## 4 Experiment
### Methodology of Stacking Models
Table 3 highlights the R\({}^{2}\) scores and RMSE values for FNN and PNN models using two different feature sets. The PyTorch NN model exhibited superior performance compared to the FNN model in terms of both R\({}^{2}\) score and RMSE for every feature set, suggesting a more optimal fit and more precise predictions. This consistent performance indicates that the PNN was somewhat more effective in this scenario. In Table 4, the performance of the FNN and PNN, as measured by R\({}^{2}\) scores across six separate runs, is compared to the implementation of TensorFlow in the FNN. Tests were conducted using the identical architecture and hyperparameters for both methods to determine if the PNN provided a statistically significant improvement over the FNN. The table illustrates that for each run, the PNN consistently outperformed the FNN in terms of R\({}^{2}\) scores. Even if the changes in R\({}^{2}\) scores appeared to be small, there may have been an overall increase in performance when employing the PyTorch-based model. \begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{**Model**} & \multicolumn{2}{c}{**Feature Set 1**} & \multicolumn{2}{c}{**Feature Set 2**} \\ \cline{2-5}  & **R\({}^{2}\) Score** & **RMSE** & **R\({}^{2}\) Score** & **RMSE** \\ \hline RF & 0.9187 & 289.87 & 0.8866 & 342.33 \\ GBR & 0.9158 & 294.89 & 0.8946 & 330.00 \\ XGB & 0.8903 & 336.68 & 0.8717 & 364.23 \\ LR & 0.8928 & 375.57 & 0.9045 & 314.21 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Performance Comparison of ML Models Without PyTorch. \begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{**Model**} & \multicolumn{2}{c}{**Feature Set 1**} & \multicolumn{2}{c}{**Feature Set 2**} \\ \cline{2-5}  & **R\({}^{2}\) Score** & **RMSE** & **R\({}^{2}\) Score** & **RMSE** \\ \hline FNN & 0.9000 & 321.47 & 0.9015 & 319.08 \\ PyTorch NN & 0.9041 & 314.83 & 0.9054 & 312.61 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Performance Metrics of Neural Network Models. The RMSE values for the two models are compared, in addition to the assessment of R\({}^{2}\) scores. The PNN performed better than the FNN in all six runs, as shown in Table 5, with a consistently lower RMSE value. When predicting rental prices, the lower RMSE of the PNN implies improved prediction accuracy and decreased error. The variations in these results indicate that the PyTorch implementation enhances the performance of models in real-world prediction tasks. A paired \(t\)-test and \(p\)-value analysis for both R\({}^{2}\) scores and RMSE values were conducted to evaluate the differences in performance between the FNN and PNN, as shown in Table 6. With a \(p\)-value of 8.29 \(\times\) 10\({}^{-5}\) and a \(t\)-statistic of \(-11.619\) for the R\({}^{2}\) scores, the results are well below the conventional threshold of 0.05, indicating that the PNN shows slight improvements over the FNN in terms of R\({}^{2}\) scores. Comparatively, the \(t\)-statistic for the RMSE was 39.035 with a \(p\)-value of 2.08 \(\times\) 10\({}^{-7}\), suggesting that the PNN outperformed the FNN according to the RMSE results. The small \(p\)-values provide solid evidence for the improved efficiency of the PyTorch model, indicating that these results were not due to random chance. The statistical tests support the conclusion that the PNN performed considerably better than the FNN with regard to both R\({}^{2}\) and RMSE, thereby verifying the advantages of using PyTorch in rental price prediction. Table 7 exhibits the R\({}^{2}\) score and RMSE values for four distinct ensemble models that integrate PNNs with other techniques, namely RF, GBR, XGB, and LR. These evaluations were conducted on both feature set 1 and feature set 2. The R\({}^{2}\) value for feature set 1 exhibited a high level of correlation between the anticipated and actual values across all methods, and the respective RMSE values indicated a high level of precision in the forecasts, with minimal error.