## 4 Construction of Time Series Based on the UTC and UTC Rapid Scale
### The Method of Constructing the TS4_1 and TS5_1 Time Series Prepared with the 1-Day Interval
The determined statistical parameters for the values of \(\Delta xb(t)\) for the period of 491 days (\(\Delta xb_{aver}=0.28\) ns and standard deviation \(\sigma=1.94\) ns) and 375 values of \(\Delta xb\) within the range from \(-1\) ns to 1 ns confirm that the adopted PCHIP function is a very good model to describe the values of \(xb\). The second subset completes the TS4_1 or TS5_1 with data groups determined on the basis of the UTCr scale (Figure 8) between days \(t_{n}\) and \(t_{nr}\) (7 days) containing elements with values determined on the basis of relation  
\[x(t)=xa(t)+xbr(t)=UTCr(t)-clock_{k}(t), \tag{10}\]  
for TS4_1 and on the basis of relation (8) for TS5_1. The values of \(xbr(t)\) are published by BIPM on day \(t_{pthr}\) (Figure 8), which can also be the prediction date (\(t_{pred}\)) (\(t_{pred}\geq t_{pthr}\)). A significant advantage of the TS4_1 and TS5_1 is the weekly introduction of new data groups calculated on the basis of the relation (8) or (10) and determination of successive \(xb_{p}(t_{pred})\) prediction values in the same month. At the time of publishing a new \"Circular T\" bulletin, a new data group for the next month \(i+1\) is created (Figure 8), based on the relation (1) for TS4_1 or (3) for TS5_1, which replaces the previous data for the relevant days on the basis of relation (8) or (10). It thus supplements the first data subset for the series TS4_1 or TS5_1 defined with a one-day interval. Figure 9: Set of \(\Delta xb(t)\) values for a period of 491 days. Assessment of the Construction Method of the Proposed Time Series Based on the UTC and UTC Rapid Scales  
The method of constructing the TS4_1 and TS5_1 series has been assessed on the example of UTC(PL) prediction based on the UTC and UTC Rapid scales by means of the GMDH-type NN. The research has been carried out over a period of 17 months, from MJD 56204 to MJD 56699, determining the first prediction in each week for days that end with digits 4 or 9. Input data for the GMDH-type NN are the TS4_1 and TS5_1 time series. In order to compare the quality of determined predictions for two methods of preparing time series in the same period of time, the TS1_1 has been  
1. Comparison of ME, MAE, and MSE\({}_{1}\) error values for three analyzed time series shows that for the TS5_1 time series, the predictions are the least loaded. 2. Predictions of the deviations determined by the GMDH-type NN for the TS4_1 time series are characterized by smaller MSE\({}_{2}\) and MSE\({}_{3}\) error values compared to the TS5_1 time series. This means better anticipation of fluctuations in the prediction variable and greater compliance of the direction of prediction changes in relation to the direction of changes in the predicted value. Hence, 64% of determined values of residuals for the TS4_1 time series are in the range of \(\pm\)5 ns, and 96% of the determined residual values for this time series are in the range of \(\pm\)10 ns. The remaining determined residual values (4%) are within \(\pm\)11 ns. 3. From the group of TS1_1, TS4_1 and TS5_1 time series examined, the best prediction quality results were obtained for TS4_1 time series. This is due to the shorter prediction horizon, that has been achieved by introducing \(xbr(t)\) deviations determined by BIPM into the construction of these time series. 4. The introduction in the second method of constructing the time series of \(xbr(t)\) deviations determined by BIPM also makes it possible for TS5_1 to achieve better-quality predictions than for TS3_1