## 5 Methods for Estimation of HFR Video Characteristics  
### Hurst Index  
In time series analysis, specific behavior is analyzed using common statistical measures, as shown in Figure 3. LRD is especially of interest in order to understand behavior of a structure/sequence, especially in the cases of video traffic analysis. Particularly, Hurst index or exponent, \(H\), is evaluated as a statistical measure to better determine the characteristics of traffic, cardiac dynamics or finance [38; 57; 58; 68; 69]. Hurst exponent \(H\) can be estimated for LRD tests in different ways via: R/S statistics, periodogram, aggregated variance method, absolute moments method, detrended fluctuation analysis, etc. [68; 69; 70; 71; 72]. The Hurst index is found in the range 0.5-1, where \(H\) equals 0.5 in the case of pure random process like Brownian motion, with no correlation between incremental signal changes. An index is applied for measuring dependence in a structure/sequence, and it can be considered as a fractal-related feature. In the case when the index is less than 0.5, \(0<H<0.5\), the process is negatively correlated. Likewise, when the Hurst index is higher than 0.5, \(0.5<H<1\), it indicates a persistent behavior with long-term positive autocorrelation, meaning that higher values are probably followed by another high value. This self-similarity expressed by \(H\) values higher than 0.5 means that LRD occurs. LRD can be described having a relatively high degree of correlation between distant data points. In a nutshell, the complexity can be evaluated using a correlation sum \(C(r)\) calculated for a range of distances, or points within a radius \(r\), where the correlation sum scales with radius as \(C(r)\sim r^{D}\), giving the exponent \(D\) as a correlation dimension [68; 72]. It is possible to estimate the dimension as the local slope of \(\log(C(r))\) versus \(\log(r)\) for a sufficiently large range of small \(r\) values. Since the variance of local slope can be relatively high, different approaches may be used for as a function of distances between considered points in a structure. In a generalized case and fractal geometry [68; 71; 72], Hurst index \(H\), as a measure of LRD, which is directly related to fractal dimension, can be described as a function of parameter denoted as \(q\) for a time series \(x(t)\), using the scaling properties given as:  
\[\frac{<|x(t+r)-x(t)|^{q}>}{<|x(t)|_{q}>}\ \ \ \sim r^{dH(q)}, \tag{1}\]  
where radius or lag averaging over the considered time window is denoted by \(<.\ >.\) Another common algorithm is Detrended Fluctuation Analysis (DFA) [71; 72; 73]. Firstly, integrated sequence \(y_{k}\) is obtained based on \(x\), and then the local trend denoted by the \begin{table}
\begin{tabular}{c c c c c c} \hline
**No.** & **Source (YUV)** & **Spatial Resolution** & **Frame Rate** & **Frame Number** & **Bit Depth** \\ \hline
1 & Beauty & 3180 \(\times\) 2160 (2160 p) & 120 fps & 600 & 8 \\
2 & Bosphorus & 3180 \(\times\) 2160 (2160 p) & 120 fps & 600 & 8 \\
3 & HoneyBee & 3180 \(\times\) 2160 (2160 p) & 120 fps & 600 & 8 \\
4 & Jockey & 3180 \(\times\) 2160 (2160 p) & 120 fps & 600 & 8 \\
5 & ReadySetGo & 3180 \(\times\) 2160 (2160 p) & 120 fps & 600 & 8 \\
6 & YachtRide & 3180 \(\times\) 2160 (2160 p) & 120 fps & 600 & 8 \\ \hline \end{tabular}
\end{table} 
Table 1: Test source video files. of boxes or subseries of length \(m\) is found. This is followed by the calculation of the root mean square fluctuation:  
\[F(m)=\sqrt{\frac{1}{N}\sum_{k=1}^{N}\left[y_{k}-y_{m}(k)\right]^{2}}. \tag{2}\]  
The index is finally calculated as the slope of linear regression, which compares \(\log(F(m))\) and \(\log(m)\). The alternative for Hurst index calculation can be periodogram method [70; 71; 74].