Ok, I've found one reference for this formula:
- Loria, J. P. and Kempf, J. G. (2004) Measurement of
Intermediate Exchange Phenomena. Methods in Molecular Biology, 278,
185-231. (http://dx.doi.org/10.1385/1-59259-809-9:185).
Specifically section 3.2.7.1 "Two-Point Data Sampling", and equation 22:
R2(tau_CP^-1) = t^-1 ln[ave(I0) / ave(I(t))] +/- t^-1 Delta_Q,
and equation 23:
Delta_Q = sqrt(Delta_I0^2 + Delta_I(t)^2).
This is apparently from the book:
- Shoemaker, D. P., Garland, C. W., and Nibler, J. W. (1989)
Experiments in Physical Chemistry. 5th ed. McGraw-Hill, New York.
Though I don't have access to that to check.
Regards,
Edward
On 1 September 2014 10:55, Edward d'Auvergne <edw...@nmr-relax.com> wrote:
> Hi Troels,
>
> The 2-point exponential error formula is currently equation 11.4 in
> the relax manual (http://www.nmr-relax.com/manual/R2eff_model.html).
> I unfortunately did not write down a reference for it. But the
> equation is in a number of dispersion papers - I just have to find it.
> Feel free to search yourself. I also simulated this error, see figure
> 11.1 in the relax manual
> (http://www.nmr-relax.com/manual/R2eff_model.html#fig:_dispersion_error_comparison)
>
> Regards,
>
> Edward
>
> On 30 August 2014 13:49, Troels Emtekær Linnet <tlin...@gmail.com> wrote:
>> Hi Edward.
>>
>> I found this old post in gwing gwong doogle Google.
>> http://thread.gmane.org/gmane.science.nmr.relax.scm/17553
>>
>> And then I see, that for a two point exponential, this i just
>> converting the values to a linear problem.
>>
>> For several time points, a good initial guess for estimating r2eff,
>> and i0, by converting to a linear problem, and solving by linear least
>> squares.
>> (This is currently the experimental 'estimate_x0_exp' in the R2eff
>> estimate module).
>> Maybe I should try some weighting on this.
>>
>> ---------
>> # Convert to linear problem.
>> # Func
>> # I = i0 exp(-t R)
>> # Convert to linear
>> # ln(I) = R (-t) + ln(i0)
>> # Compare
>> # f(I) = a x + b
>>
>> ln_I = log(I)
>> x = - 1. * times
>> n = len(times)
>>
>> # Solve by linear least squares.
>> a = (sum(x*ln_I) - 1./n * sum(x) * sum(ln_I) ) / ( sum(x**2) - 1./n *
>> (sum(x))**2 )
>> b = 1./n * sum(ln_I) - b * 1./n * sum(x)
>>
>> # Convert back from linear to exp function. Best estimate for parameter.
>> r2eff_est = a
>> i0_est = exp(b)
>> -----------
>>
>> And then I see in And then I look in lib.dispersion.two_point.py
>>
>> ---------------
>> """Calculate the R2eff/R1rho error for the fixed relaxation time data.
>>
>> The formula is::
>>
>> __________________________________
>> 1 / / sigma_I1 \ 2 / sigma_I0 \ 2
>> sigma_R2 = ------- / | -------- | + | -------- |
>> relax_T \/ \ I1(nu1) / \ I0 /
>>
>> where relax_T is the fixed delay time, I0 and sigma_I0 are the
>> reference peak intensity and error when relax_T is zero, and I1 and
>> sigma_I1 are the peak intensity and error in the spectrum of interest.
>> -------------
>>
>> Right now, I don't know where that comes from.
>>
>> My reference book:
>> An introduction to Error Analysis, Second Edition
>> John R. Taylor
>> http://www.uscibooks.com/taylornb.htm
>>
>> "You will not be surprised to learn that when the uncertainties ...
>> are independent and random, the sum can be replaced by a sum in
>> quadrature."
>>
>> So, just following you analogy, I could get sigma R2 this way.
>>
>> I will look into it and make some tests scripts.
>>
>>
>>
>>
>> Troels Emtekær Linnet
>>
>>
>> 2014-08-30 9:54 GMT+02:00 Edward d'Auvergne <edw...@nmr-relax.com>:
>>> Hi,
>>>
>>> I don't have much time to reply now, but the key is it use simple
>>> synthetic noise-free data. Try converting the 5 intensities in
>>> test_suite/shared_data/curve_fitting/numeric_topology/ into 5 Sparky
>>> peak lists with a single spin. Then test the Monte Carlo simulations
>>> and covariance matrix user functions in relax. These two relax
>>> techniques should then match the numeric results from the super-basic
>>> scripts in that directory! This would then be converted into two
>>> system tests.
>>>
>>> This was my plan, to complete in my spare time. If you want to go
>>> quickly, then feel free to follow these steps yourself rather than
>>> waiting for me to do it. I actually suggested this synthetic data
>>> testing earlier to you
>>> (http://thread.gmane.org/gmane.science.nmr.relax.devel/6807/focus=6840).
>>> Synthetic noise-free data is an essential tool for implemented and
>>> debugging any new analysis type, algorithm, protocol, etc. The key is
>>> that you know the answer you are searching for! And synthetic data is
>>> simple. Nothing should ever be implemented and debugged using real
>>> data, as a good looking result might be the consequence of a nasty
>>> bug.
>>>
>>> Regards,
>>>
>>> Edward
>>>
>>>
>>>
>>>
>>>
>>>
>>> On 30 August 2014 02:49, Troels Emtekær Linnet <tlin...@gmail.com> wrote:
>>>> Hm.
>>>>
>>>> The last idea I have, is the division by number of degree of freedom.
>>>>
>>>> So either 5-2, or 4-2.
>>>>
>>>> That should be verified by a script with many different time points.
>>>>
>>>> But then the errors for intensity gets very different.
>>>>
>>>> Hm.
>>>>
>>>> On 30 Aug 2014 01:45, "Troels Emtekær Linnet" <tlin...@nmr-relax.com>
>>>> wrote:
>>>>>
>>>>> The sentence:
>>>>>
>>>>> "then the covariance matrix above gives the statistical error on the
>>>>> best-fit parameters resulting from the Gaussian errors 'sigma_i' on
>>>>> the underlying data 'y_i'."
>>>>>
>>>>> And here I note the wording:
>>>>> "statistical error"
>>>>> "Gaussian errors"
>>>>>
>>>>> Best
>>>>> Troels
>>>>>
>>>>>
>>>>> 2014-08-29 21:21 GMT+02:00 Troels Emtekær Linnet <tlin...@nmr-relax.com>:
>>>>> > Hi Edward.
>>>>> >
>>>>> > I also think it is some math some where.
>>>>> >
>>>>> > I have a feeling, that it is the creating of Monte Carlo data with 2
>>>>> > sigma?
>>>>> > and then some log/exp calculation of R2eff.
>>>>> >
>>>>> > If the errors are 2 x times over estimated, the chi2 values are 4 as
>>>>> > small, and the
>>>>> > space should be the same?
>>>>> >
>>>>> > best
>>>>> > Troels
>>>>> >
>>>>> > 2014-08-29 17:06 GMT+02:00 Edward d'Auvergne <edw...@nmr-relax.com>:
>>>>> >> I've just added the 2D Grace plots for this to the repository (r25444,
>>>>> >> http://article.gmane.org/gmane.science.nmr.relax.scm/23194). They are
>>>>> >> also attached to the task for easier access
>>>>> >> (https://gna.org/task/index.php?7822#comment107). From these plots I
>>>>> >> see that the I0 error appears to be reasonable, but that the R2eff
>>>>> >> errors are overestimated by 1.9555.
>>>>> >>
>>>>> >> The plots are very, very different. It is clear that
>>>>> >> chi2_jacobian=True just gives rubbish. It is also clear that there is
>>>>> >> a perfect correlation in R2eff when chi2_jacobian=False. The plot
>>>>> >> shows absolutely no scattering, therefore this indicates a crystal
>>>>> >> clear mathematical error somewhere. I wonder where that could be. It
>>>>> >> may not be a factor of 2, as the correlation is 1.9555. So it might
>>>>> >> be a bug that is more complicated. In any case, overestimating the
>>>>> >> errors by ~2 and performing a dispersion analysis is not possible.
>>>>> >> This will significantly change the curvature of the optimisation space
>>>>> >> and will also have a huge effect on statistical comparisons between
>>>>> >> models.
>>>>> >>
>>>>> >> Regards,
>>>>> >>
>>>>> >> Edward
>>>>> >>
>>>>> >>
>>>>> >>
>>>>> >> On 29 August 2014 16:56, Troels Emtekær Linnet <tlin...@nmr-relax.com>
>>>>> >> wrote:
>>>>> >>> The default is now chi2_jacobian=False.
>>>>> >>>
>>>>> >>> You will hopefully see, that the errors are double.
>>>>> >>>
>>>>> >>> Best
>>>>> >>> Troels
>>>>> >>>
>>>>> >>> 2014-08-29 16:43 GMT+02:00 Edward d'Auvergne <edw...@nmr-relax.com>:
>>>>> >>>> Terrible ;) For R2eff, the correlation is 2.748 and the points are
>>>>> >>>> spread out all over the place. For I0, the correlation is 3.5 and
>>>>> >>>> the
>>>>> >>>> points are also spread out everywhere. Maybe I should try with the
>>>>> >>>> change from:
>>>>> >>>>
>>>>> >>>> relax_disp.r2eff_err_estimate(chi2_jacobian=True)
>>>>> >>>>
>>>>> >>>> to:
>>>>> >>>>
>>>>> >>>> relax_disp.r2eff_err_estimate(chi2_jacobian=False)
>>>>> >>>>
>>>>> >>>> How should this be used?
>>>>> >>>>
>>>>> >>>> Cheers,
>>>>> >>>>
>>>>> >>>> Edward
>>>>> >>>>
>>>>> >>>>
>>>>> >>>>
>>>>> >>>> On 29 August 2014 16:33, Troels Emtekær Linnet
>>>>> >>>> <tlin...@nmr-relax.com> wrote:
>>>>> >>>>> Do you mean terrible or double?
>>>>> >>>>>
>>>>> >>>>> Best
>>>>> >>>>> Troels
>>>>> >>>>>
>>>>> >>>>> 2014-08-29 16:15 GMT+02:00 Edward d'Auvergne <edw...@nmr-relax.com>:
>>>>> >>>>>> Hi Troels,
>>>>> >>>>>>
>>>>> >>>>>> I really cannot follow and judge how the techniques compare. I
>>>>> >>>>>> must
>>>>> >>>>>> be getting old. So to remedy this, I have created the
>>>>> >>>>>>
>>>>> >>>>>> test_suite/shared_data/dispersion/Kjaergaard_et_al_2013/exp_error_analysis/
>>>>> >>>>>> directory (r25437,
>>>>> >>>>>> http://article.gmane.org/gmane.science.nmr.relax.scm/23187). This
>>>>> >>>>>> contains 3 scripts for comparing R2eff and I0 parameters for the 2
>>>>> >>>>>> parameter exponential curve-fitting:
>>>>> >>>>>>
>>>>> >>>>>> 1) A simple script to perform Monte Carlo simulation error
>>>>> >>>>>> analysis.
>>>>> >>>>>> This is run with 10,000 simulations to act as the gold standard.
>>>>> >>>>>>
>>>>> >>>>>> 2) A simple script to perform covariance matrix error analysis.
>>>>> >>>>>>
>>>>> >>>>>> 3) A simple script to generate 2D Grace plots to visualise the
>>>>> >>>>>> differences. Now I can see how good the covariance matrix
>>>>> >>>>>> technique
>>>>> >>>>>> is :)
>>>>> >>>>>>
>>>>> >>>>>> Could you please check and see if I have used the
>>>>> >>>>>> relax_disp.r2eff_err_estimate user function correctly? The Grace
>>>>> >>>>>> plots show that the error estimates are currently terrible.
>>>>> >>>>>>
>>>>> >>>>>> Cheers,
>>>>> >>>>>>
>>>>> >>>>>> Edward
>>>>> >>>>>>
>>>>> >>>>>> _______________________________________________
>>>>> >>>>>> relax (http://www.nmr-relax.com)
>>>>> >>>>>>
>>>>> >>>>>> This is the relax-devel mailing list
>>>>> >>>>>> relax-devel@gna.org
>>>>> >>>>>>
>>>>> >>>>>> To unsubscribe from this list, get a password
>>>>> >>>>>> reminder, or change your subscription options,
>>>>> >>>>>> visit the list information page at
>>>>> >>>>>> https://mail.gna.org/listinfo/relax-devel
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