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DeltaCC12: Difference between revisions
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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. | |||
: <math>\Delta CC_{1/2}= CC_{1/2 overall}-CC_{1/2 i} </math> | : <math>\Delta CC_{1/2}= CC_{1/2 overall}-CC_{1/2 i} </math> | ||
If | 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> | : <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> | ||
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The unbiased sample variance from all averaged intensities of all unique reflections is calculated by: | The unbiased sample variance from all averaged intensities of all unique reflections is calculated by: | ||
<math>\sigma^2_{y} = \frac{1}{ | <math>\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>x_{i} </math> , average intensity of all observations from all frames/crystals of one unique reflection i. This is done for all reflections | With <math>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>\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 ) \backslash \frac{n}{2} </math> | <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 ) \backslash \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. As we are considering CC12, the variance <math>\sigma^2_{\epsilon i} </math> is divided by <math>\frac{n}{2} </math> and not by '''n''' as described in [https://en.wikipedia.org/wiki/Sample_mean_and_covariance#Variance_of_the_sample_mean ]. | 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. As we are considering CC12, the variance <math>\sigma^2_{\epsilon i} </math> is divided by <math>\frac{n}{2} </math> and not only by '''n''' as described in [https://en.wikipedia.org/wiki/Sample_mean_and_covariance#Variance_of_the_sample_mean ], 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 n in a resolution shell and divided by N, the total number of unique reflections. | ||
<math>\sum^N_{i} \sigma^2_{\epsilon i} \backslash N </math> | |||
===''' <math>\sigma^2_{\epsilon} </math>''' -weighted=== | ===''' <math>\sigma^2_{\epsilon} </math>''' -weighted=== | ||
to be edited | to be edited | ||