Least Squares Fitting--Exponential
LeastSquaresExp
To fit a functional form
| y=Ae^(Bx), |
(1)
|
take the logarithm of both sides
| lny=lnA+Bx. |
(2)
|
The best-fit values are then
where B=b and A=exp(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
Applying least squares fitting gives
![画像: [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].](/index.cgi/contrast/https://mathworld.wolfram.com/images/equations/LeastSquaresFittingExponential/NumberedEquation6.svg){kind=link}
Solving for a and b,
In the plot above, the short-dashed curve is the fit computed from (◇) and (◇) 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 LawExplore with Wolfram|Alpha
WolframAlpha
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Cite this as:
Weisstein, Eric W. "Least Squares Fitting--Exponential." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/LeastSquaresFittingExponential.html