# Time Scale Control Using Dynamic GMDH Neural Network Forecasting Based on Real Measurement Data
## 3 Results  
The presented research focuses on the dynamic forecasting of the Polish Time Scale UTC(PL). A GMDH-type neural network is used to implement the forecasts, and the analysis covers measurement data obtained in the period from 1 January to 31 December 2024 (MJD 60310-MJD 60675). The assessment of the quality of the UTC(PL) forecasting is carried out by analyzing the discrepancies between the forecasted values of the \(xbf(t)\) and the reference values of the differences in relation to the \(xbf(t)\), published by the BIPM for  
Figure 3: Screenshot of an example e-mail sent to the GUM with forecasts determined using a GMDH-type neural network and input data prepared based on TS1 and TS2 time series for UTC(PL) for the period 18–25 September 2024.  
the same forecast day (\(t_{forecast}\)). For this purpose, the residuum (\(r\)) values are determined, calculated according to the relationship:  
\[r(t_{forecast})=xb(t_{forecast})-xbf(t_{forecast}). \tag{7}\]  
In addition, the analysis of the \(xb(t)\) difference values for the UTC(PL) also allows its quality to be assessed. The closer the \(xb(t)\) values published in the \"Circular T\" bulletin by the BIPM are to zero, the better the accuracy and stability of a given time scale. The BIPM classifies UTC(k) time scales into the best group if the \(xb(t)\) values determined in the \"Circular T\" bulletin do not exceed \(\pm 10\) ns. The next two groups of scales are defined by values not exceeding \(\pm 20\) ns and \(\pm 50\) ns. Figure 4 illustrates the forecasting results for the TS1 and TS2 time series in the period from MJD 60310 to MJD 60675. It is important that for each day of the MJD (and specifically for each Wednesday) two forecasts are included. This is due to the specificity of the UTC(PL) forecasting process, in which weekly forecasts sent to the GUM include both the current and following Wednesday. In turn, Figure 5 presents the residual values for the MJD days ending with digits 4 and 9, for the same period. This specific selection of days is aimed at providing a direct comparison of the obtained forecast values with the values of the \(xb(t)\) differences, which are published by the BIPM in the \"Circular T\" bulletin precisely for the MJD days ending with 4 and 9. The analysis of the \(xb(t)\) values for the UTC(PL) in the presented period (Figure 4) clearly indicates the very high quality of the Polish UTC(PL) Time Scale. The values of the \(xb(t)\) differences, determined by the BIPM for the UTC(PL) on the MJD days ending with digits 4 and 9, do not exceed the limit of \(\pm 4.4\) ns. This proves the excellent quality of the UTC(PL) control, comparable to the leading UTC(k) time scales, such as the English UTC(NPL), the French UTC(OP), or the German UTC(PTB). It is worth emphasizing that the mentioned scales use primary frequency standards in the form of a caesium fountain to control clocks implementing a given UTC(k), which additionally highlights the achievements of the UTC(PL). Figure 4: Forecast values obtained using a GMDH-type neural network for the UTC(PL) scale for time series TS1 and TS2 compared with the values of \(xb(t)\) and \(xbf(t)\). Based on the calculated residual values (\(r\)), selected forecast quality measures [23] are determined, which are presented in Table 1. They included the mean error (\(ME\)) with its percentage value (_MPE_), the absolute mean error (\(MAE\)) with its percentage value (_MAPE_), the mean square error (\(MSE\)) with its components (\(MSE_{1}\), \(MSE_{2}\), \(MSE_{3}\)), and the root mean square error (\(RMSE\)). The assessment of forecast quality \(xbf(t_{\textit{ forecast}})\) is performed based on these residuals and selected forecast quality measures [23]. Based on the results presented in Figures 3 and 4 and in Table 1, the following conclusions were drawn:  
1. The best forecasting quality is obtained for the UTC(PL) TS1 case, which is the result from the comparison of all the forecast quality measures and residual values. 2.