## 2 Materials and Methods
### Use of R Statistical Environment for Estimating Youngs Moduli  
Data analysis was performed in the R statistical environment. R is a free software environment used in the academic community for modelling and graphics. In this paper, version 4.1.0 on 64-bit Windows 10 operating system was used. In addition, some extra packages were used: _leaps_ for displaying all linear models according to \(R^{2}\), _tree_ for the random tree analysis, and _randomForest_ for bagging and the random forest model. Linear regression is a very simple and straightforward approach for predicting a response variable \(Y\) based on a predictor \(X\). It is assumed that there is a linear relationship between \(Y\) and \(X\) which is described by (2)  
\[Y=\beta_{0}+\beta_{1}X, \tag{2}\]  
where \(\beta_{0}\) and \(\beta_{1}\) are constants. As a linear function is determined by precisely two different points and a dataset is always bigger, the sign \"=\" in the last equation is changed to \"\(\approx\)\". In other words, for each point, there is an error. There are many points (usually all of them) that the linear function does not pass through. We say that the response variable \(Y\) is approximately modelled by the predictor variable \(X\). This idea can be easily generalised for more than one predictor. A multiple linear regression model can be stated as follows. It is assumed that there are \(p\) predictors \(X_{1}\), \(X_{2}\),..., \(X_{p}\) and a response variable \(Y\). The model is given by the Formula (3)  
\[Y=\beta_{0}+\beta_{1}X_{1}+\beta_{2}X_{2}+\ldots+\beta_{p}X_{p}+\epsilon \tag{3}\]  
where \(\beta_{0}\), \(\beta_{1}\),..., \(\beta_{p}\) are regression coefficients and \(\epsilon\) is a random error term with zero mean. The coefficient \(\beta_{0}\) is also known as the intercept term. The interpretation of the coefficients is as follows. If \(X_{i}\) increase for one unit, holding all other predictors fixed, the regression coefficient \(\beta_{i}\) equals the average change of \(Y\). In practice, the coefficients are unknown and estimated using the least squares method [14]. In the case of one predictor (\(k=1\)), the data are organised in a set of points (\(x_{i}\), \(y_{i}\)) where \(x_{i}\) is a measurement of \(X\) and \(y_{i}\) is a measurement of \(Y\) that corresponds to \(x_{i}\). The least squares method gives \"the best\" line that fits the data--so that \(y_{i}\approx\beta_{0}+\beta_{1}\ x_{i}\), for \(i=1\),..., \(n\). In the case of two predictors (\(k=2\)), the method gives \"the best\" plane that fits the data. In the case of more than two predictors, there is a lack of intuition and imagination, but the model is still correct and interpretable in the described way. After the appropriate model is fitted, the goodness of model fit is required. There are a few measures of goodness of model fit which can be useful for model selection. The quantity which is often used as a measure of the quality of the model fit is the coefficient of determination, denoted by \(R^{2}\) and given by (4)  
\[R^{2}=(TSS-RSS)/TSS=1-RSS/TSS \tag{4}\]  
\[RSS=\sum_{i=1}^{n}(y_{i}-\hat{y_{i}})^{2} \tag{5}\]  
\[TSS=\sum_{i=1}^{n}(y_{i}-\overline{y})^{2} \tag{6}\]  
where \(RSS\) is the residual sum of squares calculated according to Formula (5), \(TSS\) is the total sum of squares calculated according to Formula (6), \(\hat{y_{i}}\) is the predicted value for \(y_{i}\) using \(x_{i}\) and a statistical method, and \(\overline{y}\) is the overall sample mean. Total variance in the response variable (\(TSS\)) can be divided into the variance explained by the fitted model (\(TSS\)-\(RSS\)) and the variance that is left unexplained (\(RSS\)). The \(R^{2}\) statistic takes the form of the proportion of the variability in the response variable that is explained by the model. If \(R^{2}\) takes the value close to 1, that indicates that a large amount of variability in the response variable has been explained by the fitted model. However, it is clear that \(R^{2}\) will increase when more variables are added to the model.