# Multivariate Analysis for Prediction of Splitting Tensile Strength in Concrete Paving Blocks
## 2 Materials and Methods
\[\hat{\beta}=\left(X^{T}X\right)^{-1}\ X^{T}\ Y \tag{17}\]  
If the errors follow normal distribution with constant variance, then the \(T\) statistic follows a student's t distribution with \(n\)-\(p\)-1 degrees of freedom and is given by  
\[T_{i}=\frac{\hat{\beta}_{i}-\beta_{i}}{se\bigl{(}\hat{\beta}_{i}\bigr{)}}\ \sim\ t_{n-p-1} \tag{18}\]  
The term \(se\bigl{(}\hat{\beta}\bigr{)}\) is the estimated standard deviation of \(\hat{\beta}_{i}\), where the null hypothesis indicates that \(H_{0}\): \(\beta_{i}=0\) and the alternative hypothesis is \(H_{a}\): \(\beta_{i}\
eq 0\). In the analysis of variance in the multivariate case, as in the case of simple linear regression, the total variability is equal to the variability explained by the model plus the unexplained variability or error. The \(F\) statistic follows an F distribution with \(p\) and \(n\)-\(p\)-1 degrees of freedom, where the null hypothesis is Ho: \(\beta_{1}=\beta_{2}=\beta_{3}=\cdots=\beta_{p}=0\) and the alternative hypothesis indicates that \(H_{a}\): _at least some of the \(\beta_{i}\
eq 0\)_.  
\[F=\frac{SSreg/p}{\frac{RSS}{(n-p-1)}}\ \sim\ F_{p,\ n-p-1} \tag{19}\]  
Adding the number of predictors increases \(R^{2}\), so \(R^{2}_{adj}\) is used. \[R^{2}_{adj}=1-\frac{RSS/(n-p-1)}{SST/(n-p)} \tag{20}\]  
**Regression trees. ** A decision tree is an algorithm in machine learning that can be used in regression and classification; that is, a white box where they are intuitive and easy to interpret. For the regression case, the tree, instead of predicting a class, predictsa value that is the average value across the training instances of the node. Instead of minimizing impurity, the regression tree minimizes the mean squared error (\(MSE\)) [27]. The cost function for regression is  
\[J(k,t_{k})=\frac{m_{left}}{m}\ MSE_{left}+\frac{m_{right}}\ MSE_{right}\ MSE_{ right} \tag{21}\]  
where \(m\) is the number of instances to the left or right, and \(MSE\) is the mean square error. \[MSE=\sum_{ie\ node}\left(g_{node}-y^{(i)}\right)^{2} \tag{22}\]  
\[\hat{y}_{node}=\frac{1}{m_{node}}\sum_{ie\ node}y^{(i)} \tag{23}\]  
The limits on the decision trees are perpendicular to the axis (orthogonal) and are sensitive to variations in training. The regression trees, according to [28], generate divisions of the database into more homogeneous groups, with a set of \"if\" and \"then\" conditions being easily interpretable with different predictors. A disadvantage that could occur is instability with minor changes in the data. The oldest and most widely used technique is CART by [29], whose methodology is to find the predictor and the dividing value of the base whose squared error is the smallest. \[SSE=\sum_{i\in S1}\left(y_{i}-\overline{y}_{1}\right)^{2}+\sum_{i\in S2}(y_{i }-\overline{y}_{2})^{2} \tag{24}\]  
\(\overline{y}_{1}\) represents the subgroup mean \(S_{1}\), \(\overline{y}_{2}\) represents the subgroup mean \(S_{2}\), and the division process continues within the sets \(S_{1}\) and \(S_{2}\) up to a stopping criterion. The predictor's relative importance can be calculated using \(SSE\), where the predictors higher up the tree or that are more frequent are the most important. **Random forests. ** In the research by [30], an algorithm called the random forest allows for predicting and reducing overfitting. The procedure consists of choosing the number of tree models to be built from 1 to m, obtaining an initial sample, and then training the tree model for each division; the predictors are randomly selected, the best one is chosen, and finally, the stopping criteria are used. Each tree model generates a prediction, and the m predictions are averaged to generate the final prediction. By randomly choosing the \(k\) variables in each division, their correlation decreases. Random forests are computationally more efficient tree by tree, and the predictors' importance can also be seen through the permutation or impurity methodology [28]. The tree bagging procedure reduces the prediction variance.