## 2 Preliminaries  
### Hurst Parameter, ACF, LRD, FD  
When the hydrologist [PERSON] spent many years analyzing the records of elevation of the Nile River in the 1950s. He found a strange phenomenon: the long-range recording of the elevation time series of the Nile River has much stronger coupling effects [10]. To quantify the level of coupling, the rescaled range (R/S) analysis method was first provided to estimate the coupling level, and [PERSON] modified the original Hurst R/S approach [11]. Later, many useful Hurst parameter estimators were provided and evaluated [12]. However, time series data with trend, seasonality, LRD and multiple scale (multiscaling) behavior cannot be captured by these conventional Hurst methods. A generalization of the Hurst exponent associated with the scaling behavior of statistically significant variables constructed from the time series is proposed in [13].  
For a time series \(X_{t}\) (\(t=1,2,3,\cdots,N\)), the ACF is defined as follows:  
\[\rho(k)=\frac{Cov(X_{t},X_{t-k})}{Var(X_{t})}, \tag{1}\]  
where \(Cov(\cdot)\) is the covariance and \(Var(\cdot)\) is the variance of the series. The behavior of the ACF at \(k=0\) determines the local properties of the realizations.  
ACF has a close relation with LRD since the asymptotic behavior of ACF at infinity (\(k\rightarrow\infty\)) quantifies the presence or absence of LRD:  
\[\rho(k)\sim ck^{-\beta},\text{ as }k\rightarrow\infty, \tag{2}\]  
where \(c\) is a constant and \(\beta\in(0,1)\). LRD also means \(\rho(k)\) is non-integral over the interval \((0,\infty)\). Besides, \(\beta\) is related to the Hurst parameter \(H\) by using  
\[\beta=2-2H. \tag{3}\]  
Therefore, the larger is the \(H\) value, the stronger is the LRD or long-range persistence [14]. Specifically, if  
\[\rho(k)\sim 1-|k|^{\alpha},\text{ as }k\to 0, \tag{4}\]  
for some \(\alpha\in(0,2]\), then the realizations of the random function have the fractal dimension  
\[D=n+1-\frac{\alpha}{2}. \tag{5}\]  
The concept of fractals is introduced by [PERSON] to describe complex systems and phenomena [15]. Generally speaking, the FD of a profile or surface is a roughness measure, with \(D\in[n,n+1)\), for a surface in \(n\)-dimensional space, with higher values indicating rougher surfaces.  
LRDs in time series or spatial data are instead associated with power-law correlations and often referred to as Hurst effects. The Hurst parameter \(H\) is a simple parameter which can characterize the level or degree of LRD. If \(H=0.5\), the time series has no statistical dependence. If \(H<0.5\), the time series is a negatively correlated process or an anti-persistent process. If \(H>0.5\), the time series is a positively correlated process [16]. The LRD processes are closely related to fGn and fBm via the fractional calculus. To capture the property of coupling or hyperbolic decaying autocorrelation, fractional calculus based LRD models have been suggested, such as autoregressive fractionally integrated moving average (ARFIMA) model and fractional integrated autoregressive conditional heteroskedasticity (FIGARCH) model [17].  
The two quantities \(D\) and \(H\) are independent of each other: the fractal dimension \(D\) is a local property, while LRD index \(H\) is a global characteristic. Nevertheless, the two notions are closely linked in much of the scientific literature. This stems from the success of self-similar models such asfractional Gaussian noise and fractional Brownian motion [18]. In addition, the asymptotic relationships Equations (2) and (4) can be expressed equivalently in terms of the spectral density and its behavior at infinity and zero, respectively [19].