Spectrum error analysis

Intensity Spectrum error analysis

See the manual

Peak heights with baseplane noise RMSD

When none of the spectra have been replicated, then the peak height errors are calculated using the RMSD of the baseplane noise, the value of which is set by the spectrum.baseplane_rmsd user function. This results in a different error per peak per spectrum. The standard deviation error measure for the peak height, sigma_I, is set to the RMSD value.

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

[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, 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.

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.

If there are in addition to the replicated spectra loaded peak intensities which only consist of a single spectrum, i .e. not all spectra are replicated, then the variances of replicated replicated spectra sets will be averaged.
This will be used for the entire experiment so that there will be only a single error value for all spins and for all spectra.

Peak heights with all spectra replicated

If all spectra are collected in duplicate (triplicate or higher number of spectra are supported), the each replicated spectra set will have its own error estimate.
The error for a single peak is calculated as when partially replicated spectra are collected, and these are again averaged to give a single error per replicated spectra set.
However as all replicated spectra sets will have their own error estimate, variance averaging across all spectra sets will not be performed.

Peak volumes with baseplane noise RMSD

The method of error analysis when no spectra have been replicated and peak volumes are used is highly dependent on the integration method.
Many methods simply sum the number of points within a fixed region, either a box or oval object. The number of points used, $N$, must be specified by another user function in this class.
Then the error is simply given by the sum of variances:

[math] \sigma_{vol}^2 = \sigma_i^2 * N [/math]

where σ_vol is the standard deviation of the volume, σ_i 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.

Note  Also for non-point summation methods, for example when line shape fitting is used to determine peak volumes, the equations above cannot be used. Hence again this category of error analysis cannot be used. This is the case for one of the three integration methods used by Sparky (http://www.cgl.ucsf.edu/home/sparky/manual/peaks.html#Integration). And if fancy techniques are used, for example as Cara does to deconvolute overlapping peaks (http://www.cara.ethz.ch/Wiki/Integration), this again makes this error analysis impossible.

Peak volumes with partially replicated spectra

When peak volumes are measured by any integration method and a few of the spectra are replicated,
then the intensity errors are calculated identically as described in the `Peak heights with partially replicated spectra' section above.

Peak volumes with all spectra replicated

With all spectra replicated and again using any integration methodology, the intensity errors can be calculated as described in the `Peak heights with all spectra replicated' section above.

See also

1. RMSD