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 ( ${ y = ax^2 + bx + c}$ ) 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

${ y = 1.1071x^2 + x + 0.5714 }$

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.