DeltaCC12

ΔCC12 is a quantity, that detects datasets/frames, that are non-isomorphous. As described in 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 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.

[math]\displaystyle{ \Delta CC_{1/2}= CC_{1/2 overall}-CC_{1/2 i} }[/math]

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]\displaystyle{ 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]\displaystyle{ \sigma^2_{y} }[/math], the variance of the average intensities across the unique reflections of a resolution shell, and [math]\displaystyle{ \sigma^2_{\epsilon} }[/math], the average of all sample variances of the mean across all unique reflections of a resolution shell.

Implementation

[math]\displaystyle{ \sigma^2_{y} }[/math]

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

[math]\displaystyle{ \sigma^2_{y} = \frac{1}{N-1} \cdot \left ( \sum^N_{i} x^2_i - \frac{\left ( \sum^N_{i}x_{i} \right )^2}{ N} \right ) }[/math]

With [math]\displaystyle{ x_{i} }[/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.

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[math]\displaystyle{ \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]\displaystyle{ \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]\displaystyle{ x_{j} }[/math] , a single observation j of all observations n of one reflection i. [math]\displaystyle{ \sigma^2_{\epsilon i} }[/math] is then divided by the factor [math]\displaystyle{ \frac{n}{2} }[/math], because the variance of the sample mean (the merged observations) is the quantity of interest. As we are considering CC12, the variance [math]\displaystyle{ \sigma^2_{\epsilon i} }[/math] is divided by [math]\displaystyle{ \frac{n}{2} }[/math] and not only by n as described in [1], because we are calculating the random errors of the merged intensities of a half-dataset. The single variance terms are then summed up for all reflections i in a resolution shell and divided by N, the total number of unique reflections.

[math]\displaystyle{ \sum^N_{i} \sigma^2_{\epsilon i} / N }[/math]

Example

An example is shown for a very simplified data file (unmerged ASCII.HKL). Only two frames/crystals are looked at and the diffraction pattern also consists only of two unique reflections with each three observations for every unique reflection.

First reflection with 6 observations:
     h     k     l       int     σ(int)  #datset
     2     0     0  9.156E+02  3.686E+00   1 
     0     2     0  5.584E+02  3.093E+00   1 
     0     0     2  6.301E+02  2.405E+01   1  
     2     0     0  9.256E+02  3.686E+00   2 
     0     2     0  2.584E+02  3.093E+00   2 
     0     0     2  7.301E+02  2.405E+01   2 

[math]\displaystyle{ x_{i} }[/math] , the average intensity of all observations from all frames/crystals of this reflection = 669.6999

[math]\displaystyle{ \sigma^2_{\epsilon i} }[/math], the unbiased sample variance of the mean of all observations of this unique reflection i = 20848.2198 (62544.6597/(n/2))

Second reflection with 6 observations:
     h     k     l       int     σ(int)  #datset
     1     1     2  2.395E+01  8.932E+01   1  
     1     2     1  9.065E+01  7.407E+00   1  
     2     1     1  5.981E+01  9.125E+00   1  
     1     1     2  3.395E+01  8.932E+01   2  
     1     2     1  9.065E+01  7.407E+00   2  
     2     1     1  1.608E+01  2.215E+01   2  

[math]\displaystyle{ x_{i} }[/math] , the average intensity of all observations from all frames/crystals of this reflection = 52.5150

[math]\displaystyle{ \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))

[math]\displaystyle{ \sigma^2_{\epsilon} }[/math] , the average of all the [math]\displaystyle{ \sigma^2_{\epsilon i} }[/math] = 10605.7733

[math]\displaystyle{ \sigma^2_{y} }[/math], the variance of all the averaged intensities = 190458.6533

As a result of these calculations CC12 =