## 4 Methodology  
### Log-Ratio Approach  
Using raw geochemical data for standard statistical methods can lead to misleading and confusing results because they are compositional data [22; 39; 40]. Compositional data refers to data that is distributed within a finite region and subject to unit and sum constraints; therefore, geochemical element concentration data are considered as typical compositional data [41]. The log-ratio transformation is a projection transformation of compositional data based on the fact that the ratio of compositional components is not affected by the \"constant sum\" constraint, and the logarithm of the ratio often follows a normal distribution. The specific definition [27; 38; 42] is as follows:  
\[S^{D}=\left\{x=(x_{1},\cdots,x_{D})^{\prime},x_{i}>0,\sum_{i=1}^{D}x_{i}=k\right\} \tag{1}\]  
where \(S^{D}\) represents the simplex space of compositional data, and \(k\) is an arbitrary constant [42]. Therefore, it is recommended to perform a suitable data transformation before analysis [42]. Log-ratio transformation of compositional data not only avoids the closure effect but also allows for analysis using unconstrained multivariate statistics [39; 42; 43]. There are three available transformations for compositional data: the centered log-ratio (CLR) transformation proposed by [PERSON], the additive log-ratio (ALR) given by [PERSON], and the isometric log-ratio (ILR) proposed by [PERSON] [27; 38; 39] with the following functions:  
\[alr(X)=\ln\frac{x_{i}}{x_{j}}(i=1,2,\cdots,D-1) \tag{2}\]  
\[clr(X)=\log\frac{x_{i}}{\sqrt[D]{\prod_{i=1}^{D}x_{i}}}(x=1,2,3,\cdots,D-1,D) \tag{3}\]  
\[ilr(X)=\sqrt{\frac{i}{i+1}}\ln\left[\frac{\left|\dot{\prod_{i=j}^{i}x_{j}} \right.}{x_{i+1}}\right](i=1,2,3\cdots,D-1) \tag{4}\]  
where \(X\) (\(n\times D\)) is a sequence of observed compositional components, \(x_{i}\) is the \(i\)-th component, \(x_{j}\) is the (\(i+1\))-th component, and \(D\) is the number of compositional components. \begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline
**Element** & **Min** & **Max** & **Mean** & **Median** & \
\begin{tabular}{c} **Standard** \\ **Deviation** \\ \end{tabular} & **Skewness** & **Kurtosis** &
\begin{tabular}{c} **Coefficient of** \\ **Variation** \\ \end{tabular} \\ \hline Au & 0 & 498 & 1.3 & 1 & 0.7 & 41.1 & 2207 & 0.57 \\ Ag & 0.02 & 0.08 & 0.04 & 0.04 & 0.01 & 61.42 & 4250 & 0.33 \\ Cu & 2.6 & 83.4 & 26.3 & 19.3 & 19 & 6.2 & 78 & 0.73 \\ Pb & 4.8 & 39.8 & 22.3 & 21.6 & 5.7 & 24.2 & 788 & 0.26 \\ Zn & 1.7 & 124.9 & 47.5 & 46.2 & 25.7 & 24.2 & 788 & 0.54 \\ Cr & 2.5 & 497.4 & 115 & 65.9 & 126.8 & 9.3 & 132 & 1.1 \\ Ni & 1.8 & 181.9 & 46.5 & 29.4 & 45.1 & 6.8 & 61 & 0.97 \\ Co & 1.3 & 46.8 & 15.1 & 11.3 & 10.5 & 2.2 & 12 & 0.7 \\ W & 0.21 & 4.36 & 1.47 & 1.19 & 0.96 & 23.94 & 755 & 0.65 \\ Mo & 0.01 & 2.16 & 0.67 & 0.54 & 0.49 & 21.85 & 664 & 0.74 \\ Cd & 0.01 & 0.23 & 0.092 & 0.08 & 0.05 & 16.55 & 518 & 0.5 \\ As & 0.64 & 14.85 & 5.37 & 4.45 & 3.15 & 16.2 & 394 & 0.59 \\ Sb & 0.1 & 1.05 & 0.42 & 0.37 & 0.21 & 8.51 & 122 & 0.49 \\ Hg & 0 & 0.03 & 0.01 & 0.01 & 0 & 34.63 & 1557 & 0.3 \\ Bi & 0.01 & 0.92 & 0.25 & 0.18 & 0.22 & 23.07 & 935 & 0.87 \\ \hline \hline \end{tabular} Note: The unit for the element Au is ppb, while the unit for other elements is ppm.  
\end{table} 
Table 1: Statistical characteristics of stream sediment elements from L\(\rm\acute{u}\)liangshan and Shuangkoushan, NQ. While alr transformation has some arbitrariness in the choice of denominator, clr transformation results in all variables summing to zero (i.e., matrix singularity issue, resulting in parallel data) [21; 43; 44]. Hence, both alr and clr transformations are not suitable for robust statistics. Instead, the ilr transformation is adopted. However, after ilr transformation, the dimension of the data is reduced by one, making it difficult for geological interpretation. Therefore, the ilr transformation is used to transform the compositional data into Euclidean space, and combined with Robust Factor Analysis (RFA) to obtain loadings and scores.