# Predictive Modeling of VO2 max Based on 20 m Shuttle Run Test for Young Healthy People
## 2 Material and Methods
### Predictive Methods  
Multiple input single output (MISO) model types were used in the study. Classic regression models, regularised regression models and artificial neural networks were used in the calculations of the models. All predictive models were calculated in R colorred software [43]. The implemented methods included:  
* The ordinary least squares (OLS) regression used a popular method of least squares, in which weights are calculated by minimizing the sum of the squared errors. The function lm to calculate the OLS was used. The criterion of performance \(J(\mathbf{w})\) takes the form: \[J(\mathbf{w})_{OLS}=\sum_{i=1}^{n}(y_{i}-\sum_{j=1}^{p}x_{ij}w_{j})^{2},\] (1) where \(n\)--total number of data, \(p\)--total number of input, \(x_{j}\)--input variable, \(y_{i}\)--output variable, \(w_{j}\) are unknown weights (parameters) of the model. * The Ridge model was calculated using the function lm.ridge from the \"MASS\" package. In Ridge regression [44], the criterion of performance includes a penalty for increased weights and takes the form: \[J(\mathbf{w},\lambda)_{Ridge}=\sum_{i=1}^{n}(y_{i}-\sum_{j=1}^{p}x_{ij}w_{j})^{ 2}+\lambda\sum_{j=1}^{p}w_{j}^{2}.\] (2) Parameter \(\lambda\geq 0\) decides the size of the penalty: the greater the value of \(\lambda\), the bigger the penalty. * Multilayer Percleptron (MLP)--to implement these methods, the RSNNS package was used [45]. Apart from the linear models, the artificial neural network was used. Two types of ANNs were applied: a multi-layer perceptron (MLP). MLPs are fully connected feedforward networks, and the most popular architecture used in applications. Training was performed by error backpropagation and logistical function as an activation function of hidden layers was used. * Artificial neural networks with a radial basis function (RBF)--the application of the RBF network is similar to MLP and used the RSNNS package [45]. The RBF are also feed-forward networks, but they have only one hidden layer. This network performs a linear combination of radially basis functions. The presented methods were used to calculate models from all variables (Table 2) and for subcollections of variables received after using selected reductive procedures of the input collection variables [46; 47]. Lasso regression is the first method analysed. Its mechanism facilitates assigning a penalty to variables and in this way they are eliminated from equations. The Lasso regression was obtained with the function enet from the \"elasticnet\" package. In Lasso regression [48], the norm \(L_{1}\) is used i.e., the sum of absolute values:  
\[J(\mathbf{w},\lambda)_{Lasso}=\sum_{i=1}^{n}(y_{i}-\sum_{j=1}^{p}x_{ij}w_{j})^ {2}+\lambda\sum_{j=1}^{p}|w_{j}|. \tag{3}\]  
Classic \"variable selection\" procedures are used when there is a high number of potential predictive variables (\(\mathbf{\Upsilon}\)) which means a high number of potentially calculable equations (solutions of input collections) [47]. The procedures introduce or eliminate individual variables from the equation and consist only of studying the subcollection of all possible equations. The following four selection procedures were used in the analysis:* Stepwise Forward Regression--The forward selection procedure begins with an equation which contains only a free expression. The first variable in the equation is the one which has the highest correlation with the **Y** variable. If the coefficient of regression of the variable differs significantly from zero, the variable remains in the equation and another variable is added. The second variable introduced into the equation is the one which has the highest correlation with **Y**. (**Y** has been adjusted for the effect of the first variable). If the regression coefficient is significant, adding the next variable is implemented in the same way. * Stepwise Backward Regression--The backward elimination procedure begins with an equation with all variables. One variable is removed in each step.