# Time Scale Control Using Dynamic GMDH Neural Network Forecasting Based on Real Measurement Data
## 3 Results
In all the analyzed cases of UTC(PL) time scale forecasting, the residual values are within \(\pm 4.4\) ns. This indicates a very high forecasting quality. 3. A comparison of the \(ME\) (and _MPE_), \(MAE\) (and \(MAPE\)), and \(MSE_{1}\) showed that the forecasts are unbiased in all cases of UTC(PL) scale forecasting. It is observed that the \begin{table}
\begin{tabular}{c c c c c} \hline
**Quality Forecast Measure** & **UTC(PL) TS1** & **UTC(PL)\({}^{\prime}\) TS1** & **UTC(PL) TS2** & **UTC(PL)\({}^{\prime}\) TS2** \\ \hline _Max_ [ns] & \(1.75\) & \(1.75\) & \(1.07\) & \(1.07\) \\ _Min_ [ns] & \(-3.74\) & \(-3.74\) & \(-4.14\) & \(-4.4\) \\ _ME_ [ns] & \(-0.28\) & \(-0.23\) & \(-0.77\) & \(-0.77\) \\ _MPE_ [\%] & \(86.8\) & \(90.9\) & \(91.4\) & \(109\) \\ _MAE_ [ns] & \(1.03\) & \(1.04\) & \(0.96\) & \(0.97\) \\ _MAP_ [\%] & \(125.1\) & \(130.3\) & \(131.1\) & \(147.3\) \\ _MSE_ [ns\({}^{2}\)] & \(1.72\) & \(1.72\) & \(1.85\) & \(2.03\) \\ \(MSE_{1}\) [ns\({}^{2}\)] & \(0.078\) & \(0.051\) & \(0.6\) & \(0.59\) \\ \(MSE_{2}\) [ns\({}^{2}\)] & \(0.006\) & \(0.014\) & \(0.01\) & \(0.005\) \\ \(MSE_{3}\) [ns\({}^{2}\)] & \(1.64\) & \(1.66\) & \(1.24\) & \(1.44\) \\ _RMS_ [ns] & \(1.31\) & \(1.31\) & \(1.36\) & \(1.42\) \\ \hline \end{tabular}
\end{table} 
Table 1: The results of the obtained forecast quality measurements. Figure 5: Residual values obtained for the UTC(PL) time scale for time series TS1 and TS2. residual values are multidirectional, with noticeable differences between the \(ME\) and \(MAE\), as well as low \(MSE_{1}\) error values. 4. For both the TS1 and TS2 time series, the \(MSE_{2}\) components are characterized by very small values. This means that the prediction of the variability of the forecasted values is very precise in relation to the variability of the observed values. 5. In all cases of forecasting the UTC(PL) time scale, regardless of the time series used (TS1 or TS2), the \(MSE_{3}\) error component constitutes the largest share in regard to the overall MSE error. Despite this, the values of the \(MSE_{3}\) components are very small, which indicates high consistency in the direction of the changes in the forecast with the direction of the changes in the forecasted value. This is a consequence of the very high stability of the UTC(PL) time scale. To sum up, the obtained forecast values, calculated residuals (\(r\)), and selected measures of forecast quality clearly indicate a very high quality of the forecasting of the Polish Time Scale UTC(PL). The forecasting procedure, based on a GMDH-type neural network and real measurement data, has proven its effectiveness. For both the TS1 and TS2 time series, the observed differences between the forecasts and the reference values, \(xb(t)\) and \(xbr(t)\) (published by the BIPM), are insignificant. Additionally, a very high agreement between the forecasts for the same day of the MJD (i.e., for the last day of forecasting in a given week and the first day of the new week) is noted. This strongly confirms the excellent forecasting properties of the developed procedure, which uses a GMDH-type neural network and real measurement data.