Hi Ifeffit people, 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? And finally, how are errors calculated in Artemis for the parameters N (coordination number), deltaE0, S02, deltaR and sigma^2? Thanks a lot JA
Hi JA, Oh dear, I seem to have gotten a rule named after me that I didn't even know I'd put down in print. As I recall, I got it from an offhand comment by Bruce, and he may have gotten it from Rossner and Krappe... :) Here's the deal in a nut-shell: The Nip you give is the Nyquist criterion. Or it's the Nyquist criterion +1, or the Nyquist criterion +2. In any case, the Nyquist criterion was developed for a signal. If I want to give you information in a periodic signal, how much information can I give you using a certain bandwidth and time? That's what Nyquist tells you. But there's no reason that Nature had to be so kind as to pack the maximum possible amount of information into an EXAFS signal. Some of the information may be redundant in some sense. So the true number of independent points might be somewhat less than Nip. The 2/3 is purely made up, as far as I know. It's like saying the fit is pretty close to the data when the EXAFS R-factor is less than 0.02. That 0.02 is arbitrary, but works as a rule of thumb. I myself don't always pay attention to the 2/3 that I have occasionally mentioned. I'm pretty sure I published stuff that's had more free parameters than that. In sum, I think it's bad to think that there is some magic number of free parameters, below which you're OK, and above which you're not. The fewer free parameters you can get away with, the better. And you should always give the k-space and r-space ranges in your publications, along with the number of free parameters, so that people can draw their own conclusions. If I see a published fit with 15 independent points by the Nyquist criterion and 14 free parameters, I'm not going to discount it completely, but I will be somewhat more skeptical than if it had only 4 free parameters, all else being equal. Hope that helps. --Scott Calvin Sarah Lawrence College At 02:29 PM 9/13/2006, Juan Antonio Maciá Agulló wrote:
Hi Ifeffit people,
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?
Thank very much Scott and Bruce for your answers, and sorry for the joking new "Scott's rule". JA
Hi Juan, I liked the rule. It motivated Scott to give a quick reply. And the rest of us got a smile out of it. Cheers, Shelly
-----Original Message----- From: ifeffit-bounces@millenia.cars.aps.anl.gov [mailto:ifeffit- bounces@millenia.cars.aps.anl.gov] On Behalf Of Juan Antonio Maciá Agulló Sent: Thursday, September 14, 2006 1:08 PM To: XAFS Analysis using Ifeffit Subject: Re: [Ifeffit] Short questions
Thank very much Scott and Bruce for your answers, and sorry for the joking new "Scott's rule".
JA
Hi Shelly, It was an unintentional good method for a quick answer and for several smiles :-). Thank you very much everybody for your interesting XAFS tutorials available. Best regards, JA
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@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/
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
participants (5)
-
Bruce Ravel -
Juan Antonio Maciá Agulló -
Kelly, Shelly D. -
Mamadou Diallo -
Scott Calvin