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CC1/2: Difference between revisions
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== why CC<sub>1/2</sub> can be negative == | == why CC<sub>1/2</sub> can be negative == | ||
There is a mathematical reason, explained in §4.1 of [https://cms.uni-konstanz.de/index.php?eID=tx_nawsecuredl&u=0&g=0&t=1475179096&hash=5cf64234a23a794a1894c5408384c57208d7b602&file=fileadmin/biologie/ag-strucbio/pdfs/Assman2016_JApplCryst.pdf Assmann, G., Brehm, W. and Diederichs, K. (2016) Identification of rogue datasets in serial crystallography (2016) J. Appl. Cryst. 49, 1021-1028.] | There is a mathematical reason, explained in §4.1 of [https://cms.uni-konstanz.de/index.php?eID=tx_nawsecuredl&u=0&g=0&t=1475179096&hash=5cf64234a23a794a1894c5408384c57208d7b602&file=fileadmin/biologie/ag-strucbio/pdfs/Assman2016_JApplCryst.pdf Assmann, G., Brehm, W. and Diederichs, K. (2016) Identification of rogue datasets in serial crystallography (2016) J. Appl. Cryst. 49, 1021-1028.] | ||
==CC<sub>1/2</sub> calculation== | |||
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. | |||
== Implementation == | |||
===''' <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: | |||
<math>\sum^N_{i} \sigma^2_{\epsilon i} / N </math> | |||
---- | |||
===''' <math>\sigma^2_{y} </math>'''=== | |||
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. | |||
== 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. | |||
<pre> | |||
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 | |||
</pre> | |||
<math>x_{i} </math> , the average intensity of all observations from all frames/crystals of this reflection = 669.6999 | |||
<math>\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)) | |||
<pre> | |||
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 | |||
</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)) | |||
<math>\sigma^2_{\epsilon} </math> , the average of all the <math>\sigma^2_{\epsilon i} </math> = 10605.7733 | |||
<math>\sigma^2_{y} </math>, the variance of all the averaged intensities = 190458.6533 | |||
As a result of these calculations CC12 = | |||