# 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  
The DNN part of the proposed method consists of TCN, MGU, LSTM, Multihead Attention, and FCNN layers. The Temporal Convolutional Network (TCN) is used to model long-range dependencies, while the Minimal Gated Unit (MGU) serves as a more efficient replacement for traditional gated units. Long Short-Term Memory (LSTM) layers handle long-term dependencies, and Multihead Attention mechanisms focus on the most relevant input features. Fully Connected Neural Networks (FCNNs) perform feature mapping and regression. In this architecture, both the Minimal Gated Unit (MGU) and Long Short-Term Memory (LSTM) layers are utilized to leverage their complementary strengths. MGU is designed for efficiency, providing faster computation with fewer parameters. LSTM excels at capturing more complex long-term dependencies. By integrating both units, the model achieves a balance between computational efficiency and the ability to retain intricate patterns over time. This allows for the DNN to adapt to varying levels of temporal complexity in the data, ensuring both speed and depth in sequence modeling tasks. The proposed deep neural network \(\boldsymbol{\mathcal{N}}\) is shown in Figure 2. Initially, the two-dimensional acceleration data \((251,1)\) is preprocessed and converted into a three-dimensional tensor and then fed into the Temporal Convolutional Network (TCN) module. The TCN module consists of three blocks, each with three layers. Through a trial-and-error process, the kernel size in the first and second TCN blocks is set to 20, while it is set to 10 in the third block. The number of filters in the blocks is selected as 64, 36, and 16, respectively. The following equation is used to determine the minimum kernel size required in a TCN:  
\[RF=1+2\times(\text{kernel\_size}-1)\times\sum_{i=0}^{n-1}\text{ dilation\_rate}^{i} \tag{20}\]  
where:  
Figure 1: Load estimation generation using RFADNN. * Receptive field RF is the maximum length of input data that the model can consider at once. * kernel_size is the size of the kernel (filter) used in each convolutional layer. * dilation_rate\({}^{i}\) is the dilation rate at the \(i^{th}\) layer, which exponentially increases with the layer level. * \(n\) is the total number of layers. The dilation rate for each layer is determined by Dilation rate(layer) = 2(layer level). The acceleration data, represented by a tensor of dimensions (None, 251, 16), undergoes initial processing in the TCN module before entering the Minimal Gated Unit (MGU) [27] module, as depicted in Figure 2. Structured into three layers with hidden units of 251, 64, and 32 respectively, the MGU module outputs a tensor (None, 251, 32) that encapsulates the dynamics among acceleration data points. This output is then conveyed to the LSTM [28] module for enhanced analysis. The term \"None\" signifies an undefined or unspecified dimension size in the tensor's shape. The dimension expressed by \"None\" entirely stems from the usage procedure of the libraries used for implementation. These equations in Table 2 describe the different gates and state updates in an LSTM cell. The forget gate (\(f_{t}\)) controls which information to discard from the cell state, while the input gate (\(i_{t}\)) decides how much new information to add. The output gate (\(o_{t}\)) determines how much of the updated cell state is used to compute the hidden state (\(h_{t}\)). The cell state (\(c_{t}\)) is updated using a combination of the old state and the new candidate state (\(c_{t}\)). In the equations, \(W_{f}\), \(W_{i}\), \(W_{o}\), and \(W_{c}\) represent weight matrices that are learned during training. These matrices are used to combine the input vector \(x_{t}\) and the previous hidden state \(h_{t-1}\) for each gate. Each weight matrix corresponds to a specific gate:  
* \(W_{f}\): forget gate weight matrix. * \(W_{i}\): input gate weight matrix.