Exponential least squares fitting with a positive amplitude 
uses the model
 |
(1)
|
For data with 
and 
not all identical, taking the logarithm
of both sides gives
 |
(2)
|
Minimizing the sum of squared residuals in the logarithms gives
 |  |  |
(3)
|
 |  |  |
(4)
|
where 
and 
.
In general, this does not minimize the sum of squared errors in 
. An alternative is to minimize the weighted sum
 |
(5)
|
Applying least squares fitting gives
 |
(6)
|
 |
(7)
|
![[sum_(i=1)^(n)y_i sum_(i=1)^(n)x_iy_i; sum_(i=1)^(n)x_iy_i sum_(i=1)^(n)x_i^2y_i][a; b]=[sum_(i=1)^(n)y_ilny_i; sum_(i=1)^(n)x_iy_ilny_i].](/images/equations/LeastSquaresFittingExponential/NumberedEquation6.svg) |
(8)
|
Solving for 
and 
,
 |  |  |
(9)
|
 |  |  |
(10)
|
where again 
and 
. In particular, 
is the logarithm of the
exponential amplitude.
For small relative residuals, 
approximates 
. Thus the weights 
approximately weight the squared residuals in 
by 
. Using weights 
instead approximates equal weighting of the squared residuals in 
.
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).
