## 3 Results
### Out-of-Sample Test  
For the _Out-of-Sample_ testing, _XGBoost_ achieved the lowest error (median _RMSE_ score of 9.47) among the eleven regression methods. The three closest regression methods to _XGBoost_ are _Random Forest_ (median _RMSE_ of 9.67), _Spline Continued Fraction_ (median _RMSE_ of 10.99), and _Gradient Boosting_ (median _RMSE_ of 12.66). _Stochastic Gradient Descent_ without parameter estimation performed the worst among all regression methods used in the experiment, and due to the unreasonable high error observed in the runs we have omitted it from further analysis.  
Statistical Significance of the Results of the Out-of-Sample Test  
To evaluate the significance of the results obtained by the different regression methods for the _Out-of-Sample_ test, we applied a [PERSON] test for repeated measure [20] for the 100 runs. We computed the ranking of the methods for each of the runs based on the _RMSE_ score obtained in the test distribution of the _Out-of-Sample_ settings. This helps to determine whether the experiment's techniques were consistent in terms of performance. The statistical test found a \(p\)-value = \(1.9899\times 10^{-183}\), which rejects the null hypothesis, i.e., _\"all the algorithms perform the same\"_; thus, we proceeded with the post hoc test.  
We applied [PERSON]'s post hoc test on the ranking of the ten regressors computed for the _RMSE_ scores obtained for 100 runs of the _Out-of-Sample_ test. In Figure 2a, the \(p\)-values obtained for the test are plotted as a heat map. It can be seen that there exist no significant  
\begin{table}
\begin{tabular}{c c c} \hline \hline \multirow{2}{*}{**Regressor**} & \multicolumn{2}{c}{**Median RMSE Score \(\pm\) Std**} \\ \cline{2-3}  & _Out-of-Sample_ & _Out-of-Domain_ \\ \hline Spln-CFR & 10.989 \(\pm\) 0.382 & **36.327 \(\pm\) 1.187** \\ xg-b & **9.474 \(\pm\) 0.190** & 37.264 \(\pm\) 0.947 \\ rf & 9.670 \(\pm\) 0.197 & 38.074 \(\pm\) 0.751 \\ grad-b & 12.659 \(\pm\) 0.178 & 39.609 \(\pm\) 0.619 \\ l-regr & 17.618 \(\pm\) 0.187 & 41.265 \(\pm\) 0.466 \\ krnl-r & 17.635 \(\pm\) 0.163 & 41.427 \(\pm\) 0.464 \\ mlp & 19.797 \(\pm\) 5.140 & 41.480 \(\pm\) 9.640 \\ ada-b & 18.901 \(\pm\) 0.686 & 47.502 \(\pm\) 0.743 \\ l-svr & 26.065 \(\pm\) 7.838 & 47.985 \(\pm\) 1.734 \\ lasso-l & 34.234 \(\pm\) 0.267 & 74.724 \(\pm\) 0.376 \\ sgd-r 1 & N.R. & N.R. \\ \hline \hline \end{tabular} \({}^{\dagger}\) The _Stochastic Gradient Descent_ Regressor (sgd-r) without parameter estimation predicted unreasonable high values and had an extreme predicted error measure. Hence, we do not report (N.R.) the performance of sgd-r, and have omitted it from further analysis.  
\end{table}
Table 1: Results from 100 runs of the proposed Spline Continued Fraction and ten regression methods all trained on the same dataset, with the median of the Root Mean Squared Error (_RMSE_) and standard deviation as the uncertainty of error.  
differences (NS) between the performances of _Spline Continued Fraction_ (Spnl-CFR) and those of rf and grad-b.  
Additionally, we generated the Critical Difference (CD) diagram proposed in [21] to visualize the differences among the regressors for their median ranking. The CD plot uses the Nyemeni post hoc test and places the regressors on the \(x\)-axis of their median ranking. It then computes the _critical difference_ of the rankings between them and connects those which are closer than the critical difference with a horizontal line, denoting them as statistically _'non-significant'_.  
We plot the CD graph in Figure 1(b) using the implementation from the Orange data mining toolbox [22] in Python. The Critical Difference (CD) is found to be 1.25. It can be seen that xg-b ranked first among the regressors, with no significant difference', while rf ranked second. The median ranking of the proposed _Spline Continued Fraction_ was third, withno significant differences in the performance rankings of rf and grad-b.