Intensities scaled to the local average

Shmueli, U.; Wilson, A. J. C.

doi:10.1107/97809553602060000554

International
Tables for
Crystallography
Volume B
Reciprocal space
Edited by U. Shmueli

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International Tables for Crystallography (2006). Vol. B. ch. 2.1, p. 198 | 1 | 2 |

Section 2.1.6.3. Intensities scaled to the local average

U. Shmueli^a ^* and A. J. C. Wilson^b ^‡

^a School of Chemistry, Tel Aviv University, Tel Aviv 69 978, Israel, and ^bSt John's College, Cambridge, England
Correspondence e-mail: ushmueli@post.tau.ac.il

2.1.6.3. Intensities scaled to the local average

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When the $[G_{i}]$ 's are a subset of the $[H_{i}]$ 's, the beta distributions of the second kind are replaced by beta distributions of the first kind, with means and variances readily found from Table 2.1.5.1. The distribution of such a ratio is chiefly of interest when Y relates to a single reflection and Z relates to a group of m intensities including Y. This corresponds to normalizing intensities to the local average. Its distribution is $[p(I/\langle I \rangle)\;{\rm d}(I/\langle I \rangle) = \beta_{1}(I/n\langle I \rangle\hbox{; } 1,n-1)\;{\rm d}(I/n\langle I \rangle) \eqno(2.1.6.18)]$ in the acentric case, with an expected value of $[I/\langle I \rangle]$ of unity; there is no bias, as is obvious a priori. The variance of $[I/\langle I \rangle]$ is $[\sigma^{2} = {{n-1}\over{n+1}}, \eqno(2.1.6.19)]$ which is less than the variance of the intensities normalized to an `infinite' population by a fraction of the order of [2/n] . Unlike the variance of the scaling factor, the variance of the normalized intensity approaches unity as n becomes large. For intensities having a centric distribution, the distribution normalized to the local average is given by $[{p(I/\langle I \rangle)\;{\rm d}(I/\langle I \rangle) = \beta_{1}[I/n\langle I \rangle\hbox{; } 1/2,(n-1)/2]\;{\rm d}(I/n\langle I \rangle),} \eqno(2.1.6.20)]$ with an expected value of $[I/\langle I \rangle]$ of unity and with variance $[\sigma^{2} = {{2(n-1)}\over{n+2}}, \eqno(2.1.6.21)]$ less than that for an `infinite' population by a fraction of about [3/n] .

Similar considerations apply to intensities normalized to $[\Sigma]$ in the usual way, since they are equal to those normalized to $[\langle I \rangle]$ multiplied by $[\langle I \rangle/\Sigma]$ .

References

International Tables for Crystallography (2006). Vol. B. ch. 2.1, p. 198