# A Piecewise Linear Regression Model Ensemble for Large-Scale Curve Fitting
## 6 Empirical Results
### CPU-Time versus Accuracy
\begin{table}
\begin{tabular}{c c c c c c c c c c c c} \hline \hline  & \multicolumn{3}{c}{**PR1 (MSE)**} & \multicolumn{3}{c}{**PR2 (MSE)**} & \multicolumn{3}{c}{**PR3 (MSE)**} & \multicolumn{3}{c}{**PR4 (MSE)**} \\ \hline  & \(M\) & **SLS** & **STS** & **TTS** & **SLS** & **STS** & **TTS** & **SLS** & **STS** & **TTS** & **SLS** & **STS** & **TTS** \\ \hline MS & 2 & 52.22 & 52.27 & \(1.31\times 10^{-2}\) & 63.99 & 64.03 & \(1.02\times 10^{-2}\) & 16.00 & 16.00 & \(3.31\times 10^{-3}\) & 149.71 & 149.64 & \(1.82\times 10^{-2}\) \\  & 5 & 52.22 & 52.27 & \(1.11\times 10^{-2}\) & 63.99 & 64.03 & \(8.30\times 10^{-3}\) & 16.00 & 16.00 & \(3.28\times 10^{-3}\) & 149.71 & 149.64 & \(1.70\times 10^{-2}\) \\  & 10 & 52.21 & 52.27 & \(1.09\times 10^{-2}\) & 63.99 & 64.03 & \(7.77\times 10^{-3}\) & 16.00 & 16.00 & \(3.03\times 10^{-3}\) & 149.71 & 149.64 & \(1.73\times 10^{-2}\) \\  & 20 & 52.22 & 52.27 & \(1.10\times 10^{-2}\) & 63.99 & 64.03 & \(7.82\times 10^{-3}\) & 16.00 & 16.00 & \(3.04\times 10^{-3}\) & 149.71 & 149.64 & \(1.76\times 10^{-2}\) \\  & 50 & 52.21 & 52.27 & \(1.04\times 10^{-2}\) & 63.99 & 64.03 & \(7.09\times 10^{-3}\) & 16.00 & 16.00 & \(2.93\times 10^{-3}\) & 149.71 & 149.63 & \(1.72\times 10^{-2}\) \\ \hline MC1 & 2 & 52.21 & 52.27 & \(9.02\times 10^{-3}\) & 63.99 & 64.03 & \(8.23\times 10^{-3}\) & 16.00 & 16.00 & \(2.74\times 10^{-3}\) & 149.70 & 149.64 & \(1.57\times 10^{-2}\) \\  & 5 & 52.21 & 52.26 & \(5.41\times 10^{-3}\) & 63.99 & 64.03 & \(5.45\times 10^{-3}\) & 16.00 & 16.00 & \(1.98\times 10^{-3}\) & 149.70 & 149.63 & \(9.94\times 10^{-3}\) \\  & 10 & 52.21 & 52.26 & \(3.78\times 10^{-3}\) * & 63.98 & 64.03 & \(4.75\times 10^{-3}\) & 16.00 & 16.00 & \(1.73\times 10^{-3}\) & 149.70 & 149.63 & \(8.08\times 10^{-3}\) \\  & 20 & 52.21 & 52.26 & \(3.26\times 10^{-3}\) & 63.98 & 64.03 & \(4.24\times 10^{-3}\) & 16.00 & 16.00 & \(1.65\times 10^{-3}\) & 149.70 & 149.63 & \(7.29\times 10^{-3}\) \\  & 50 & 52.21 & 52.26 & \(3.09\times 10^{-3}\) & 63.98 & 64.03 & \(4.04\times 10^{-3}\) & 16.00 & 16.00 & \(1.56\times 10^{-3}\) & 149.70 & 149.63 & \(6.74\times 10^{-3}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Average MSE over the different datasets (SLS, STS, and TTS) in all the problems applying the proposed methods with different values of \(M\) and \(n=1\) M (calculated from 30 replications). Note that MS, MC2, and MC3 include a final OLS step. The last row represents the results of the reference algorithm (RA) using a sample of \(N=10\) M. The values with * represent the first value of \(M\), for each problem and method, in which the error over the TTS is lower than the one obtained with RA. Values given in parentheses indicate the best method for each problem, obtained with minimum \(M\). In terms of accuracy, only MC2 and MC3 are competitive alternatives to the reference algorithm in all the problems, with MC3 always being the best option. It can be observed that MC2, for some different values of \(M\), is able to outperform the reference algorithm in all the problems. Observing Table 3, it can be seen that these improvements are only due to a variance reduction since it always presents more bias than the RA. Furthermore, Tables 2 and 3 confirm the lack of stability of MC2 (see, e.g., PR3). MC3 always has better results when combining only two sub-models, and regarding bias and variance, it also presents good results in the tested problems.