A comparative study on different machine learning approaches with periodic items for the forecasting of GPS satellites clock bias

A comparative study on different machine learning approaches with periodic items for the forecasting of GPS satellites clock bias

Data description

The study utilized GPS data collected between January 1 and January 10, 2024, with a 5-minute sampling interval and precise clock bias information sourced from the IGS data center for experimental purposes. To maintain data continuity, 6 satellites with consistent data were selected due to missing data from certain satellites.

There are five types of clocks in Table 1 for GPS satellites, including Block IIR Rb, Block IIR-M Rb, Block III-A Rb, Block IIF Rb, and Block IIF Cs.

Table 1 Types of GPS satellite clock during the data collection period (rb and cs represent Rubidium and Cesium, respectively).

Data analysis and modeling

The first step to getting a better understanding of the satellite clock bias data, the satellite clock bias of G-1 since January 1, 2024 until January 10, 2024, is displayed in Fig. 3.

Fig. 3
figure 3

Satellite clock bias sequence of G-1.

In polynomial function-based modeling, the 2-order polynomial functions are used to model the satellite clock error series and get the trend term of the satellite clock difference. The trend term of the clock bias signal is shown in Fig. 4a. By removing the trend term of the satellite clock difference, the periodic term of the satellite clock difference can be obtained.

To verify the nonstationary and time-dependent behavior of satellite clock bias time series data, the 2D plot for the periodic term of the original acquired signal is shown in the time domain and frequency domain respectively. The periodic term of the clock bias signal is shown in Fig. 4b. The power spectra of the periodic term after the Fast Fourier Transform (FFT) are shown in Fig. 4c. The high-frequency part for the periodic term of the clock bias signal is negligible for being too weak, therefore, the spectra are truncated, and only the low-frequency signal from 0 to 0.002 Hz is kept. Also, it can be seen from the spectra diagram that the single difference series has obvious periodicity. The results show that the polynomial function is not suitable for modeling the long-term satellite clock series.

Fig. 4
figure 4

Trend term and periodic term of satellite clock error series for G-1 (a) 2D plot in the time domain of trend term (b) 2D plot in the time domain of periodic term (c) 2D plot in the frequency domain of the periodic term.

Four deep learning models will be used to handle the periodic Items of satellite clock bias time-series datasets. Indeed, LSTM and its improved versions GRU possesses a desirable feature that permits tracking of the trend of satellite clock bias and explains the most variability in the data. Thus, the efficiency of the GRU model for satellite clock bias forecasting is promising and manifested. Also, It should be noted that the WNN model is very efficient in extracting relevant information from more lengthy time-dependent data.

Forecasting results

In this study, we focus on the univariate time-series data of satellite clock bias from six considered satellites G1, G2, G4, G5, G10, and G24. Now, the forecasting quality of the previously designed models will be verified using unseen testing data. The testing data consists of satellite clock bias recorded in the six considered satellites on January 10 2024.

In polynomial function-based modeling, the 2 order polynomial functions are used to model the satellite clock error series. The results show that the 2-order polynomial function is not suitable for modeling the long-term satellite clock series. The satellite clock error prediction for the six considered satellites is conducted using a quadratic polynomial model with periodic terms(QPMwPT). The predicted values for the six considered satellites are then compared with the IGS precision clock difference, the quadratic polynomial model with periodic terms demonstrated MAE values of 0.2181 ns for the six satellites clock bias data, and the results are presented in Table 2. For illustrative purposes, the predicted sequences of the quadratic polynomial model with periodic terms are compared with the quadratic polynomial model. The average prediction accuracy of forecasting has improved by approximately 13.51% during 1-day forecasting.

Table 2 The RMSE and MAE values for the six satellites using prediction models.
Fig. 5
figure 5

(af) Forecasting results of the six consider satellites using quadratic polynomial model with periodic terms.

Figure 5 illustrates the forecasted satellite clock bias using a quadratic polynomial model with periodic terms from January 1st to January 9, 2024. These algorithms offer promising predictions of the satellite clock bias for the six satellites under consideration.

This section will compare the forecasting performances of several neural networks time series forecasting models namely BPNN, WNN, LSTM and GRU. First, each model is trained with the training measurements. Then, we forecast each variable using the trained models for the unseen testing dataset. The training data consist of univariate time series data of satellite clock bias from January 1, 2024, through January 9, 2024. Here, the challenge in this study is to investigate the performance of these four deep learning models in the presence of relatively small data. Parameters of the constructed BPNN, WNN, LSTM and GRU models based on training datasets are presented in Table 3.

Table 3 Parameter settings of the studied approaches.

As for the new satellite clock bias, we applied the four deep learning models to forecast the newly single difference series. Figure 6a–f shows the forecasting results of the six consider satellites using the four deep learning models. These models show good forecasting performance in the testing stage.

Fig. 6
figure 6

(af) Forecasting results of the six consider satellites.

To more clearly compare the forecasting results, Fig. 7 displays the forecasting results of the satellite clock bias on the six considered satellites using BPNN, WNN, LSTM and GRU based forecasting models. The forecasting results of the BPNN, WNN, LSTM, and GRU based models follow the overall trend of the recorded satellite clock bias, indicating that the NNbased models can capture the time-dependent in the satellite clock bias data. This fact is due to their high capability to model non-linear and time-dependent data. Furthermore, the results reveal the potential in using deep learning models even in the presence of small data.

Fig. 7
figure 7

(af) Forecasting results of the satellite clock bias on the six considered satellites.

The performance of BPNN, WNN, LSTM, and GRU models in terms of RMSE and MAE when applied to the satellite clock bias data from three satellites are summarized in Table 2. The predicted sequences generated by these four machine learning models are compared against those produced by a Quadratic Polynomial (QP) model. Notably, the average forecasting accuracy has improved significantly, with enhancements of approximately 39.45%, 57.57%, 27.28%, and 29.14% for the respective models during one-day forecasting. It can be easily seen that the WNN model outperformed the other models by providing good forecasting performance with lower RMSE and MAE values colorer to 1 explaining most of the variance ithe data, for all data stations. For illustration, the WNN model achieved MAE values of 0.0204, 0.1493, 0.1176, 0.1474, 0.1002, and 0.0162 ns for satellite clock bias data from G1, G2, G4, G5, G10, and the G24, respectively. The WNN can capture almost all variability in data and provide more accurate forecasting in comparison to the other models. All other models perform moderate forecasting performance in terms of RMSE and MAE, and show poor performance in terms of explained variance. This is maybe due to their need for more data in the training to capture the dynamics of satellite clock bias. The worst model is BPNN because of its simplicity and followed by its extended versions of WNN, LSTM, and GRU models.

Overall, this study provided a comparison between deep learning models in forecasting the satellite clock bias from six different satellites. This work highlights the potential of BPNN, WNN, LSTM and GRU methods to be used for forecasting satellite clock bias, even when applied to small satellite clock bias training data sets. Essentially, these deep learning models are able to capture time-variant properties and relevant patterns of past data and forecast the future tendency of satellite clock bias time-series data. The forecasting results show the superiority of the WNN model by achieving higher accuracy compared to the other models for one-step forecasting. This can be attributed to the great capacity of the WNN in capturing process nonlinearity and dealing with small times series data. Compared to the other herein forecasting models, the WNN presents the advantage of the lower dimensionality because of using reduced hidden units number. This last permits the extraction of the discriminative features.

Generally speaking, there is no obvious answer here that one of them is better than the other, but for this application, the WNN model outperforms the other herein forecasting approaches used for the satellite clock bias prediction.

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