## 2 Materials and Methods
### Use of R Statistical Environment for Estimating Youngs Moduli
The process starts with the whole dataset and searches every distinct value of every predictor to find the predictor \(X_{k}\) and split value \(s\) that partitions the data into two groups \(S_{1}=\{X\,|\,X_{k}<s\}\) and \(S_{2}=\{X\,|\,X_{k}\geq s\}\) such that the overall sums of squares are minimised (15)  
\[Y=\beta_{0}+\beta_{1}X_{1}+\beta_{2}X_{2}+\ldots+\beta_{p}X_{p}+\epsilon \tag{3}\]3  
where \(\overline{y}_{1}\) and \(\overline{y}_{2}\) are the averages of the training set outcomes within groups \(S_{1}\) and \(S_{2}\), respectively [15]. The process continues within each group in the same manner until the number of observations in the splits falls below some threshold (usually, the threshold is 5). That kind of splitting is a type of greedy algorithm known as recursive binary splitting. At the end of the recursive process, the data are divided into \(N\) disjoint subsets \(S_{1}\), \(S_{2}\),..., \(S_{N}\) called terminal nodes or leaves. Each of the terminal nodes is presented by the average of the response variable of observations settled in that node. After the tree has been grown, it can be very large and complex and is likely to overfit the training set. The tree is then pruned back to a smaller tree (a subtree of the full tree constructed by the above procedure). The structure of the regression tree constructed by the described procedure highly depends on the training set. Bagging, short for bootstrap aggregation, is a general approach that uses the combination of bootstrapping and regression trees to produce a suitable prediction model. First, different training datasets are generated from the original data. On every bootstrapped training dataset, a regression tree is structured. For the given values of predictors, \(M\) predictions of the response variable \((\hat{y}_{1},\ \hat{y}_{2},\ \ldots,\ \hat{y}_{M})\) are obtained and then averaged to give the bagged model's prediction (16). \[Y=\beta_{0}+\beta_{1}X_{1}+\beta_{2}X_{2}+\ldots+\beta_{p}X_{p}+\epsilon \tag{3}\]4  
The described approach usually improves predictive performance for unstable models, like regression tree models. In the case of a large number of bootstrap samples, bagging is a computationally exhaustive method. Another disadvantage is that trees from different bootstrap samples may have similar structures to each other (especially at the top of the trees) due to a relationship between one strong predictor and response. This feature is known as tree correlation, and it prevents bagging from optimally reducing the variance of the predicted values \((\hat{y}_{1},\ \hat{y}_{2},\ \ldots,\ \hat{y}_{M})\)[15]. Adding randomness to the construction procedure can reduce the correlation among predictors. Random forest is such a method which at each split randomly selects \(m\) predictors from the whole set of \(p\) predictors. The number \(m\) is known as the random forests' tuning parameter, and the recommended choice is \(m=p/3\)_or_\(\sqrt{(}p)\) (in this paper, \(p=6\), so \(m=2\)). Since there is a collection of bagged trees, results of the random forest method and bagging are less interpretable than the results of a single tree model. However, measures of predictor importance can be constructed by combining measures of importance from the individual models across the set of \(M\) constructed trees. The total amount that the \(RSS\) is decreased due to splits over a given predictor is averaged over all \(M\) trees. A large value is interpreted as an important predictor.