# A New BDS-2 Satellite Clock Bias Prediction Algorithm with an Improved Exponential Smoothing Method
## 2 Theory of Improved Exponential Smoothing Method Prediction
### Exponential Smoothing Prediction Model  
The ES model is a predicting method that is a weighted average of the clock bias time series observed in the past. The basic idea is that each time series has its inherent characteristics, that is, there is a certain basic mathematical model. However, the time series of actual observed clock bias not only embodies the above basic ideas but also reflects the characteristics of its random variation. In general, the impact of historical clock bias time series on future clock bias decreases with time gradually. Therefore, a more practical approach should be that the ES value of clock bias at the time \(t\) is equal to the weighted average of the actual observed value of clock bias at the time \(t\) and the ES value of clock bias at the time \(t-1\). In other words, the influence of the maximum and minimum is eliminated in the historical clock bias data to obtain the smoothed value of the clock bias time series. These smoothed clock biases are used as the forecast value of the clock bias at the future time.  
In the whole forecasting process, the ES model can continuously correct new forecast values using forecast errors, i.e., the principle of \"error feedback\" can be utilized to correct the sequences of the forecasted clock bias, to boost the accuracy of the forecasted values of clock bias. The detailed process of the ES model is calculated as follows.  
The prediction model for the ES3 method can be written as [37]  
\[F_{t+m}=A_{t}+B_{t}\cdot m+\frac{1}{2}\cdot C_{t}\cdot m^{2} \tag{9}\]  
where,  
\[\left\{\begin{array}{l}A_{t}=3\cdot S_{t}^{\prime}-3\cdot S_{t}^{\prime\prime} +S_{t}^{\prime\prime\prime}\\ B_{t}=\frac{\alpha^{2}\cdot\left[\left(6\cdot 5\cdot 4\right)\cdot S_{t}^{\prime}- \left(10-8\cdot 4\right)\cdot S_{t}^{\prime\prime}-\left(4-3\cdot 4\right)\cdot S_{t}^{ \prime\prime\prime}\right]}{2\cdot\left(1-\alpha\right)^{2}}\\ C_{t}=\frac{\alpha^{2}\cdot\left(S_{t}^{\prime}-2\cdot S_{t}^{\prime\prime}+S _{t}^{\prime\prime}\right)}{\left(1-\alpha\right)^{2}}\\ m=1,2,\cdots\end{array}\right.\]  
and \(F_{t+m}\) represents the forecast value of the observed clock bias sequences at the time \(t+m\).  
Since the ES method is a recursive calculation formula, for the convenience of calculation, it is usually expressed as follows:  
\[S_{t}^{\prime}=S_{t}^{\prime\prime}=S_{t}^{\prime\prime\prime}=x_{1} \tag{10}\]  
The most critical problem of the ES prediction model is how to determine an appropriate smoothing coefficient \(\alpha\) accurately. Since different \(\alpha\) will have a larger influence on the forecast result, the value of \(\alpha\) should be chosen between the interval \((0,1)\) according to the specific situation of the clock bias sequences. The value of \(\alpha\) reflects the degree of correction. The larger the value of \(\alpha\), the larger the degree of correction; the smaller the value of \(\alpha\), the smaller the degree of correction. Generally, when there is a big random fluctuation in the time series data of the clock bias, a big \(\alpha\) should be selected to keep up with the recent changes in the clock bias quickly. When the time series of the clock bias is relatively stable, a smaller \(\alpha\) should be selected. To address this conundrum that the smoothing coefficient \(\alpha\) is hard to determine in the ES forecast model, the principle of the TNTF is adopted to optimize the smoothing coefficient \(\alpha\) to achieve a better forecasting results in this contribution.  
The ES1 prediction model is applicable to time series with less pronounced change trends. The ES2 prediction model is suitable for time series like linear variation trends. The ES3 prediction model is mainly applied to time series with quasi-nonlinear variation trends. Considering the characteristics of clock bias time series changes, we analyze the forecasting performance of the ES2 model, the ES3 model, and the forecasting models using different modeling strategies for SCBs.