# Research and Application of Slag-Nanosilica Stabilizer for Silt Subgrade
## 4 Results and Discussion
### Stabilizer Mix Ratio Design
#### 4.1.3 Grey Correlation Analysis  
In this study, the grey correlation analysis method was used to determine the correlation degree and weight of various factors in the test so as to determine the optimal composition ratio of the stabilized soil. The specific steps were as follows:  
1. The reference sequence, which is the optimal value of each factor attribute, was determined. For the samples of \(k\)th factors, the reference sequence is usually expressed as follows: \[X^{*}=\{X_{1}^{*},X_{2}^{*},\ldots X_{k}^{*}\},\ (k=1,\ 2,\ \ldots\ n),\] (1) where \(X_{k}^{*}\) is the optimal value of the \(k\)th factor. The 7 d compressive strength of standard curing, 28 d compressive strength of standard curing, 7 d compressive strength of water bath curing, and 28 d compressive strength of water bath curing were taken as the evaluation criteria for the stabilized soil. Among the four evaluation criteria, the maximum value was the best value, so the results were \(X_{1}^{*}=3.12\), \(X_{2}^{*}=5.25\), \(X_{3}^{*}=1.52\), and \(X_{4}^{*}=3.43\). 2. Each factor was quantified without dimension. Because changes in each factor were different, it was necessary to eliminate the dimension of each factor and make it unified. There are many dimensionless quantitative processing methods. In this study, the method of deviation standardization was adopted. The formula is as follows: \[Y_{ij}=\frac{X_{ij}-X_{jmin}}{X_{jmax}-X_{jmin}},\] (2) \[Y_{ij}=\frac{X_{jmax}-X_{ij}}{X_{jmax}-X_{jmin}},\] (3) where \(X_{jmax}\) and \(X_{jmin}\) represent the optimal and worst value of the \(j\)th factor, respectively. The larger the factor, the better it is to use Equation (2), while the smaller the factor, the better it is to use Equation (3). \(X_{ij}\) represents the \(j\)th factor value of the \(i\)th test group, and \(Y_{ij}\) represents the dimensionless value of the \(j\)th factor of the \(i\)th test group. After dimensionless quantization, the optimal value and the worst value were 1 and 0, respectively. The remaining results are given in Table 8. \begin{table}
\begin{tabular}{c c c c c} \hline \hline \(\mathbf{Y^{*}}\) & \(\mathbf{1.000\ 0}\) & \(\mathbf{1.000\ 0}\) & \(\mathbf{1.000\ 0}\) & \(\mathbf{1.000\ 0}\) \\ \hline \(Y_{1}\) & 0.000 0 & 0.000 0 & 0.000 0 & 0.000 0 \\ \(Y_{2}\) & 0.338 6 & 0.259 5 & 0.120 0 & 0.419 8 \\ \(Y_{3}\) & 0.220 5 & 0.200 0 & 0.280 0 & 0.475 3 \\ \(Y_{4}\) & 0.362 2 & 0.405 4 & 0.720 0 & 0.216 0 \\ \(Y_{5}\) & 0.496 1 & 0.789 2 & 0.810 0 & 0.265 4 \\ \(Y_{6}\) & 0.685 0 & 1.000 0 & 1.000 0 & 1.000 0 \\ \(Y_{7}\) & 1.000 0 & 0.648 6 & 0.320 0 & 0.666 7 \\ \(Y_{8}\) & 0.944 9 & 0.535 1 & 0.330 0 & 0.475 3 \\ \(Y_{9}\) & 0.803 1 & 0.497 3 & 0.670 0 & 0.635 8 \\ \hline \hline \end{tabular}
\end{table} 
Table 8: Grey system normalization results. 3. To find the correlation coefficient between the reference sequence and the comparison sequence, the degree of difference between the geometric shapes of the curves is called correlation, so the difference value between the curves can be used as a measure of correlation degree. The formula is as follows: \[\xi_{ij}=\frac{\underset{i}{\text{minmin}}\big{|}Y^{*}-Y_{ij}\big{|}+\rho\times \underset{i}{\text{maxmax}}\big{|}Y^{*}-Y_{ij}\big{|}}{\big{|}Y^{*}-Y_{ij} \big{|}+\rho\times\underset{i}{\text{maxmax}}\big{|}Y^{*}-Y_{ij}\big{|}},\] (4) where \(\xi_{ij}\) represents the correlation coefficient of the \(j\)th factor of the \(i\)th test group, and \(\rho\) is the resolution coefficient, \(\rho\in(0,1)\). The smaller the \(\rho\), the larger the difference between the correlation coefficients and stronger the distinguishing ability. \(\rho\) is usually 0.5. \(Y_{ij}\) represents the dimensionless value of the \(j\)th factor of the \(i\)th test group. \(\underset{i}{\text{minmin}}\big{|}Y^{*}-Y_{ij}\big{|}\) represents the second-order minimum difference of the value of the \(j\)th factor of the \(i\)th test group after dimensionless quantization.