Least Squares Method for Exponential Curve Fitting

The Least Squares Method for Exponential Fitting fits a curve of the form $y = a e^{bx}$ by transforming non-linear growth or decay data into a linear relationship using natural logarithms.

Linear Transformation ( $\ln y = \ln a + b x$ ):

Y' = A + b \cdot x \quad (Y' = \ln y, A = \ln a)

Where:

$A = \ln a \implies a = e^A$ .
$b$ is the exponential growth or decay rate coefficient.

Example of Exponential Curve Fitting

Given the following 4 observation pairs:

X	1	2	3	4
Y	1.6	4.5	13.8	40.2

Step 1 & 2: Log Transform & Statistics

\sum x = 10, \quad \sum Y' = 8.2926

\sum x^2 = 30, \quad \sum x Y' = 26.1276

Step 3: Solve for Rate ( $b$ ) and Intercept ( $a$ )

b = 1.0792, \quad A = -0.625 \implies a = e^{-0.625} \approx 0.5353

Conclusion

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

Therefore, using Exponential Least Squares Fitting, the curve equation of best fit is $y \approx 0.5353 \cdot e^{1.0792x}$ .

Interactive Exponential Curve Fitting Calculator

Least Squares Non-Linear Curve Fitting

PreviousFitting Parabola

Unit 5•24 of 28

NextZ-Test (Significance)

Least Squares Method for Exponential Curve Fitting

The Least Squares Method for Exponential Fitting fits a curve of the form $y = a e^{bx}$ by transforming non-linear growth or decay data into a linear relationship using natural logarithms.

Linear Transformation ( $\ln y = \ln a + b x$ ):

Y' = A + b \cdot x \quad (Y' = \ln y, A = \ln a)

Where:

$A = \ln a \implies a = e^A$ .
$b$ is the exponential growth or decay rate coefficient.

Example of Exponential Curve Fitting

Given the following 4 observation pairs:

X	1	2	3	4
Y	1.6	4.5	13.8	40.2

Step 1 & 2: Log Transform & Statistics

\sum x = 10, \quad \sum Y' = 8.2926

\sum x^2 = 30, \quad \sum x Y' = 26.1276

Step 3: Solve for Rate ( $b$ ) and Intercept ( $a$ )

b = 1.0792, \quad A = -0.625 \implies a = e^{-0.625} \approx 0.5353

Conclusion

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

Therefore, using Exponential Least Squares Fitting, the curve equation of best fit is $y \approx 0.5353 \cdot e^{1.0792x}$ .

Interactive Exponential Curve Fitting Calculator

Least Squares Non-Linear Curve Fitting

PreviousFitting Parabola

Unit 5•24 of 28

NextZ-Test (Significance)

Least Squares Method for Exponential Curve Fitting

Example of Exponential Curve Fitting

Step 1 & 2: Log Transform & Statistics

Step 3: Solve for Rate (bbb) and Intercept (aaa)

Conclusion

Interactive Exponential Curve Fitting Calculator

Least Squares Non-Linear Curve Fitting

NETZ Engine

Least Squares Method for Exponential Curve Fitting

Example of Exponential Curve Fitting

Step 1 & 2: Log Transform & Statistics

Step 3: Solve for Rate (bbb) and Intercept (aaa)

Conclusion

Interactive Exponential Curve Fitting Calculator

Least Squares Non-Linear Curve Fitting

Step 3: Solve for Rate ( $b$ ) and Intercept ( $a$ )

Step 3: Solve for Rate ( $b$ ) and Intercept ( $a$ )