Bruce I enjoyed reading your US$0.02. It is actually worth US$100.0 Mamadou Bruce Ravel wrote:
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