Statistics - Quadratic Regression Equation
Quadratic regression is deployed to figure out an equation of the parabola which can best fit the given set of data. It is of following form:
${ y = ax^2 + bx + c \ where \ a \ne 0}$
Least square method can be used to find out the Quadratic Regression Equation. In this method, we find out the value of a, b and c so that squared vertical distance between each given point (${x_i, y_i}$) and the parabola equation (${ y = ax^2 + bx + 2}$) is minimal. The matrix equation for the parabolic curve is given by:
$ {\begin{bmatrix} \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{bmatrix}
\begin{bmatrix} a \\ b \\ c \end{bmatrix}
=
\begin{bmatrix} \sum {x_i}^2{y_i} \\ \sum x_iy_i \\ \sum y_i \end{bmatrix}
}$
Correlation Coefficient, r
Correlation coefficient, r determines how good a quardratic equation can fit the given data. If r is close to 1 then it is good fit. r can be computed by following formula.
${ r = 1 - \frac{SSE}{SST} \ where \\[7pt]
\ SSE = \sum (y_i - a{x_i}^2 - bx+i - c)^2 \\[7pt]
\ SST = \sum (y_i - \bar y)^2
}$
Generally, quadratic regression calculators are used to compute the quadratic regression equation.
Example
Problem Statement:
Compute the quadratic regression equation of following data. Check its best fitness.
| x | -3 | -2 | -1 | 0 | 1 | 2 | 3 |
|---|
| y | 7.5 | 3 | 0.5 | 1 | 3 | 6 | 14 |
|---|
Solution:
Compute a quadratic regression on calculator by putting the x and y values. The best fit quadratic equation for above points comes as
${ y = 1.1071x^2 + 0.5714x }$
To check the best fitness, plot the graph.
So the value of Correlation Coefficient, r for the data is 0.99420 and is close to 1. Hence quadratic regression equation is best fit.
