Weighted Exponential Regression Calculator
Fit the exponential model y = a e to the bx power to frequency weighted (x, y) data using weighted least squares. Get coefficients a and b, the coefficient of determination, the correlation, total weight, and a scatter plot with the fitted curve.
Input
Enter frequency weighted (x, y) data, one point per line, to fit the exponential model y = a e to the bx power by weighted least squares.
One point per line. Separate x, y, weight with a comma or space. If the weight is omitted it is treated as 1. Keep y and weight greater than 0.
Result
Regression equation
y = 2.085051 e^(0.374277 x)
Coefficient a
2.085051
Coefficient b
0.374277
R² (determination)
0.999301
Correlation r
0.99965
Total weight
20
Data points
5
Scatter plot and regression curve
Data points and fitted values
| x | y | Weight | Fitted |
|---|---|---|---|
| 0 | 2.1 | 3 | 2.085 |
| 1 | 3 | 5 | 3.032 |
| 2 | 4.4 | 4 | 4.408 |
| 3 | 6.5 | 6 | 6.408 |
| 4 | 9.1 | 2 | 9.318 |
How it works
- The model is y = a e to the bx power, where a is a positive value greater than 0. Taking the natural log of both sides gives ln y = ln a + b x, which reduces to a straight line regression in x.
- Each data point carries a frequency (weight). Weighted least squares uses these weights so that points with larger frequencies influence the fit more strongly.
- Using the weighted means xbar and Ybar, b is the weighted covariance divided by the weighted variance, ln a is Ybar minus b times xbar, and a is its exponential.
- The coefficient of determination and the correlation are computed in the log transformed space Y = ln y, weighted by frequency. Note that they differ from values measured in the original y space.
- Enter y values greater than 0 and weights greater than 0. If all x values are identical the slope is undefined and the fit cannot be computed.
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Weighted Exponential Regression Calculator