ab Exponential Regression (y=ab^x)

Fit y=ab^x to your (x,y) data by least squares. Get the coefficient a, base b, and R squared with a scatter plot and regression curve.

Input

Enter one (x, y) pair per line to fit the exponential model y = ab^x by least squares. Only positive y values are supported.

Data points (x, y)

One pair per line, written as x,y with a comma or x y with a space.

Regression equation

y = 2.006 × 1.503^x

Coefficient a

2.005848

Base b

1.503325

R squared

0.999836

Correlation

0.999918

Data points

Scatter plot and regression curve

Data points and predicted values

The model is y = a × b^x, where both a and b are assumed positive. Taking the natural logarithm of both sides gives ln y = ln a + (ln b)×x, so Y = ln y becomes a linear regression on x (linearization).
The linearized form Y = p + q×x is solved by least squares, then converted back with a = exp(p) and b = exp(q). Because y is logged, enter positive y values only.
R squared and the correlation coefficient are computed on the linearized x and ln y. The closer R squared is to 1, the better the fit on the logarithmic scale.
The natural base form y = a × e^(kx) relates to this model by b = e^k. You can convert between them with k = ln b and b = e^k.
Enter one (x, y) pair per line, separated by a comma as x,y or by a space as x y. Blank lines are ignored.

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