## 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  
One of the factors influencing the quality of prediction by means of the GDMH-type NN is the prediction horizon. The delay in the publication of \(xb(t)\) deviations by BIPM means that the first prediction can be made between the 10 th and 20 th day of the month. This has a negative impact on the result of the prediction obtained, and consequently on maintaining the best possible compliance of UTC(k) with UTC. BIPM launched \"A Rapid UTC\" project in 2012, the purpose of which is to accelerate the transmission of information about the divergence of UTC(k) in relation to UTC. Based on the data from clocks and time transfer systems sent daily to BIPM, an auxiliary UTC Rapid (UTCr) scale is calculated for each day every week [17]. When determining the UTC scale, the weights of individual clocks in the current month are assigned with the values that have been determined for the UTC scale in the previous month. Based on the determined UTC scale, the [UTCr-UTC(k)] deviations for the previous week are published every Wednesday on the BIPM ftp server. They are determined according to the relation  
\[xbr(t)=UTCr(t)-UTC_{k}(t). \tag{8}\]  
The publication by BIPM of the \(xbr(t)\) values every Wednesday enables the first prediction of [UTCr-UTC(PL)] to be determined each week, for the MJD day that ends with digit 4 or 9 with a shorter prediction horizon of three to seven days, compared to the prediction horizon for time series built on the basis of the UTC scale. This makes it possible to introduce another method of constructing time series (Figure 8), the values of which are determined with a one-day interval. The basis for the preparation of these time series are the \(xb(t)\) and \(xbr(t)\) deviations determined on the basis of the UTC and UTCr scales, respectively. The proposed time series (TS4_1 and TS5_1) consist of two subsets of elements prepared according to the principle presented in Figure 8. The first subset of the data group, from 1 to \(i\) months, is determined on the basis of relation (1) for the TS4_1 and relation (3) for the TS5_1. The values of \(xb(t)\) appearing in both relations are calculated on the basis of the PCHIP function. The thus determined values of the TS4_1 and TS5_1 are known from day \(t_{0}\) to day \(t_{n}\) for which the last \(x(t_{n})\) value is known before the publication day \(t_{pub}\). From the properties of the PCHIP function, quoted in Section 3.2, it follows that the determined values of \(xb(t)\) falling on MJD days that end with digits 4 and 9 are exactly equal to the values of \(xb(t)\) determined by BIPM. It is very important because the values of predictions for these days are the basis for correcting UTC(k). However, the values of \(xbr(t)\) are determined by BIPM on the basis of the UTC Rapid scale, and therefore for MJD days that end with the digits 4 and 9, they will not always reach the values equal to \(xb(t)\)  
Figure 8: A method of constructing a time series based on the \(xb(t)\) and \(xbr(t)\) values published by BIPM. In order to compare the values of \(xb(t)\) determined on the basis of the PCHIP function and xbr(t), their differences are calculated from the relation  
\[\Delta xb(t)=xb(t)-xbr(t). \tag{9}\]  
Figure 9 shows an exemplary set of \(\Delta\)xb values for the period of 491 days (from MJD 56204 to MJD 56694). In the analyzed period of time, there are 99 MJD days that end with the digits 4 and 9, but only for five MJD days (56274, 56499, 56544, 56559, and 56659) does \(\Delta xb=0\). However, for all 491 days, the values \(\Delta\)xb = 0 are only for 13 days. Figure 9 also shows that the discrepancies between the values of xb and xbr are within the interval of \(-5.7\) ns to 6.2 ns for the entire analyzed period, and for 99 MJD days ending with the digits 4 and 9 in the interval of \(-3.7\) ns to 4.0 ns.