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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.
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