## 2 Materials and Methods
### Methods
#### 2.3.3 Characterization of Fractal Parameters  
Self-similarity, in which a fractal object can be subdivided into parts, each of which may be a small-scale copy of the whole or statistically similar, is a central feature of fractals [47]. The multiple fractals in complex systems can therefore characterize the non-uniformity and diversity of the system's performance at different scales to reveal the complexity and inhomogeneity that exists within the system at different scales. Researchers have proposed various methods for estimating the fractal dimension, including the box counting and triangular prism methods, and fractional Brownian motion [48]. The box-counting method is relatively simple, provides reliable results, and is widely used in many fields [49; 50]; thus, this method was used for the multiple fractal analysis in this study.  
(1) Box counting is a commonly used method by which the fractal dimension can be calculated [51], with small boxes with different side lengths (r) used to cover a study area. In this study, boxes were considered to be two-dimensional, \(\rm{N_{ij}}\) indicated the number of data points in the (i,j) box, and N was the total number of data points in the whole area. The distribution probability \(\rm{P_{ij}}\) of the (i,j) box is expressed using  
\[P_{ij}=\frac{N_{ij}}{N} \tag{2}\]  
(2) The number of boxes N(r) required at different scales are calculated by varying the box size (r). The matching function \(\rm{\chi_{q}(r)}\), which is a weighted summation of the \(q\)th power of \(\rm{P_{ij}(r)}\), is then defined using the mathematical expression:  
\[x_{q}(r)=\sum_{i,j}p_{ij}(r)^{q} \tag{3}\]  
where \(\rm{q}\in(-\infty,+\infty)\) is the order of the statistical moments characterizing the degree of multifractal inhomogeneity. When \(\rm{q}>>1\), the value of \(\rm{P_{ij}(r)}\) with a higher probability in \(\rm{\chi_{q}(r)}\) plays a major role; when \(\rm{q}<<-1\), \(\rm{P_{ij}(r)}\) with a lower probability plays a major role. All integers between \(\rm{q}\in(-10,10)\) were utilized in this study.  
(3) The mass index \(\rm{\tau(q)}\) is a characteristic function of fractal behavior and can be obtained from the slope of the \(\rm{log\chi_{q}(r)}\) versus the logr curve:  
\[\rm{\tau(q)}=\lim_{\rm{\tau}\to 0}\frac{log\chi_{q}^{(r)}}{log\ r} \tag{4}\]  
The multifractal spectrum is meaningful only if \(\rm{log\chi_{q}(r)}\) is linear (i.e., scale-invariant) with respect to logr. If \(\rm{\tau(q)}\) is a straight line with respect to \(\rm{q}\), the object is monofractal; however, if \(\rm{\tau(q)}\) is an upward convex monotonically increasing curve with respect to \(\rm{q}\), the object is multifractal.  
(4) The generalized fractal dimension \(\rm{D_{q}}\) is calculated using \(\rm{\tau(q)}\):  
\[D_{q}=\frac{\rm{\tau(q)}}{q-1} \tag{5}\]  
When \(\rm{q}=0\), \(\rm{D_{q}}=\rm{D_{0}}\), i.e., the ordinary Hausdorff dimension; when \(\rm{q}=1\), \(\rm{D_{q}}=\rm{D_{1}}\), thus representing the information dimension, which is expressed using  
\[D_{1}=-\lim_{\rm{\tau}\to 0}\sum_{i,j}p_{ij}logp_{ij} \tag{6}\]  
(5) The Legendre transform between the generalized fractal dimension and the multifractal spectrum is then satisfied, and the multifractal spectrum is obtained from \(\rm{\tau(q)}\) and the corresponding transform of \(\rm{q}\):  
\[\rm{f(\alpha)}=q\alpha(q)-\rm{\tau(q)} \tag{7}\]\[\alpha(q)=\frac{d\tau(q)}{dq} \tag{8}\]  
The relationship between the independent variables \(q\), \(\tau\), \(f(\alpha)\), and \(\alpha\) is constructed using the above two equations. The series \(f(\alpha)\) refers to the series of sequential spectra obtained in which \(\alpha\) denotes the localized dimensionality of the manifold and the localized region, and changes constantly with r.