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CC1/2: Difference between revisions
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== Implementation == | == Implementation == | ||
===''' <math>\sigma^2_{\epsilon} </math>''' | ===''' <math>\sigma^2_{\epsilon} </math>'''=== | ||
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: | 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> | <math>\sigma^2_{\epsilon i} = \frac{1}{n-1} \cdot \left ( \sum^n_{j} x^2_{j,i} - \frac{\left ( \sum^n_{j}x_{j,i} \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: | With <math>x_{j,i} </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 (intensities of 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>\sum^N_{i} \sigma^2_{\epsilon i} / N </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}{N-1} \cdot \left ( \sum^N_{i} \overline{x}^2 - \frac{\left ( \sum^N_{i} \overline{x} \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> | ||
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. | With <math>\overline{x}_{i}= \sum^n_{j} x_{j,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. | ||