UNIT 5 • CURVE FITTING & STATISTICS

Least Squares
Exponential Curve Fitting

Exponential Curve Fitting transforms non-linear growth or decay relationship $y = a e^{bx}$ into a straight line by taking natural logarithms, allowing normal equations to find optimal growth rates $b$ and initial scale $a$ !

LOGARITHMIC LINEARIZATION FORMULAEXPONENTIAL BEST FIT

y = a \cdot e^{b x} \implies \ln y = \ln a + b x \implies Y' = A + b x

1. Log TransformationLinear variable

Y' = \ln y

and intercept

A = \ln a

2. Normal System

N A + b \sum x = \sum Y'

and

A \sum x + b \sum x^2 = \sum x Y'

3. Re-ExponentiationRecover original multiplier:

a = e^A

Step-by-Step Worked Example

GUIDED STORY

OUR PROBLEM TO SOLVE

Fit an exponential curve $y = a e^{bx}$ to the 4 observation points:

x	1	2	3	4
y	1.6	4.5	13.8	40.2

STEP 01 • LOG TRANSFORM & STATISTICAL SUMS

x	y	Y' = ln(y)	x²	x · Y'
1	1.6	0.47000	1	0.47000
2	4.5	1.50408	4	3.00816
3	13.8	2.62467	9	7.87401
4	40.2	3.69387	16	14.77548
Sum = 10	-	Sum = 8.29262	Sum = 30	Sum = 26.12765

STEP 02 • SOLVE LINEAR NORMAL EQUATIONS

4 A + 10 b = 8.29262

10 A + 30 b = 26.12765

b = \frac{4(26.12765) - 10(8.29262)}{4(30) - (10)^2} = \frac{104.5106 - 82.9262}{20} = 1.0792

A = \frac{8.29262 - 10(1.0792)}{4} = -0.6249 \implies a = e^{-0.6249} \approx 0.5353

FINAL RESULT

Exponential Model: y ≈ 0.5353 · e^(1.0792x)

\boxed{y = 0.5353 \cdot e^{1.0792 x}}

Using logarithmic least squares regression, the best fitting exponential growth curve is y ≈ 0.5353 · e^(1.0792x)!

LABORATORY

Interactive Live Calculator & Visualizer

NON-LINEAR CURVE FITTING LABORATORY

Least Squares Fitting Engine

PreviousFitting Parabola

Unit 5•24 of 28

NextZ-Test (Significance)

∑

f(x)∫dxπ

NETZ Engine

Evaluating numerical algorithm

UNIT 5 • CURVE FITTING & STATISTICS

Least Squares
Exponential Curve Fitting

LOGARITHMIC LINEARIZATION FORMULAEXPONENTIAL BEST FIT

y = a \cdot e^{b x} \implies \ln y = \ln a + b x \implies Y' = A + b x

1. Log TransformationLinear variable

Y' = \ln y

and intercept

A = \ln a

2. Normal System

N A + b \sum x = \sum Y'

and

A \sum x + b \sum x^2 = \sum x Y'

3. Re-ExponentiationRecover original multiplier:

a = e^A

Step-by-Step Worked Example

GUIDED STORY

OUR PROBLEM TO SOLVE

Fit an exponential curve $y = a e^{bx}$ to the 4 observation points:

x	1	2	3	4
y	1.6	4.5	13.8	40.2

STEP 01 • LOG TRANSFORM & STATISTICAL SUMS

x	y	Y' = ln(y)	x²	x · Y'
1	1.6	0.47000	1	0.47000
2	4.5	1.50408	4	3.00816
3	13.8	2.62467	9	7.87401
4	40.2	3.69387	16	14.77548
Sum = 10	-	Sum = 8.29262	Sum = 30	Sum = 26.12765

STEP 02 • SOLVE LINEAR NORMAL EQUATIONS

4 A + 10 b = 8.29262

10 A + 30 b = 26.12765

b = \frac{4(26.12765) - 10(8.29262)}{4(30) - (10)^2} = \frac{104.5106 - 82.9262}{20} = 1.0792

A = \frac{8.29262 - 10(1.0792)}{4} = -0.6249 \implies a = e^{-0.6249} \approx 0.5353

FINAL RESULT

Exponential Model: y ≈ 0.5353 · e^(1.0792x)

\boxed{y = 0.5353 \cdot e^{1.0792 x}}

Using logarithmic least squares regression, the best fitting exponential growth curve is y ≈ 0.5353 · e^(1.0792x)!

LABORATORY

Interactive Live Calculator & Visualizer

NON-LINEAR CURVE FITTING LABORATORY

Least Squares Fitting Engine

PreviousFitting Parabola

Unit 5•24 of 28

NextZ-Test (Significance)

Least Squares Exponential Curve Fitting

Step-by-Step Worked Example

Exponential Model: y ≈ 0.5353 · e^(1.0792x)

Interactive Live Calculator & Visualizer

Least Squares Fitting Engine

NETZ Engine

Least Squares Exponential Curve Fitting

Step-by-Step Worked Example

Exponential Model: y ≈ 0.5353 · e^(1.0792x)

Interactive Live Calculator & Visualizer

Least Squares Fitting Engine

Least Squares
Exponential Curve Fitting

Least Squares
Exponential Curve Fitting