# Network Traffic Prediction Incorporating Prior Knowledge for an Intelligent Network
## 4 Predictive Models  
### Flow Characterization Module  
Realistic network traffic series are usually non-stationary, although these traffic series tend to have obvious self-similarity. The self-similarity property is reflected in the fact that the local and overall time-series are related. Traditional network traffic models are only applicable to small-scale bursty traffic, but the variation characteristics of network traffic data have some similarities under large-scale conditions. Self-similar traffic characteristics differ from short correlated traffic characteristics in that they can reflect similar abrupt changes in the network traffic time-series at different time scales over long periods. The Hurst exponent is based on the asymptotic process of a rescaling range, defined as a function of the period of the time-series, and is used to describe the self-similarity of time-series with long correlation properties, defined as follows: For a given series of length \(n\), the sample mean is, the sample variance is, and we have:  
\[\frac{R}{S}(n)=\frac{1}{s(n)}[\max(0,W_{1},W_{2},\ldots,W_{n})-\min(0,W_{1},W_ {2},\ldots,W_{n})] \tag{2}\]  
As \(n\rightarrow\infty\), we have \(E(\frac{R}{S}(n))=cn^{H}\), where \(c\) is a constant, and the above equation yields the Hurst exponent H. In particular, \(R\) refers to the range of the deviations from the mean, \(S\) to the sum of standard deviations or variances over the sequence, and \(W\) to the cumulative sums.  
After calculation of the Hurst exponent, it can be seen that the time-series shows trends of averaging, regression, aggregation, and so on. The Hurst exponent can be used to measure the long-term memory and fractality of the time-series. As few assumptions are made about the underlying system, the Hurst exponent has wide applicability in time-series. The value of the Hurst exponent can be used to classify time-series trends according to three features: The higher the value, the smoother, less volatile, and less rough the time-series. When its value is between 0 and 1, the time-series has different properties.  
When \(0<H<0.5\), the time-series shows a negative correlation trend, and the time series has violent fluctuations.  
When \(0.5<H<1\), the time-series shows correlation over a long period, which indicates that the network business flow is correlated over time.  
Figure 2: Network traffic prediction framework incorporating prior knowledge (FPK-Net).  
When \(H=0.5\), the time-series shows the process of Brownian motion, and the correlation coefficients between the series are 0 and independent of each other.  
There are several methods for calculating the Hurst exponent and, in this paper, we choose the rescaling range analysis method to measure the network traffic data. Let a time-series \(\{X_{1},X_{2},X_{3},\ldots,X_{N}\}\) of length N be divided equally into several shorter time-series, with length on the order of \(n=N,N/2,N/4\). The average readjustment range for each sub-sequence is then calculated as follows:  
* Calculate the average value of each subsequence \(m=\frac{1}{n}\sum\limits_{i=1}^{n}X_{i}\).
* Create mean-adjusted series \(Y_{t}=X_{t}-m,t=1,2,\ldots,n\).
* Generate cumulative deviation series \(z_{t}=\sum\limits_{i=1}^{t}Y_{i},t=1,2,\ldots,n\)
* Calculation range \(R_{t}=\max(Z_{1},Z_{2},\ldots,Z_{t})-\min(Z_{1},Z_{2},\ldots,Z_{t}),t=1,2,\ldots,n\)
* Calculation range \(S_{t}=\sqrt{\frac{1}{t}\sum\limits_{i=1}^{t}\left(X_{i}-u\right)^{2},t=1,2, \ldots,n}\)
* Calculating the rescaling range \(\left(R/S\right)_{t}=R_{t}/S_{t},t=1,2,\ldots,n\)  
By taking the logarithm of both sides of the equation, we can obtain the relationship \(\left(R/S\right)_{t}=\log c+H\log t\), where c is a constant. By plotting the relationship \(\left(R/S\right)\) with \(t\) on the logarithmic axis and depicting the points, all points \(\left(\log t,\log(R/S)\right)\) can be found to lie on an almost straight line; therefore, the slope of the regression line can be approximated by the Hurst exponent.