## 4 Discussion  
The Pearson correlation coefficient is employed to measure the linear relationship between two variables. Numerically, a correlation coefficient closer to 1 or \(-1\) indicates a stronger linear correlation, while proximity to 0 suggests a weaker correlation. This holds significant implications for under-standing the interdependence, trends, and mutual influences among variables. The Pearson correlation coefficient was applied to measure the variable correlation. The principle of calculation is shown in Equation (9), and the results are shown in Figure 4. The Pearson correlation coefficient between the temperature and unconfined compressive strength of permafrost was \(-0.590\), while the Pearson correlation coefficient between the strain rate and unconfined compressive strength was \(0.370\). According to the Pearson correlation coefficient correlation standards, the temperature and strength exhibited a moderately linear correlation, potentially indicating a negative linear relationship due to the negative coefficient. The strain rate and strength showed a weak correlation, suggesting a relatively weak positive correlation between the strain rate and strength, necessitating further assessment of their relationship based on predictions from the XGB model. \[Pearson=\frac{\sum_{i=1}^{n}{(x_{i}-\overline{x})(y_{i}-\overline{y})}}{\sqrt{ \sum_{i=1}^{n}{(x_{i}-\overline{x})}^{2}}}\sqrt{\sum_{i=1}^{n}{(y_{i}-\overline {y})}^{2}} \tag{9}\]  
where \(x_{i}\) and \(y_{i}\) are the input and output values of the \(i\)-th sample, \(\overline{x}\) and \(\overline{y}\) are the input and output averages of the overall sample, respectively. To explore the impact of temperature and strain rate on the variation in unconfined compressive strength in clay, pulverized soil, and sandy soil, a dataset was created due to the limited availability of experimental data. This dataset was based on common soil physical properties and experimental conditions that resembled actual working conditions. Additionally, the study aimed to analyze the strength differences among clay, silt, and sand after freezing. The soil particle size distribution was represented as the relative content percentage of particles, as outlined in Table 4.  
Figure 4: Heat map of correlation. From Figure 5, it can be observed that the unconfined compressive strength of the three types of frozen soil exhibited a similar trend, increasing with decreasing temperature and increasing strain rate. The most rapid increase in unconfined compressive strength occurred within the S1 and S3 ranges, while the strength increase was relatively gradual during the S2 stage. In the temperature range of \(-2\) to \(-4\)\({}^{\circ}\)C (S1), the soil reached the freezing point of free water, leading to rapid freezing of the free water and consequently a significant increase in strength. In the temperature range of \(-4\) to \(-16\)\({}^{\circ}\)C (S2), the linear relationship between strength and temperature weakened. However, in the range of \(-16\) to \(-20\)\({}^{\circ}\)C (S3), the strength experienced a substantial increase due to the complete freezing of weakly bound water within the pores. When considering strain rates between \(10^{-5}\) s\({}^{-1}\) and \(10^{-4}\) s\({}^{-1}\), as well as between \(10^{-3}\) s\({}^{-1}\) and \(10^{-2}\) s\({}^{-1}\), the strength values exhibited similar levels with only minor increments. During this stage, the dominant factor influencing the strength was the strain rate, where higher strain rates led to increased soil density and higher unconfined compressive strength. When the temperature dropped below \(-20\)\({}^{\circ}\)C, it further affected the freezing process of strongly bound water within the soil body. However, due to the limited range of experimental data, the influence of strongly bound water on the strength of frozen soil was not taken into consideration.