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Quadratic regression is the process of finding the equation of a parabola, that best fits your dataset.
You can identify a quadratic regression by the plotting of your scatterplots. If the scatterplots are in a shape looking like a “U” (concave up), or the scatterplot are plotted in a shape like an up-side down U like “∩” (concave down), then you can say that you have a Quadratic regression at your hand which is best fitting your data. The shape of the scatterplots are not always complete. So you might see a half U or just a 3/4th of it.
As being an extension of simple linear regression, we can say that the major difference between the both is that in linear regression, a straight line can be drawn using only two points too. But when it comes to Quadratic regression, as it is a parabolic curve, you need as much points as possible to plot a perfect curve. Due to this disadvantage of Quadratic regression, it is generally more costly than simple linear regression.
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y = ax2 + bx + c
a = { [ Σ x2 y * Σ xx ] – [Σ xy * Σ xx2 ] } / { [ Σ xx * Σ x2 x2] – [Σ xx2 ]2 }
b = { [ Σ xy * Σ x2 x2 ] – [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,
a, b, c are the coefficients of Quadratic regression.
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 ]
a, b, c are the coefficients of Quadratic regression.
Let’s take an example to understand how the formula works around the give data.
Example: draw a second degree polynomial with polynomial regression for the given dataset.
x values | y values |
5 | 3 |
6 | 2 |
4 | 4 |
In our case, n=3.
x | y | x2 | x3 | x4 | xy | x2y |
5 | 3 | 25 | 125 | 625 | 15 | 75 |
6 | 2 | 36 | 216 | 1296 | 12 | 72 |
4 | 4 | 16 | 64 | 256 | 16 | 64 |
∑x | ∑y | ∑x2 | ∑x3 | ∑x4 | ∑xy | ∑x2y |
15 | 9 | 77 | 405 | 2177 | 43 | 211 |
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Σ x x = [ Σ x 2 ] – [ ( Σ x )2 / n ]
∑xx = [77] – [(15)2 / 3]
∑xx = 77 – [225/3]
∑xx = 77 – 75
∑xx = 2
Σ x y = [ Σ x y ] – [ ( Σ x * Σ y ) / n ]
∑xy = [43] – [(15 x 9) /3]
∑xy = 43 – [135/3]
∑xy = 43 – 45
∑xy = -2
Σ x x2 = [ Σ x 3 ] – [ ( Σ x 2 * Σ x ) / n ]
∑xx2 = [405] – [(77 x 15) / 3]
∑xx2 = 405 – [1155 / 3]
∑xx2 = 405 – 385
∑xx2 = 20
Σ x2 y = [ Σ x 2 y] – [ ( Σ x 2 * Σ y ) / n ]
∑x2y = [211] – [(77 x 9) / 3]
∑x2y = 211 – [693 / 3]
∑x2y = 211 – 231
∑x2y = -20
Σ x2 x2 = [ Σ x 4 ] – [ ( Σ x 2 )2 / n ]
∑x2x2 = [2177] – [(77)2 / 3]
∑x2x2 = 2177 – [5929 / 3]
∑x2x2 = 2177 – 1976
∑x2x2 = 201
a = { [ Σ x2 y * Σ xx ] – [Σ xy * Σ xx2 ] } / { [ Σ xx * Σ x2 x2] – [Σ xx2 ]2 }
a = {[-20 * 2] – [-2 * 20]} / {[2 * 201] – [20]2}
a = {(-40) – (40)} / {(402) – (400)}
a = -80 / 2
a = -40
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b = { [ Σ xy * Σ x2 x2 ] – [Σ x2y * Σ xx2 ] } / { [ Σ xx * Σ x2x 2] – [Σ xx2 ]2 }
b = {[-2 * 201] – [-20 * 20]} / {[2 * 201] – [20]2}
b = {[-402] – [-400]} / {(402) – (400)}
b = -2 / 2
b = -1
c = [ Σ y / n ] – { b * [ Σ x / n ] } – { a * [ Σ x 2 / n ] }
c = [9 / 3] – {-1 * (15 / 3)} – {-40 * (77 / 3)}
c = 3 – [-1 * 5] – [-40 * 25.66]
c = 3 – (-5) – (-1026.4)
c = 1034.4
y = ax2 + bx + c
y = -40x2 + (-1x) + 1034.4
y = -40x2 – x + 1034.4
Hence, the Quadratic regression equation of your parabola is y = -40x2 – x + 1034.4
Apart from this, there are various online Quadratic regression calculators that make your task easy and save all these steps and give the Quadratic regression equation straight away.
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Hindol Basu
GM, Voxco Intelligence
English
