# Linear Algorithms for Radioelectric Spectrum Forecast
## 3 Case Study and Experiment Procedure
### Design of GARCH Algorithm  
When analyzing in detail the large amount of acquired information, the existence of standard deviation was observed; therefore the GARCH algorithm was used to forecast the behavior of measured series. Stochastic models ARIMA and SARIMA are methods for univariate modeling. The main difference among former models and GARCH model lies in the constant variance assumption. Even though for the developed algorithm there is stationarity in original signal from Monday to Friday, for this case the fifth difference is developed because there is a greater degree of stationarity. \begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline \(p\) & \(d\) & \(q\) & \(P\) & \(D\) & Q & \begin{tabular}{c} **Low Occupancy** \\ **Channel AIC** \\ \end{tabular}  & \ \begin{tabular}{c} **Medium Occupancy** \\ **Channel AIC** \\ \end{tabular}  & \begin{tabular}{c} **High Occupancy** \\ **Channel AIC** \\ \end{tabular}  \\ \hline
1 & 0 & 5 & 0 & 0 & 1 & \(-8.24\) & \(-30.6\) & \(-50.82\) \\
1 & 0 & 5 & 1 & 0 & 0 & \(-8.3\) & \(-32.7\) & \(-51.7\) \\
1 & 1 & 5 & 0 & 0 & 1 & \(-14.1\) & \(-46.9\) & \(-76.2\) \\
1 & 0 & 5 & 1 & 0 & 1 & \(-8 In Figure 9 the difference for each channel is presented. Channel measurements are converted into returns by logarithmic transformation. The logarithmic returns are defined in Equation (10),  
\[r_{t}=ln\frac{P_{t}}{P_{t-1}} \tag{10}\]  
where \(P_{t}\) is power value in time \(t\) and \(P_{t-1}\) is power value in time \(t_{t-1}\). Now we present a formal statistical test in order to establish the presence of ARCH effects in the data and correlation. H = 0 implies that there exist no significant correlation as well as H = 1 indicates that there exists a significant correlation. In Tables 3 and 4, all the \(p\) values show that Ljung-Box-Pierce Q-Test and Engle ARCH Test in lag 10, 15 and 20 are significant, revealing the presence of ARCH effects (heteroskedasticity), indicating that GARCH modeling is appropriate. The GARCH model selection for each channel was done by fulfilling \(\alpha_{i}+\beta_{i}<1\) criterion, so the model is stationary, and then taking into account the more proximate values to zero of MAE, MAPE and SMAPE from Tables 5-7. Therefore, the selected models for low, medium and high channel are GARCH(2,2), GARCH(0,2) and GARCH(0,1), respectively. Parameters for low channel model were estimated and are presented in Table 8. GARCH(2,2), where \(\alpha_{1}+\alpha_{2}+\beta_{1}+\beta_{2}<1\) is fulfilled. Thus, the model according to Table 8 is presented in Equations (11) and (12),  
\[r_{t}=-96.112+\epsilon_{t} \tag{11}\]  
\[\sigma_{t}^{2}=0.003516+0.098255\sigma_{t-1}^{2}+0.90062\sigma_{t-2}^{2}+0.000 29573\epsilon_{t-1}^{2} \tag{12}\]  
For medium channel, GARCH(0,2), model values presented in Table 9 are estimated. Therefore, Equations (13) and (14) are obtained,  
\[r_{t}=-95.061+\epsilon_{t} \tag{13}\]  
\[\sigma_{t}^{2}=5+0.085692\epsilon_{t-1}^{2}+0.088298\epsilon_{t-2}^{2} \tag{14}\]  
For high channel, GARCH(0,1), the following parameters were obtained, as shown in Table 10.  
Then the model is described in Equations (15) and (16),  
\[r_{t}=-94.585+\epsilon_{t} \tag{15}\]  
\[\sigma_{t}^{2}=5+0.86058\epsilon_{t-1}^{2} \tag{16}\] \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Parameter** & **Estimated Value** & **Standard Error** & **\(t\) Value** \\ \hline \(\mu\) & \(-94.585\) & 0.0026236 & \(-36\),051.8702 \\ \(\alpha_{0}\) & 5 & 0.015341 & 325.9324 \\ ARCH(1) & 0.86058 & 0.0044771 & 192.2169 \\ \hline \hline \end{tabular}
\end{table} 
Table 10: Parameters estimation for high channel model. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Parameter** & **Estimated Value** & **Standard Error** & **\(t\) Value** \\ \hline \(\mu\) & \(-96.112\) & 0.0019308 & \(-49\),778.3308 \\ \(\alpha_{0}\) & 0.003516 & 0.00041447 & 8.4833 \\ GARCH(1) & 0.098255 & 0.19212 & 0.5114 \\ GARCH(2) & 0.90062 & 0.19201 & 4.6905 \\ ARCH(1) & 0.00029573 & 0.00018772 & 1.5753 \\ ARCH(2) & 0 & 0.00020886 & 0 \\ \hline \hline \end{tabular}
\end{table}