Anatoly --
I think I'm siding with Scott on this, but maybe I'm no understanding
all your points. As far as I can tell, Tadij did not say that only N
would be varied in an anlysis -- The question was how to constrain the
total coordination number from a set of scattering paths: Shelly and
Scott answered that, pretty well I think.
Now, I agree that if 7 oxygen paths make up a "first shell" with a
narrow range of distances (though this was not actually clear from the
original question, it does seem likely), there's no way you'll be able
to measure 6 independent coordination numbers. But if there were 2
paths / atom types that made up a first shell, than relative occupancy
can certainly be determined. As a concrete example, determining the
number of Rb-Br and Rb-Cl neighbors in a salt solution is almost
certainly possible. ;). If the two atoms are the same species, they
just need to be sufficiently separated in distance.
On 12/17/05, Anatoly Frenkel
Hi Scott,
Thank you for pointing out at Vince's work on ferrites. It is a nice demonstration of the effect of site occupancy by dopants on EXAFS. In Vince's APL 68, 2082 (1996), they found the distribution of Zn between A and B sites (octahedral and tetrahedral) by including MS, as you described. However, he varied only one parameter in the fit to the MS range - site occupancy, keeping all other fixed (p. 2083). As such, it is not a non-linear least square fit, but a linear least square fit of a (linear) combination of fixed functions, and thus the chi-squared of such fit must have a single minimum - either within the range of x, or at the boundary (x=0 or 1, as they obtained in most cases).
I don't think it's important whether a linear or non-linear algorithm is used for the analysis, especially when compared to what was varied in the analysis (well, for some problems a linear approach is not possible, but even when a linear approach is possible it's not obvious that it's better than a non-linear approach). Whether or not there is a single minimum or not (and, related, what would make different minima distinguishable) is also more a matter of the problem, and not the algorithm used to find a solution. For a simple problem with 1 variable for relative intensities of two basis functions, it is very likely to have one minimum, no matter whether a linear or non-linear approach is used (as an aside, 1 variable does not always mean one minimum, but we're usually dealing with well-behaved problems). In any event, claiming that linear v. non-linear algorithm makes a difference is a dangerous and/or silly position to take for someone analyzing XAFS data. We have to use non-linear methods -- do you find this troubling?
What Tadej is describing seems more like a general, non-linear least squares fit problem, that has N minima in chi squared of which N-1 are false.
False? What does that mean? Sure, there may be multipe minima in chi-square. If so, either these can be distinguished because some have lower chi-squares than others, or that cannot be distinguished because the chi-squares are close. I assume that by using the word False that you somehow don't believe statistics or the model used. In my view, since one is making a model to describe a distribution of atoms, "False" is a useles concept -- all models are false in the sense that they are simplifications of reality, but some models are better than others. In this example, if 6 relative coordination numbers were used, I would expect the error bars to blow up, which means local minima in chi-square would be within the estimated errors, and there would not be N-1 "False minima", but 1 very broad minima. Perhaps you have a counter-example?
It is counter-intuitive to assume that the contributions of EXAFS from each inequivalent Nb can be fixed in the fit except for its fractional occupancy. What about sigma2 of different MS paths?
I don't think Tadij said this. But it does sound like what Harris et al may have done (ie, if they only had one unknown in their model, they must have made some assertions abou the other parameters. Does that not bother you as much??
... Thus, you obviously reduce the number of variables but you may exclude the true, physical, minimum of chi squared from your parameter space.
Again, I think I do not understand. What does "True, physical, minimum of chi squared" mean? You make a model, you do a fit, you understand the statistical parameters and consider making other models. In the end you pick the model that fits your data best and has the most sensible interpretation. I would consider "True, physical" to mean the complete description of the distribution of atoms sampled. A typical measurement samples the local coordination of around 10^9 atoms. If we have good data, we may be able to fit as many as 10 parameters to describe the first shell of the partial pair distribution function of these 10^9 atoms. So "True" and "Physical" are a bit far removed from our ability to see with our data. We must make model for the distribution and compare these models to our data. Generally, one picks the model that best matches the data as the most likely model. It's often called "best fit", but rarely called "True". Are you worried about the situation in which there are minima in chi-square that are clearly distinct (outside the error bars) and for which the model with significantly lower chi-square is a worse physical explanation (or disagrees with other measurements)? This seems to be a common fear, but I find it to be largely unfounded. Can you (or anyone) give an example of this? There seem to be a lot of stories about this, but most of the stories I've heard end up being at least partially due to ignoring (or not even trying to estimate) the error bars (and often partially due to someone wanting to believe something that their data doe not support). Of course, you can certainly pick multiple models and get different results. Is that a problem?? Since I'm not understanding your concerns with the way analysis is done with Ifeffit, perhaps you can clarify these concerns. Thanks, --Matt