# Multivariate Analysis for Prediction of Splitting Tensile Strength in Concrete Paving Blocks
## 3 Results
### Neural Networks  
4.1 Regression Using Neural Networks for the First Group of Predictors (Thickness, Width, Length, Mass of the Fresh Paving Block, and Percentage of Water Absorption)  
An artificial neural network model without hidden layers is made, with a normalized layer to be able to eliminate drawbacks regarding the units of measurement of the variables in different scales; an output neuron is taken into consideration since the variable to be predicted (tensile splitting strength) is continuous. It is trained with 100 epochs, and the learning rate is 0.1 for each learning stage, taking into account the gradient descent. The model performance or loss is a mean square error of 0.112198 using the ELU (Exponential Linear Unit) activation function for the neural network with no hidden layer. Figure 17 shows the neural network training calculating the loss for different epochs; the red line indicates the errors for the training dataset, and the blue line for the test dataset. Principal component analysis is applied to specify the number of optimal hidden layers in the neural network. Figure 18 shows that the cumulative variance proportion reaches 0.9999 with two principal components. So, two hidden layers are needed to explain 99.99% of the variability. For the first hidden layer, the number of neurons is determined by iterating from 2 to 10 neurons, calculating the mean square error (loss), which is lower with 10 neurons. The training process stabilizes with 40 epochs, which can be evidenced in Figure 19, whose result of the model for one hidden layer of 10 neurons through the ELU activation function gives a mean square error of 0.114271; the structure of the model can be seen in Figure 20. For the second hidden layer, the optimal number of neurons is defined by iterating the  
Figure 17: Training the artificial neural network with no hidden layer for the first group of predictors. Figure 18: Cumulative variance proportion according to the number of principal components for the first group of predictors. The values in the graph indicate the cumulative variance for different numbers of principal components. results of the mean square error with the activation function ELU; the optimal number of neurons is ten, and the training process is visualized in Figure 21, where the error for the training dataset is in red and blue for the test dataset, the mean square error of the model is 0.156214, and the architecture is visualized in Figure 22. The variables enter the model, creating paths through interconnected networks of nodes that represent neurons, each node processes the input, generating an output through the activation function, and during training, the weights are adjusted to minimize the prediction error. 4.2 Regression Using Neural Networks for the Second Group of Predictors (Density of the Fresh Paving Blocks and Percentage of Water Absorption)  
The neural network model without hidden layers, only with the normalization layer and one output neuron, using an ELU activation function, is trained with 100 epochs, giving a mean square error performance of 0.112402. Figure 23 shows the training with the errors of the training dataset and test dataset. Using principal component analysis, it is determined that one hidden layer explains 99.9% of the variability; this can be seen in Figure 24, and the number of neurons is determined by iteration from 2 to 10 using the ELU activation function. The model's performance results in a mean square error of 0.116783 with ten neurons for one hidden layer. Figure 19: Training of the artificial neural network for one hidden layer with ten neurons for the first group of predictors. Figure 20: The architecture of the artificial neural network for one hidden layer with ten neurons for the first group of predictors. The circles represent neurons, and the lines are the connections in the neural network.