## 6 Regressor Evaluation and Prediction Error Analysis  
In this section, we want to verify the efficacy of the created dataset as training for the ML predictors of the four previously defined target variables (i.e., total energy consumption, maximum system power, mean system power and runtime). Specifically, we consider decision tree regressor, random forest regressor, and gradient boosting decision tree regressor. We remark that this work aims to provide the scientific community with a detailed dataset of HPC job runs, useful for the further analysis of the environmental footprint of linear solvers. As such, proposing novel ML models for energy consumption prediction is out of the scope of the paper.  
Table 4 details the performance metrics for the chosen regression models in predicting the job's energy performance. For each model, we used GridSearchCV to optimise hyper-parameters, with negative mean absolute error as the evaluation criteria. The table also includes metrics like RMSE, MAE, and MAPE, along with the optimal hyperparameter value for depth.  
When considering all the evaluation metrics, the Random Forest Regressor emerges as the best overall performer for predicting the total energy consumption and the maximum power. Instead, the gradient boosting decision tree regressor shows the strongest performance for predicting the average power and runtime.  
The histograms in Figure 10 depict the distribution of prediction errors for the four chosen targets. The _x axis_ of each graph represents the prediction error value, while the _y axis_ represents the error frequency, that is, the number of job runs tested that returned that error value. The models were trained on 70% of the data, and the remaining 30% was used for the testing. The errors for total energy consumption and runtime show a similar trend, with a significant peak centred around zero. The errors for the power targets (maximum and average power) exhibit a wider distribution, ranging from 0 to 40 W in absolute value. However, considering the average values of the power targets (with a maximum of approximately 2130 W and 1960 W on average), this error range remains very good.  
\begin{table}
\begin{tabular}{c l c c c c} \hline \hline
**TARGET** & **REGRESSOR** & **BEST DEPTH** & **RMSE** & **MAE** & **MAPE** \\ \hline Total Energy & _Decision tree_ & 19 & 0.013 & 0.004 & 0.076 \\ \cline{2-6} Min value: 0.000470 kWh & _Random forest_ & 13 & 0.010 & 0.003 & 0.062 \\ \cline{2-6} Max value: 0.938590 kWh & _GBDT_ & 5 & 0.010 & 0.003 & 0.119 \\ \hline Max power & _Decision tree_ & 10 & 12.676 & 7.966 & 0.024 \\ \cline{2-6} Min value: 150.000000 W & _Random forest_ & 11 & 10.241 & 6.947 & 0.020 \\ \cline{2-6} Max value: 440.000000 W & _GBDT_ & 6 & 10.339 & 7.089 & 0.021 \\ \hline Mean power & _Decision tree_ & 9 & 7.297 & 5.188 & 0.024 \\ \cline{2-6} Min value: 115.000000 W & _Random forest_ & 11 & 6.530 & 4.632 & 0.022 \\ \cline{2-6} Max value: 330.158420 W & _GBDT_ & 5 & 5.942 & 4.319 & 0.020 \\ \hline Runtime & _Decision tree_ & 10 & 22.238 & 7.296 & 0.065 \\ \cline{2-6} Min value : 6.000000 s & _Random forest_ & 12 & 19.651 & 7.033 & 0.060 \\ \cline{2-6} Max value: 2097.000000 s & _GBDT_ & 6 & 14.637 & 5.560 & 0.065 \\ \hline \hline \end{tabular}
\end{table}
Table 4: Summary table of regressor prediction evaluation of the dataset.