# Vertical Force Monitoring of Racing Tires: A Novel Deep Neural Network-Based Estimation Method
## 5 Performance Evaluations  
The mean absolute error (MAE), Percentage Normalized Root Mean Square Wrror (NRMS%), coefficient of determination (R squared), and Root Mean Square Error (RMSE) were used to evaluate the performance of the proposed method compared to other methods. The formulas are as follows:  
MAE \[= \frac{1}{n}\sum_{i=1}^{n}|F_{i}-\hat{F}_{i}|\] (26) NRMS% \[= \left(\frac{\sqrt{\frac{1}{n}\sum_{i=1}^{n}(F_{i}-\hat{F}_{i})^{2 }}}{\hat{F}}\right)\times 100\%\] (27) \[R^{2} = 1-\frac{\sum_{i=1}^{n}(F_{i}-\hat{F}_{i})^{2}}{\sum_{i=1}^{n}(F_ {i}-\hat{F})^{2}}\] (28) RMSE \[= \sqrt{\frac{1}{n}\sum_{i=1}^{n}(F_{i}-\hat{F}_{i})^{2}}\] (29)  
where,  
* \(F_{i}\): actual measured force;
* \(\hat{F}_{i}\): values estimated by the model;
* \(\hat{F}\): the average of the actual values;
* \(n\): the number of samples. The proposed method is compared with traditional LSTrefM and dilated convolution methods, as well as with the study presented in [18], based on the performance criteria stated above. The results of the comparison are given in Tables 4-6 for several metrics. Figures 13-32 illustrate the regions of the data that the model attends to and their impact on estimation accuracy. In Figures 13-22, the attention heatmaps generated after 250 epochs of training are shown, while Figures 23-32 presents the attention maps after 1000 epochs of training. It can be observed that the attention weights in the heatmaps after 1000 epochs are concentrated on fewer points compared to the results after 250 epochs. In the heatmaps, the regions highlighted in yellow represent areas where the attention is highly focused, indicating high attention scores, while regions in blue represent areas with minimal attention scores. This shift towards more concentrated attention with increased training likely enables the model to focus on the most relevant features, leading to improved estimation accuracy. This more concentrated focus likely contributes to more accurate estimations,  
Figure 12: Automatic labeling: True Negative (TN) data segment and corresponding attention heatmap. as the model learns to prioritize the most relevant parts of the sequence as the number of epochs increases. In each graph, the original data obtained under different vertical loads (150 kg, 180 kg, 210 kg,... 420 kg) are presented alongside the attention heatmaps generated by the attention mechanism. In the comparative analysis conducted, the proposed technique demonstrated superior performance across all evaluated performance metrics, including MAE, RMSE, R\({}^{2}\), and NRMS%, consistently outperforming all other methods compared in the study. This establishes the proposed technique as the most effective approach among the methods analyzed. In terms of memory footprint, it ranked second after the 1D Convolution-BiGRU [20] method. However, with an average estimation time of 3.44 ms, it exhibited inferior performance compared to other methods. Nonetheless, the critical importance placed on achieving high accuracy in performance metrics outweighs concerns regarding estimation speed. The baseline LSTM model shows an estimation error with an MAE of 1.816 kgf on the dataset. Notably, the proposed RFADNN method reduced the error to an MAE of about 0.77 kgf, as detailed in the same table. Figure 33 illustrates the performance of the proposed method in estimating the load exerted on the test tire, as shown in Figure 4, under various internal inflation pressures and at different rotational speeds. The graphs indicate instances of sudden divergence between the estimated errors and the actual values. This discrepancy is attributed to mechanical vibrations in the section of the tire testing machine that applies force to the tire.