Exponential Regression Calculator

Fit y = a e^(b x) to your (x, y) data points using least squares. Shows the coefficients a and b, R-squared, the correlation coefficient, a scatter plot with the fitted curve, and predicted values.

Input

Enter one (x, y) data point per line to fit the exponential regression y = a e^(b x) by least squares. Separate values with a comma or a space.

Data points (x, y)

One pair per line, for example 1, 2.7. Separate with a comma or space. Only positive y values are supported.

Result

Exponential regression equation

y = 0.9955 e^(1.0012 x)

Coefficient a

0.9955

Coefficient b

1.0012

R-squared

0.999997

Correlation r

0.999999

Scatter plot and fitted curve

Data points and predictions

x	Observed y	Predicted y
1	2.7	2.709339
2	7.4	7.373497
3	20.1	20.067054
4	54.6	54.612711
5	148.4	148.629103

How it works

The exponential model y = a e^(b x) becomes linear after taking the natural logarithm of both sides: ln y = ln a + b x. Setting Y = ln y and X = x turns it into a simple linear regression, so least squares on ln y gives the slope b and intercept ln a, and the coefficient a is recovered as a = e^(ln a).
Because natural logarithms are used, every y value must be positive. Points with y of zero or below cannot be processed. You also need at least two distinct x values, since identical x values leave the slope undefined.
R-squared and the correlation coefficient r are evaluated in the linearized space (x on the horizontal axis, ln y on the vertical axis). An R-squared close to 1 indicates that the data fits the exponential curve well.
A positive b gives a rising exponential curve and a negative b gives a falling one. The coefficient a equals the value of y at x = 0. Review the data points, the fitted curve, and the table of predicted values together.

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