# Multivariate Analysis for Prediction of Splitting Tensile Strength in Concrete Paving Blocks
## 2 Materials and Methods
The ensemble method is the algorithm that analyzes the predictions as a whole, obtaining the predictions of each individual tree with different random subsets [27]. Analyzing the predictions together will yield better results than just one prediction. Random forests are trained by bagging with max_samples. **Principal component analysis. ** It is a dimension reduction technique that occupies the orthogonal transformation so that a group of correlated \(n\)-dimensional variables can maintain their variability information in other uncorrelated \(k\)-dimensional ones. The general process consists first of data standardization so that the base has a mean of zero and a variance of one. The covariance matrix, correlation matrix, eigenvectors, and eigenvalues are calculated. The first eigenvectors representing the most significant variability are chosen [31]. Research carried out by [32] indicates that this technique was developed by [PERSON] and [PERSON] independently. The technique linearly transforms multivariate data into a new uncorrelated set of variables. The eigenvectors are vectors that do not change position when a data transformation occurs and represent the axis of maximum variance, called the principal component. According [33], principal components are commonly defined as the matrix multiplication between the eigenvectors of the correlation matrix (A) and the standardized variables (\(X^{*}\)). \[z=A^{T}\ X^{*} \tag{25}\]The principal components calculated by covariance have a drawback, and it is the sensitivity to the units of measurement, for which it is carried out using the correlation matrix with the standardized variables, since each variable has a different unit of measurement. Also, the sizes of the variances of the principal components have the same implications in correlation matrices as in covariance matrices. One of the properties of the principal components using the correlation matrix is that they do not depend on the absolute values of the correlation. According to the study in [34], principal component analysis can determine the number of hidden layers in artificial neural networks, which represents sufficient variability for statistical analysis. On the other hand, the study by [35] shows the number of hidden layers in neural networks through principal component analysis to predict a continuous variable, giving good results through quality measures with optimal performance. **Artificial neural networks. ** Neural networks were inspired by the biological capacity of the brain. The perceptron is a different neuron called a threshold logic unit, where the inputs are numbers just like the outputs, and each connection has a weight [27]. A fully connected layer has the outputs \(h_{W,b}(X)\), where \(X\) is the input matrix (instance rows and feature columns), \(W\) is the weight matrix (rows per input neurons and columns per artificial neuron), \(b\) is the polarization vector (connection weights of the bias neuron and the artificial neurons), and is the activation function. \[h_{W,b}(X)=(X\,W+b) \tag{26}\]  
Learning has a rule; the perceptron connections are strengthened when the error is reduced, receiving one instance at a time and making the predictions. Perceptron learning is performed by \(w_{i,j}(\mathit{next\ step})\):  
\[T_{i}=\frac{\hat{\beta}_{i}-\beta_{i}}{se\bigl{(}\hat{\beta}_{i}\bigr{)}}\ \sim\ t_{n-p-1} \tag{18}\]0  
where \(w_{i,j}\) is the weight of the connection between the input and output of the neurons, \(x_{i}\) is the input value of the instance, \(\hat{y}_{j}\) is the output value of the instance, \(y_{j}\) is the target output value, and \(\eta\) is the learning rate. The perceptron convergence theorem tells us that the algorithm converges to a solution if the instances are linearly separable. The multilayer perceptron, MLP, has an input layer (lower layer), hidden layers, and an output layer (upper layer). If the artificial neural network (ANN) has more than one hidden layer, it is called a deep neural network (DNN).