*2.1. Linear and Quadratic Modeling Equations*

The linear model is relatively simple with a form given in Equation (1).

$$
\hat{y} = \beta\_0 + \sum\_{i=1}^{n} \beta\_i \mathbf{x}\_i \tag{1}
$$

where *y*ˆ is the predicted value for output variable *y*, the *x<sup>i</sup>* terms are the input values (there are *n* different inputs with subscripts *i*) and the *β* values are fitted parameters. Considering a quadratic expression, there will be a number of additional terms:

$$\mathcal{Y} = \beta\_0 + \sum\_{i=1}^{n} \beta\_i \mathbf{x}\_i + \sum\_{i=1}^{n} \sum\_{j=i}^{n} \beta\_{ij} \mathbf{x}\_i \mathbf{x}\_j \tag{2}$$

Including the linear terms from Equation (1) in addition to pair-wise combinations of different inputs, which can lead to a large number of terms and a large number of additional parameters *βij*, which need to be fitted.
