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.
[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:
2 0 0 9.156E+02 3.686E+00 1532.7 1573.4 0.4 0.04149 1
0 2 0 5.584E+02 3.093E+00 1532.7 1516.6 0.7 0.04129 1
0 0 2 6.301E+02 2.405E+01 1570.7 1562.9 0.9 0.02624 1
2 0 0 9.256E+02 3.686E+00 1532.7 1573.4 1.4 0.04149 2
0 2 0 2.584E+02 3.093E+00 1532.7 1516.6 1.7 0.04129 2
0 0 2 7.301E+02 2.405E+01 1570.7 1562.9 1.9 0.02624 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:
1 1 2 2.395E+01 8.932E+01 1558.2 1508.9 0.4 0.05239 1
1 2 1 9.065E+01 7.407E+00 1539.1 1507.3 0.2 0.05473 1
2 1 1 5.981E+01 9.125E+00 1538.8 1507.4 0.9 0.05470 1
1 1 2 3.395E+01 8.932E+01 1558.2 1508.9 1.4 0.05239 2
1 2 1 9.065E+01 7.407E+00 1539.1 1507.3 1.2 0.05473 2
2 1 1 1.608E+01 2.215E+01 1519.8 1516.6 1.3 0.04126 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 =