# Multivariate Analysis for Prediction of Splitting Tensile Strength in Concrete Paving Blocks
## 5 Conclusions  
The study proposes different models to predict the tensile splitting strength through the explanatory variables in the concrete paving block production process and the percentage of water absorption in a company in Quito-Ecuador. The first group of predictor variables includes thickness, width, length, mass of the fresh paving block, and percentage of water absorption. The second group of predictor variables includes the density of the fresh paving block and the percentage of water absorption. The \(R\) programming language is used to carry out descriptive and inferential statistical analysis, multivariate models, and three-dimensional graphs, with the advantage and freedom that programming generates to deepen the investigation, allowing one to understand the behavior of the data and models. Additionally, _Python_ is used with the Anaconda distribution to use the _Keras_ and _TensorFlow_ packages with the _articulate_ library.  
The variable with the greatest influence on the prediction of indirect tensile strength is the percentage of water absorption, which significantly influenced the random forest study shown in Figures 12 and 14, which is corroborated by Equation (33) of multiple linear regression, where the coefficient is greater. The study allowed us to determine the capacity of the developed models, errors, and their practical advantages, with the conclusion that the multiple linear regression makes it easier to apply the values in the equation to obtain the punctual prediction with simple calculations; the regression tree allowed us to follow a path-specific conditional according to the values of predictors, while random forests require the use of software for their application, and neural networks, having greater flexibility to learn behaviors, such as nonlinear ones, require software, due to its high predictive capacity.  
Table 2 shows that the best model to predict the tensile splitting strength in the first group of predictors is multiple linear regression, with a mean square error (MSE) of 0.110086 and an adjusted coefficient of determination of 0.7974, followed by the neural network without hidden layers, with an MSE of 0.112198. The best model for the second group of predictors is the neural network without hidden layers, with a mean square error (MSE) of 0.112402, followed by multiple linear regression, with an MSE of 0.115044 and an adjusted coefficient of determination of 0.7897. The worst method used for prediction is the regression tree, with a mean square error of 0.165174 for the first group of predictors and 0.139050 for the second. Also, the regression tree method has the worst ability to avoid overfitting and greater risk of suffering from it, which can be seen in Figures 8 and 11. Therefore, it is concluded that it is possible to predict the tensile splitting strength through the predictor variables of the first and second groups, allowing us to determine in advance the results inferred to the population from the production process and, with the water absorption test, to guarantee the quality of tensile splitting strength of the paving block.  
Conceptualization, V.R.B.-R., W.J.Y.-V. and E.P.H.-G.; methodology, W.J.Y.-V. and E.P.H.-G.; software, V.R.B.-R. and E.P.H.-G.; validation, W.J.Y.-V. and E.P.H.-G.; formal analysis, V.R.B.-R., W.J.Y.-V. and E.P.H.-G.; investigation, V.R.B.-R.; writing--review and editing, V.R.B.-R.; visualization, V.R.B.-R.; supervision, W.J.Y.-V. and E.P.H.-G.; project administration, W.J.Y.-V. All authors have read and agreed to the published version of the manuscript.  
This work was supported by Carehi State Polytechnic University (www.upec.edu.ec).