[Ifeffit] Breaking down correlationships between parameters
Rana, Jatinkumar Kantilal
jatinkumar.rana at helmholtz-berlin.de
Mon Mar 23 07:54:04 CDT 2015
Hi Chris,
The term N*S02 is fitted for each path of the FEFF calculation. So my question is, even if we know N with a great certainty for some path, how can we vary both N and S02 for other paths ? or Did I understand it wrong ?
Best regards,
Jatin
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Today's Topics:
1. Re: Breaking down correlationships between parameters
(Rana, Jatinkumar Kantilal)
2. Re: Breaking down correlationships between parameters
(Chris Patridge)
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Message: 1
Date: Mon, 23 Mar 2015 11:00:19 +0000
From: "Rana, Jatinkumar Kantilal"
<jatinkumar.rana at helmholtz-berlin.de>
To: "ifeffit at millenia.cars.aps.anl.gov"
<ifeffit at millenia.cars.aps.anl.gov>
Subject: Re: [Ifeffit] Breaking down correlationships between
parameters
Message-ID: <DA4C7D90F90BC249A03505C87F2EB8AB230EE777 at didag1>
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Hi Scott,
Thank you for your comments. Can you please elaborate a little bit more on this "In cases like that, both N for all paths but one and S02 can be fit without 100% correlation."
Best regards,
Jatin
-----Original Message-----
From: ifeffit-bounces at millenia.cars.aps.anl.gov [mailto:ifeffit-bounces at millenia.cars.aps.anl.gov] On Behalf Of ifeffit-request at millenia.cars.aps.anl.gov
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Today's Topics:
1. Re: Breaking down correlationships between parameters
(Scott Calvin)
2. Re: Breaking down correlationships between parameters
(Matt Newville)
3. Re: Breaking down correlationships between parameters
(Rana, Jatinkumar Kantilal)
----------------------------------------------------------------------
Message: 1
Date: Sun, 22 Mar 2015 13:44:28 -0400
From: Scott Calvin <scalvin at sarahlawrence.edu>
To: XAFS Analysis using Ifeffit <ifeffit at millenia.cars.aps.anl.gov>
Subject: Re: [Ifeffit] Breaking down correlationships between
parameters
Message-ID: <E0C66911-7AD2-448C-9F64-F03E262D04D7 at slc.edu>
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One side-comment from me:
On Mar 22, 2015, at 12:52 PM, Matt Newville <newville at cars.uchicago.edu<mailto:newville at cars.uchicago.edu>> wrote:
N and S02 are always 100% correlated (mathematically, not merely by the finite k range).
Matt is saying that N and S02 are always 100% correlated for a single path. But in some situations you might know N for one path but not others. For example, you might know that the absorbing atom is octahedrally coordinated to oxygen but not be as certain as to next-nearest neighbors, or that there are copper atoms on the corners of a simple cubic lattice with a mixture of atoms at other positions. In cases like that, both N for all paths but one and S02 can be fit without 100% correlation.
The degeneracy of multiple-scattering paths can often be constrained in terms of the coordination numbers for direct-scattering paths, which can further reduce (not ?break?) the correlation.
In terms of the main question, I agree with Matt: I don?t think there?s much point in using the line-crossing technique nowadays; fitting using multiple k-weights simultaneously accomplishes the same thing but is a bit easier to interpret statistically.
?Scott Calvin
Sarah Lawrence College
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Message: 2
Date: Sun, 22 Mar 2015 15:56:20 -0500
From: Matt Newville <newville at cars.uchicago.edu>
To: XAFS Analysis using Ifeffit <ifeffit at millenia.cars.aps.anl.gov>
Subject: Re: [Ifeffit] Breaking down correlationships between
parameters
Message-ID:
<CA+7ESbq_s0X3L9rPDDnwAdhx7h6Ptx5-8guCUFTM2OjwkYEerA at mail.gmail.com>
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On Sun, Mar 22, 2015 at 12:44 PM, Scott Calvin <scalvin at sarahlawrence.edu>
wrote:
> One side-comment from me:
>
> On Mar 22, 2015, at 12:52 PM, Matt Newville
> <newville at cars.uchicago.edu>
> wrote:
>
> N and S02 are always 100% correlated (mathematically, not merely by
> the finite k range).
>
>
> Matt is saying that N and S02 are always 100% correlated *for a single
> path*. But in some situations you might know N for one path but not
> others. For example, you might know that the absorbing atom is
> octahedrally coordinated to oxygen but not be as certain as to
> next-nearest neighbors, or that there are copper atoms on the corners
> of a simple cubic lattice with a mixture of atoms at other positions.
> In cases like that, both N for all paths but one and S02 can be fit without 100% correlation.
>
>
Yes, I completely agree with Scott -- this is a good point that I
neglected. In addition to looking at multiple shells, one might also
consider using temperature or pressure dependence to separate N*S02 and
sigma2. Those aren't without assumptions, and still don't remove the
inherent correlation, but are useful approaches.
The degeneracy of multiple-scattering paths can often be constrained in
> terms of the coordination numbers for direct-scattering paths, which
> can further reduce (not ?break?) the correlation.
>
> In terms of the main question, I agree with Matt: I don?t think
> there?s much point in using the line-crossing technique nowadays;
> fitting using multiple k-weights simultaneously accomplishes the same
> thing but is a bit easier to interpret statistically.
>
> ?Scott Calvin
> Sarah Lawrence College
>
> _______________________________________________
> Ifeffit mailing list
> Ifeffit at millenia.cars.aps.anl.gov
> http://millenia.cars.aps.anl.gov/mailman/listinfo/ifeffit
>
>
--Matt
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Message: 3
Date: Mon, 23 Mar 2015 09:16:02 +0000
From: "Rana, Jatinkumar Kantilal"
<jatinkumar.rana at helmholtz-berlin.de>
To: "ifeffit at millenia.cars.aps.anl.gov"
<ifeffit at millenia.cars.aps.anl.gov>
Subject: Re: [Ifeffit] Breaking down correlationships between
parameters
Message-ID: <DA4C7D90F90BC249A03505C87F2EB8AB230EE755 at didag1>
Content-Type: text/plain; charset="utf-8"
Hi Matt,
Thank you very much for your detailed explanation. As you pointed out that this approach ignores the statistical significance of fits and assumes that all fits are "good" fits. Also, the point that this approach yields a value of the parameter which is only slightly less correlated with the other one, but not completely removes the correlation. It makes it really clear to me that how this approach works and what are the pros and cons.
Well, I myself has never tried this approach of minimizing the correlation between N*S02 and sigma2, but I read a lot about it in the literature. With my limited knowledge about the method, I could not judge this approach, although I had my own doubts.
I truly appreciate your efforts in providing me a deeper insight into this approach.
Best regards,
Jatin
-----Original Message-----
From: ifeffit-bounces at millenia.cars.aps.anl.gov [mailto:ifeffit-bounces at millenia.cars.aps.anl.gov] On Behalf Of ifeffit-request at millenia.cars.aps.anl.gov
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1. Re: Breaking down correlationships between parameters
(Matt Newville)
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Message: 1
Date: Sun, 22 Mar 2015 11:52:30 -0500
From: Matt Newville <newville at cars.uchicago.edu>
To: XAFS Analysis using Ifeffit <ifeffit at millenia.cars.aps.anl.gov>
Subject: Re: [Ifeffit] Breaking down correlationships between
parameters
Message-ID:
<CA+7ESbqcyQHf9XUh2uhk=Lv09An7E9LXxqDTX-X6kRJHy9PFzw at mail.gmail.com>
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Hi Jatin,
On Sat, Mar 21, 2015 at 10:41 AM, Rana, Jatinkumar Kantilal < jatinkumar.rana at helmholtz-berlin.de> wrote:
> Hi Matt,
>
> Thanks a lot for your prompt reply. The method I am referring to is
> not the multiple k-weight fits by constraining N*S02. My apologies for
> not being clear enough. Let's do it again. I am actually referring to
> an approach where we take an advantage of a different k-dependence of
> various parameters to breakdown correlations between them. For
> example, S02 and sigma2. S02 is k-independent and Sigma2 has k^2 dependence.
>
>
Yes, I am familiar with this approach, and I understand that this is what you are using. What I am saying is that this does not work nearly as well as (sometimes) claimed, and is sort of cheating. It ignores the measures of statistical significance.
In this case, to breakdown correlation between S02 and sigma2,
The correlation between N*S02 and sigma2 is inherent to the finite k-range of the EXAFS signal. It cannot be "broken", though it might be reduced.
> one can assume a series of S02 values and perform fits using a single
> k-weight each time (say k-weight 1,2 and 3) and record corresponding
> sigma2 values.
Let us say for k-weight =1, a series of preset S02 values will result in a
> series of corresponding sigma2 values refined in fits, which can be
> plotted as a straight line in sigma2 vs. S02 plot.
OK, one can fit sigma2 with a series of preset values on N*S02. That's fine. But it does NOT lead to an infinitely thin line of sigma2 vs.
N*S02. Each sigma2 value on that line has a width, corresponding to its
uncertainty. In fact, the line you produce nicely demonstrates and
measures the correlation of N*S02 and sigma2 as the slope of this line.
> Similar straight lines can be obtained for fits using k-weight = 2 and
> then 3.
Now, these three lines may intersect at or near some point, which will
> determine the "true" value of parameters independent of k-weight.
The different lines (each with finite thickness) will give a *range of
values* for N*S02 and sigma2, not a single value.
The biggest problem with this approach is that it ignores the relative goodness-of-fits (let's just assume that is 'chi-square' for the purpose of
this discussion) for the fits along these lines. Some fits are better
than others, and this approach completely ignores that fact, and equally importantly ignores the fact that there is a range of values for chi-square
that are consistent with "good". If you include these values, your
linear plot will become contours of chi-square as a function of N*S02 and
sigma2. And, yes, by using different k-weights and k-ranges and so on you
can get overlapping contour plots which may reduce the correlation a small amount when looked at as an ensemble. And you can find a best set of values for N*S02 and sigma2, but *each* of these will have an uncertainty.
So, you can use this approach to find a good value for N*S02, but it is not breaking the correlation. You can do this by hand. Or you can just do a
fit with datasets with different k-weights and k-ranges. When you do this
as a fit, you will see that the correlation is still fairly large.
Also, just to be clear, this is absolutely not a "true" value. It is a measured value. Not at all the same thing.
One can then constrain S02 to a value obtained from the point of
> intersection of three lines and vary sigma2 in a fit.
Well, one can certainly set N*S02 to some value and fit sigma2. As I said earlier, this ignores the correlation of N*S02 and sigma2, but does not remove that correlation.
> In this particular case, however, the advantage is, S02 does not
> depend on changes inside sample and we have very good estimate of its
> range (say 0.7
> - 1.0).
>
> Now suppose instead of S02 (which i now set to a reasonable value), I
> am interested in determining N, but it is highly correlated with
> sigma2. Each time when disorder in the sample increases, the sigma2
> increases and due to its high correlation, N is also overestimated. On
> the other hand, when the disorder in the sample decreases, the sigma2
> decreases and I can have a "true" estimation of N in the sample. Can I
> still apply the above mentioned approach to break the correlationship between N and sigma2 and get a "true"
> estimation of N, even if disorder is high in my samples ? or it is
> simply not possible due to the fact that both N and sigma2 varies with
> changes inside the sample.
>
>
N and S02 are always 100% correlated (mathematically, not merely by the finite k range). So, to the extent that the approach works at all, you can use it for "N" or "S02". Really, the approach is comparing N*S02 and sigma2, in one case you asserted a value of "N" and projected all changes to "S02" -- you can equally assert "S02" and project all changes to "N".
To be clear, this is not going to find the "true" value of anything, because no analysis is ever going to find the "true" value -- it's going to find a measured value.
Finally, the correlation of N*S02 and sigma2 does not imply a bias in the values for N*S02. N*S02 is NOT overestimated because it is highly correlated with sigma2.
Hope that helps,
--Matt
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Helmholtz-Zentrum Berlin f?r Materialien und Energie GmbH
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Gesch?ftsf?hrung: Prof. Dr. Anke Rita Kaysser-Pyzalla, Thomas Frederking
Sitz Berlin, AG Charlottenburg, 89 HRB 5583
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Message: 2
Date: Mon, 23 Mar 2015 07:15:05 -0400
From: Chris Patridge <patridge at buffalo.edu>
To: XAFS Analysis using Ifeffit <ifeffit at millenia.cars.aps.anl.gov>
Subject: Re: [Ifeffit] Breaking down correlationships between
parameters
Message-ID: <29942DF9-8043-424E-BDCC-9CA19EB4AC9B at buffalo.edu>
Content-Type: text/plain; charset=utf-8
I think Scott was pointing out that first neighbors may be known with high certainty and therefore you can set this value thereby removing it and slightly reducing the correlations.
Chris
Sent from my iPhone
> On Mar 23, 2015, at 7:00 AM, Rana, Jatinkumar Kantilal <jatinkumar.rana at helmholtz-berlin.de> wrote:
>
> Hi Scott,
>
> Thank you for your comments. Can you please elaborate a little bit more on this "In cases like that, both N for all paths but one and S02 can be fit without 100% correlation."
>
> Best regards,
> Jatin
>
> -----Original Message-----
> From: ifeffit-bounces at millenia.cars.aps.anl.gov
> [mailto:ifeffit-bounces at millenia.cars.aps.anl.gov] On Behalf Of
> ifeffit-request at millenia.cars.aps.anl.gov
> Sent: 23 March, 2015 10:16
> To: ifeffit at millenia.cars.aps.anl.gov
> Subject: Ifeffit Digest, Vol 145, Issue 41
>
> Send Ifeffit mailing list submissions to
> ifeffit at millenia.cars.aps.anl.gov
>
> To subscribe or unsubscribe via the World Wide Web, visit
> http://millenia.cars.aps.anl.gov/mailman/listinfo/ifeffit
> or, via email, send a message with subject or body 'help' to
> ifeffit-request at millenia.cars.aps.anl.gov
>
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>
> When replying, please edit your Subject line so it is more specific than "Re: Contents of Ifeffit digest..."
>
>
> Today's Topics:
>
> 1. Re: Breaking down correlationships between parameters
> (Scott Calvin)
> 2. Re: Breaking down correlationships between parameters
> (Matt Newville)
> 3. Re: Breaking down correlationships between parameters
> (Rana, Jatinkumar Kantilal)
>
>
> ----------------------------------------------------------------------
>
> Message: 1
> Date: Sun, 22 Mar 2015 13:44:28 -0400
> From: Scott Calvin <scalvin at sarahlawrence.edu>
> To: XAFS Analysis using Ifeffit <ifeffit at millenia.cars.aps.anl.gov>
> Subject: Re: [Ifeffit] Breaking down correlationships between
> parameters
> Message-ID: <E0C66911-7AD2-448C-9F64-F03E262D04D7 at slc.edu>
> Content-Type: text/plain; charset="utf-8"
>
> One side-comment from me:
>
> On Mar 22, 2015, at 12:52 PM, Matt Newville <newville at cars.uchicago.edu<mailto:newville at cars.uchicago.edu>> wrote:
>
> N and S02 are always 100% correlated (mathematically, not merely by the finite k range).
>
> Matt is saying that N and S02 are always 100% correlated for a single path. But in some situations you might know N for one path but not others. For example, you might know that the absorbing atom is octahedrally coordinated to oxygen but not be as certain as to next-nearest neighbors, or that there are copper atoms on the corners of a simple cubic lattice with a mixture of atoms at other positions. In cases like that, both N for all paths but one and S02 can be fit without 100% correlation.
>
> The degeneracy of multiple-scattering paths can often be constrained in terms of the coordination numbers for direct-scattering paths, which can further reduce (not ?break?) the correlation.
>
> In terms of the main question, I agree with Matt: I don?t think there?s much point in using the line-crossing technique nowadays; fitting using multiple k-weights simultaneously accomplishes the same thing but is a bit easier to interpret statistically.
>
> ?Scott Calvin
> Sarah Lawrence College
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> ------------------------------
>
> Message: 2
> Date: Sun, 22 Mar 2015 15:56:20 -0500
> From: Matt Newville <newville at cars.uchicago.edu>
> To: XAFS Analysis using Ifeffit <ifeffit at millenia.cars.aps.anl.gov>
> Subject: Re: [Ifeffit] Breaking down correlationships between
> parameters
> Message-ID:
>
> <CA+7ESbq_s0X3L9rPDDnwAdhx7h6Ptx5-8guCUFTM2OjwkYEerA at mail.gmail.com>
> Content-Type: text/plain; charset="utf-8"
>
> On Sun, Mar 22, 2015 at 12:44 PM, Scott Calvin
> <scalvin at sarahlawrence.edu>
> wrote:
>
>> One side-comment from me:
>>
>> On Mar 22, 2015, at 12:52 PM, Matt Newville
>> <newville at cars.uchicago.edu>
>> wrote:
>>
>> N and S02 are always 100% correlated (mathematically, not merely by
>> the finite k range).
>>
>>
>> Matt is saying that N and S02 are always 100% correlated *for a
>> single path*. But in some situations you might know N for one path
>> but not others. For example, you might know that the absorbing atom
>> is octahedrally coordinated to oxygen but not be as certain as to
>> next-nearest neighbors, or that there are copper atoms on the corners
>> of a simple cubic lattice with a mixture of atoms at other positions.
>> In cases like that, both N for all paths but one and S02 can be fit without 100% correlation.
> Yes, I completely agree with Scott -- this is a good point that I
> neglected. In addition to looking at multiple shells, one might also
> consider using temperature or pressure dependence to separate N*S02 and
> sigma2. Those aren't without assumptions, and still don't remove the
> inherent correlation, but are useful approaches.
>
> The degeneracy of multiple-scattering paths can often be constrained
> in
>> terms of the coordination numbers for direct-scattering paths, which
>> can further reduce (not ?break?) the correlation.
>>
>> In terms of the main question, I agree with Matt: I don?t think
>> there?s much point in using the line-crossing technique nowadays;
>> fitting using multiple k-weights simultaneously accomplishes the same
>> thing but is a bit easier to interpret statistically.
>>
>> ?Scott Calvin
>> Sarah Lawrence College
>>
>> _______________________________________________
>> Ifeffit mailing list
>> Ifeffit at millenia.cars.aps.anl.gov
>> http://millenia.cars.aps.anl.gov/mailman/listinfo/ifeffit
> --Matt
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>
> ------------------------------
>
> Message: 3
> Date: Mon, 23 Mar 2015 09:16:02 +0000
> From: "Rana, Jatinkumar Kantilal"
> <jatinkumar.rana at helmholtz-berlin.de>
> To: "ifeffit at millenia.cars.aps.anl.gov"
> <ifeffit at millenia.cars.aps.anl.gov>
> Subject: Re: [Ifeffit] Breaking down correlationships between
> parameters
> Message-ID: <DA4C7D90F90BC249A03505C87F2EB8AB230EE755 at didag1>
> Content-Type: text/plain; charset="utf-8"
>
> Hi Matt,
>
> Thank you very much for your detailed explanation. As you pointed out that this approach ignores the statistical significance of fits and assumes that all fits are "good" fits. Also, the point that this approach yields a value of the parameter which is only slightly less correlated with the other one, but not completely removes the correlation. It makes it really clear to me that how this approach works and what are the pros and cons.
>
> Well, I myself has never tried this approach of minimizing the correlation between N*S02 and sigma2, but I read a lot about it in the literature. With my limited knowledge about the method, I could not judge this approach, although I had my own doubts.
>
> I truly appreciate your efforts in providing me a deeper insight into this approach.
>
> Best regards,
> Jatin
>
> -----Original Message-----
> From: ifeffit-bounces at millenia.cars.aps.anl.gov
> [mailto:ifeffit-bounces at millenia.cars.aps.anl.gov] On Behalf Of
> ifeffit-request at millenia.cars.aps.anl.gov
> Sent: 22 March, 2015 18:00
> To: ifeffit at millenia.cars.aps.anl.gov
> Subject: Ifeffit Digest, Vol 145, Issue 40
>
> Send Ifeffit mailing list submissions to
> ifeffit at millenia.cars.aps.anl.gov
>
> To subscribe or unsubscribe via the World Wide Web, visit
> http://millenia.cars.aps.anl.gov/mailman/listinfo/ifeffit
> or, via email, send a message with subject or body 'help' to
> ifeffit-request at millenia.cars.aps.anl.gov
>
> You can reach the person managing the list at
> ifeffit-owner at millenia.cars.aps.anl.gov
>
> When replying, please edit your Subject line so it is more specific than "Re: Contents of Ifeffit digest..."
>
>
> Today's Topics:
>
> 1. Re: Breaking down correlationships between parameters
> (Matt Newville)
>
>
> ----------------------------------------------------------------------
>
> Message: 1
> Date: Sun, 22 Mar 2015 11:52:30 -0500
> From: Matt Newville <newville at cars.uchicago.edu>
> To: XAFS Analysis using Ifeffit <ifeffit at millenia.cars.aps.anl.gov>
> Subject: Re: [Ifeffit] Breaking down correlationships between
> parameters
> Message-ID:
>
> <CA+7ESbqcyQHf9XUh2uhk=Lv09An7E9LXxqDTX-X6kRJHy9PFzw at mail.gmail.com>
> Content-Type: text/plain; charset="utf-8"
>
> Hi Jatin,
>
>
>> On Sat, Mar 21, 2015 at 10:41 AM, Rana, Jatinkumar Kantilal < jatinkumar.rana at helmholtz-berlin.de> wrote:
>>
>> Hi Matt,
>>
>> Thanks a lot for your prompt reply. The method I am referring to is
>> not the multiple k-weight fits by constraining N*S02. My apologies
>> for not being clear enough. Let's do it again. I am actually
>> referring to an approach where we take an advantage of a different
>> k-dependence of various parameters to breakdown correlations between
>> them. For example, S02 and sigma2. S02 is k-independent and Sigma2 has k^2 dependence.
> Yes, I am familiar with this approach, and I understand that this is what you are using. What I am saying is that this does not work nearly as well as (sometimes) claimed, and is sort of cheating. It ignores the measures of statistical significance.
>
> In this case, to breakdown correlation between S02 and sigma2,
>
>
> The correlation between N*S02 and sigma2 is inherent to the finite k-range of the EXAFS signal. It cannot be "broken", though it might be reduced.
>
>
>> one can assume a series of S02 values and perform fits using a single
>> k-weight each time (say k-weight 1,2 and 3) and record corresponding
>> sigma2 values.
>
> Let us say for k-weight =1, a series of preset S02 values will result
> in a
>> series of corresponding sigma2 values refined in fits, which can be
>> plotted as a straight line in sigma2 vs. S02 plot.
>
>
> OK, one can fit sigma2 with a series of preset values on N*S02. That's fine. But it does NOT lead to an infinitely thin line of sigma2 vs.
> N*S02. Each sigma2 value on that line has a width, corresponding to its
> uncertainty. In fact, the line you produce nicely demonstrates and
> measures the correlation of N*S02 and sigma2 as the slope of this line.
>
>
>> Similar straight lines can be obtained for fits using k-weight = 2
>> and then 3.
>
> Now, these three lines may intersect at or near some point, which will
>> determine the "true" value of parameters independent of k-weight.
>
>
> The different lines (each with finite thickness) will give a *range of
> values* for N*S02 and sigma2, not a single value.
>
> The biggest problem with this approach is that it ignores the relative goodness-of-fits (let's just assume that is 'chi-square' for the purpose of
> this discussion) for the fits along these lines. Some fits are better
> than others, and this approach completely ignores that fact, and equally importantly ignores the fact that there is a range of values for chi-square
> that are consistent with "good". If you include these values, your
> linear plot will become contours of chi-square as a function of N*S02 and
> sigma2. And, yes, by using different k-weights and k-ranges and so on you
> can get overlapping contour plots which may reduce the correlation a small amount when looked at as an ensemble. And you can find a best set of values for N*S02 and sigma2, but *each* of these will have an uncertainty.
>
> So, you can use this approach to find a good value for N*S02, but it is not breaking the correlation. You can do this by hand. Or you can just do a
> fit with datasets with different k-weights and k-ranges. When you do this
> as a fit, you will see that the correlation is still fairly large.
>
> Also, just to be clear, this is absolutely not a "true" value. It is a measured value. Not at all the same thing.
>
> One can then constrain S02 to a value obtained from the point of
>> intersection of three lines and vary sigma2 in a fit.
>
>
> Well, one can certainly set N*S02 to some value and fit sigma2. As I said earlier, this ignores the correlation of N*S02 and sigma2, but does not remove that correlation.
>
>
>> In this particular case, however, the advantage is, S02 does not
>> depend on changes inside sample and we have very good estimate of its
>> range (say 0.7
>> - 1.0).
>>
>> Now suppose instead of S02 (which i now set to a reasonable value), I
>> am interested in determining N, but it is highly correlated with
>> sigma2. Each time when disorder in the sample increases, the sigma2
>> increases and due to its high correlation, N is also overestimated.
>> On the other hand, when the disorder in the sample decreases, the
>> sigma2 decreases and I can have a "true" estimation of N in the
>> sample. Can I still apply the above mentioned approach to break the correlationship between N and sigma2 and get a "true"
>> estimation of N, even if disorder is high in my samples ? or it is
>> simply not possible due to the fact that both N and sigma2 varies
>> with changes inside the sample.
> N and S02 are always 100% correlated (mathematically, not merely by the finite k range). So, to the extent that the approach works at all, you can use it for "N" or "S02". Really, the approach is comparing N*S02 and sigma2, in one case you asserted a value of "N" and projected all changes to "S02" -- you can equally assert "S02" and project all changes to "N".
>
> To be clear, this is not going to find the "true" value of anything, because no analysis is ever going to find the "true" value -- it's going to find a measured value.
>
> Finally, the correlation of N*S02 and sigma2 does not imply a bias in the values for N*S02. N*S02 is NOT overestimated because it is highly correlated with sigma2.
>
> Hope that helps,
>
> --Matt
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