# A Novel Artificial Neural Network to Predict Compressive Strength of Recycled Aggregate Concrete
## 2 Materials and Methods
Finally, to delimit the errors existing between estimated and observed data, and consequently choose the best option among the four tested, we used the mean absolute error (_MAE_) Equation (1), complemented with the standard deviation, the mean absolute percentage error (_MAPE_) Equation (2), and the root mean square error (_RMSE_) Equation (3), as well as the correlation coefficient (R\({}^{2}\)):  
\[MAE=\frac{\sum_{i=1}^{n}\left|t_{i}-o_{i}\right|}{n} \tag{1}\]  
\[MAPE=\frac{100}{n}\sum_{i=1}^{n}\left|\frac{t_{i}-o_{i}}{o_{i}}\right| \tag{2}\]  
\[RMSE=\sqrt{\frac{\sum_{i=1}^{n}\left(t_{i}-o_{i}\right)^{2}}{n}} \tag{3}\] \begin{table}
\begin{tabular}{c c c c c c c c c c c} \hline \hline  & **C** & **FA** & **W** & **SP** & **FNA** & **CAN** & **RCA** & **FM of** & **WA** & **SSD** & **TM** \\  & **kg/m3** & **kg/m3** & **kg/m3** & **\%** & **kg/m3** & **kg/m3** & **kg/m3** & **FNA** & **\%** & **g/cm3** & **mm** \\ \hline Range & 180 \(\ldots\) & 60 \(\ldots\) & 130 \(\ldots\) & 0.09 \(\ldots\) & 536 \(\ldots\) & 210 \(\ldots\) & 43 \(\ldots\) & 2.1 \(\ldots\) & 0.71 \(\ldots\) & 2.36 \(\ldots\) & 10 \(\ldots\) 25 \\  & 702 & 305 & 271 & 5.07 & 870 & 1237 & 1171 & 2.64 & 8.2 & 2.66 & 10 \(\ldots\) 25 \\ n & 177 & 67 & 177 & 60 & 177 & 118 & 101 & 177 & 177 & 177 & 177 \\ Mean & 369.33 & 137.98 & 187.78 & 1.20 & 670.73 & 936.86 & 742.27 & 2.20 & 2.80 & 2.55 & 19.51 \\ Std. Dev. & 87.40 & 75.59 & 24.77 & 0.91 & 55.73 & 485.51 & 452.45 & 0.15 & 1.98 & 0.09 & 6.52 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Summarized values of the dataset used. where \(t\) and \(o\) are the predicted and observed data, respectively, and \(n\) is the total amount of data. In the same way, the results obtained with the ANN have been compared with others from classical regression methods, such as Gaussian Process Regression (Matern 5/2), Support-Vector Machines (SVM), and Linear Regression. Additionally, to determine the impact of the input parameters on the compressive strength values, new ANN have been carried out eliminating some of them. The following alternatives have been proposed: (i) unify the amount of cement (C) and fly ash (FA) in a single input parameter called the cementitious binder [42]; (ii) do not consider the superplasticizer input (SP), since the use of superplasticizer significantly reduces the amount of water (W) in the mix and both parameters could be related [43]; (iii) within the properties of the aggregates, the fineness modulus (FM) that indicates the average size of the sand particles; and iv) the maximum particle sizes \({}^{\text{TM}}\) of the coarse fractures may be candidates not to be taken into account in the new ANN. Figure 3: Distribution of errors for the training methods tested. (**a**): LM-15; (**b**): LM-20; (**c**) BR-15; (**d**): BR-20. Figure 2: Correlations between target and output data for the four cases tested. (**a**): LM-15; (**b**): LM-20; (**c**) BR-15; (**d**): BR-20. All the results obtained are good enough for estimating the compression strength of concrete with recycled aggregates. Furthermore, the BR training method has shown to be better than LM for obtaining predicted data perhaps due to the heterogeneity of the input data, which is justified by the different dosages and properties of the tested concretes. Especially, for the case of BR-20, 170 of 177 predicted compression resistances (96.05%) fall within a difference interval from \(-3\) to +5 units away from target data, while in the case of BR-15, 176 outputs are between \(-7\) and +8. In the case of LM-15, 170 outputs cover a range from \(-8\) to +7 units of differences from targets, and in the case of LM-20, 160 outputs cover a range from \(-5\) to +6. The test carried out with 20 hidden layers and Bayesian Regularization training method has been shown to be the best for all the metrics calculated. [PERSON] et al. [24] also developed an ANN model to predict the 28-day compressive strength of recycled aggregate concrete.