## 3 Results and Discussion
### KNN Algorithms  
In this section, we analyse the behaviour of the KNN algorithms in terms of predicting the compressive strength of LWAC. Two similarity metrics were used, the Euclidean and Manhattan distances, and the neighbourhood size \(K\) was varied from 1 to 50. Figure 8 illustrates the influence of the type of distance used in the KNN algorithm. The mean values of the RMSE, MAE and MAPE for the test datasets of the 10 random models are shown in Figure 8a-c, respectively. The best results were achieved with the Manhattan distance and values of \(K\) of between five and seven. For these values of \(K\), Figure 9 displays the RMSE of the 10 random runs of the validation process. For the training dataset, the best results were obtained with \(K=5\), with a mean RMSE of 3.8173, a mean MAE of 2.8956 and a mean MAPE of 15.3229%. For the test data, the best mean RMSE and MAE were obtained with \(K=7\) (4.3365 and 3.3134, respectively) and the best mean MAPE with \(K=5\) (see Table 5). Table 6 displays several goodness-of-fit measures for the model with these values of \(K\), obtained both without validation and after the validation process. The same results were obtained with both Monte Carlo and repeated \(k\)-fold cross-validation. The best model achieved a coefficient of determination of 0.812 and an RMSE of 3.8863. The best model without validation had a coefficient of determination of 0.8365 and an RMSE of 3.6246. A comparison of these models with those presented in [19] (see Table 3) and the ANN models in [6] (see Table 4) shows that the KNN algorithms (using the Manhattan distance) outperformed the MLR models (RMSE = 4.332) in predicting the compressive strength of LWAC. However, KNN showed slightly lower performance than the best tree model in [19] (RMSE = 3.808) and the best ANN model in [6] (RMSE = 3.745). Figure 8: Comparison of KNN models, \(1\leq K\leq 50\); mean values of statistical measures on the test data for the 10 random models (75:25 ratio). **(a)** RMSE; **(b)** MAE; **(c)** MAPE. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Model** & **Measures** & \(K=5\) & \(K=7\) \\ \hline Without & \(R^{2}\) & 0.8365 & 0.8115 \\ validation & MAE & 2.7562 & 2.9255 \\  & MAPE & 14.5033 & 15.5158 \\  & RMSE & 3.6246 & 3.8916 \\ \hline With validation & \(R^{2}\) & 0.8038 & 0.7930 \\ (mean of 10 models) & MAE & 3.0006 & 3.1044 \\  & MAPE & 15.9617 & 16.6643 \\  & RMSE & 3.9694 & 4.0775 \\ \hline With validation & \(R^{2}\) & 0.8120 & 0.7974 \\ (best model) & MAE & 2.9567 & 3.0597 \\  & MAPE & 15.6981 & 16.3948 \\  & RMSE & 3.8863 & 4.0340 \\ \hline \hline \end{tabular}
\end{table} 
Table 6: Statistical measures for KNN models with the Manhattan distance. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Model** & **St. Measures** & **Median** & **Mean** & **Confidence Interval (95\%)** \\ \hline Manhattan & MAE & 3.3871 & 3.3158 & [3.1676, 3.4640] \\ distance & MAPE & 17.7148 & 17.8784 & [16.7483, 19.0085] \\ \(K=5\) & RMSE & 4.3912 & 4.3852 & [4.1707, 4.5997] \\ \hline Euclidean & MAE & 3.4199 & 3.4384 & [3.3320, 3.5448] \\ distance & MAPE & 18.0707 & 18.5220 & [17.6788, 19.3652] \\ \(K=5\) & RMSE & 4.4556 & 4.4921 & [4.3254, 4.6588] \\ \hline Manhattan & MAE & 3.3981 & 3.3318 & [3.1932, 3.4704] \\ distance & MAPE & 17.5864 & 17.9849 & [16.8763, 19.0935] \\ \(K=6\) & RMSE & 4.3845 & 4.3733 & [4.1709, 4.5757] \\ \hline Euclidean & MAE & 3.4646 & 3.4592 & [3.3369, 3.5815] \\ distance & MAPE & 18.3423 & 18.6691 & [17.6019, 19.7363] \\ \(K=6\) & RMSE & 4.4300 & 4.4808 & [4.3001, 4.6615] \\ \hline Manhattan & MAE & 3.3550 & 3.3134 & [3.1579, 3.4689] \\ distance & MAPE & 17.4759 & 17.9372 & [16.8324, 19.0420] \\ \(K=7\) & RMSE & 4.3177 & 4.3365 & [4.1150, 4.5580] \\ \hline Euclidean & MAE & 3.4203 & 3.4474 & [3.3160, 3.5788] \\ distance & MAPE & 18.2648 & 18.6589 & [17.5924, 19.7254] \\ \(K=7\) & RMSE & 4.4611 & 4.4920 & [4.3151, 4.6689] \\ \hline \hline \end{tabular}
\end{table}