# Study on Classification Method of Soil Liquefaction Potential Based on Decision Tree
## 5 LR liquefaction Probability Equation  
### Construction  
In this study, the LR model was selected to construct the liquefaction probability equation. After deleting repeated sites from the databases published by [PERSON] et al. and [PERSON] et al., database B was obtained, and this database was used to fit the liquefaction probability equation and construct the liquefaction potential grade criteria, too. The database published by [PERSON] et al. was used as database C for the verification of the probability equation and the liquefaction potential grade criteria. The database used to construct the equations was database B, which consisted of 473 sets of data. Based on the binary logistic model, the CSR was taken as the transition variable during the fitting process. In other words, in the process of equation fitting, the CSR was used as the fitting parameter. Since the CSR was obtained using parameters such as \(d_{s}\) and \(d_{w}\), the final equation does not  
Figure 1: Ranking of independent variable importance. reflect the CSR, but instead reflects these parameters. The shear wave velocity \(V_{s}\) was used to represent the anti-liquefaction capability of the saturated soil. To enhance the weight of the ground motion intensity in the equation fitting, \(a_{\text{max}}\) was selected as the weight of the CSR. The following equation was used for the logistic regression:  
\[P_{L}=\frac{1}{1+\text{EXP}(-(\theta_{0}+\theta_{1}V_{S}+(\theta_{2}*a_{\text{ max}}+\theta_{3})\text{ln}(\text{CSR})))} \tag{1}\]  
where \(\theta_{0}\), \(\theta_{1}\), and \(\theta_{2}\) are undetermined coefficients. The dependent variable, \(P_{L}\), was a \"yes or no liquefaction\" result, where a liquefied site was assigned a value of 1 and a non-liquefied site was assigned a value of 0. The CSR expression was as follows:  
\[\text{CSR}=\frac{\tau_{av}}{\tau_{v0}^{\prime}}=0.65\frac{a_{\text{max}}}{g} \frac{\sigma_{av}}{\sigma_{v0}^{\prime}}\tau_{d} \tag{2}\]  
For the convenience of calculation, the equation for \(r_{d}\) was simplified to \(r_{d}=1-0.008d_{s}\). After simplification, the CSR calculation formula was as follows:  
\[\text{CSR}=0.65a_{\text{max}}(1-0.008d_{s})\bigg{(}\frac{\gamma_{s}d_{s}}{ \gamma_{s}d_{s}-\gamma_{w}(d_{s}-d_{w})}\bigg{)} \tag{3}\]  
where \(\gamma_{s}\) is the sand bulk density, and \(\gamma_{w}\) is the water bulk density. Assuming that the soil bulk density was 19 \(\text{kN}/\text{m}^{3}\), Equation (3) was substituted into Equation (1), and the following equations were finally obtained after fitting:  
\[P_{L} =\bigg{(}1+\text{EXP}\bigg{(}-(3.3a_{\text{max}}+1.78)\text{ln} \bigg{(}0.65a_{\text{max}}\frac{19d_{s}}{9d_{s}+10d_{w}}(1-0.008d_{s})\bigg{)}+0.038V_{S}-9.5\bigg{)}\bigg{)}^{-1} \tag{4}\] \[V_{\text{scr}} =\bigg{(}\text{ln}\bigg{(}\frac{1-P_{L}}{P_{L}}\bigg{)}+(3.3a_{ \text{max}}+1.78)\text{ln}\bigg{(}0.65a_{\text{max}}\bigg{(}\frac{19d_{s}}{9d_ {s}+10d_{w}}\bigg{)}(1-0.008d_{s})\bigg{)}+9.5\bigg{)}\Bigg{/}_{0.038} \tag{5}\]  
where \(P_{L}\) is the liquefaction probability of the saturated soil, \(V_{s}\) is the measured shear wave velocity (m/s), \(a_{\text{max}}\) is the peak ground acceleration (g), \(d_{s}\) is the buried depth of the liquefiable layer (m), \(d_{w}\) is the groundwater depth (m), and \(V_{\text{scr}}\) is the critical shear wave velocity for the preset liquefaction probability \(P_{L}\). When the measured shear wave velocity (\(V_{s}\)) was less than \(V_{\text{scr}}\), the soil layer could be identified as undergoing liquefaction for the corresponding liquefaction probability level; otherwise, it did not undergo liquefaction. When this equation was used in the fitting sample, the success rates varied with the preset liquefaction probability, as shown in Figure 2. When \(P_{L}=50\%\) was taken as the critical liquefaction probability, the success rate was used to represent the accuracy of the prediction, which was about 70% for the liquefied and non-liquefied sites. As the preset liquefaction probability was varied, the prediction accuracy varied monotonically.