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

Appendix B 264
The Marquardt-Levenberg algorithm gives a recipy for finding the minimum of 2 using
a combination of the method of steepest descent (following the gradient of the hypersurface
defined by 2 as a function of the parameters 1 and, near the minimum
of 2 , a parabolic Taylor series expansion of the fitting function around the minimum).
At the end of a fit, we shall usually report the (± 2 standard deviations), the
standard deviation of the ’s, and the so-called reduced- 2 ,whichis 2 divided by the
number of degrees of freedom, = − , i.e.,
"
#
2 = 1 X [ − ( )] 2
(B.5)
−
2
=1
I Example 1: Exponential decay. Suppose we measure the time dependent concentration
() of a molecule in some quantum state, which is populated at some time 0
by a laser pulse and then decays monoexponentially. As a model fit function, we use
the expression
() = 1 exp [− 2 ( − 3 )] + 4 + 5
(B.6)
The parameters we are interested in are 2 (the decay rate coefficient) and 1 (the
initial amplitude or initial concentration). The parameter 3 just describes the trigger
delay time (= 0 ). We have also added the parameters 4 and 5 to describe a constant
background term (due to some offset of our experimental signal) and a linear drift in
this background term (perhaps due to some experimental instability, such as a gradual
temperature increase in the lab). Our figure of merit is then
" #
X
2 [ − ()] 2
=
(B.7)
=1
The (in general nonlinear) conditions for the minimum of 2 , from which we determine
the parameters ,are
" #
2
= X [ − ()] 2
X
∙ ¸
[ − ()] ()
= −2
=0
1 1
2
=1
2
=1 1
(B.8)
2
2
= (B.9)
(B.10)
Figure B.1 below shows a simple two-parameter fit (parameters 1 and 1) tosucha
signal.
2