software to fit optical spectra - Quantum Materials Group

influenced by the number of parameters. It rather depends on the adequacy of the model to the
experimental data and the success of the initial approximation.
2.1.2. Simultaneous fitting of several datasets of different types
So far we considered the fitting of only one dataset by a single model. It is rather
straightforward to extend the discussion to a case, when several datasets of different
experimental types have to be fitted simultaneously with several models.
Let us consider Q datasets, while an ν -th dataset ( ν = 1KQ
) contains N ν datapoints:
ν ν ν
{ x i , yi
σ i } ( i = 1K
Nν
). Suppose that an ν -th dataset should be fitted by its own model
fν ( x,
p1,
K,
pM
) . Although in this notation all models depend formally on the same set of
parameters, it does not imply that every model really depends on all parameters. In other
words, some derivatives ∂ fν / ∂pk
may be equal to zero by definition. It is important, however,
that different models may depend on the same parameters.
Our goal is to fit several datasets simultaneously. For each dataset a separate chi-square
term can be written:
χ
Equation 2-11
⎛ y − f
⎝
( x , p K p
≡ ∑ ⎜
=
ν N ν
ν
2
i ν i 1
ν
ν
i 1 σ i
M
) ⎞
⎟
⎠
We can compose the total chi-square to be minimized:
2
χ
=
Equation 2-12
Q
∑
ν = 1
w
2
ν χν
2
Here w ν are the ‘weights’ of individual chi-square terms that have to be adjusted, as discussed
below. The definitions of β k (Equation 2-2) and α kl (Equation 2-6) should be modified
accordingly:
Q Nν
ν
ν
yi
− fν
( xi
, p1,
K, pM
) ∂fν
ν
β k = wν ∑
( x , 1,
,
2
i p K p
ν
( σ ) ∂p
∑
ν = 1 i= 1
i
k
Q Nν
1 ∂fν
ν
∂fν
ν
α kl = ∑ wν ∑ ( x , 1,
, ) ( , 1,
, )
2
i p K pM
xi
p K pM
.
ν
ν = 1 i= 1 ( σ i ) ∂pk
∂pl
The remaining part of the LM algorithm goes exactly as in 2.1.1.
The weight coefficients w ν deserve special remarks. Rigorously speaking, they should be
equal to 1, provided that the spreads of all data points are statistically independent. However,
due to the systematic error bars, this assumption is obviously not correct. For instance, the shift
of the reflectivity coefficient, caused by the reference mirror imperfection, is not very different
for two spectrally close data points. The second problem is that the error bars ν
σ i are not
always well known. Therefore, it is often necessary to ‘tune’ the weight coefficients in order to
Guide to RefFIT Page 12
M
) ,