Curvilinear Regression – Problems & Solutions

Curvilinear regression is used when the relationship between variables is not linear (not a straight line).

Common forms:

1. Quadratic (Parabolic):

$$Y = a + bX + cX^2$$

2. Exponential:

$$Y = a b^X$$

3. Power function:

$$Y = aX^b$$

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🔵 Problem 1: Quadratic (Parabolic) Regression

🔹 Given Data

X	-2	-1	0	1	2
Y	7	2	1	2	7

Fit equation:

$$Y = a + bX + cX^2$$

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✅ Step 1: Normal Equations

For quadratic regression:

$$\sum Y = na + b\sum X + c\sum X^2$$ $$\sum XY = a\sum X + b\sum X^2 + c\sum X^3$$ $$\sum X^2Y = a\sum X^2 + b\sum X^3 + c\sum X^4$$

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✅ Step 2: Compute Required Sums

Prepare table:

X	Y	X²	X³	X⁴	XY	X²Y
-2	7	4	-8	16	-14	28
-1	2	1	-1	1	-2	2
0	1	0	0	0	0	0
1	2	1	1	1	2	2
2	7	4	8	16	14	28

Now totals:

$$\sum X = 0$$ $$\sum X^2 = 10$$ $$\sum X^3 = 0$$ $$\sum X^4 = 34$$ $$\sum Y = 19$$ $$\sum XY = 0$$ $$\sum X^2Y = 60$$ $$n = 5$$

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✅ Step 3: Substitute into Normal Equations

$$19 = 5a + 0 + 10c$$ $$19 = 5a + 10c$$

$$0 = 0 + 10b + 0$$ $$10b = 0$$ $$b = 0$$

$$60 = 10a + 0 + 34c$$

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✅ Step 4: Solve Remaining Equations

From (1):

$$5a + 10c = 19$$

From (3):

$$10a + 34c = 60$$

Multiply (1) by 2:

$$10a + 20c = 38$$

Subtract from (3):

$$(10a + 34c) - (10a + 20c)$$ $$14c = 22$$ $$c = 1.571$$

Substitute into (1):

$$5a + 10(1.571) = 19$$ $$5a + 15.71 = 19$$ $$5a = 3.29$$ $$a = 0.658$$

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✅ Final Equation

$$Y = 0.658 + 1.571X^2$$

(Since $b = 0$)

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🔵 Problem 2: Exponential Regression

🔹 Given

X	1	2	3	4
Y	2	4	8	16

Fit:

$$Y = ab^X$$

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✅ Step 1: Take Logarithm

$$\log Y = \log a + X\log b$$

Let:

$$Y' = \log Y$$

Now we fit a straight line:

$$Y' = A + BX$$

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Compute log values

X	Y	log Y
1	2	0.301
2	4	0.602
3	8	0.903
4	16	1.204

We can see:

Slope $B ≈ 0.301$

Intercept $A ≈ 0$

So:

$$\log b = 0.301$$ $$b = 2$$ $$a = 1$$

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✅ Final Equation

$$Y = 2^X$$

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🔑 Key Points for Exams

• Quadratic regression requires 3 normal equations

• If data is symmetric around zero → $b = 0$

• Exponential & power models are converted into linear form using logarithm

• Used when scatter diagram shows curve pattern