## 3 Results and Discussion
### Partial Least Square Regression Prediction Analysis of SSCFC's Fluidity and Compressive Strength
#### 3.5.2 Principal Component Analysis of PLSR Model  
This study employs partial least squares regression (PLSR) to thoroughly analyze factors' effects on SSCFC's fluidity and compressive strength, while establishing predictive models. To enhance model reliability, the optimal number of principal components was first determined by analyzing three key metrics: (1) Cross-validated R-squared for the \(h\)-th component (Qh\({}^{2}\)), (2) Root Mean Square Error of Prediction (RMSEP), and (3) Variable Importance in Projection (VIP) scores. The determination of the principal component number through these three metrics effectively minimizes overfitting risks while ensuring precise mix proportion optimization. Table 11 comprehensively presents the Qh\({}^{2}\) values (Cross-validated R-squared for the \(h\)-th component) for varying principal component counts, including the computational parameters of the residual sum of squares (SS) and prediction residual error sum of squares (PRESS).The calculation of Qh\({}^{2}\) values follows the formula presented below. \[Qh^{2}=1-\frac{PRESS}{SS\times(h-1)} \tag{16}\]  
Note: SS is the residual sum of squares; PRESS is the cross-residual sum of squares. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Principal Component Number** & **SS** & **PRESS** & **Qh\({}^{2}\)** \\ \hline
1 & 3593.891 & 6224.127 & 1.000 \\
2 & 664.646 & 2066.568 & 0.425 \\
3 & 664.461 & 2021.070 & \(-\)2.041 \\ \hline \hline \end{tabular}
\end{table} 
Table 11: Cross-validity analysis. The analysis first focused on the \(Q\)\(l\)\({}^{2}\) values. As shown in Table 11, the Qh\({}^{2}\) values for three principal components drop sharply to \(-\)2.041, suggesting that extracting the third component would lead to overfitting. In contrast, when the number of principal components is one or two, the Qh\({}^{2}\) values remain at 1.000 and 0.425 (both exceeding the threshold), demonstrating that the first two components progressively capture essential variable correlations without accumulating redundant information. Subsequent analysis examined the RMSEP. As illustrated in Figure 12, increasing the number of components from one to two reduced the RMSEP from 25.88 to 13.83 (46.5% decrease), demonstrating the second component's substantial information extraction capacity. In contrast, introducing a third component only marginally lowered the RMSEP to 13.82 (<0.1% change), revealing minimal informational gain. This saturation effect confirms that the model performance stabilizes at two components, beyond which overfitting risks escalate. Selecting two principal components thus achieves an optimal balance between predictive accuracy and model simplicity. The final analysis focused on the VIP values. As shown in Figure 13 and Table 12, when increasing the number of principal components from one to two, the VIP values of the three independent variables exhibited significant changes: the W/C ratio decreased by 0.216, the SS replacement rate increased by 0.578, and the ceramicste replacement rate increased by 0.269. This demonstrated that the second component enhanced variable correlations. However, when further increasing to three components, the VIP values remained essentially stable, with minimal variation rates. This analysis of VIP values provides additional validation for selecting two principal components as optimal, as the third component contributed negligible improvements while potentially increasing model complexity. \begin{table}
\begin{tabular}{c c c c} \hline \hline \multirow{2}{*}{**Factor**} & **One Principal** & **Two Principal** & **There Principal** \\  & **Component** & **Components** & **Components** \\ \hline W/C & 1.716 & 1.500 & 1.499 \\ SS & 0.228 & 0.806 & 0.806 \\ ceramicste & 0.050 & 0.319 & 0.321 \\ \hline \hline \end{tabular}
\end{table} 
Table 12: Summary of projected importance indicators (VIP). Figure 12: Line chart of RMSEP.