Weighted ab Exponential Regression
Fit the exponential model y = a times b to the power x to frequency weighted data points. Weighted least squares on log transformed values gives the coefficient a, base b and coefficient of determination, with a scatter plot sized by frequency.
Input
Enter x, y and frequency (weight) per line, separated by commas or spaces. A missing frequency is treated as 1. Only positive y values are supported.
For example, enter one point per line like 0, 2.1, 3.
Result
Regression equation
y = 2.089578 × 1.444733 ^ x
Coefficient a
2.089578
Base b
1.444733
R squared
1
Total frequency
15
Number of points
5
Scatter plot and fitted curve
Data and fitted values
| No. | x | y | Frequency | Fitted |
|---|---|---|---|---|
| 1 | 0 | 2.1 | 3 | 2.09 |
| 2 | 1 | 3 | 5 | 3.019 |
| 3 | 2 | 4.4 | 2 | 4.361 |
| 4 | 3 | 6.3 | 4 | 6.301 |
| 5 | 4 | 9.1 | 1 | 9.104 |
How it works
- Fits the exponential model y = a times b to the power x, where a is the value at x equal to 0 and b is the multiplier for each unit increase in x, to frequency weighted data points.
- Taking the natural log of both sides linearizes the model to ln y = ln a + (ln b) x, then weighted least squares is solved in log space. The frequency w is treated as the weight of a point observed w times.
- The base b is obtained as exp of the log space slope, and a as exp of the log space intercept.
- R squared is computed from the residual variation relative to the total variation in log space.
- Only positive y values are supported because the log is taken. Enter positive numbers for frequencies.
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Weighted ab Exponential Regression