Hi
Bruce,
A side comment on a topic you’ve raised before. You said to Shaofeng
that:
On Jul 26, 2016, at 8:39 AM, Bruce Ravel <bravel@bnl.gov> wrote:
It also
means that the uncertainty is
such that you can support your
conclusion. While the red line might
overplot the blue with
sigma^2=0.0003+0.001, that may be a troubling
result because sigma^2 is not
positive definite!
I’m not convinced that it should be disturbing at all if a fit for sigma2
yields a result that is not positive definite.
Suppose, for a moment, that the true sigma2 for a scattering path is 0.0003
Å2, and that data is being analyzed up to k = 9
Å-1. The EXAFS equation tells us that the effect of sigma2 on chi(k)
is quite modest in that case, and is also relatively insensitive to the precise
value of sigma2. According to the EXAFS equation, at the top of the data range,
where it’s effect is greatest, the sigma2 factor, e^(-2k^2 sigma2), is
multiplying the amplitude by 0.95. Suppose further that the uncertainty was +/-
0.0005 Å2. That would imply the sigma2 factor was at the top of the
range was as small as 0.88 or as large as 1.03. It doesn’t seem terribly
different to me than a fit which yields an S02 of 0.95 +/- 0.08. Or, in a case
where coordination number is expected to be either 4, 6, or a mixture of the
two, N = 5.7 +/- 0.5. The latter result is not generally considered troubling,
even though the range implied by the uncertainty overlaps with values (N > 6)
that might be considered wildly implausible for the system being studied.
The comparison to coordination number or S02 is not perfect, because sigma2
is sensitive to the difference between the amplitude of chi(k) at low k and at
high k, whereas S02 or N correspond to a uniform suppression. Still, in a case
when the true value of sigma2 was 0.0003 Å2, the difference between
the chi(k) amplitude at the bottom of the k-range and the top is quite
modest, and might reasonably be statistically indistinguishable from no
difference at all, particularly if it is for one path in a multi-path fit, if
the data is somewhat noisy, or if the k-range is small.
To borrow a similar example from another specialty in physics, for quite
some time, measurements of the square of the electron neutrino mass often
yielded results that were not positive definite. This was taken neither as
evidence that the neutrino mass was imaginary, nor that the data was bad.
I worry that sending the message that it is troubling to get a result for
sigma2 that is not positive definite can lead to beginners rejecting such fits,
thus introducing a bias toward larger values of sigma2. Such a user might prefer
a model that generates sigma2 = 0.0020 Å2 +/- 0.0018 to one that gave
0.0003 Å2 +/- 0.0005, for example. To me, all else being equal, the
second result is better, because it is more precise.
Of course, a result that was negative definite, such as sigma2 = -0.0009
Å2 +/- 0.0004, would indeed be troubling, and good evidence that
something about the model or the data was problematic.
—Scott Calvin
Sarah Lawrence College
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