# A Hybrid Forecasting Model for Self-Similar Traffic in LEO Mega-Constellation Networks
## 4 Network Traffic Forecasting Based on ARIMA-BPNN
### ARIMA-Based IMF Forecasting
#### 4.1.2 Order Determination and Parameter Estimation for the ARIMA  
Multi-order IMFs are adopted as the input of the ARIMA in this paper. In traditional ARIMA(\(p\), \(d\), \(q\)), the autocorrelation coefficient (ACF) and partial autocorrelation coefficient (PACF) are commonly utilized to determine the autoregressive order \(p\) and moving-average order \(q\). However, considering satellite network traffic has the properties of self-similarity and LRD, there is commonly a slight fluctuation near zero, making theoretical truncation difficult to obtain for ACF or PACF. To address this problem, this paper takes the Akaike information criterion (AIC) as the objective function and twice the standard deviation, i.e., \(2\sigma_{ACF}\) and \(2\sigma_{PACF}\), as the threshold. An adaptive fixed-order optimization search operator (AFOS) is designed to reduce the interpretation error. The pseudo code of ARIMA\((p,d,q)\) is shown in Algorithm 1. ```
1 Input:\(IMF_{i,d}\)order of IMF:\(n\)
2 Output: ARIMA(\(p\), \(d\), \(q\)), \(\zeta_{i}(t)\), \(IMF_{i}^{\prime}(t)\)
3 for\(i=1;i<n;i++\)do
4\(IMF_{i,d}(t)\leftarrow\) IMF\({}_{i}(t),d\gets 0\)
5 ADF test for IMF\({}_{i,d}(t)\)
6\(ACF_{i,d}(s),2\sigma_{ACF},PAC_{i,d}(s),2\sigma_{PACF},s\in\{1,2,3,4,5\}\)
7\(s_{1}\gets s_{p,\max}\), \(s_{2}\gets s_{q,\max}\)
8 if\(s_{2}=5\)then
9\(q_{\max}\gets 0\)
10 else
11\(q_{\max}=s_{q,\max}-1\)
12 end if
13 if\(s_{1}=5\)then
14\(p_{\max}\gets 0\)
15 else
16\(p_{\max}=s_{p,\max}-1\)
17 end if
18\(ARIMA(p_{i},d_{i},q_{i}),p_{i}\in\{1,2,\cdots,p_{\max}\},q_{i}\in\{1,2,\cdots,q_{\max}\}\)
19\(\phi_{i},i\in\{1,2,\cdots,p\};\theta_{j},i\in\{1,2,\cdots,q\};\sigma_{x,p,d_{ q}}^{2}\)
20\(AIC_{p,d_{q}}\gets T\cdot\ln\left(\sigma_{x,p,d_{q}}^{2}\right)+2(p+q+1)\)
21\(ARIMA(p,d,q)\gets ARIMA(p_{i},d_{i},q_{i})\big{|}_{\min\{AIC\}}\)
22\(IMF_{i}^{\prime}(t)\leftarrow\) d - order differential reduction of \(IMF_{i,d}^{\prime}(t)\)
23\(\zeta_{i}(t)\gets IMF_{i}^{\prime}(t)-IMF_{i}^{\prime}(t)\)
24 end
```  
**Algorithm 1**Order Determination and Parameter Estimation for ARIMA  
The modeling steps of ARIMA(\(p\), \(d\), \(q\)) are as follows:  
**Step 1:** Set the order of IMF as \(I\) and \(i=1\). Set the total number of IMFs as \(I\). **Step 2:** Set the difference order \(d=0\). \(IMF_{i,d}(t)=IMF_{i}(t)\). Take \(IMF_{i,d}(t)\) as the input sequence for modeling. **Step 3:** Perform the ADF test for \(IMF_{i,d}(t)\) to determine whether the input sequence is a stationary sequence. If yes, perform step 5, otherwise, perform step 4. **Step 4:** Perform the difference operation on \(IMF_{i,d}(t)\). \(d=d+1\). Assign the result of the difference operation to \(IMF_{i,d}(t)\). Perform step 5. **Step 5:** Perform the white noise test based on LB statistics to determine whether the input noise is white noise. If yes, perform step 17, otherwise, perform step 6. **Step 6:** Calculate the autocorrelation coefficient \(ACF_{i,d}(s)\), corresponding standard deviation \(2\sigma_{ACF}\), partial autocorrelation coefficient \(PACF_{i,d}(s)\) and corresponding standard deviation \(2\sigma_{PACF}\). \(s\in\{1,2,3,4,5\}\). **Step 7:** s turns from 1 to 5. Judge whether \(ACF_{i,d}(s)\geq 2\sigma_{ACF}\) and \(PACF_{i,d}(s)\geq 2\sigma_{PACF}\) hold. **Step 8:** Judge whether \(s_{q,\max}=5\) holds. If yes, \(q_{\max}=0\), otherwise, \(q_{\max}=s_{q,\max}-1\). **Step 9:** Judge whether \(s_{p,\max}=5\) holds. If yes, \(p_{\max}=0\), otherwise, \(p_{\max}=s_{p,\max}-1\). **Step 10:** Permutation and combination (\(p\), \(q\)) to construct ARIMA(\(p\), \(d\), \(q\)) models. **Step 11:** Calculate parameters in ARIMA(\(p\), \(d\), \(q\)) by maximum likelihood estimation (MLE). **Step 12:** Calculate the \(AIC_{p,d,q}\) of different ARIMA(\(p\), \(d\), \(q\)) using the AIC criterion:  
\[AIC_{p,d,q}=T\ln\left(\sigma_{x,p,d,q}^{2}\right)+2(p+q+1) \tag{11}\]  
**Step 13:** Take the ARIMA(\(p\), \(d\), \(q\)) with the minimum \(AIC_{p,d,q}\) as the optimal model. **Step 14:** Calculate the fitting sequence \(IMF_{i,d}^{Fit}(t)\) of \(IMF_{i,d}(t)\) by the optimal model.