## 2 Preliminaries
### fBm  
The fBm with Hurst index \(H(0<H<1)\) is defined as the stochastic integral, for \(t\geq 0\)  
\[B_{H}(t)=\frac{1}{\Gamma(H+1/2)}\left\{\int_{-\infty}^{0}\left[(t-s)^{H-1/2}-(- s)^{H-1/2}\right]dW(s)+\int_{0}^{t}(t-s)^{H-1/2}dW(s)\right\}, \tag{6}\]  
where \(W\) denotes a Wiener process defined on \((-\infty,+\infty)\).  
The Hurst parameter \(H\) determines the type of fBm and the degree of self-similarity. When \(H=0.5\), fBm is reduced to the conventional Brownian motion. The fBm process \(B_{H}(t)\) has the following covariance function [20]  
\[cov(B_{H}(s),B_{H}(t))=\frac{\sigma^{2}}{2}\left(|t|^{2H}+|s|^{2H}+|t-s|^{2H} \right). \tag{7}\]  
The mean value of fBm is \(E(B_{H}(t))=0\) and the variance function of fBm is \(Var[B_{H}(t)]=\frac{\sigma^{2}}{2}|t|^{2H}\). Figure 3 illustrates 5000 points of fBm and fGn with different Hurst parameters.  
The fractional Gaussian noise (fGn) is the increment sequence of fBm. The relationship of white Gaussian noise (wGn) and fBm can be defined with Riemann-Liouville fractional integral [21]:  
\[B_{H}(t)=\frac{1}{\Gamma(H+1/2)}\int_{0}^{t}(t-\tau)^{H-1/2}\omega(\tau)d\tau, \tag{8}\]  
where \(H>0\) and \(\omega(t),t>0\) is the one-sided white Gaussian noise.  
Figure 3: Fractional Brownian motion (fBm) and fractional Gaussian noise (fGn). In the left panel, from top to bottom, are fBm with \(H=0.5\), \(0.6\), \(0.7\), and \(0.8\). In the right panel, from top to bottom, are fGn with \(H=0.5\), \(0.6\), \(0.7\), and \(0.8\). With the increase of Hurst exponent \(H\), the coupling effects of fBm and fGn are strengthened.  
As discussed above, the constant-order fractional processes with a constant Hurst parameter \(H\) can be used to accurately characterize the long memory process and the short-range dependent stochastic processes [22]. However, the multiscaling or multifractal characteristic of stochastic processes cannot be captured by the constant Hurst parameter. Therefore, the multifractional or multifractal processes are the extension of fractional processes by generalizing the constant Hurst parameter \(H\) to the case where \(H\) is indexed by a time-dependent local Holder exponent \(H(t)\) with time variant Hurst parameter \(H(t)\) to describe complex or chaotic phenomena in several fields of sciences [23; 24]. The performance and the robustness of 12 sliding-windowed Hurst estimators for multifractional processes are reviewed by [PERSON] et al. [25].  
Nonetheless, not all signals are time series, such as the graphic images in geomorphology, urban geography, cartology, etc. Therefore, some novel techniques on fractal image compression and fractal encoding are proposed in [26; 27]. In addition, a generalization of the Hurst estimation approach with \(q\)-th order moments of the distribution of the increments are used to characterize the statistical evolution of the series in [13]. A process with a constant \(H(q)=H\) shows the characteristic of the monofractal process. For a process with \(H(q)\) depending on the order \(q\), the process is commonly called multi-scaling (or multifractal). In this case, a multifractal spectrum can be defined with the local Holder exponent quantifies the scaling properties of the process at a given point in time. In Section 3, one of the useful algorithms called multifractal detrending fluctuation analysis (MFDFA) will be introduced to analyze the complex signals which exhibit a local self-similarity property, multifractional processes with variable local Holder exponent \(H(q)\).