## 4 MS-FARIMA Algorithm and Orbit Extrapolation Improvement with MSEC
### Ms-Farima  
FARIMA is a prediction model used to handle long-term time-series data. It combines autoregressive moving average (ARMA) models with fractional differencing to acquire long-term dependencies in data [28; 29]. Traditional FARIMA algorithms can only predict single types of continuous data, and their accuracy decreases when an RE plays a role in the data. As the employed ephemeris data are of different types and are discrete, the MS-FARIMA algorithm is expected to improve the error. The MS-FARIMA algorithm design is shown in Figure 5.  
Firstly, we assume multiple discrete data sequences \(\{a[0],\ a[1],\ldots,a[\mathrm{i}-1],a[i],\ldots,a[n]\}\) and calculate \(K\) sets of the error values via \(K\) sets of predicted data sequences and \(K\) sets of accurate data sequences. A fusion operator \(A(\cdot)\) is defined to establish and eliminate the common RE among the multiple data sets.  
Next, after eliminating the RE, a \(FARIMA(p,d,q)\) model is constructed for the multiple sets of discrete data, which can be expressed as  
\[\Phi(B)\Delta^{d}X_{t}=\Theta(B)a_{t} \tag{24}\]  
where \(B\) is a defined delay operator, \(BX_{t}=X_{t-1}\), and \(a_{t}\) is an independent, normally distributed white noise sequence. The coefficient polynomial for the differenced autoregressive term is \(\Phi(B)=1-\phi_{1}B-\cdots-\phi_{p}B^{p}\), where \(p\) is the autoregressive order. The coefficient polynomial for the moving average term is \(\Theta(B)=1-\theta_{1}B-\cdots-\theta_{q}B^{q}\), whereas \(q\) is the moving average order.  
\(\Delta^{d}\) is the fractional differencing operator, which is specifically expressed as follows:  
\[\Delta^{d}=\left(1-B\right)^{d}=\sum_{k=0}^{\infty}\binom{d}{k}(-B)^{k} \tag{25}\]  
where  
\[\binom{d}{k}=\frac{\Gamma(d+1)}{\Gamma(k+1)\Gamma(d-k+1)} \tag{26}\]  
\(\Gamma\) represents the Gamma function, \(d\) is the fractional differencing coefficient, and \(k\) is the summation index variable. When considering long-term dependence, the value of \(d\) lies within \((0,0.5)\).The larger the value, the more pronounced the long-term dependence of the series.  
The specific procedure for the application of the \(\mathit{FARIMA}(p,d,q)\) prediction model to multiple sets of discrete data is shown as follows.  
1. Determine the differencing order \(d\) to achieve stationarity in the data series. For the discrete data series \(a[i]\), set the differencing order \(d\) that satisfies \(\Delta^{d}=(1-B)^{d}\). Through multiple differencing processes, obtain a stationary data series \(a^{\prime}[i]\);
2. Determine the coefficients \(p\) for the autoregressive terms and \(q\) for the moving average terms. Using the above formulas, employ the Akaike information criterion (AIC) to determine the orders \(p\) and \(q\). The values of \(p\) and \(q\) are obtained when the AIC value is minimized;
3. Fit the model parameters using maximum likelihood estimation (MLE). This step results in the final \(\mathit{FARIMA}(p,d,q)\) prediction model;
4. Predict the values for each data set using the derived prediction model. Apply the model to each group of data to obtain the prediction sequences for the respective discrete data sets.  
Figure 5: MS-FARIMA framework.  
The MS-FARIMA algorithm leverages the complementarity among multiple data sets to eliminate the impact of RE on the predictions. Consequently, this algorithm offers higher prediction accuracy compared to traditional FARIMA algorithms.