A Novel Deep Neural Network-Based Estimation Method
## 3 Random Forest Assisted Deep Neural Network Model
### Deep Neural Network Model of the Proposed Method
\begin{table}
\begin{tabular}{l l} \hline \hline
**Gate/State** & **Equation** \\ \hline Forget gate & \(f_{t}=\sigma(W_{f}[h_{t-1},x_{t}]+b_{f})\) \\ \hline Input gate & \(i_{t}=\sigma(W_{i}[h_{t-1},x_{t}]+b_{i})\) \\ \hline Output gate & \(o_{t}=\sigma(W_{o}[h_{t-1},x_{t}]+b_{o})\) \\ \hline Cell state candidate & \(c_{t}=\tanh(W_{c}[h_{t-1},x_{t}]+b_{c})\) \\ \hline Cell state update & \(c_{t}=f_{t}\odot c_{t-1}+i_{t}\odot\c_{t}\) \\ \hline Hidden state & \(h_{t}=o_{t}\odot\tanh(c_{t})\) \\ \hline \hline \end{tabular}
\end{table} 
Table 2: LSTM Equations. Figure 2: Proposed deep neural network model (\(\mathcal{N}\)) in RFADNN. * \(W_{0}\): output gate weight matrix. * \(W_{c}\): cell state candidate weight matrix. The functions used in MGU equations are as follows: Similarly, in the MGU equations, \(W_{f}\) and \(W_{h}\) are weight matrices:
* \(W_{f}\): forget gate weight matrix in MGU. * \(W_{h}\): weight matrix for the hidden state candidate update. These weights are also learned during training and are used to transform both the input and the hidden states for computing the forget gate and hidden state updates. MGU reduces the computational cost by combining the functions of LSTM's three gate mechanisms into a single forget gate. This leads to a model that is computationally more efficient and resource-saving, particularly advantageous when working with large datasets. Below are explanations of the functions used in LSTM and MGU, Tables 2 and 3, respectively:  
* Sigmoid function (\(\sigma\)): The sigmoid function outputs values between 0 and 1, which determine how much information to pass through the gate. * Tanh function (tanh): Used for cell state candidates and hidden states, outputs values between \(-1\) and 1. * Element-wise multiplication (\(\odot\)): Used for controlling which parts of the cell state are updated. (\(L_{p}\)). The weighted loss (\(L_{w}\)) is particularly designed to adjust the model's focus on errors based on their magnitude, thereby ensuring a balanced learning approach across both small and large discrepancies. The weighted loss is defined as follows:  
\[L_{w}=\sum_{i=1}^{n}w_{i}\cdot|F_{i}-\hat{F}_{i}|, \tag{21}\]  
where the dynamic weights \(w_{i}\) are computed based on the magnitude of the error, expressed as follows:  
\[w_{i}=(|F_{i}-\hat{F}_{i}|+\epsilon)^{\alpha} \tag{22}\]  
In this formulation, \(w_{i}\) represents the dynamic weights assigned to each error based on its magnitude. The term \(\epsilon=1\times 10^{-6}\) is included to ensure numerical stability, preventing division by zero or extremely small values. The parameter \(\alpha=0.5\) controls the sensitivity of the weighting, allowing for the model to balance the focus on larger errors without over-penalizing smaller ones. The log-cosh function is used in the outlier loss (\(L_{s}\)) to reduce the influence of outliers. For small errors, this function behaves similarly to the Mean Squared Error (MSE), but as the error increases, it gradually transitions to a behavior similar to the Mean Absolute Error (MAE). This ensures that the model does not disproportionately react to large deviations, maintaining a balanced response to all errors. \[L_{s}=\lambda\cdot\log(\cosh(F-\hat{F})) \tag{23}\]  
In this formula, \(\lambda\) is a scaling factor, and the log-cosh function [29] effectively smooths the loss for larger errors, preventing the model from overfitting to these extreme values.