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Spectrum error analysis

77 bytes added, 14:33, 3 November 2015

Fix for the symbols and maths - all $x$ usage has been replaced by <math> tags or direct HTML conversion.

=== Peak heights with partially replicated spectra ===

When spectra are replicated, the variance for a single spin at a single replicated spectra set is calculated by the formula

~~$$ \sigma^2 = \frac{\sum( I_i - I_{av} )}{n -1 } $$~~

where $\sigma^2$ is the variance, $\sigma$ is the standard deviation, $n$ is the size of the replicated spectra set with i being the corresponding index, $I_i$ is the peak intensity for spectrum $i$ , and $I_{av}$ is the mean over all spectra i .e. the sum of all peak intensities divided by $n$.

<math>\sigma^2 = \frac{\sum( I_i - I_{av} )}{n -1 }</math> where σ2 is the variance, σ is the standard deviation, ''n'' is the size of the replicated spectra set with ''i'' being the corresponding index, ''Ii'' is the peak intensity for spectrum ''i'' , and ''Iav'' is the mean over all spectra i.e. the sum of all peak intensities divided by ''n''. As the value of $''n$ '' in the above equation is '''always very low''' since normally only a couple of spectra are collected per replicated spectra set, '''the variance of all spins''' is '''averaged''' for a '''single replicated spectra set'''.

Although this results in all spins having the same error, the accuracy of the error estimate is significantly improved.

Then the error is simply given by the sum of variances:

<math>

\sigma_{vol}^2 = \sigma_i^2 * N

</math>

~~$$\sigma_{vol}^2 = \sigma_i^2 * N $$~~where ~~$\sigma_{~~''σvol}$ '' is the standard deviation of the volume, ~~$\sigma_{~~''σi}$ '' is the standard deviation of a single point assumed to be equal to the RMSD of the baseplane noise, and $''N$ '' is the total number of points used in the summation integration method. For a box integration method, this converts to the

Nicholson, Kay, Baldisseri, Arango, Young, Bax, and Torchia (1992) Biochemistry, 31: 5253-5263 equation:

$$<math>\sigma_{vol} = \sigma_i * \sqrt{(n*m)}$$</math>

where $''n$ '' and $''m$ '' are the dimensions of the box.~~ ~~

{{note|there are a number of programs, for example peakint (http://hugin.ethz.ch/wuthrich/software/xeasy/xeasy_m15.html) that not use all points within the box. And if the number N can not be determined, this category of error analysis is not possible.}}

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