[Ifeffit] Chi^2-Map question

Matt Newville newville at cars.uchicago.edu
Thu Mar 4 08:27:00 CST 2004

Hi Norbert,
> I want to check with chi^2 maps if I can separate 2 shells, e.g. if a peak is 
> comprising two O shells or only one. For this I take the one-shell fit, half 
> the coordination number and then vary the distance of the shells against each 
> other (all other values taking from the best single shell fit). If I find two 
> contributions, I should see 2 mirror-symmetric minima centered at the 
> corresponding DR values, e.g. (0,-0.1) and (-0.1,0) if I work with the same 
> reference. 
> These minima are separated by a saddle point which is more or less pronounced. 
> In the case of some Fe complexes I simulated, it is clearly visible, while 
> for my gold oxides, it is hardly seen but still there. If I take a system 
> where I know there IS only one shell (e.g. Au metal) and do the map, I find 
> no saddle point and correspondingly no 2 minima.
> Does anyone know a relation between the height of a saddle point and the 
> physical significance of the map? I would be glad to learn more about, but 
> the statistic books I know don't deal with such a problem (and I am chemist 
> and not a mathematician)... 

If I understand correctly, you're making two-parameter maps of
chi-square, and see multiple minima with saddle points between them.  
I think the way to deal with this is to just use the usual
chi-square signficance tests:  The larger chi-square is, the less
likely the result.  If the saddle point is "low enough", the two
minima are really not distinct solutions, and the error bars have to
include both possibilities.  If the saddle point is "high enough"
there are two distinct solutions.

The main issue deciding what "low enough" means is that we don't
really know the uncertainty in our data.  Ifeffit uses the
time-honored cheat of asserting that increasing chi-square by
reduced-chi-square sets the error bars.  With that, the cut-off
would mean changing chi-square by approximately reduced-chi-square.

Do I categorically recommend using this simple cheat for your
application without further thought?  No way!!!!  It might be OK,
but you should look into this more carefully.  For Ifeffit, it's
important to always give a reasonable estimate of the uncertainties.
You're doing something more complicated.  I hope you might be able
to help Ifeffit do something better than what it does!!

That's probably not a very complete answer, but I hope it helps,


PS:  You probably don't want a mathematician (I'd guess most
     chemists would know more statistics than most mathematicians).  
     An experimental psychologist or population biologist is 
     probably a better bet.

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