DeltaCC12: Difference between revisions

ΔCC12 is a quantity, that detects datasets/frames, that are non-isomorphous. As described in [https://scripts.iucr.org/cgi-bin/paper?zw5005 Assmann and Diederichs (2016)], Δcc12 is calculated with the σ-τ method. This method is a way to calculate the Pearson correlation coefficient for the special case of two sets of values (intensities) that randomly deviate from their true values, but is not influenced by a random number sequence as shown in [https://www.ncbi.nlm.nih.gov/pmc/articles/PMC3457925/ Karplus and Diederichs (2012)]. For the σ-τ method CC12 is calculated for all datasets/frames, which will be called CC12_overall (?) and CC12 is calculated for all datasets/frames except for one dataset i, which is omitted from calculations and denoted as CC12_i. The difference of the two quantities is Δcc12.

If ΔCC12 is > 0 -CC12overall is bigger than CC12i- that means if omitting dataset i from calculations, a lower CC12 results, which is why we want to keep it. Thus it is improving the whole merged dataset. If ΔCC12 is < 0, -CC12overall is smaller than CC12i- that means that by omitting dataset i from calculations a higher CC12 results, which is why we want to exclude it from calculations, because it is impairing the whole merged dataset. CC12 is calculated by:

 

: <math>CC_{1/2}=\frac{\sigma^2_{\tau}}{\sigma^2_{\tau}+\sigma^2_{\epsilon}} =\frac{\sigma^2_{y}- \frac{1}{2}\sigma^2_{\epsilon}}{\sigma^2_{y}+ \frac{1}{2}\sigma^2_{\epsilon}} </math>

 

This requires calculation of <math>\sigma^2_{y} </math>, the variance of the average intensities across the unique reflections of a resolution shell, and <math>\sigma^2_{\epsilon} </math>, the average of all sample variances of the mean across all unique reflections of a resolution shell.

 

 

===''' <math>\sigma^2_{\epsilon} </math>''' - unweighted===

 

The average of all sample variances of the mean across all unique reflections of a resolution shell is obtained by calculating the sample variance of the mean for every unique reflection i by:

 

<math>\sigma^2_{\epsilon i} =  \frac{1}{n-1} \cdot \left ( \sum^n_{j} x^2_{j} - \frac{\left ( \sum^n_{j}x_{j} \right )^2}{ n} \right )    / \frac{n}{2} </math>

 

With <math>x_{j} </math> , a single observation j of all observations n of one reflection i. <math>\sigma^2_{\epsilon i} </math> is then divided by the factor  <math>\frac{n}{2} </math>, because the variance of the sample mean (the merged observations) is the quantity of interest. The division by n/2 takes care of providing the variance of the mean (merged) intensity of the half-datasets, as defined in [https://en.wikipedia.org/wiki/Sample_mean_and_covariance#Variance_of_the_sample_mean ]. These "variances of means" are averaged over all unique reflections of the resolution shell:

 

 

 

 

 

The unbiased sample variance from all averaged intensities of all unique reflections is calculated by:

 

<math>\sigma^2_{y} = \frac{1}{N-1} \cdot \left ( \sum^N_{i} \overline{x}^2 - \frac{\left ( \sum^N_{i} \overline{x} \right )^2}{ N} \right ) </math>

 

With <math>\overline{x}= \sum^n_{j} x_{j}</math> , average intensity of all observations from all frames/crystals of one unique reflection i. This is done for all reflections N in a resolution shell.

 

 

 

<pre>

    0    2    0  5.584E+02  3.093E+00  1  

</pre>

 

 

    2     1    1  1.608E+01  2.215E+01  2 

</pre>

<math>x_{i} </math> , the average intensity of all observations from all frames/crystals of this reflection = 52.5150

 

<math>\sigma^2_{\epsilon i} </math>, the unbiased sample variance of the mean of all observations of this unique reflection i = 363.3267 (1089.9803/(n/2))