## 3 Construction of Time Series Based on the UTC Scale  
### The Method of Constructing the TS1_5 and TS2_5 Time Series Prepared with the 5-Day Interval  
An important problem of applying NNs for predicting the [UTC-UTC(k)] deviations is the proper preparation of training and testing data sets. Compared to the applied analytical models [3; 4; 5; 6], NNs for predicting deviations require a much larger number of time series elements [22] in the training and testing process to achieve the best possible quality of predicting the [UTC-UTC(k)] deviations. The paper [23] presents the results of the research that determined the minimum number of time series elements, ensuring the required quality of UTC(PL) prediction. Predicting the values of [UTC-UTC(k)] deviations by means of NNs for the selected UTC(k) scale is based on a time series \(x(t)\), the values of which (Figure 2) are determined at times \(t_{0}\) to \(t_{n}\), where \(t_{0}\) is the day of the first known value of the time series, and \(t_{n}\) is the day of the last known value of the time series in the month preceding the prediction. The values of the time series elements for the last month preceding the prediction designation may be determined not earlier than on day \(t_{pub}\). This is the date when BIPM publishes the [UTC-UTC(k)] deviation values for previous month. They are designated for MJD days that end with digits 4 and 9, i.e., with a 5-day interval (\(t_{n}-t_{n-1}\) = 5). Figure 1: GMDH Shell solver application view. The time series \(x(t)\) characterizes the time instability of the clock implementing UTC(k) with respect to UTC. Its values are calculated based on the relation  
\[x(t)=xa(t)+xb(t)=UTC(t)-clock_{k}(t), \tag{1}\]  
where  
\(xa(t)\)**--**historical results of phase time measurements between 1 pps (pulse per second) signals from UTC(k) and the clock realizing UTC(k), determined with a one-day interval according to the relationship:  
\[xa(t)=UTC_{k}(t)-clock_{k}(t), \tag{2}\]  
\(xb(t)\)**--**values of [UTC-UTC(k)] deviations determined by BIPM for UTC(k) in relation to UTC, which are designated as MJD days that end in digits 4 and 9 according to the formula  
\[xb(t)=UTC(t)-UTC_{k}(t). \tag{3}\]  
The time \(t_{prad}\) is the day when the deviation for UTC(k) is predicted by the GMDH-type NN. Striving to achieve the highest possible accuracy of UTC(k) prediction also forces the prediction process to keep the prediction horizon as low as possible, i.e., the time interval \(t_{prad}-t_{n}\), which depends on the publication date (\(t_{pub}\)) of the \(xb(t)\) deviation values. Taking into account the time \(t_{pub}\) and the need to make a prediction of \(xb_{prad}(t_{prad})\) deviation for the next MJD day that ends with digits 4 or 9, the prediction of deviation is carried out between the 10 th and 20 th day of the following month. Establishing predictions for the following days (4 and 9) and in the following months for the same training data will cause, regardless of the analytical and NN methods used for prediction, a significant increase in prediction errors. Practically, it means that one value of the predicted \(xb_{prad}(t_{prad})\) deviation is determined in a given month. However, to determine the prediction in the next month, it is necessary to supplement the \(x(t)\) time series with a new group of data \(i+1\) (Figure 3), taking into account the new values of \(xb(t)\) deviations from the month preceding the prediction, published by BIPM. Such action means the necessity to re-run the full process of determining the new prediction value for the next month. The basis for the preparation of training data for the NN in the form of the TS1_5 time series is the relationship (1). These data are determined with a 5-day interval. The TS1_5  
Figure 3: A method of constructing a time series based on the \(xb(t)\) values published by BIPM. Figure 2: Illustration of time series \(x(t)\). time series is the first group of data on the basis of which GMDH-type NN training and prediction of \(x_{pred}\) values has been performed.