Least Squares Fitting--Exponential

To fit a functional form

$$y=Ae^{Bx},$$ (1)

take the Logarithm of both sides

$$\ln y=\ln A+B\ln x.$$ (2)

The best-fit values are then

$$a = {\sum \ln y\sum x^2-\sum x\sum x\ln y\over n\sum x^2-\left({\sum x}\right)^2}$$ (3)

$$b = {n\sum x\ln y-\sum x\sum \ln y\over n\sum x^2-\left({\sum x}\right)^2},$$ (4)

where $B\equiv b$ and $A\equiv\mathop{\rm exp}\nolimits (a)$.

This fit gives greater weights to small $y$ values so, in order to weight the points equally, it is often better to minimize the function

$$\sum y(\ln y-a-bx)^2.$$ (5)

Applying Least Squares Fitting gives

$$a\sum y+b\sum xy=\sum y\ln y$$ (6)

$$a\sum xy+b\sum x^2y=\sum xy\ln y$$ (7)

$$\left[{\matrix{\sum y& \sum xy\cr \sum xy& \sum x^2y\cr}}\ri... ...ght] = \left[{\matrix{\sum y\ln y\cr \sum xy\ln y\cr}}\right].$$ (8)

Solving for $a$ and $b$,

$$a = {\sum(x^2y)\sum(y\ln y)-\sum(xy)\sum(xy\ln y)\over \sum y\sum(x^2y)-\left({\sum xy}\right)^2}$$ (9)

$$b = {\sum y\sum(xy\ln y)-\sum(xy)\sum(y\ln y)\over \sum y\sum(x^2y)-\left({\sum xy}\right)^2}.$$ (10)

In the plot above, the short-dashed curve is the fit computed from (3) and (4) and the long-dashed curve is the fit computed from (9) and (10).

See also Least Squares Fitting, Least Squares Fitting--Logarithmic, Least Squares Fitting--Power Law