Poisson and normal statistics for diffraction

The normal situation for diffraction data is that the observed signal is a photon count. Therefore it follows a Poisson distribution. If we have a count value $C_0$ that follows a Poisson distribution, we can assume immediately that the average is equal to $C_0$ and the s.d. is $\sqrt{C_0}$ . I.e., repeated experiments would give values $n$ distributed according to the normalized distribution

$$P(n)={\ensuremath{\displaystyle{\frac{{\ensuremath{\displaystyle{C_0^n{\ensuremath{\mathrm{e}}}^{-C_0} }}}}{{\ensuremath{\displaystyle{ n!}}}}}}}$$

This obeys

$$\mathop{\sum}_{n=0}^{+\infty} P(n)=1\ ;$$

$$\langle n\rangle=\mathop{\sum}_{n=0}^{+\infty} nP(n)=C_0\ ;$$

$$\langle n^2\rangle=\mathop{\sum}_{n=0}^{+\infty} n^2 P(n)=C_0^2+C_0\ ;$$

The standard deviation comes then to

$$\sigma_{C_0}=\sqrt{\langle n^2\rangle-\langle n\rangle^2}=\sqrt{C_0}$$

When the data have to be analyzed, one must compare observations with a model which gives calculated values of the observations in dependence of a certain set of parameters. The best values of the parameters (the target of investigation) are the one that maximize the likelihood function [4,5]. The likelihood function for Poisson variates is pretty difficult to use; furthermore, even simple data manipulations are not straightforward with Poisson variates (see Sec. 5.2.6). The common choice is to approximate Poisson variates with normal variates, and then use the much easier formalism of normal distribution to a) do basic data manipulations and b) fit data with model. To the latter task, in fact, the likelihood function is maximized simply by minimizing the usual weighted-$\chi ^2$ [4] :

$$\chi^2 = \mathop{\sum}_{j=1}^{N_{\mathrm{obs}}} {\ensuremath{\dis... ...\left({F_j-O_j}\right)}}^2 }}}}{{\ensuremath{\displaystyle{ \sigma_j^2 }}}}}}}$$

where $O_j$ are the experimentally observed values, $F_j$ the calculated model values, $\sigma_j$ the s.d.s of the observations.

Substituting directly the counts (and derived s.d.s) for the observations in the former :

$$\chi_{(0)}^2 = \mathop{\sum}_{j=1}^{N_{\mathrm{obs}}} {\ensuremat... ...remath{\left({F_j-C_j}\right)}}^2 }}}}{{\ensuremath{\displaystyle{ C_j }}}}}}}$$

is the most common way. It is slightly wrong to do so, however [6], the error being large only when the counts are low. There is also a divergence for zero counts. In fact, a slightly modified form [6] exists, reading

$$\chi_{(1)}^2 = \mathop{\sum}_{j=1}^{N_{\mathrm{obs}}} {\ensuremat... ...\right)}}}\right)}}}\right)}}^2 }}}}{{\ensuremath{\displaystyle{ C_j+1 }}}}}}}$$

Minimizing this form of $\chi ^2$ is equivalent - to an exceptionally good approximation [6]- to maximizing the proper Poisson-likelihood.