## 2 Materials and Method
### Sensitivity Analysis via Principal Component Regression  
This study employs PCR to analyze the sensitivity of UCS to interdependent geotechnical properties in lime-cement-stabilized high-plasticity clay soils. The complex interplay between stabilization mechanisms and mechanical performance--mediated by variables such as void ratio, porosity, compression index, preconsolidation stress, and compressibility modulus--necessitates a multivariate approach beyond traditional univariate analyses. PCR addresses multicollinearity by transforming correlated predictors into orthogonal principal components, which are then used to model UCS as a response variable. This methodological framework enables the quantification of relationships between microstructural modifications (e.g., pore volume reduction, cementitious bonding) and macroscopic strength development. Sensitivity coefficients derived from PCR, as defined in Equation (6), quantify the relative influence of each PC on UCS, revealing both dominant and counterintuitive relationships. By isolating the contributions of PCs linked to specific physicochemical mechanisms (e.g., flocculation, pozzolanic reactions), this approach identifies key drivers of strength enhancement while accounting for nonlinear interactions inherent to stabilized soils. The resulting insights provide a mathematical basis for optimizing stabilizer dosages and treatment protocols, enhancing the efficiency of soil stabilization in geotechnical applications. \[S_{PCI}=\beta_{i}\times\sqrt{\lambda_{i}} \tag{6}\]  
This equation weights the regression coefficient by the square root of the eigenvalue. This captures both the impact of the component on UCS. \((\beta_{i})\) and the component's importance in explaining overall soil variability (\(\sqrt{\lambda_{i}}\)). Higher absolute values indicate a stronger influence on UCS. The percentage of UCS variance is explained by each principal component as in Equation (7). \[VC_{PCI}(\%)=\frac{\beta_{i}^{2}\times\lambda_{i}}{\sum\limits_{j=1}^{k}\beta _{j}^{2}\times\lambda_{j}}\times 100\% \tag{7}\]  
where  
\(VC_{PCI}\) (\(\%\)) = percentage of variance in UCS explained by \(PC_{i}\)  
This metric quantifies how much of the total explained variance in UCS can be attributed to each principal component. For example, the 46% contribution of PC1 (Soil Compressibility) indicates that nearly half of the explainable variation in UCS comes from this component. Backward transformation to original variables, to interpret results in terms of original soil properties, sensitivities are mapped back as in Equation (8). This transformation reveals how changes in individual soil properties (like void ratio or lime content) affect UCS. The equation accounts for both direct and indirect effects through all principal components. \[S_{X_{j}}=\sum\limits_{i=1}^{k}S_{PCI}\times W_{ii} \tag{8}\]  
where  
\(S_{X_{j}}=\) sensitivity of UCS to original soil property \(X_{j}\)  
\(W_{ii}=\) loading of property j on principal component i  
Normalized sensitivity coefficients, to make sensitivities comparable across different properties as in Equation (9). This dimensionless coefficient expresses how many standard deviations UCS changes when a soil property changes by one standard deviation. Values near \(+1\) or \(-1\) indicate strong positive or negative influence. \[NSC_{X_{j}}=\frac{S_{X_{j}}\times\sigma_{X_{j}}}{\sigma_{UCS}} \tag{9}\]  
where  
\(NSC_{X_{j}}=\) normalized sensitivity coefficient  
\(\sigma_{X_{j}}=\) standard deviation of variable Xj  
\(\sigma_{UCS}=\) standard deviation of UCS  
Elasticity coefficients, to express sensitivity in percentage terms as in Equation (10). Elasticity coefficients indicate the percentage change in UCS expected from a 1% change in a soil property. For example, an elasticity of 0.5 for lime content would mean that increasing lime by 10% would increase UCS by approximately 5%.