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Quadratic Regression

Aquadraticregression is the process of finding the equation of theparabolathat best fits a set of data. As a result, we get an equation of the form:

$y=a{x}^2+bx+c$where$a\ne 0$.

The best way to find this equation manually is by using the least squares method. That is, we need to find the values of$a,b,\text{and}c$such that the squared vertical distance between each point$\left(x_i,y_i\right)$and the quadratic curve$y=a{x}^2+bx+c$is minimal.

The matrix equation for the quadratic curve is given by:

$\left[\begin{array}{ccc}\sum x_i{}^4& \sum x_i{}^3& \sum x_i{}^2\\ \sum x_i{}^3& \sum x_i{}^2& \sum x_i\\ \sum x_i{}^2& \sum x_i& n\end{array}\right]\left[\begin{array}{c}a\\ b\\ c\end{array}\right]=\left[\begin{array}{c}\sum x_i{}^2{y}_i\\ \sum x_i{y}_i\\ \sum y_i\end{array}\right]$

The relative predictive power of a quadratic model is denoted by$R^2$.

This can be obtained using the formula:

$R^2=1-\frac{\text{SSE}}{\text{SST}}$where$\text{SSE}=\sum {\left(y_i-a{x}_i{}^2-b{x}_i-c\right)}^2$and$\text{SST}={\sum \left(y_i-\bar{y}\right)}^2$

The value of$R^2$varies between$0$and$1$. The closer the value is to$1$, the more accurate the model is.

But these are very tedious calculations. So, we will use a graphing calculator to automatically calculate the curve.