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CC1/2: Difference between revisions
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===''' <math>\sigma^2_{\epsilon} </math>'''=== | ===''' <math>\sigma^2_{\epsilon} </math>'''=== | ||
With <math>x_{j,i} </math> , a single observation j of all observations n of one reflection i, 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: | With <math>x_{j,i} </math> , a single observation j of all observations n of one reflection i, the average of all sample variances of the mean across all unique reflections of a resolution shell is obtained by calculating the unbiased sample variance of the mean for every unique reflection i by: | ||
<math>\sigma^2_{\epsilon i} = \frac{1}{n_{i}-1} \cdot \left ( \sum^{n_{i}}_{j} x^2_{j,i} - \frac{\left ( \sum^{n_{i}}_{j}x_{j,i} \right )^2}{n_{i}} \right ) / \frac{n_{i}}{2} </math> | <math>\sigma^2_{\epsilon i} = \frac{1}{n_{i}-1} \cdot \left ( \sum^{n_{i}}_{j} x^2_{j,i} - \frac{\left ( \sum^{n_{i}}_{j}x_{j,i} \right )^2}{n_{i}} \right ) / \frac{n_{i}}{2} </math> | ||
<math>\sigma^2_{\epsilon i} </math> is divided by the factor <math>\frac{n}{2} </math>, because the variance of the sample mean (intensities of the merged observations) is the quantity of interest. The division by '''n/2''' takes care of providing the variance of the mean ([https://en.wikipedia.org/wiki/Sample_mean_and_covariance#Variance_of_the_sample_mean ]) (merged) intensity of the '''half'''-datasets, as defined in [https://www.ncbi.nlm.nih.gov/pmc/articles/PMC3457925/ Karplus and Diederichs (2012)]. These "variances of means" are averaged over all unique reflections of the resolution shell: | <math>\sigma^2_{\epsilon i} </math> is divided by the factor <math>\frac{n}{2} </math>, because the variance of the sample mean (intensities of the merged observations) is the quantity of interest. The division by '''n/2''' takes care of providing the variance of the mean ([https://en.wikipedia.org/wiki/Sample_mean_and_covariance#Variance_of_the_sample_mean ]) (merged) intensity of the '''half'''-datasets, as defined in [https://www.ncbi.nlm.nih.gov/pmc/articles/PMC3457925/ Karplus and Diederichs (2012)]. These "variances of means" are averaged over all unique reflections of the resolution shell: | ||
<math>\sigma^2_{\epsilon}=\sum^N_{i} \sigma^2_{\epsilon i} / N </math> | |||
---- | |||
<math>\sigma^2_{\epsilon}=\sum^ | |||
If the standard deviations <math>\sigma_{int} </math> for the single observations are considered as weights for the CC<sub>1/2</sub> calculation, with <math>\sum^{n_{i}}_{j}w_{i}=\sum^{n_{i}}_{j}\frac{1}{\sigma_{int}^2} </math> the unbiased '''weighted''' sample variance of the mean for every unique reflection i is obtained by: | |||
<math>\sigma^2_{\epsilon i\_w} = \frac{n_{i}}{n_{i}-1} \cdot \left ( \frac{\sum^{n_{i}}_{j}w_{j,i} x^2_{j,i}}{\sum^{n_{i}}_{j}w_{j,i}} - \frac{\left ( \sum^{n_{i}}_{j}w_{j,i}x_{j,i} \right )^2}{\sum^{n_{i}}_{j}w_{j,i}} \right ) / \frac{n_{i}}{2} </math> | |||
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Let N be the number of reflections. With <math>\overline{x}_{i}= \sum^n_{j} x_{j,i}</math> , the unbiased sample variance from all averaged intensities of all unique reflections is calculated by: | Let N be the number of reflections. With <math>\overline{x}_{i}= \sum^n_{j} x_{j,i}</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}_{i}^2 - \frac{\left ( \sum^N_{i} \overline{x}_{i} \right )^2}{ N} \right ) </math> | <math>\sigma^2_{y} = \frac{1}{N-1} \cdot \left (\sum^N_{i} \overline{x}_{i}^2 - \frac{\left ( \sum^N_{i} \overline{x}_{i} \right )^2}{ N} \right ) </math> | ||