## 2 Materials and Methods
### Methods
#### 2.2.2 Multiple Linear Regression  
MLR is one of the most used statistical techniques to model the relationship between an outcome (dependent variable) as a function of predictors (multiple independent variables). This function provides a way to understand and quantify how each independent variable influences the dependent variable, as well as to make predictions about the dependent variable given a specific set of values for the independent variables. There is an extensive body of literature where multiple linear regression (MLR) has been used as an alternative method to reduce the high computational costs of FEM. [PERSON] and [PERSON] (2024) [16] propose a methodology to quantify the uncertainty of the ultimate strength of rectangular plates due to the welding initial deflection using FEM and a Multiple Regression model.  
MLR extends the concept of simple linear regression by allowing for more than one predictor variable. This technique assumes that the dependent variable \(Y\) can be modelled as a linear combination of \(p\) predictor variables \(x_{1}\), \(x_{2}\), \(\ldots\), \(x_{p}\), plus an intercept term 0. The coefficients \(i\) represent the weights of each independent variable in the resulting \(Y\).  
Mathematically, this relationship can be expressed as:  
\[Y=\beta_{0}+\sum_{i=1}^{p}\beta_{i}X_{i}+\epsilon \tag{6}\]  
where the term \(\epsilon\) represents the error existing between the prediction by the model and the real data. Equation (6) presents a generalised formulation of linear regression, where each predictor is associated with a linear term. However, more complex relationships are possible, where coefficients are determined for interactions between different pairs of predictors or even for all pairs, including squared terms. These enhancements can offer several advantages over standard MLR models, improving the model's accuracy and its ability to capture complex relationships between variables. Squared terms and interaction terms are useful when the effect of a predictor on the dependent variable changes at different levels of the independent variable or the effect depends on the level of another predictor, respectively (see Equation (7)). Furthermore, the incorporation of squared and interactions terms becomes the model more flexible and can better fit the data. This results in the reduction in residual errors and a more accurate representation of the underlying relationships, leading to more reliable predictions for new data.  
\[Y=\beta_{0}+\sum_{i=1}^{p}\beta_{i}X_{i}+\sum_{i=1}^{p}\beta_{ii}X_{i}^{2}+ \sum_{i=1}^{p-1}\sum_{j=i+1}^{p}\beta_{ij}X_{i}X_{j}+e \tag{7}\]  
where  
* **Y** is the dependent variable (the maximum stress),
* \(\beta_{0}\) is the intercept term,
* \(\beta_{i}\) are the regression coefficients representing the effect of each independent variable \(X_{i}\) on the dependent variable,
* \(\beta_{ii}\) are the quadratic coefficients associated with \(X_{i}\),
* \(\beta_{ij}\) are the interaction coefficients between \(X_{i}\) and \(X_{j}\), and
* \(\epsilon\) is the error term, which captures the variability not explained by the model  
Replacing FEM with regression models for stress calculation in stiffened plates could offer several advantages: (i) computational efficiency: Once trained, regression models can provide faster calculations compared to FEM, especially for repetitive analyses, (ii) ease of use: Regression models could be more accessible to engineers without extensive FEM expertise, (iii) resource optimisation: Reduced computational requirements could lead to more efficient resource allocation [17; 18; 19; 20; 21].