Quadratic Regression – Definition, Formula, Example

How to Calculate Quadratic Regression Equation? - Tutorial

Quadratic Regression Definition:

Quadratic regression is a type of multiple linear regression by which the equation of a parabola of 'best fit' is found for a set of data.

Formula:
Quadratic Regression Equation(y) = a x^2 + b x + c a = { [ Σ x2 y * Σ xx ] - [Σ xy * Σ xx2 ] } / { [ Σ xx * Σ x2x 2] - [Σ xx2 ]2 } b = { [ Σ xy * Σ x2x2 ] - [Σ x2y * Σ xx2 ] } / { [ Σ xx * Σ x2x 2] - [Σ xx2 ]2 } c = [ Σ y / n ] - { b * [ Σ x / n ] } - { a * [ Σ x 2 / n ] }
Where ,
Σ x x = [ Σ x 2 ] - [ ( Σ x )2 / n ] Σ x y = [ Σ x y ] - [ ( Σ x * Σ y ) / n ] Σ x x2 = [ Σ x 3 ] - [ ( Σ x 2 * Σ x ) / n ] Σ x2 y = [ Σ x 2 y] - [ ( Σ x 2 * Σ y ) / n ] Σ x2 x2 = [ Σ x 4 ] - [ ( Σ x 2 )2 / n ] x and y are the Variables. a, b, and c are the Coefficients of the Quadratic Equation n = Number of Values or Elements Σ x= Sum of First Scores Σ y = Sum of Second Scores Σ x2 = Sum of Square of First Scores Σ x 3 = Sum of Cube of First Scores Σ x 4 = Sum of Power Four of First Scores Σ xy= Sum of the Product of First and Second Scores Σ x2y = Sum of Square of First Scores and Second Scores
Example:

Draw a second degree polynomial with polynomial regression for the given data set.

x valuesy values
23
46
64

To find quadratic equation, coefficients of the quadratic equation

Step 1:

Count the given number of values (i.e), n = 3

Step 2:

Then find the following values

 Σ x   Σ y   Σ x 2   Σ x 3   Σ x 4   Σ xy    Σ x2y  
12 13 56 288 1568 54 252
Step 3:

Substitute the given values in the formula Σ x x = [ Σ x 2 ] - [ ( Σ x )2 / n ] Σ x x = [ 56 ] - [ 12 * 12 / 3 ] Σ x x = 8 Σ x y = [ Σ x y ] - [ ( Σ x * Σ y ) / n ] Σ x y = [ 54 ] - [ ( 12 * 13 ) / 3 ] Σ x y = 2 Σ x x2 = [ Σ x 3 ] - [ ( Σ x 2 * Σ x ) / n ] Σ x x2 = [ 288 ] - [ ( 56 * 12 ) / 3 ] Σ x x2 = 64 Σ x2 y = [ Σ x 2 y] - [ ( Σ x 2 * Σ y ) / n ] Σ x2 y = [ 252 ] - [ ( 56 * 13 ) / 3 ] Σ x2 y = 9.333 Σ x2 x2 = [ Σ x 4 ] - [ ( Σ x 2 )2 / n ] Σ x2 x2 = [ 1568 ] - [ 56 * 56 / 3 ] Σ x2 x2 = 522.6666

Step 4:

Calculate the value of a

a = { [ Σ x2 y * Σ xx ] - [Σ xy * Σ xx2 ] } / { [ Σ xx * Σ x2x 2] - [Σ xx2 ]2 } a = { [ 9.333 * 8 ] - [ 2 * 64 ] } / { [ 8 * 522.6666 ] - [64 * 64 ]} a = { 74.664 - 128 }/{4181.33 - 4096 } a = - 53.36 / 85.33 a = - 0.625

Step 5:

Calculate the value of b

b = { [ Σ xy * Σ x2x2 ] - [Σ x2y * Σ xx2 ] } / { [ Σ xx * Σ x2x 2] - [Σ xx2 ]2 } b = { [ 2 * 522.6666 ] - [ 9.333 * 64 ] } / { [ 8 * 522.6666 ] - [ 64 * 64 ] } b = { 1045.333 - 597.312 } / { 4181.33 - 4096 } b = 448.012 / 85.33 b = 5.25

Step 6:

Calculate the value of c

c = [ Σ y / n ] - { b * [ Σ x / n ] } - { a * [ Σ x 2 / n ] } c = [ 13 / 3 ] - { 5.25 * [ 12 / 3 ] } - { - 0.625 * [ 56 / 3 ] } c = 4.333 - 21 + 11.66 c = - 5.0003

Step 7:

Substitute the value of a,b and c in the Quadratic Regression Equation,

y = a x^2 + b x + c y = - 0.625 x^2 + 5.25 x - 5.0003

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