# Multivariate Analysis for Prediction of Splitting Tensile Strength in Concrete Paving Blocks
## 2 Materials and Methods
** The tensile splitting strength test is carried out using a hydraulic press, giving the measured load at failure in newtons, and the strength is calculated by applying the following formula indicated in the standard NTE INEN 3040 (2016). \[T=0.637\times k\times\frac{P}{S} \tag{2}\]  
where \(T\) is the paving block strength in MPa, \(P\) is the measured load at failure in newtons, \(S\) is the area of failure plane in mm\({}^{2}\) that results from the multiplication of the measured failure length and the thickness at the failure plane of the paving block, and \(k\) = 0.87 for a thickness of 60 mm. **Mahalanobis Distance. ** The importance of identifying outlier data in a database is that these can distort the statistical analysis, and therefore, the distance to the centroid and the shape are taken into account; the Mahalanobis distance takes these two premises into account [23]. The study by [24] indicates that in the multivariate field with Gaussian data, the Mahalanobis distance follows a chi-square distribution, where \(p\) means degrees of freedom and represents the number of variables. The Mahalanobis distance measures the amount of the standard deviation of an observation or individual from the mean of a distribution, considering correlations for multivariate analysis. The Mahalanobis distance transforms to a Euclidean distance when the covariance matrix is the identity matrix [25]. A multivariate normal distribution is defined as follows:  
\[f(X)=\bigg{(}\frac{1}{2\pi}\bigg{)}^{p/2}*|\Sigma|^{-1/2}*exp\bigg{\{}-\frac{1 }{2}(x-\mu)^{T}\Sigma^{-1}(x-\mu)\bigg{\}}, \tag{3}\]  
where \(\Sigma\) is the covariance matrix, and \(\mu\) is the mean vector. If \(X\) is a vector with \(p\) variables which follows a multivariate normal distribution \(X\sim N_{p}(\mu,\Sigma)\), then the Mahalanobis distance square \(D^{2}\) follows a chi-squared distribution with \(p\) degrees of freedom \(D^{2}\sim X_{p}^{2}\). Mahalanobis represents the distance between each data point and its center of mass and is defined by the following formula:  
\[D^{2}=\left(X-\mu\right)^{T}\Sigma^{-1}\left(X-\mu\right) \tag{4}\]  
**Simple linear regression. ** Simple linear regression allows one to relate two variables: variable Y, called response or dependent, and variable \(X\), predictor or explanatory. The regression of the two random variables is given by the expected value of \(Y\) when \(X\) takes a specific value (\(X=x\)). If we consider linear regression with intercept \(\beta_{0}\), slope \(\beta_{1}\), and represents the random error of \(Y_{i}\), Ref. [26] explains that the residuals \(\ell_{i}\) are \(y_{i}-\hat{y}_{i}\), where \(\hat{y}_{i}\) is the fitted value of y.  
\[Y_{i}=E(Y|X=x)+e_{i}=\beta_{0}+\beta_{1}x+e_{i} \tag{5}\]  
The popular method for obtaining \(\beta_{1}\) and \(\beta_{0}\) is _RSS_ ordinary least squares to minimize the difference between the observed and predicted values. The minimization is carried out by differentiating RSS concerning the coefficients \(b_{0}\) and \(b_{1}\) and setting it equal to 0. \[RSS=\sum_{i=1}^{n}\ell_{i}^{2}=\sum_{i=1}^{n}(y_{i}-b_{0}+b_{1}x_{i})^{2} \tag{6}\]  
The structural assumptions of the regression model are the linearity that explains that \(Y\) depends on \(x\) through linear regression, and the homoscedasticity indicates that the variance of the errors when \(X=x\) must be common, better explained as \(Var(e|X=x)=\sigma^{2}\), the normality assumption indicates that the errors must follow a normal distribution with 0 mean and variance \(\sigma^{2}\), and finally the independence of the errors.