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 Ų, 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 Ų. 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 Ų, 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 Ų +/- 0.0018 to one that gave 0.0003 Ų +/- 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 Ų +/- 0.0004, would indeed be troubling, and good evidence that something about the model or the data was problematic.

—Scott Calvin

Sarah Lawrence College