Prediction of blasting-induced fragmentation in Meydook copper mine using empirical, statistical, and mutual information models

Abstract

One of the most important aims of blasting in open pit mines is to reach desirable size of fragmentation. Prediction of fragmentation has great importance in an attempt to prevent economic drawbacks. In this study, blasting data from Meydook mine were used to study the effect of different parameters on fragmentation; 30 blast cycles performed in Meydook mine were selected to predict fragmentation where six more blast cycles are used to validate the results of developed models. In this research, mutual information (MI) method was employed to predict fragmentation. Ten parameters were considered as primary ones in the model. For the sake of comparison, Kuz-Ram empirical model and statistical modeling were also used. Coefficient of determination (R ²), root mean square error (RMSE), and mean absolute error (MAE) were then used to compare the models. Results show that MI model with values of R ², RMSE, and MAE equals 0.81, 10.71, and 9.02, respectively, is found to have more accuracy with better performance comparing to Kuz-Ram and statistical models.

Appendix

Example 1

Calculation of the weight factor of burden (B) and rock quality design (RQD) on D ₈₀ through mutual information method for five data sets is as follows:

At first, we should calculate the entropy function of parameters. The number of bin is arbitrary. In this example, bin is equal to 5. The values of parameters are as follows:

$$ {\displaystyle \begin{array}{l}X=B=\left[5.5,6.5,6.5,5.5,6.5\right]\\ {}Z=\mathrm{RQD}=\left[25,40,30,25,35\right]\\ {}Y={D}_{80}=\left[45.03,92.88,71.03,14.31,69.02\right]\end{array}} $$

Entropy

$$ {\displaystyle \begin{array}{l}\mathrm{for}\ X\ \mathrm{and}\ \mathrm{bin}=5\kern0.5em \Rightarrow \mathrm{interval}\ \mathrm{classification}=\frac{6.5-5.5}{5}=0.2\\ {}X=\left\{{X}_1=\left[5.5-5.7\right),{X}_2=\left[5.7-5.9\right),{X}_3=\left[5.9-6.1\right),{X}_4=\left[6.1-6.3\right),{X}_5=\left[6.3-6.5\right]\right\}\\ {}P\left({X}_1\right)=\frac{2}{5},P\left({X}_2\right)=P\left({X}_3\right)=P\left({X}_4\right)=0,P\left({X}_5\right)=\frac{3}{5}\\ {}H(X)=-\sum \limits_{i=1}^5P\left({X}_i\right)\cdot {\mathrm{log}}_e^{P\left({X}_i\right)}=-\left(\frac{2}{5}{\mathrm{log}}_e^{\frac{2}{5}}+0+\frac{3}{5}{\mathrm{log}}_e^{\frac{3}{5}}\right)=0.673\\ {}\to \to \\ {}\mathrm{for}\ Z\kern0.5em \mathrm{and}\ \mathrm{bin}=5\kern0.5em \Rightarrow \kern0.5em \mathrm{interval}\ \mathrm{classification}=\frac{40-25}{5}=3\\ {}Z=\left\{{Z}_1=\left[25-28\right),{Z}_2=\left[28-31\right),{Z}_3=\left[31-34\right),{Z}_4=\left[34-37\right),{Z}_5=\left[37-40\right]\right\}\\ {}P\left({Z}_1\right)=\frac{2}{5},P\left({Z}_2\right)=\frac{1}{5},P\left({Z}_3\right)=0,P\left({Z}_4\right)=\frac{1}{5},P\left({Z}_5\right)=\frac{1}{5}\\ {}H(Z)=-\sum \limits_{i=1}^5P\left({Z}_i\right)\cdot {\mathrm{log}}_e^{P\left({Z}_i\right)}=-\left(\frac{2}{5}{\mathrm{log}}_e^{\frac{2}{5}}+\frac{1}{5}{\mathrm{log}}_e^{\frac{1}{5}}+0+\frac{1}{5}{\mathrm{log}}_e^{\frac{1}{5}}+\frac{1}{5}{\mathrm{log}}_e^{\frac{1}{5}}\right)=1.3321\\ {}\to \to \\ {}\mathrm{for}\ Y\ \mathrm{and}\ \mathrm{bin}=5\Rightarrow \mathrm{interval}\ \mathrm{classification}=\frac{92.88-14.31}{5}=15.72\simeq 16\\ {}Y=\left\{{Y}_1=\left[14.31-30.31\right),{Y}_2=\left[30.31-46.31\right),{Y}_3=\left[46.31-62.31\right),{Y}_4=\left[62.31-78.31\right),{Y}_5=\left[78.31-94.31\right]\right\}\\ {}P\left({Y}_1\right)=\frac{1}{5},P\left({Y}_2\right)=\frac{1}{5},P\left({Y}_3\right)=0,P\left({Y}_4\right)=\frac{2}{5},P\left({Y}_5\right)=\frac{1}{5}\\ {}H(Y)=-\sum \limits_{i=1}^5P\left({Y}_i\right)\cdot \mathrm{lo}{\mathrm{g}}_e^{P\left({Y}_i\right)}=-\left(\frac{1}{5}{\mathrm{log}}_e^{\frac{1}{5}}+\frac{1}{5}{\mathrm{log}}_e^{\frac{1}{5}}+0+\frac{2}{5}{\mathrm{log}}_e^{\frac{2}{5}}+\frac{1}{5}{\mathrm{log}}_e^{\frac{1}{5}}\right)=1.3321\\ {}\end{array}} $$

Joint entropy calculation

Table 7 Joint distribution of (Z,Y)

$$ H\left(Z,Y\right)=-\sum_{z,y}P\left(Z,Y\right)\cdot {\mathrm{log}}_e^{P\left(Z,Y\right)}=-\left(\frac{1}{5}{\mathrm{log}}_e^{\frac{1}{5}}+\frac{1}{5}{\mathrm{log}}_e^{\frac{1}{5}}+\frac{1}{5}{\mathrm{log}}_e^{\frac{1}{5}}+\frac{1}{5}{\mathrm{log}}_e^{\frac{1}{5}}+\frac{1}{5}{\mathrm{log}}_e^{\frac{1}{5}}\right)=1.609 $$

Table 8 Joint distribution of (X,Y)

$$ H\left(X,Y\right)=-\sum_{x,y}P\left(X,Y\right)\cdot {\mathrm{log}}_e^{P\left(X,Y\right)}=-\left(\frac{1}{5}{\mathrm{log}}_e^{\frac{1}{5}}+\frac{1}{5}{\mathrm{log}}_e^{\frac{1}{5}}+\frac{1}{5}{\mathrm{log}}_e^{\frac{1}{5}}+\frac{2}{5}{\mathrm{log}}_e^{\frac{2}{5}}\right)=1.3221 $$

Mutual information

$$ {\displaystyle \begin{array}{l}I\left(X;Y\right)=H(X)+H(Y)-H\left(X,Y\right)=0.673+1.3321-1.3321=0.673\\ {}I\left(Z;Y\right)=H(Z)+H(Y)-H\left(Z,Y\right)=1.3321+1.3321-1.6094=1.0548\\ {}{a}_B=\frac{0.673}{0.673+1.0548}=0.39\\ {}{a}_{\mathrm{RQD}}=\frac{1.0548}{0.673+1.0548}=0.61\end{array}} $$