DeltaCC12: Difference between revisions

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Gassmann

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 == Implementation ==
-===''' <math>\sigma^2_{y} </math>'''===
+===''' <math>\sigma^2_{\epsilon} </math>''' - unweighted===
-The unbiased sample variance from all averaged intensities of all unique reflections is calculated 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_{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>
+<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_{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.
+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 i in a resolution shell and divided by N, the total number of unique reflections.
-----
+<math>\sum^N_{i} \sigma^2_{\epsilon i} / N </math>
-===''' <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. 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 i in a resolution shell and divided by N, the total number of unique reflections.
+===''' <math>\sigma^2_{y} </math>'''===
-<math>\sum^N_{i} \sigma^2_{\epsilon i} / N </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}</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.