An exponential regression calculator is a powerful mathematical tool used to analyze and model data that exhibits exponential growth or decay.
Imagine you’re studying the growth of a bacterial colony. You measure the number of bacteria at different time intervals:
Time (hours) | Number of Bacteria |
---|---|
0 | 100 |
2 | 250 |
4 | 620 |
6 | 1550 |
An exponential regression calculator would use this data to determine the best-fitting exponential function, which might look like: y = 100 * e^(0.46x), where y is the number of bacteria and x is the time in hours.
Exponential Regression Calculation Chart
x (Time) | y (Observed) | y’ (Predicted) | (y – y’)^2 |
---|---|---|---|
0 | 100 | 100.00 | 0.00 |
2 | 250 | 248.70 | 1.69 |
4 | 620 | 618.30 | 2.89 |
6 | 1550 | 1537.46 | 157.29 |
Sum of (y - y')^2 = 161.87
R-squared = 0.9999 (indicating an excellent fit)
Exponential Regression Formula
The exponential regression formula is typically expressed as:
y = a * e^(bx)
Where:
- y is the dependent variable
- x is the independent variable
- a is the y-intercept (initial value when x = 0)
- b is the growth rate
- e is Euler’s number (approximately 2.71828)
For our bacterial growth example: y = 100 * e^(0.46x)
This formula tells us that the bacterial population starts at 100 and grows exponentially with a rate of 0.46 per hour.
How to Find Exponential Regression?
To find the exponential regression, follow these steps:
- Collect data: Gather paired (x, y) data points.
- Linearize the data: Take the natural log of y values to transform the exponential equation into a linear form: ln(y) = ln(a) + bx
- Perform linear regression: Use the transformed data to find the best-fitting line.
- Calculate a and b: a = e^(y-intercept), and b = slope
- Form the exponential equation: y = a * e^(bx)
For example, using bacterial growth data:
Transform y values: ln(100) = 4.61, ln(250) = 5.52, ln(620) = 6.43, ln(1550) = 7.35
Perform linear regression on (x, ln(y)) points
Find y-intercept (4.61) and slope (0.46)
Calculate a = e^4.61 ≈ 100, b = 0.46
Final equation: y = 100 * e^(0.46x)