# 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 MC2 & 2 & 52.22 & 52.27 & \(1.31\times 10^{-2}\) & 63.99 & 64.03 & \(1.23\times 10^{-2}\) & 16.00 & 16.00 & \(3.14\times 10^{-3}\) & 149.70 & 149.64 & \(1.53\times 10^{-2}\) \\  & 5 & 52.21 & 52.26 & \(6.27\times 10^{-3}\) & 63.98 & 64.03 & \(5.14\times 10^{-3}\) & 16.00 & 16.00 & \(1.94\times 10^{-3}\) & 149.70 & 149.63 & \(9.55\times 10^{-3}\) \\  & 10 & 52.21 & 52.26 & \(3.13\times 10^{-3}\) * & 63.98 & 64.02 & \(3.60\times 10^{-3}\) & 16.00 & 16.00 & \(1.39\times 10^{-3}\) & 149.70 & 149.63 & \(7.84\times 10^{-3}\) \\  & 20 & 52.21 & 52.26 & \(1.84\times 10^{-3}\) & 63.98 & 64.02 & \(2.64\times 10^{-3}\) * & 16.00 & 16.00 & \(1.21\times 10^{-3}\) * & 149.70 & 149.63 & \(6.96\times 10^{-3}\) * \\  & 50 & 52.21 & 52.26 & \(1.15\times 10^{-3}\) & 63.98 & 64.02 & \(1.73\times 10^{-3}\) & 16.04 & 16.04 & \(3.89\times 10^{-2}\) & 149.69 & 149.63 & \(5.77\times 10^{-3}\) \\ \hline MC3 & 2 & 52.21 & 52.26 & \((1.75\times 10^{-3}\) * & 63.98 & 64.02 & \((1.48\times 10^{-3}\) *) & 16.00 & 16.00 & \((1.09\times 10^{-3}\) *) & 149.69 & 149.63 & \((4.45\times 10^{-3}\) *) \\  & 5 & 52.20 & 52.26 & \(1.12\times 10^{-3}\) & 63.98 & 64.02 & \(1.06\times 10^{-3}\) & 16.00 & 16.00 & \(8.92\times 10^{-4}\) & 149.69 & 149.63 & \(4.89\times 10^{-3}\) \\  & 10 & 52.20 & 52.26 & \(9.26\times 10^{-4}\) & 63.98 & 64.02 & \(9.17\times 10^{-4}\) & 16.00 & 16.00 & \(7.97\times 10^{-4}\) & 149.69 & 149.63 & \(4.57\times 10^{-3}\) \\  & 20 & 52.20 & 52.26 & \(7.81\times 10^{-4}\) & 63.98 & 64.02 & \(8.46\times 10^{-4}\) & 16.00 & 16.00 & \(8.10\times 10^{-4}\) & 149.69 & 149.63 & \(4.23\times 10^{-3}\) \\  & 50 & 52.20 & 52.26 & \(7.09\times 10^{-4}\) & 63.98 & 64.02 & \(7.51\times 10^{-4}\) & 16.00 & 16.00 & \(7.21\times 10^{-4}\) & 149.69 & 149.63 & \(4.21\times 10^{-3}\) \\ \hline RA & - & 52.21 & 52.26 & \(4.53\times 10^{-3}\) & 63.98 & 64.02 & \(2.86\times 10^{-3}\) & 16.00 & 16.00 & \(1.36\times 10^{-3}\) & 149.68 & 149.63 & \(7.06\times 10^{-3}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 2: _Cont._  
\begin{table}
\begin{tabular}{c c c c c c c c c c} \hline \hline  & \multicolumn{3}{c}{**PR1 (Bias–Var)**} & \multicolumn{3}{c}{**PR2 (Bias–Var)**} & \multicolumn{3}{c}{**PR3 (Bias–Var)**} & \multicolumn{3}{c}{**PR4 (Bias–Var)**} \\ \hline  & \(M\) & \(Bias^{2}\) & \(Var\) & \(Bias^{2}\) & \(Var\) & \(Bias^{2}\) & \(Var\) & \(Bias^{2}\) & \(Var\) \\ \hline MS & 2 & \(5.28\times 10^{-3}\) & \(2.16\times 10^{-2}\) & \(5.50\times 10^{-3}\) & \(1.54\times 10^{-2}\) & \(2.78\times 10^{-3}\) & \(3.97\times 10^{-3}\) & \(1.21\times 10^{-2}\) & \(2.51\times 10^{-2}\) \\  & 5 & \(4  
Regarding the CPU-time, Table 4 shows the results for every problem and method. To establish an upper bound, the columns \(t_{sub}\) represent the total calculation times of all the sub-models sequentially. Note that this time can be reduced by increasing the number of threads and fitting these models in parallel, with nearly linear speed-ups obtainable. It can be observed that those columns are equal in every method and problem, since all the sub-models are calculated for every method in the same way. Joining the results presented in this table with the previous error tables, the method MC2 was able to outperform the reference algorithm with different values of \(M\) depending on the problem. This means that choosing a high enough number of sub-samples, we can obtain accurate results. However, MC3 with only two sub-samples was able to overcome the RA results, and therefore, since the combination method itself is not very computationally expensive, we can obtain better results even in a lower time.