## 3 Results
### Feature Matrix and Regression Algorithm Analysis
#### 3.2.2 Ridge Regression Model Analysis  
The adjusted R\({}^{2}\) was 0.7363, the root-mean-square error (RMSE) was 0.068, the \(p\)-value was \(2.089\times 10^{-16}\), which was obtained by the leave-one-out method when the final feature set and regression model were determined, and the mean absolute error of each individual was \(0.054\pm 0.041\) (since the slope S is a ratio, we used the absolute error instead of the relative error). Eight of them had an absolute error greater than 0.1 and an average of 0.126. We used the questionnaires filled out by participants and exercise experimental data to find the key factors that cause large errors (see Table 3). In the experimental data of these eight individuals, some individuals had an exhaustive heart rate that was more than 15 bpm lower than the HR\({}_{\text{max}}\) calculated by the formula but had completed at least five levels of load. This shows that their cardiopulmonary exercise ability is stronger than that of the same type of people, and that the highest experimental level can no longer make them reach the HR\({}_{\text{max}}\). However, this situation may not be reflected through the collected static human body data, and it was not covered in the questionnaire either. There were also individuals whose exhausted heart rate was 15 bpm or more than 15 bpm lower than the maximum HR\({}_{\text{max}}\) calculated by the formula and their completed load level was less than five. The load level they accomplished was lower and their exhausted heart rate was far from the theoretical HR\({}_{\text{max}}\), indicating that there may be other influencing factors in the experiment, such as poor subjective cooperation of the participants or errors in the experimental data collected. These are problems with exercise experimental data, indicating that the slope S obtained by linear regression of those individuals may deviate from the true value. Using an inaccurate slope S as the dependent variable to train the multiple regression model would undoubtedly cause a large error. In other individuals, some had fatty livers (which has a certain impact on exercise ability [12]), and some never exercised. The information obtained from the questionnaire shows that the static body data in the feature matrix used are not comprehensive enough and are only applicable to normal people, which may cause large errors. After excluding individuals with large errors, the model was re-trained, and the adjusted R\({}^{2}\) was 0.8336, the RMSE was 0.049, the \(p\)-value was \(4.43\times 10^{-18}\), and the average prediction absolute error was \(0.041\pm 0.028\). Finally, the following regression equation was obtained:  
\[ \begin{split} Slope=1.7015-2.676\times 10^{-2}\times LLLM-1.654 \times 10^{-3}\times STI\\ -6.143\times 10^{-3}\times GS-0.5188\times WHR\end{split}  \tag{2}\]  
We substituted the test set data into the equation and achieved an adjusted R\({}^{2}\) of 0.7068 and a \(p\)-value of \(6.92\times 10^{-5}\). Then, relying on the information reflected in the lifestyle questionnaire and the experimental data, we eliminated two cases of large error data. The adjusted R\({}^{2}\) value after correction was 0.8247, the \(p\)-value was \(4.046\times 10^{-5}\), and the predicted value was significantly correlated with the \begin{table}
\begin{tabular}{c c c} \hline \hline
**Number of Individuals** & **Feature** & **Predicted Result** \\ \hline
2 & Exhausted HR is more than 15 bpm lower than & \\  & the formula HR\({}_{\text{max}}\) (the load level is 5 or more) & \\ \hline
2 & Exhausted HR is more than 15 bpm lower than & \\  & the formula HR\({}_{\text{max}}\) (the load level is less than 5) & Uncertain \\ \hline
2 & Fatty liver & Lower \\ \hline
1 & Never exercise & Lower \\ \hline
1 & Unknown & \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Features of the samples with high prediction errors mined from existing questionnaires and exercise data. measured value, which shows that the above formula can predict the slope S of the general population.