Weighted Power Regression

Fit a power model y = a x^b to frequency-weighted data points using least squares, showing the coefficient, exponent, and R-squared with a scatter plot.

Input

Enter (x, y, frequency) on each line to fit the power model y = a x^b with frequency-weighted least squares. A line without a frequency is treated as frequency 1.

Data points (x, y, frequency)

One point per line. x and y must be positive. Example: 2, 5.5, 5

Result

Regression equation

y = 2.0249 x^1.4198

Coefficient a

2.024853

Exponent b

1.419835

R-squared

0.9998

Total frequency

Data points

Scatter plot and regression curve

Data and fitted values

No.	x	y	Frequency	Fitted
1	1	2	3	2.0249
2	2	5.5	5	5.4176
3	3	9.6	2	9.6345
4	4	14.4	4	14.495
5	5	19.8	1	19.8983

How it works

Enter each data point as (x, y, frequency) on its own line. A line without a frequency is treated as frequency 1.
Taking the natural logarithm of y = a x^b gives ln y = ln a + b ln x, reducing the problem to a straight-line regression in X = ln x and Y = ln y.
The exponent and coefficient are found with frequency-weighted least squares: b = sum w (X - mean X)(Y - mean Y) / sum w (X - mean X) squared, and a = exp(mean Y - b times mean X).
R-squared is computed in log space from the weighted total and residual variation as R-squared = 1 - residual sum of squares / total variation. A value close to 1 indicates a good fit.
Both x and y must be positive. Points with values of zero or below cannot be used because the logarithm is undefined there.

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