# Multivariate Analysis for Prediction of Splitting Tensile Strength in Concrete Paving Blocks
## 4 Discussion  
Based on the results and literature, it can be noted that regression trees are not a robust technique from the data for the prediction of tensile splitting strength. However, the neural network allows nonlinear behaviors to be learned, and based on the study carried out, quality can be guaranteed in terms of tensile splitting strength, knowing the explanatory variables in the production process, which is a contribution compared to related works, such as those that relate the components of the mixture, or a predictor variable which explains in a limited way the population behavior of the tensile splitting strength. Tensile splitting strength prediction using neural networks was studied by [22], resulting in a mean square error of 0.141; neural network models should be compared with different layers and numbers of neurons for better modelling, explaining the activation function used. Table 2 indicates a mean square error of 0.112198 for the neural network model without layers analyzed in the first group of predictors. The graphs presented in this research are a contribution to the understanding of the prediction method used and the behavior of the data, as in Figure 5, where the prediction of the response variable tensile splitting strength forms a plane whose value point moves three-dimensionally according to Equation (33), where the predictor variables create the projected dimension through linear behavior, which is very characteristic of multiple linear regression. Figure 10 allows us to graphically understand in three dimensions the conditions of the second group of predictor variables in the regression tree, where each rung predicts the tensile splitting strength. The prediction using random forest is represented in Figure 15 with an optimal number of 144 trees. Figures 25 and 26 show the nonlinear behavior for the prediction using artificial neural networks for the second group of predictor variables. It is observed in Equation (32) for the first group of predictors that the variable with the most significant influence for the prediction in MLR is the thickness followed by the percentage of water absorption; for random forests, the importance is shown in Figure 12, with the percentage of water absorption coming in first place, followed by the mass of the fresh paving block, the thickness, width, and length. For the second group of predictors, Equation (33) indicates a more significant influence on the percentage of water absorption, showing this importance in Figure 14, followed by the density of the fresh paving block. It is evident in Figure 1 that the density of the fresh paving block and the percentage of water absorption has a high correlation with the tensile splitting strength and indicates that the behavior of the data is linear for the prediction, which can be distinguished in Figures 3 and 5, which show the three-dimensional representation with its projection plane using multiple linear regression, and can be confirmed in Figures 18 and 24, where it can be seen that one principal component explains more than 90% of the accumulated variance in the dataset for the two groups of predictors. The overfitting was controlled by simplifying the model, regularizing and stopping the training just in time, taking into account the principle of parsimony; a low test error is achieved, generalizing the predictions well. Figures 1-3 indicate the behavior of the data. For the regression trees of the first group of predictors, Figure 8 shows a low test error, with eight nodes, and is the optimal training stopping point to avoid overfitting, since with more nodes and higher model complexity, the errors of training decrease, but validation errors increase, which makes new data poorly predicted.