## 3 Construction of Time Series Based on the UTC Scale
### Assessment of the Construction Method of the Proposed Time Series Based on the UTC Scale  
The method of constructing the proposed TS1_5, TS2_5, TS1_1, TS2_1, and TS3_1 time series has been assessed on the example of predicting the UTC(PL) scale by means of the GMDH-type NN. Prediction has been carried out for all time series in the period of five months, from January to May 2008 (from MJD 54479 to MJD 54599) for days that end with the digits 4 or 9. The prediction horizon is from 10 to 20 days. Input data for the NN for all time series are available from 4 January 2006. Figure 5: Exemplary values of \(x(t)\) calculated from (1) for \(xb(t)\) determined with the 1-day (\(x\_1\)) and 5-day (\(x\_5\)) interval for a period of one month for UTC(PL). The basis for the evaluation of all proposed time series are the residual values, determining the discrepancy between the predicted \(xb_{p}(t_{pred})\) deviation and \(xb(t_{pred})\) deviation published by BIPM on the same prediction day (\(t_{pred}\)), which are calculated from the relation  
\[r(t_{pred})=xb(t_{pred})-xb_{p}(t_{pred}), \tag{7}\]  
and selected measures of prediction quality determined on their basis [24; 25]. The calculated values of the residuals for the five analyzed time series are presented in Figure 6. The values of the residuals for each time series are included in the following intervals:  
* from \(-\)5.05 ns to 6.95 ns for TS1_5,
* from \(-\)4.63 ns to 7.74 ns for TS2_5,
* from \(-\)0.60 ns to 3.08 ns for TS1_1,
* from \(-\)0.07 ns to 0.40 ns for TS2_1,
* from \(-\)15.91 ns to 6.89 ns for TS3_1. Table 1 presents selected measures of prediction quality [24; 25]: mean error (ME), absolute mean error (MAE), mean square error (MSE) with its components (MSE\({}_{1}\), MSE\({}_{2}\), MSE\({}_{3}\)) and the root of mean square error (RMSE). Based on the presented results (Figure 6) and the calculated values of prediction quality measures (Table 1), it can be concluded that: \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Measure of Quality of Prediction** & **TS1\_5** & **TS2\_5** & **TS1\_1** & **TS2\_1** & **TS3\_1** \\ \hline \(ME\) [ns] & 0.45 & 1.61 & 0.55 & 0.1 & \(-\)3.5 \\ \(MAE\) [ns] & 3.32 & 3.52 & 0.8 & 0.13 & 6.41 \\ \(MSE\) [ns2] & 16 & 19 & 2.0 & 0.04 & 74 \\ \(MSE_{1}\) [ns2] & 0.21 & 2.56 & 0.31 & 0.01 & 12.8 \\ \(MSE_{2}\) [ns2] & 5.15 & 6.73 & 0.01 & 0.01 & 2.25 \\ \(MSE_{3}\) [ns2] & 10.9 & 9.97 & 1.67 & 0.02 & 59.4 \\ \(RMSE\) [ns] & 4.04 & 4.39 & 1.41 & 0.19 & 8.6 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Obtained values of selected quality prediction measures for the TS1_5, TS2_5, TS1_1, TS2_1, and TS3_1. Figure 6: Obtained \(r\) residuals values for predicting \(xb_{p}(t)\) for the TS1_5, TS2_5, TS1_1, TS2_1, TS3_1. 1. The smallest residual values have been obtained for the TS1_1 and TS2_1. This is due to the application of the PCHIP interpolation function, which makes it possible to prepare data for the GMDH-type NN with a one-day interval. 2. Comparison of ME, MAE, and MSE\({}_{1}\) error values for all time series shows that for the TS3_1 time series the predictions are the most loaded. 3. The deviations predictions determined for TS1_1 and TS2_1 are characterized by small values of the MSE\({}_{2}\) error component. This allows a good prediction of the variability of the predicted values in relation to the variability of the values observed for both time series, which results from the density of training data in a given month, thanks to application of the PCHIP function. 4. In the case of results obtained for TS2_1 time series, the smallest value of the MSE\({}_{3}\) error component has been obtained in relation to the other time series. This is due to the decomposition of the TS1_1 time series, which resulted in a better compliance of the direction of changes in the prediction and the direction of changes in the predicted value for the TS2_1 time series. 5.