# Variable Selection in Time Series Forecasting Using Random Forests
## 3 Results
### Simulations  
We present the results from the application of the methods to the simulated time series. Each simulation experiment lasts 30 h, while the computation time for all the methods excluding RF is some minutes. In particular, we present the boxplots of the absolute errors and the errors. Furthermore, we \begin{table}
\begin{tabular}{c c c} \hline Experiment & Model & Parameters \\ \hline \(1\) & ARMA(1, 0) & \(\phi_{1}=0.6\) \\ \(2\) & ARMA(1, 0) & \(\phi_{1}=-0.6\) \\ \(3\) & ARMA(2, 0) & \(\phi_{1}=0.6,\phi_{2}=0.2\) \\ \(4\) & ARMA(2, 0) & \(\phi_{1}=-0.6,\phi_{2}=0.2\) \\ \(5\) & ARMA(0, 1) & \(\phi_{1}=0.6\) \\ \(6\) & ARMA(0, 1) & \(\phi_{1}=-0.6\) \\ \(7\) & ARMA(0, 2) & \(\theta_{1}=0.6,\theta_{2}=0.2\) \\ \(8\) & ARMA(0, 2) & \(\theta_{1}=-0.6,\theta_{2}=-0.2\) \\ \(9\) & ARMA(1, 1) & \(\phi_{1}=0.6,\theta_{1}=0.6\) \\ \(10\) & ARMA(1, 1) & \(\phi_{1}=-0.6,\theta_{1}=-0.6\) \\ \(11\) & ARMA(2, 2) & \(\phi_{1}=0.6,\phi_{2}=0.2\) \\ \(12\) & ARIMA(0, 0.40, 0) & \(\phi_{1}=0.6\) \\ \(13\) & ARIMA(1, 0.40, 0) & \(\phi_{1}=0.6\) \\ \(14\) & ARIMA(0, 0.40, 1) & \(\phi_{1}=0.6\) \\ \(15\) & ARIMA(1, 0.40, 1) & \(\phi_{1}=0.6,\theta_{1}=0.6\) \\ \(16\) & ARIMA(2, 0.40, 2) & \(\phi_{1}=0.6,\phi_{2}=0.2\) \\ \hline \end{tabular}
\end{table} 
Table 5: Simulation experiments, their respective models and their defined parameters. See Section 2.1.1 for the definitions of the parameters. Figure 2: Map of locations for the 135 stations used in the analysis. present the medians of the absolute errors, medians of the squared errors and the regression coefficients. The rf methods were optimized by minimizing the RMSE, therefore, the most representative figures of their predictive performance are the ones presenting the squared errors. Given that the information presented by the boxplots of the squared errors (available in the supplementary information) is minimal, we mostly present here results related to the absolute errors of the forecasts. The two approaches yield equivalent results, as shown in the present sections. Additionally, the primary aim of the present study is the optimization of the predictive performance of the RF, therefore, the results of the methods described in Section 2, should be assessed using various methods. Detailed results for all the experiments can be found in the analysis.html file in Appendix A. In Figure 3, we present a typical example in which we test the methods when applied to the 1000 time series simulated from the ARMA(1, 0) model with \(\phi_{1}=0.6\). In this typical example, we observe that all methods perform similarly with respect to the absolute and squared errors. The rf30 method is the best amongst the rf methods, while arifma has the best performance and naive2 has the worst performance. The arifma method is approximately 5% better than the best rf methods. The naive1 and theta methods perform similarly to the best rf methods. The simplest rf method, i.e., rf1, performs well. In fact, its performance is comparable to the best rf methods, i.e., rf2 Oimp, rf25, and rf30. Regarding the use of important variables, introduced with the rf20 imp and the rf50 imp methods, their performance is similar to that of the rf20 and rf50 methods, respectively. When looking at the regression coefficients, the pattern of the performance of the methods is clear. The rf methods are better when using fewer predictor variables. The naive1 method performs better than all methods, followed by the theta and arifma methods. In Figures 4-6, we present the boxplots of the absolute errors for the cases of the ARMA and ARFIMA models. It seems that in most cases, the rf methods with less predictor variables are better  
Figure 3: Barplots of the medians of the absolute errors, medians of the squared errors and regression coefficients when forecasting the 101 st value of 1000 simulated time series from an ARMA(1, 0) model with \(\phi_{1}=0.6\). compared to the rf methods with many predictor variables according to the median of the absolute errors.