# Research on Hard Rock Pillar Stability Prediction Based on SABO-LSSVM Model
## 2 Materials and Methods  
### Pillar Dataset  
The mine pillar sample dataset used in this study was sourced from [PERSON]'s database (2019). The dataset consists of 162 complete mine pillar samples, including 58 \"stable\" samples, 37 \"unstable\" samples, and 67 \"failed\" samples. The mine pillar samples in this database primarily consist of large sulfide pillars with a rock mass rating (RMR) ranging from 60% to 85%. The structural characteristics of the pillars were not considered to be stability-influencing factors. The data sources for the dataset included Westmin Resources Ltd., [PERSON] (1988), [PERSON] et al. (1984), Hedley and Grant (1972), and [PERSON] (1992) (2019). The original dataset features include the width (W) and height (H) of the mine pillar, aspect ratio (K) of the pillar, average pillar constraint (C\({}_{\text{pav}}\)), uniaxial compressive strength of the rock (UCS), and pillar stress (\(\sigma\)). The pillar stress was calculated using sublevel-area theory or numerical simulation methods. [PERSON] et al. (2020) added pillar strength (\(\sigma_{\text{p}}\)) as a feature in their mine pillar stability prediction. They used [PERSON] empirical formula to calculate pillar strength, which is shown in Equation (1):  
\[\frac{\sigma_{\text{p}}}{0.44\text{UCS}}=0.68+0.52\text{tan}\left(\cos^{-1} \left(\frac{1-0.46\left(\log_{10}\left(K+0.75\right)\right)^{\frac{1.4}{K}}}{1 +0.46\left(\log_{10}\left(K+0.75\right)\right)^{\frac{1.4}{K}}}\right)\right) \tag{1}\]  
[PERSON] and [PERSON] et al. (2019) conducted an in-depth study on the mechanics of hard rock pillars and re-evaluated the Lunder pillar database. They found that the aspect ratio of pillars varied with changes in brittleness when the strength trend underwent a transition, which can be approximated by the uniaxial compressive strength (UCS). Based on this observation, they developed a new empirical formula for pillar strength and achieved better pillar classification results. In their formula, for pillar samples with 50 MPa < UCS < 350 MPa and \(0<W/H<3\), the recommended formula for estimating the pillar strength is as follows:  
\[\frac{\sigma_{p}}{UCS}=0.405+\left(\frac{0.2408-0.405}{1+e^{\left(\frac{W}{H} -\frac{UCS}{105}-0.11\right)/0.3}}\right)+\left(0.096+\left(\frac{0.063-0.096} {1+e^{\left(\frac{W}{H}-\frac{UCS}{105}-0.11\right)/0.3}}\right)\right)\frac{ W}{H} \tag{2}\]For pillar samples with UCS > 300 MPa and W/H > 2, it is recommended to use the following conservative estimation formula to calculate the pillar strength:  
\[\frac{\sigma_{p}}{UCS}=0.462+\left(\frac{0.227-0.462}{1+e^{\left(\frac{W}{H}- \frac{UCS}{195}-1.42\right)/0.71}}\right)+\left(0.0871+\left(\frac{0.0245-0.0871} {1+e^{\left(\frac{W}{H}-\frac{UCS}{195}-1.42\right)/0.71}}\right)\right)\frac{W }{H} \tag{3}\]  
In the mine pillar dataset used in this study, all the samples satisfied the conditions for using the first formula. Therefore, when the pillar strength feature (\(\sigma_{p}\)) was added, the calculation was performed using this formula. The boxplots of the complete dataset are shown in Figure 1. In the boxplots, the upper and lower edges of the box represent the upper and lower quartiles of the data group, the horizontal line inside the box represents the median, the small square represents the mean, the lines extending from the box represent the upper and lower limits of the data, and the black circles represent the outliers. Figure 1: _Cont._  
The graph shows that the distribution of most feature data across different categories was relatively balanced. However, for the three sets of features, pillar width (W), pillar height (H), and uniaxial compressive strength (UCS), the medians of each category's data showed noticeable deviations. This indicates that the data distribution exhibited skewness, which was mainly due to the significant differences in the sizes of the different mine pillar samples. However, structural characteristics were not considered as stability-influencing factors [26].