Physical Chemistry 3: — Chemical Kinetics — - Christian-Albrechts ...

Appendix B 263
Appendix B: The Marquardt-Levenberg non-linear least-squares
fitting algorithm
With the omnipresence of the computer, the Marquardt-Levenberg algorithm (Bevington
1992, Press 1992) has become an extremely important data analysis tool. It is generally
themethodofchoiceincaseswhereitisnotpossibletomakesimplelinearplotsto
determine rate constants.
I Problem: Experimentally, we measure the values of some quantity at the points ,
, . Supposewetake measurements.Nowsupposewewouldliketodescribeour
data points by some model. Towards these ends, we take a physically reasonable
model function
= ( ;( 1 ))
(B.1)
which we would like to fit to our data in such a way that () describes the data
points in “the best possible way”. The model function depends directly on the
variables , and it also depends parametrically on nonlinear parameters .In
kinetics, for example, is usually some concentration, e.g. , which depends on time
i.e., = (). The parameters may be the rate constants (or decay times )
and the initial concentration 0 . Our job is to adjust and optimize these parameters
by somehow “fitting” them. The method of choice is, of course, “least-squares fitting”:
We start by defining the so-called figure-of-merit or goodness-of-fit function 54
2 =
"
#
X [ − ( )] 2
=1
2
(B.2)
Here, the and the independent variables , are the measured quantities, the
are the experimental uncertainties of the ,and ( ) is the calculated value
of the model fit function at the point { } (the parameters 1 are
omitted here in ). What we have to do is to adjust the parameters in the model
function such that 2 reaches a minimum, i.e. we have to search the -dimensional
parameter space for the (global) minimum of 2 .
I Solution: The Marquardt-Levenberg algorithm is a method to determine this minimum
of 2 as function of the fit parameters in the model fit function.
The minimum of 2 as function of the fit parameters a is defined by the conditions
"
#
2
= X [ − ( )] 2
=0 (B.3)
2
=1
X
∙ ¸
[ − ( )] ( )
= −2
(B.4)
2
=1
Thus, taking the partial derivatives of 2 with respect to each of the parameters
,weobtainasetof coupled equations in the unknown parameters . These
equations are, in general, nonlinear in the .
54 Deutsch: Fehlerquadratsumme.