# Time Scale Control Using Dynamic GMDH Neural Network Forecasting Based on Real Measurement Data
## 2 Materials and Methods
The process of training the GMDH-type neural network is repeated every week after the publication of new \(xbr(t)\) data by the BIPM for the combination of the three discussed parameters for the two prepared time series. Depending on the selection of the neuron's transition function, the network structure consists of one hidden layer and several neurons in this layer (for the neuron's transition function in the form of a linear function), several hidden layers, and a total of several dozen neurons in these layers for the neuron's transition function in the form of a quadratic polynomial. An additional procedure affecting the speed of training and in order to achieve relatively small topology sizes is data pre-processing, which consists of determining the differences between neighboring data, and operations using trigonometric functions applied to these data. The input data prepared using an appropriate method (TS1 time series or TS2 time series) is entered into the GMDH-type neural network. Depending on the time series of the input of the GDMH-type neural network, the value of the difference in the \(xb(t)\) is forecasted. When the data prepared in the form of the TS1 time series is fed as input into the GMDH-type neural network, the forecast for this series is obtained at the output of the GMDH-type neural network (\(x_{1p}(t_{forecast})\)). After taking into account the value of \(xa(t)\) measured on the day of the forecast calculation, i.e., the phase time value of the clock implementing the UTC(PL) scale, the difference forecast \(xbf(t_{forecast})\) is calculated according to the relationship:  
\[xbf(t_{forecast})=x_{1}(t_{forecast})-xa(t_{forecast}). \tag{6}\]  
In the case of input into the GMDH-type neural network prepared in the form of a TS2 time series, the obtained forecast value is simultaneously the forecasted value of the difference in regard to \(xbf(t_{forecast})\). The obtained forecast values for \(xbf(t_{forecast})\) can be the basis for the correction and dynamic control of the Polish Time Scale UTC(PL). After the forecasting process is complete for the entire upcoming week (from Wednesday to the following Wednesday), the forecast results are sent by e-mail to the management and employees of the Time and Length Department at the GUM responsible for controlling the UTC(PL) time scale. Based on the received forecasts, the employee of the Time and Length Department at the GUM decides whether the time scale requires a correction in regard to the femtostepper. An example e-mail message in Polish, with forecasts determined using a GMDH-type neural network for input data prepared on the basis of the TS1 and TS2 time series for the UTC(PL) for the period from 18 September 2024 to 25 September 2024, sent to the GUM on September 18 at 7:40 p.m., is presented in \[xa(t)=UTC_{PL}(t)-clock_{PL}(t), \tag{1}\]0  
The time of sending the message to the GUM is representative of the very efficient process carried out in terms of the author's work on the dynamic forecasting of the Polish Time Scale UTC(PL), using real measurement data. Performing the activities related to the appropriate preparation of the input data for the GMDH-type neural network in the form of the TS1 and TS2 time series, conducting the process of training the GMDH-type neural network, and determining the forecasts for the UTC(PL) for the entire upcoming week takes less than an hour.