[Ifeffit] about sigma square

[Bruce Ravel]

Thanks, Scott.

When I started writing that email, it was my intention to discuss 
correlations between sigma^2 and other parameters.  By the time I got to 
the end of the email, I had forgotten all about it.

You closing comment about how a weird, indefensible value for sigma^2 
(or any parameter) is a hint that something is flawed in the model is 
another very important point.

So, Shaofeng, please consider what Scott has to say.

B

On 07/26/2016 09:39 AM, Scott Calvin wrote:
> 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 at bnl.gov
>> <mailto:bravel at 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 Å^2 , 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 Å^2 . 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
> Å^2 , 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 Å^2  +/-
> 0.0018 to one that gave 0.0003 Å^2  +/- 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
> Å^2  +/- 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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-- 
  Bruce Ravel  ------------------------------------ bravel at bnl.gov

  National Institute of Standards and Technology
  Synchrotron Science Group at NSLS-II
  Building 743, Room 114
  Upton NY, 11973

  Homepage:    http://bruceravel.github.io/home/
  Software:    https://github.com/bruceravel
  Demeter:     http://bruceravel.github.io/demeter/