# A Hybrid Forecasting Model for Self-Similar Traffic in LEO Mega-Constellation Networks
## 3 Self-Similarity Analysis of Satellite Network Traffic
### EMD-Based Satellite Network Traffic Decomposition  
Self-similarity results in the long-range dependence (LRD) property of LEO mega-constellation network traffic, i.e., the autocorrelation function of network traffic decreases as a hyperbolic function with increasing time intervals. Although the existing neural-network-based traffic forecasting models are capable of indicating the LRD property, the computation complexity of these models is too considerable to be utilized in LEO mega-constellation networks, where on-board computation and storage resources are extremely limited. Compared with neural network models, statistics-based models outperform them in terms computation complexity, whereas these models can only indicate linear laws of network traffic. The computational complexity can be reduced if the LRD traffic can be  
Figure 3: Traffic superposition of ON/OFF sources. converted into linear components, from which the ARIMA can be adopted to mine the linear laws of the network traffic. Commonly employed decomposition methods include wavelet [29] and empirical mode decomposition (EMD) [30]. The wavelet's basis is of vital importance to the performance of the wavelet, which limits its application. In contrast, EMD can effectively decompose non-linear and non-stationary data into multi-order IMFs, which can better reflect the inherent characteristics of network traffic [31]. Therefore, EMD is utilized to decompose satellite network traffic in this paper. EMD can decompose LRD traffic into multiple single frequency components and residuals, i.e.,  
\[y(t)=\sum_{i}IMF_{i}+r(t) \tag{3}\]  
where \(y(t)\) denotes network traffic, \(IMF_{i}\) denotes the \(i\)-th components obtained by decomposition and \(r(t)\) denotes the residual. Thus, satellite network traffic with LRD properties is converted into several single frequency components, i.e., IMFs, that can be effectively processed by linear models. Meanwhile, the computational complexity is extremely reduced. The essence of EMD is to determine the basic oscillation pattern of valid signals [32]. Commonly, the IMFs obtained via EMD satisfy the following conditions. * The number of signal polarization points \(\psi\) and the number of zero points \(\zeta\) satisfy \(|\psi-\zeta|\leq 1\). * The mean value of the envelope defined by the local maximum and local minimum of the signal is zero. The first condition specifies that the signal forms of IMFs are consistent with the traditional narrow-band smooth Gaussian process, which can be characterized as  
\[y(t)=a(t)e^{i\phi(t)} \tag{4}\]  
where \(a(t)\) and \(\phi(t)\) are the envelope and phase of the signal, respectively. The second condition ensures the symmetry of IMFs. The specific steps of EMD are shown in Table 2.  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Shifting** & **IMF\({}_{1}\)** & **IMF\({}_{2}\)** & **IMF\({}_{i}\)** & **IMF\({}_{n}\)** \\ \hline \(x[n]_{|L}\) & \(r_{1}[n]_{|L}=x[n]-c_{1}[n]\) &... & \(r_{n-1}[n]_{|L}=r_{n-2}[n]-c_{n-1}[n]\) \\ \(P=CS\Big{(}x[n]_{minimal}\Big{)}_{|L}\) & \(P=CS\Big{(}r_{1}[n]_{minimal}\Big{)}_{|L}\) & \(P=CS\Big{(}r_{n-1}[n]_{minimal}\Big{)}_{|L}\) & \(P=CS\Big{(}r_{n-1}[n]_{minimal}\Big{)}_{|L}\) \\ \(Q=CS\Big{(}x[n]_{maximal}\Big{)}_{|L}\) & \(Q=CS\Big{(}r_{1}[n]_{maximal}\Big{)}_{|L}\) & \(Q=CS\Big{(}r_{n-1}[n]_{maximal}\Big{)}_{|L}\) & \(Q=CS\Big{(}r_{n-1}[n]_{maximal}\Big{)}_{|L}\) \\ \(m_{11}[n]=(P+Q)/2\) & \(m_{21}[n]=(P+Q)/2\) & \(m_{21}[n]=(P+Q)/2\) & \(m_{n1}[n]=(P+Q)/2\) \\ \hline \(h_{1  
**Step 2:** Let the average value of the envelope be \(m(t)\), the remaining signal \(h(t)\) can be indicated by \(h(t)=x(t)-m(t)\). \(h(t)\) will be taken as \(IMF_{1}\) if it satisfies the above two conditions. Otherwise, \(h(t)\) will be taken to replace \(x(t)\) and the above steps will be repeated. **Step 3:** After \(k\) iterations, the difference between the obtained signal and the mean value of the envelope is \(h_{1,k}(t)\). The difference obtained in the \(k-1\) iteration is indicated as \(h_{1,k-1}(t)\).