## 3 Methodology
### Identification of Triggering Factor  
Most research on the triggering factors of slope failure were based on specific slip events [65, 66, 67]. These post-disaster studies explained the initiation mechanism of various failure modes. Identifying the triggering factors of the failure mode before the disaster occurs is more instructive for engineering slopes. However, the exploration of this issue is not enough. Statistical methods have achieved very good application effects in the field of disaster sensitivity analysis [68, 69, 70, 39]. These studies emphasize the statistical characteristics of data and provide a novel perspective for investigating the influence of multi-factors on slope stability. Thus, this study attempts to apply statistical methods to identify triggering factors.  
Exploratory factor analysis (EFA) is a kind of multivariate statistical analysis method. EFA assumes that the correlations among variables (e.g., correlations among FOS under different working conditions) can be represented by a subset of hidden common factors [71].  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multirow{2}{*}{**Gradients**} & \multicolumn{2}{c}{**Pre-Tension**} & \multicolumn{2}{c}{**Cohesion**} & \multicolumn{2}{c}{**Earthquake**} \\  & \multicolumn{2}{c}{**(kN)**} & & **c (kPa)** & **Influence Coefficients** \\ \hline  & **Type 1** & **Type 2** & **Rock Mass** & **Bedding Plane** & \\ \hline
1 & 892.5 & 1147.5 & 272.65 & 85.5 & 0.01 \\
2 & 735 & 945 & 243.95 & 76.5 & 0.03 \\
3 & 525 & 675 & 215.25 & 67.5 & 0.05 \\ \hline \hline \end{tabular}
\end{table}
Table 5: Gradients of material parameters.  
Therefore, EFA is primarily used to reduce the dimension of variables by linearly combining the variables. However, the common factor cannot account for all the variations of each variable [72]. Therefore, each variable has a proportion of unique variations. Any variable can be described as a linear combination of the common factors and the unique variations [73; 74]:  
\[W_{i}=a_{1}F_{1}+\cdots+a_{n}F_{n}+U_{i} \tag{6}\]  
where \(W_{i}\) is the \(i\)th given factor, \(F_{1}\)-\(F_{n}\) are \(n\) common factors, \(a_{1}\)-\(a_{n}\) are factor loadings (correlation coefficient between a common factor and a variable), and \(U_{i}\) is the unique variation of \(W_{i}\). The contribution rate of variation described by a common factor represents the significance of this common factor. The factor loading measures the significance of a variable when using this variable to characterize a specific factor [70]. Most importantly, it allows one to determine how well the variable aligns with common factors and allows the identification of the most important variable according to the factor loading [75]. The commonality of a variable is the percentage of variations of this variable explained by all common factors. A commonality of a variable closer to 1 indicates a better explanation [70]. It should be noted that before the EFA analysis, the Kaiser-Meyer-Olkin (KMO) and Bartlett test of sphericity need to be implemented on the raw data to verify its feasibility [76].  
First, the FOS data for each model underwent the KMO test and Bartlett test of sphericity. Next, we quantified the common factors using a scree plot to identify the number of common factors by observing the elbow point [75; 77]. We then used the principal component extraction method to obtain common factors [71]. To aid interpretation, we employed varimax rotation to regenerate the common factors [78]. Finally, we determined the triggering factors of each model by comparing the factor load values. The flow chart of the EFA is shown in Figure 7.  
Figure 7: The flowchart of EFA.