# Urban Quarry Ground Vibration Forecasting: A Matrix Factorization Approach
## 6 Experiments and Results
### Relationship between the Amount of Data and Prediction Accuracy  
In this section, we examined the relationship between the amount of data used in the measurement data matrix and the prediction accuracy of the proposed method. Weighted Non-negative Matrix Factorization (WNMF) is a form of unsupervised learning. It can perform matrix factorization (i.e., prediction) even with a small amount of data. However, previous studies have not clearly shown how much the amount of data affects the predic \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Number of Rank** & **Nor-ANN** & **Nor-1-11** & **Nor-None** \\ \hline
1 & 0.509 & 0.313 & 0.359 \\
2 & 0.520 & 0.299 & 0.332 \\
3 & 0.517 & 0.297 & 0.318 \\
4 & 0.264 & 0.366 & 0.272 \\
5 & 0.184 & 0.357 & 0.269 \\
6 & 0.245 & 0.448 & 0.362 \\
7 & 0.143 & 0.273 & 0.222 \\
8 & 0.225 & 0.341 & 0.275 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Prediction results of PPV using WNMF (comparison between Nor-ANN and Nor-1-11, Nor-None). \begin{table}
\begin{tabular}{c c c c} \hline \hline \multirow{2}{*}{**Cases**} & \multicolumn{3}{c}{**Range of Error**} & \multirow{2}{*}{**Difference**} \\ \cline{2-3} \cline{5-5}  & **Min** & **Max** & \\ \hline Matrix100 & \(-0.226\) & 0.410 & 0.636 \\ Matrix80 & \(-0.334\) & 0.188 & 0.522 \\ Matrix60 & \(-0.310\) & 0.106 & 0.417 \\ Matrix40 & \(-0.784\) & 0.899 & 1.683 \\ Matrix20 & 0.476 & 1.273 & 0.798 \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Range of error for Nor-ANN, Nor-1-11, and Nor-None. tion accuracy of WNMF. For this purpose, we prepared several measurement data matrices with different numbers of measurement points (rows) and conducted PPV prediction using WNMF. This allows us to verify the prediction accuracy of the proposed method according to the amount of data used in the measurement data matrix. Specifically, against the Matrix100 (100 rows \(\times\) 9 columns) which uses all measurement data, we prepared conditions of Matrix20 (20 rows \(\times\) 9 columns), Matrix40 (40 rows \(\times\) 9 columns), Matrix60 (60 rows \(\times\) 9 columns), and Matrix80 (80 rows \(\times\) 9 columns). The measurement points (rows) to be deleted were randomly selected. Note that the measurement data of 100 points used in Matrix100 is one of the benchmarks for the amount of data required to create a PPV prediction model using ANN (refer to Section 2.2). Therefore, by comparing the prediction accuracy of PPV using Matrix100 and other conditions, we can clarify the effectiveness of the proposed method. As for the experimental method, as in the previous experiment, 10 PPV data points were randomly extracted from each measurement data matrix condition and replaced with unmeasured values, and PPV prediction was performed using WNMF. When executing WNMF, the rank number was changed from 1 to 8, and the prediction accuracy for each rank was calculated. When comparing each condition, the rank with the highest prediction accuracy was used. RMSE was used as the evaluation metric for prediction accuracy. As a result of WNMF, higher prediction accuracy was shown when using a smaller amount of data (60, 80 points) than the amount of data required for ANN learning (100 points of measurement data). On the other hand, low prediction accuracy was shown when using an extremely small amount of data (20, 40 points). Table 6 shows the PPV prediction results using WNMF. From Table 6, Matrix100, Matrix80, and Matrix40 showed the smallest RMSE at rank 7, Matrix60 at rank 6, and Matrix20 at rank 4. Therefore, this rank was used for comparison of prediction accuracy. As a result of comparing the prediction accuracy of Matrix100 with other conditions, Matrix80 and Matrix60 showed a smaller RMSE than Matrix100. On the other hand, conditions with extremely small amounts of data, Matrix40 and Matrix20, showed a larger RMSE than Matrix100. Table 6 shows the results of comparing the error range of each condition. As a result, Matrix80 and Matrix60 showed a smaller error range than Matrix100.