## 4 Results and Discussion  
### The Question of _ilr-RPCA-Back clr_  
The extraction of geochemically combined elements has attracted much attention in geochemical prospecting, with various methods available. Currently, the methods of combination element identification are mainly PCA, RPCA, FA, RFA, and so on. These methods require the use of logarithmic transformations or logarithmic ratio transformations (alr, _clr_, _ilr_, and so on) to eliminate or attenuate the effect of component data closure as much as possible. Here, the widely used _ilr-RPCA-back clr_ method is discussed. Question 1: How does the process of element exchange impact the compositional variations of geochemical elements? The _ilr_ transform method itself belongs to the asymmetric transformation, and the correspondence between the variables before and after the transformation is disrupted. This disruption is primarily due to changes in the order of elements in the _ilr_ formula. Consequently, when applying this method to stream sediment data, different combinations of elements can be obtained through the same transformation. Based on the experiments with the six elements related to the original data of the porphyry copper deposit, the influence of the order change of different elements on _ilr_ transformation is discussed. As shown in the figure, the Ag, Mo, and W elements in Figure 3a represent the first principal component and exhibit strong positive correlations (Table 2). However, in Figure 3b, these same elements show poor representation and weak correlation (Table 2). In Figure 3c, the Ag-Mo-W and Au-Cu elements, respectively, demonstrate negative loadings on the first principal component and positive loadings on the second principal component (Table 2). Figure 3d reveals that Ag and Mo elements maintain a strong positive correlation as they represent the first principal component. From these results, it can be concluded that further research is required to investigate the application of this method for determining geochemical elements. Question 2: Score and load transform to \(clr\) space; has the element correspondence changed? \[clr(Y)=log\frac{x_{i}}{\sqrt[]{\prod_{j=1}^{D}x_{i}}}(i=1,2,3,\cdots,D-1,D), \tag{5}\] \begin{table}
\begin{tabular}{c c c c c c c c c c c} \hline \hline \multicolumn{2}{c}{**Sequence 1 (Figure 3a)**} & \multicolumn{3}{c}{**Sequence 2 (Figure 3b)**} & \multicolumn{3}{c}{**Sequence 3 (Figure 3c)**} & \multicolumn{3}{c}{**Sequence 4 (Figure 3d)**} \\ \hline Elements & PC1 & PC2 & Elements & PC1 & PC2 & Elements & PC1 & PC2 & Elements & PC1 & PC2 \\ Bi & 0.17 & 0.86 & W & \(-\)0.08 & 0.42 & W & 0.29 & 0.24 & W & 0.26 & \(-\)0.31 \\ W & \(-\)0.33 & 0.04 & Ag & \(-\)0.57 & \(-\)0.36 & Au & \(-\)0.44 & \(-\)0.38 & Bi & \(-\)0.47 & \(-\)0.66 \\ Au & 0.29 & \(-\)0.42 & Bi & 0.30 & 0.58 & Cu & \(-\)0.69 & 0.20 & Mo & 0.34 & 0.07 \\ Cu & 0.68 & \(-\)0.19 & Au & 0.22 & \(-\)0.49 & Mo & 0.29 & \(-\)0.20 & Cu & \(-\)0.52 & 0.30 \\ Ag & \(-\)0.38 & \(-\)0.21 & Cu & 0.58 & \(-\)0.30 & Ag & 0.34 & \(-\)0.52 & Au & \(-\)0.16 & 0.61 \\ Mo & \(-\)0.42 & \(-\)0.08 & Mo & \(-\)0.44 & 0.14 & Bi & 0.21 & 0.67 & Ag & 0.55 & \(-\)0.01 \\ Cumulative Proportion & 0.34 & 0.58 & Cumulative Proportion & 0.32 & 0.61 & Cumulative Proportion & 0.37 & 0.63 & Cumulative Proportion & 0.31 & 0.58 \\ \hline \end{tabular}
\end{table} 
Table 2: ILR-RPCA-back CLR Statistical Table of different elements in sequence. Figure 3: Biplots of different elements in sequence ((a): Bi, W, Au, Cu, Ag, Mo; (b): W, Ag, Bi, Au, Cu, Mo; (c): W, Au, Cu, Mo, Ag, Bi; (d): W, Bi, Mo, Cu, Au, Ag). \[u_{i}=\sqrt{\frac{i}{i+1}}\left[ \begin{array}{cc}\frac{1}{i},\cdots,\frac{1}{i} \\ i\ \ \ elements\end{array}