[Ifeffit] Short questions

[Bruce Ravel]

On Wednesday 13 September 2006 13:29, Juan Antonio Maciá Agulló wrote:
> I have a couple of short questions for you. I used the "Scott Calvin's
> rule" (number of variables < 2/3*Nip) to calculate the maximum number of
> allowed free parameters but I read that some people use the Nyquist
> theorem, which are the differences between them? and, which one is more
> correct?

Hmmm... neat-o.  Some random fraction of the Nyquist criterion is now
known as "Scott Calvin's rule".  Cool ;-)

Scott mostly covered this in his answer.  I just wanted to add my
US$0.02.

The Fourier-based analysis we do in EXAFS takes many of its ideas from
signal processing.  In the cannonical signal processing problem, we
measure a time series -- for instance the signal coming from a radio
station.  We can do a Fourier transform of that time series and get a
frequency spectrum -- the notes in the music that the radio station is
broadcasting.  If we wanted to do some kind of analysis on the signal
we receive from the radio station, we can ask how much data could we
hope to extract from the signal.  Well, that quantity has something to
do with how long (in time) we measure the signal -- if we measure for
10 minutes we will have more information than if we measure for 5
minutes.  So the information content is somehow proportional to
delta_T (the amount of time spent measuring the time sequence).  If we
then choose to analyze only a narrow range of frequency -- say one
hertz to either side of middle C -- then we will be examining less
information than if we examine an entire octave of the signal.  So the
information content is somehow proportional to delta_f (the width of
the frequency band we examine).  This is the Nyquist criterion: the
information content in an analysis of a time sequence is proportional
to delta_T * delta_f.  It turns out the proportionality constant is
2/pi.

In EXAFS, chi(k) is analogous to the time sequence and chi(R) is
analogous to the frequency spectrum.  So the information content of
the EXAFS signal is, at most, 2 * delta_k * delta_R / pi.  That is the
Nip number computed by Ifeffit based on the range of the Fourier
transform and the range of the fit.  In EXAFS the data are not ideally
packed -- that is, EXAFS is not a sum of pure sine waves -- and the
data are often quite noisy.  So real data may not support Nip worth of
variable parameters.  What you called the "Calvin rule" is just a
crude rule of thumb stating that one should be uncomfortable when the
number of parameters starts getting close to the Nip because your real
data may not support the independent evaluation of that many
parameters.

> And finally, how are errors calculated in Artemis for the parameters N
> (coordination number), deltaE0, S02, deltaR and sigma^2?

Errors are NOT calculated in Artemis (or in Ifeffit for that matter)
for the path parameters, N, deltaE0, S02, deltaR and sigma^2.  Errors
are calculated for the guess parameters.  The path parameters are
written in terms of the guess (and set and def) parameters, possibly
by rather complicated math expressions.  If you want to know the
uncertainties in the evaluations of the path parameters, you need to
propagate the errors in the fitting parameters through those math
expressions.  Sadly, the software does not do that for you at this
time.

If you are asking how the errors in the guess parameters are computed,
well Ifeffit uses a Levenberg-Marquardt non-linear minimization.  This
involves the evaluation of a covarience matrix.  The uncertainties are
the diagonal elements -- with the caveat that they are scaled by the
square root of reduced chi-square.  That rescaling is conceptually
identical to asserting that the fit is good and that the reduced
chi-square should have been equal to 1.  Any decent book on statistics
for the physical sciences will explain the L-M method, including the
covarience matrix, in excruciating detail.

HTH,
B


-- 
 Bruce Ravel  ---------------------------------------------- bravel at anl.gov

 Molecular Environmental Science Group, Building 203, Room E-165
 MRCAT, Sector 10, Advanced Photon Source, Building 433, Room B007

 Argonne National Laboratory         phone and voice mail: (1) 630 252 5033
 Argonne IL 60439, USA                                fax: (1) 630 252 9793

 My homepage:    http://cars9.uchicago.edu/~ravel 
 EXAFS software: http://cars9.uchicago.edu/~ravel/software/exafs/