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876 lines
31 KiB
TeX
Executable File
876 lines
31 KiB
TeX
Executable File
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\subsection{Introduction}
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\subsubsection{Notation}
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[I use symbol $\TT$ for the diffraction angle $2\theta$]
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\subsubsection{Observables}
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The physical observable of interest in any scattering experiment is [1-3] the differential cross section
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\[
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\DSF{\DD{}\bf{\sigma}}{\DD{}\Omega}
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\]
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as a function of direction $\Omega$.
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To measure that directly we should operate with zero-width point detectors, with instant measurement and unit incident intensity.
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Practically
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the quantity we can actually measure - putting a detector in a position covering a certain
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solid angle for a certain time with a certain incident intensity - is
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\[
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{I_0}\Delta t \Delta\Omega\DSF{\DD{}\bf{\sigma}}{\DD{}\Omega}
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\]
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If $\Delta t$, $\Delta\Omega$ are small and known and $I_0$ is separately monitored,
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we can (have to) normalize the observations by simply dividing them out.
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Specifically for the powder diffraction field, historically, this is not usually done because
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- as it is normally true with anode sources and point detectors and usual procedures -
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the counting times $\Delta t$, the solid angle width $\Delta\Omega\propto \Delta \TT$
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and the incident intensity $I_0$ are considered
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constant and therefore go into some 'global scaling' constant that is usually considered arbitrary.
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However, as we have more sophisticated acquisition methods,
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we may need revert to the original approach and consider the
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counts divided by time and angular width as the real observable.
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\subsection{Basic binning}\label{sec:11}
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\begin{itemize}
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\item[1.\ ]{
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We have several patterns, say $P$. Each $k$-th pattern, for $k=1,\ldots,P$, is
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constituted by $N_k$ angular intervals in the diffraction angle $2\theta\equiv\TT$:
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\[
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b_{k,j}=\lrs{\TT_{k,j}^{-},\TT_{k,j}^{+}},\qquad j=1,\ldots,N_k
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\]
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of center
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\[
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\hat{b}_{k,j}=\DSF{\TT_{k,j}^{+}+\TT_{k,j}^{-}}{2}
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\]
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and width
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\[
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\lrv{b_{k,j}}=\TT_{k,j}^{+}-\TT_{k,j}^{-}
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\]
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To each interval is associated a counting $C_{k,j}$, an efficiency correction factor $e_{k,j}$, a
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monitor $m_{k,j}$ (ionization chamber times acquisition time). All 'bad' intervals have been already flagged down and discarded.
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Efficiency corrections and monitors are supposed to be normalized to a suitable value.
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Note that intervals $b_{k,j}$ might have multiple overlaps and might not cover an compact angular
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range. }
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%
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\item[2.\ ]{Following Mighell's statistics[6] and normal scaling procedures, we first
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transform those numbers into associated intensities, intensity rates and relevant s.d.:
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\[
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I_{k,j}=\DSF{e_{k,j}}{m_{k,j}}\lrb{C_{k,j}+\min\lrb{1,C_{k,j}}}
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\]
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\[
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\sigma_{I_{k,j}}=\DSF{e_{k,j}}{m_{k,j}}\sqrt{\lrb{C_{k,j}+1}}
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\]
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\[
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r_{k,j}=\DSF{I_{k,j}}{\lrv{b_{k,j}}}=\DSF{e_{k,j}}{m_{k,j}\lrv{b_{k,j}}}\lrb{C_{k,j}+\min\lrb{1,C_{k,j}}}
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\]
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\[
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\sigma_{r_{k,j}}=\DSF{\sigma_{I_{k,j}}}{\lrv{b_{k,j}}}=\DSF{e_{k,j}}{\lrv{b_{k,j}}m_{k,j}}\sqrt{\lrb{C_{k,j}+1}}
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\]
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}
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\item[3.\ ]{
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We set up the final binned grid,
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composed of $M$ binning intervals
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\[
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B_\ell=[\TT_0+(\ell-1)B, \TT_0+\ell B],\qquad \ell=1,\ldots,M
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\]
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all contiguous and each having the same width \[\lrv{B_\ell}=B\] and each centered in
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\[\hat{B}_\ell=\TT_0+(\ell-1/2)B,\]
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covering completely the angular range between $\TT_0$ and $\TT_{max}=\TT_0+MB$.
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}
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\item[4.\ ]{
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For bin $\ell$, we consider only and all the experimental intervals
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\[
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b_{k,j}\qquad\text{such\ that}\qquad \lrv{ b_{k,j}\cap B_\ell } > 0.
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\]
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More restrictively, one may require to consider only and all the experimental intervals
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\[
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b_{k,j}\qquad\text{such\ that}\qquad \hat{b}_{k,j}\in B_\ell .
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\]
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}
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\item[5.\ ]{
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In order to estimate the rate in each $\ell$-th bin,
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we use all above selected rate estimates concerning bin $B_\ell$ and we get
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a better one with the weighted average method. \\
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In the weighted average method, we suppose to have a number $N_E$ of estimates $O_n$
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of the same observable $O$,
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each one with a known s.d. $\sigma_{O_n}$ and each (optionally) repeated with a frequency
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$\nu_n$.
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Then
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\[
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\langle O\rangle =\DSF{
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\mathop{\sum}_{n=1}^{N_E}\nu_n
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O_n\sigma_{O_n}^{-2}
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}{
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\mathop{\sum}_{n=1}^{N_E}\nu_n
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\sigma_{O_n}^{-2}
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}
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\]
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%
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Clearly the place of the frequencies in our case can be taken by coefficients
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\[
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\DSF{\lrv{ b_{k,j}\cap B_\ell }}{B}
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\]
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that weigh the $k,j$-th estimate by its relative extension within bin $B_\ell$.
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}
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\item[6.\ ]{
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Now
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we can simply accumulate registers
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\[
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X_\ell=\mathop{\sum_{k,j}}_{ \lrv{ b_{k,j}\cap B_\ell } > 0}
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\DSF{\lrv{ b_{k,j}\cap B_\ell }}{B}\ r_{k,j}\ \lrb{\sigma_{r_{k,j}}}^{-2}
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\]
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and
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\[
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Y_\ell=\mathop{\sum_{k,j}}_{ \lrv{ b_{k,j}\cap B_\ell } > 0}
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\DSF{\lrv{ b_{k,j}\cap B_\ell }}{B}\ \lrb{\sigma_{r_{k,j}}}^{-2}
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\]
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so that we can extract an intensity rate estimate (counts per unit diffraction angle and per unit time at constant incident intensity) as
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\[
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R_\ell=\DSF{X_\ell}{Y_\ell};
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\]
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\[
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\sigma_{R_\ell}=\DSF{1}{\sqrt{Y_\ell}}.
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\]
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Now optionally we can transforms rates in intensities (multiplying
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both $R_\ell$ and $\sigma_{R_\ell}$ by $B$).
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We can use any other scaling factor $K$ as we wish instead of $B$.
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The best cosmetic scaling is the one where
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\[
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\mathop{\sum}_{\ell=1}^M\DSF{KR_\ell}{K^2\sigma_{R_\ell}^2}=
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\DSF{1}{K}
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\mathop{\sum}_{\ell=1}^M\DSF{R_\ell}{\sigma_{R_\ell}^2}=M
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\]
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as if the intensities were simply counts.
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Therefore $K$ is given by
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\[
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K=\DSF{
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1
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}{
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M
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}\mathop{\sum}_{\ell=1}^M\DSF{R_\ell}{\sigma_{R_\ell}^2}
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\]
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In output then we give 3-column files
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with columns
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\[
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\hat{B}_\ell, \quad KR_\ell, \quad K\sigma_{R_\ell}
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\]
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}
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\end{itemize}
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\subsubsection{Special nasty cases}
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Here we explore some special cases to see the robustness
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of the method.
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1) If no experimental observation contributes to bin $B_\ell$ according to one of the criteria
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above, then we shall find $X_\ell=0$ and especially $Y_\ell=0$. The latter condition is
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valid as an exclusion condition
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(meaning that we discard that point and we do not perform further operations on it,
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neither do we output it).
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2) if only one experimental observation - call it interval $b$, dropping indices - contributes
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to bin $B_\ell$,
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then we have
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\[
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X_\ell=\DSF{\lrv{ b\cap B_\ell }}{B}\
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\DSF{e(C+1)}{m|b|}\
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\lrb{
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\DSF{|b|m}{e\sqrt{C+1}}
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}^{2}
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=\DSF{\lrv{ b\cap B_\ell }}{B}\DSF{|b|m}{e}
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\]
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\[
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Y_\ell=\DSF{\lrv{ b\cap B_\ell }}{B}\
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\lrb{
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\DSF{|b|m}{e\sqrt{C+1}}
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}^{2}
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=\DSF{\lrv{ b\cap B_\ell }}{B}\DSF{|b|^2m^2}{e^2(C+1)}
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\]
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and so
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\[
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R_\ell=\DSF{X_\ell}{Y_\ell}=\DSF{e(C+1)}{m|b|}
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\]
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that is the experimental rate as in pixel $b$;
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\[
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\sigma_{R_\ell}=\DSF{1}{\sqrt{Y_\ell}}=
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\sqrt{\DSF{B}{\lrv{ b\cap B_\ell }}}
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%
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\DSF{e\sqrt{(C+1)}}{|b|m}
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\]
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that is the same s.d. that can be calculated directly for $b$, augmented by factor
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\[
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\sqrt{\DSF{B}{\lrv{ b\cap B_\ell }}}
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\]
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that takes into account the extrapolation error.
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\subsection{Advanced binning}\label{sec:2}
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There are more advanced (and more complex) methods that take more carefully into account the real position of the centers $\hat{b}_{j,k}$ w.r.t. $\hat{B}_\ell$.
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If we find out that it is the case we may develop them too.
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\subsection{Poisson and normal statistics for diffraction}
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The normal situation for diffraction data
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is that the observed signal is a photon count.
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Therefore it follows a Poisson distribution.
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If we have a count value $C_0$ that follows a Poisson distribution,
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we can assume immediately that the average is equal to $C_0$ and the s.d. is $\sqrt{C_0}$.
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I.e., repeated experiments would give values $n$
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distributed according to the normalized distribution
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\[
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P(n)=\DSF{C_0^n\EE^{-C_0}
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}{
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n!}
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\]
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This obeys
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\[
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\mathop{\sum}_{n=0}^{+\infty}
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P(n)=1\ ;
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\]
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\[
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\langle n\rangle=\mathop{\sum}_{n=0}^{+\infty}
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nP(n)=C_0\ ;
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\]
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\[
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\langle n^2\rangle=\mathop{\sum}_{n=0}^{+\infty}
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n^2 P(n)=C_0^2+C_0\ ;
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\]
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The standard deviation comes then to
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\[
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\sigma_{C_0}=\sqrt{\langle n^2\rangle-\langle n\rangle^2}=\sqrt{C_0}
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\]
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When the data have to be analyzed, one must compare observations with a model
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which gives calculated values of the observations in dependence of a certain set of
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parameters. The best values of the parameters (the target of investigation)
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are the one that maximize the likelihood function [4,5]. The likelihood function for
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Poisson variates is pretty difficult to use; furthermore, even simple data manipulations
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are not straightforward with Poisson variates (see \sref{sec:3}). The common choice is to approximate
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Poisson variates with normal variates, and then use the much easier formalism
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of normal distribution to a) do basic data manipulations and b) fit data with model.
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To the latter task, in fact, the likelihood function is maximized simply by minimizing
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the usual weighted-$\chi^2$[4] :
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\[
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\chi^2 = \mathop{\sum}_{j=1}^{N_{\mathrm{obs}}}
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\DSF{\lrb{F_j-O_j}^2
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}{
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\sigma_j^2
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}
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\]
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where $O_j$ are the experimentally observed values, $F_j$ the calculated model values,
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$\sigma_j$ the s.d.s of the observations.
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Substituting directly the counts (and derived s.d.s) for the observations in the former :
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\[
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\chi_{(0)}^2 = \mathop{\sum}_{j=1}^{N_{\mathrm{obs}}}
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\DSF{\lrb{F_j-C_j}^2
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}{
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C_j
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}
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\]
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is the most common way. It is \emph{slightly} wrong to do so, however [6],
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the error being large only when the counts are low.
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There is also a divergence for zero counts.
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In fact, a slightly modified form [6] exists, reading
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\[
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\chi_{(1)}^2 = \mathop{\sum}_{j=1}^{N_{\mathrm{obs}}}
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\DSF{\lrb{F_j-\lrb{C_j+\min\lrb{1,C_j}}}^2
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}{
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C_j+1
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}
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\]
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Minimizing this form of $\chi^2$ is equivalent - to an exceptionally good approximation [6]-
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to maximizing the proper Poisson-likelihood.
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\subsection{Average vs. weighted average}
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\subsubsection{Simple average}
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Suppose we have $N_{\mathrm{obs}}$ Poisson-variate experimental evaluations
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$C_j,\quad j=1\ldots N_{\mathrm{obs}}$,
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of the same quantity $x$.
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There are different ways to obtain from all $N_{\mathrm{obs}}$ data values a single estimate of the observable which is better than
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any of them. The most straightforward and the best is the simple average
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\[
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x=\langle x\rangle=\DSF{1}{ N_{\mathrm{obs}}}
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\mathop{\sum}_{j=1}^{N_{\mathrm{obs}}}C_j\ .
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\]
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As the sum of Poisson variates is a Poisson variate, the standard deviation
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\[
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\sigma_x=\sqrt{\langle x^2\rangle-\langle x\rangle^2}=\sqrt{
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\DSF{1}{ N_{\mathrm{obs}}}
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\mathop{\sum}_{j=1}^{N_{\mathrm{obs}}}C_j^2-\lrb{
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\DSF{1}{ N_{\mathrm{obs}}}
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\mathop{\sum}_{j=1}^{N_{\mathrm{obs}}}C_j
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}
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}
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\]
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can be evaluated more comfortably as
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\[
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\sigma_x=\DSF{1}{ N_{\mathrm{obs}}}\sqrt{ \mathop{\sum}_{j=1}^{N_{\mathrm{obs}}}C_j }
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=\sqrt{\DSF{\langle x\rangle}{N_{\mathrm{obs}}}}
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\]
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\subsubsection{Zero-skipping average}
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In some cases, in order to avoid possible singularities,
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values $C_j=0$ are skipped. Then if $N_{\mathrm{obs}}^*$ is the number of non-zero data points,
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we can evaluate the 'zero-skipping' average as
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\[
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x=\langle x\rangle^*=\DSF{1}{ N_{\mathrm{obs}}^*}
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\mathop{\sum}_ {\stackrel{1\leqslant j\leqslant N_{\mathrm{obs}}}{{C_j>0}}}
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C_j=\DSF{1}{ N_{\mathrm{obs}}^*}
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\mathop{\sum}_{j=1}^{N_{\mathrm{obs}}}C_j = \DSF{N_{\mathrm{obs}}}{N_{\mathrm{obs}}^*}\langle x\rangle
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\]
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The standard deviation is then
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\[
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\sigma_{x^*}= \DSF{N_{\mathrm{obs}}}{N_{\mathrm{obs}}^*}\sigma_x = \sqrt{\DSF{N_{\mathrm{obs}}}{N_{\mathrm{obs}}^*}}
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\sqrt{\DSF{\langle x\rangle}{N_{\mathrm{obs}}^*}}=\sqrt{\DSF{\langle x\rangle^*}{N_{\mathrm{obs}}^*}}
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\]
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Note that the s.d. is evaluated exactly as if the non-zero $C_j$ were the only observations,
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whilst the average is overestimated by the fraction of zero-counting events.
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\subsubsection{Weighted average: definition and relationship with $\chi^2$}
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A weighted average is the result of the special case of a data fitting to a model function which is a constant.
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It is easy to see that minimizing w.r.t $x$
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\[
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\chi^2 = \mathop{\sum}_{j=1}^{N_{\mathrm{obs}}}
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\DSF{\lrb{x-O_j}^2
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}{
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\sigma_j^2
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}
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\]
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yields
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\[
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x= \langle x \rangle_{\!\mathrm{w}}=\DSF{
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\mathop{\sum}_{j=1}^{N_{\mathrm{obs}}}
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\DSF{O_j
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}{
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\sigma_j^2
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}
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}{
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\mathop{\sum}_{j=1}^{N_{\mathrm{obs}}}
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\DSF{1
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}{
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\sigma_j^2
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}
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}
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\]
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The good-faith s.d. (square-root of twice the inverse of the second derivative of $\chi^2$ at the minimum)
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is then
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\[
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\sigma_{\langle x \rangle_{\!\mathrm{w}}} = \DSF{
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1
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}{\sqrt{
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\mathop{\sum}_{j=1}^{N_{\mathrm{obs}}}
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\DSF{1
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}{
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\sigma_j^2
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}
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}}
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\]
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I use the term 'good-faith' to indicate the case when it is really appropriate to use a constant as a model functions,
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i.e. when the observations are truly different observations of the same observable.
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When this is not the case but we do not know what to do better we can at least increase the s.d.
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In fact, there is a correction factor for the s.d., given - in this case - by
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\[
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\mathsf{GoF}=
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\sqrt{
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\DSF{
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\mathop{\sum}_{j=1}^{N_{\mathrm{obs}}}
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\DSF{O_j^2
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}{
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\sigma_j^2
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}
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-\DSF{
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\lrs{
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\mathop{\sum}_{j=1}^{N_{\mathrm{obs}}}
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\DSF{O_j
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}{
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\sigma_j^2
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}
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}^2
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}{ \mathop{\sum}_{j=1}^{N_{\mathrm{obs}}}
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\DSF{1
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}{
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\sigma_j^2
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} }
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}{
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N_{\mathrm{obs}}-1
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}
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}
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\]
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so that
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\[
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{\sigma}_{\langle x \rangle_{\!\mathrm{w}}}^{\mathrm{corrected}} = \mathsf{GoF}\ \sigma_{\langle x \rangle_{\!\mathrm{w}}}
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\]
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Specializing now to the two cases above,
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\subsubsection{Straight Poisson (zero-skipping) weighted average}
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When $O_j=C_j$ and $\sigma_j^2=C_j$
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\[
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\langle x \rangle_{\!\mathrm{w(1)}}=\DSF{
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{N_{\mathrm{obs}}}
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}{
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\mathop{\sum}_{j=1}^{N_{\mathrm{obs}}}
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\DSF{1
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}{
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C_j
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}
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}
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\]
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Here we need to eliminate the singularity when $C_j=0$. In order to do so, we skip data points which are zero.
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Then if $N_{\mathrm{obs}}^*$ is the number of non-zero data points,
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\[
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\langle x \rangle_{\!\mathrm{w(1)}}=\DSF{
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{N_{\mathrm{obs}}^*}
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}{
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\mathop{\sum}_{\stackrel{1\leqslant j\leqslant N_{\mathrm{obs}}}{{C_j>0}}}
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\DSF{1
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}{
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C_j
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}
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}
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\]
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\[
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\sigma_{\langle x \rangle_{\!\mathrm{w(1)}}} = \DSF{
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1
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}{\sqrt{
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\mathop{\sum}_{\stackrel{1\leqslant j\leqslant N_{\mathrm{obs}}}{{C_j>0}}}
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\DSF{1
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}{
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C_j
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}
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}}=\sqrt{\DSF{\langle x \rangle_{\!\mathrm{w(1)}}}{
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N_{\mathrm{obs}}^*
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}}
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\]
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\[
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\mathsf{GoF}_{(1)}=
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\sqrt{
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\DSF{
|
|
\mathop{\sum}_{\stackrel{1\leqslant j\leqslant N_{\mathrm{obs}}}{{C_j>0}}}
|
|
\!\!\!\!C_j
|
|
%
|
|
-\DSF{
|
|
\lrs{
|
|
N_{\mathrm{obs}}^*
|
|
}^2
|
|
}{ \mathop{\sum}_{\stackrel{1\leqslant j\leqslant N_{\mathrm{obs}}}{{C_j>0}}}
|
|
\DSF{1
|
|
}{
|
|
C_j
|
|
} }
|
|
}{
|
|
N_{\mathrm{obs}}^*-1
|
|
}
|
|
}
|
|
=\sqrt{
|
|
\DSF{N_{\mathrm{obs}}^*}{N_{\mathrm{obs}}^*-1}
|
|
\lrb{
|
|
\langle x\rangle^*-\langle x \rangle_{\!\mathrm{w(1)}}
|
|
}
|
|
}
|
|
\]
|
|
where $\langle x\rangle^*$ is the simple average of the non-zero data points; and of course
|
|
\[
|
|
{\sigma}_{\langle x \rangle_{\!\mathrm{w(1)}}}^{\mathrm{corrected}} = \mathsf{GoF}_{(1)}\ \sigma_{\langle x \rangle_{\!\mathrm{w(1)}}}
|
|
\]
|
|
|
|
\subsubsection{Mighell-Poisson weighted average}
|
|
|
|
When $O_j=C_j+\min\lrb{1,C_j}$ and $\sigma_j^2=C_j+1$
|
|
\[
|
|
\langle x \rangle_{\!\mathrm{w(2)}}=\DSF{
|
|
{N_{\mathrm{obs}}^*}
|
|
}{
|
|
\mathop{\sum}_{\stackrel{1\leqslant j\leqslant N_{\mathrm{obs}}}{{C_j>0}}}
|
|
\DSF{1
|
|
}{
|
|
C_j+1
|
|
}
|
|
}
|
|
\]
|
|
\[
|
|
\sigma_{\langle x \rangle_{\!\mathrm{w(2)}}} = \DSF{
|
|
1
|
|
}{\sqrt{
|
|
\mathop{\sum}_{\stackrel{1\leqslant j\leqslant N_{\mathrm{obs}}}{{C_j>0}}}
|
|
\DSF{1
|
|
}{
|
|
C_j+1
|
|
}
|
|
}}=\sqrt{\DSF{\langle x \rangle_{\!\mathrm{w(2)}}}{
|
|
N_{\mathrm{obs}}^*
|
|
}}
|
|
\]
|
|
\[
|
|
\mathsf{GoF}_{(2)}=
|
|
\sqrt{
|
|
\DSF{
|
|
\mathop{\sum}_{\stackrel{1\leqslant j\leqslant N_{\mathrm{obs}}}{{C_j>0}}}
|
|
\!\!\!\!C_j+N_{\mathrm{obs}}^*
|
|
%
|
|
-\DSF{
|
|
\lrs{
|
|
N_{\mathrm{obs}}^*
|
|
}^2
|
|
}{ \mathop{\sum}_{\stackrel{1\leqslant j\leqslant N_{\mathrm{obs}}}{{C_j>0}}}
|
|
\DSF{1
|
|
}{
|
|
C_j+1
|
|
} }
|
|
}{
|
|
N_{\mathrm{obs}}^*-1
|
|
}
|
|
}
|
|
=\sqrt{
|
|
\DSF{N_{\mathrm{obs}}^*}{N_{\mathrm{obs}}^*-1}
|
|
\lrb{
|
|
\langle x\rangle^*-\langle x \rangle_{\!\mathrm{w(2)}}+1
|
|
}
|
|
}
|
|
\]
|
|
where $\langle x\rangle^*$ is the simple average of the non-zero data points; and of course
|
|
\[
|
|
{\sigma}_{\langle x \rangle_{\!\mathrm{w(2)}}}^{\mathrm{corrected}} = \mathsf{GoF}_{(2)}\ \sigma_{\langle x \rangle_{\!\mathrm{w(2)}}}
|
|
\]
|
|
|
|
|
|
|
|
|
|
|
|
|
|
\subsubsection{Comparison}
|
|
|
|
We have seen four different ways to take an average -
|
|
two simple averages (the second skipping zero values)
|
|
and two weighted averages (using straight Poisson and Poisson-Mighell [6] $\chi^2$ formulations).
|
|
We know that the simple average (not skipping zeros) is the best possible result. However,
|
|
there are inconveniences with it. If for instance we need to scale our data before averaging, then the
|
|
simple average is no more usable (it will give the correct average but a bad estimate of the s.d.) .
|
|
In any case, the passage to normal statistics (using Mighell's correction) needs to be done before or later.
|
|
Therefore a comparison is due in order to ascertain
|
|
how wrong can it be using the different methods.
|
|
|
|
We have to give a measure of what is negligible first.
|
|
The relative error is a measure of the smallest relative variation of an estimate $x$ that is \emph{not} negligible:
|
|
\[
|
|
\epsilon_x = \DSF{\sigma_x}{x}
|
|
\]
|
|
We shall then consider negligible
|
|
(w.r.t. $x$) terms whose relative magnitude is $O(\epsilon_x^2)$.
|
|
As the s.d. of $x$ is $\propto\epsilon_x$, we may not discard terms $O(\epsilon_x^2)$ on the s.d.;
|
|
there instead we may neglect terms $O(\epsilon_x^3)$.
|
|
|
|
|
|
\subsubsection{Analytical comparison of averages}
|
|
|
|
First we give an analytical comparison between simple average and Mighell-Poisson weighted average
|
|
for $N_{\mathrm{obs}}=2$.
|
|
If the two events are $C_1$ and $C_2$, then
|
|
\[
|
|
\langle x \rangle=\DSF{C_1+C_2}{2}; \qquad \sigma_x=\DSF{\sqrt{C_1+C_2}}{2}
|
|
\]
|
|
For the M-P weighted average,
|
|
\[
|
|
\langle x \rangle_{\mathrm{w(2)}}=\DSF{2(C_1+1)(C_2+1)}{C_1+C_2+2}; \qquad
|
|
\sigma_{\langle x \rangle_{\mathrm{w(2)}}}=\sqrt{\DSF{(C_1+1)(C_2+1)}{C_1+C_2+2}}
|
|
\]
|
|
|
|
Now, supposing that the common 'true' value of $C_1,C_2$ is $\lambda$,
|
|
we use the Poisson distribution to compare the expectation values of the two results. The expectation value of the simple average is
|
|
\[
|
|
E\lrb{\langle x \rangle} = \mathop{\sum}_{m,n=0}^{+\infty}
|
|
\DSF{n+m}{2}P(n)P(m)=\mathop{\sum}_{m=0}^{+\infty}
|
|
\DSF{m}{2}\DSF{\lambda^m\EE^{-\lambda}}{m!}
|
|
+\mathop{\sum}_{n=0}^{+\infty}
|
|
\DSF{n}{2}\DSF{\lambda^n\EE^{-\lambda}}{n!}
|
|
=\lambda
|
|
\]
|
|
As expected, the simple average gives the true value.
|
|
For its variance,
|
|
\[
|
|
E\lrb{\sigma_x^2} = \mathop{\sum}_{m,n=0}^{+\infty}
|
|
\DSF{n+m}{4}P(n)P(m)=\DSF{
|
|
\lambda}{2};\qquad E\lrb{\sigma_x} =\sqrt{\DSF{\lambda}{2}}
|
|
\]
|
|
|
|
|
|
In order to evaluate the difference with the M-P weighted average, we rewrite the latter as
|
|
\[
|
|
\langle x \rangle_{\mathrm{w(2)}}=\langle x \rangle + 1 -\DSF{(C_1-C_2)^2}{4(\langle x \rangle+1)}
|
|
\]
|
|
and calculate the expectation value of the last term:
|
|
\[
|
|
E\lrb{\DSF{(C_1-C_2)^2}{4(\langle x \rangle+1)}} =
|
|
\mathop{\sum}_{m,n=0}^{+\infty}
|
|
\DSF{(n-m)^2}{2(n+m+2) }P(n)P(m)=\DSF{\EE^{-2\lambda}}{2}
|
|
\mathop{\sum}_{m,n=0}^{+\infty}
|
|
\DSF{(n-m)^2}{(n+m+2) }\DSF{\lambda^{n+m}}{n!m!}
|
|
\]
|
|
Rearranging the sums with $s=n+m$, $s=0\ldots +\infty$; $n-m=s-2k$, $k=0\ldots s$,
|
|
we get
|
|
\[
|
|
E\lrb{\DSF{(C_1-C_2)^2}{4(\langle x \rangle+1)}} =
|
|
\DSF{\EE^{-2\lambda}}{2}
|
|
\mathop{\sum}_{s=0}^{+\infty}
|
|
\mathop{\sum}_{k=0}^{s}
|
|
\DSF{(s-2k)^2(s+1)}{(s+2)! }{\lambda^{s}}
|
|
\binom{s}{k}=\DSF{1}{2}-\DSF{1}{2\lambda}+\DSF{1-\EE^{-2\lambda}}{4\lambda^2}
|
|
%{n!m!}
|
|
\]
|
|
So, the relative difference between averages is
|
|
\[
|
|
\DSF{E\lrb{\langle x \rangle_{\mathrm{w(2)}}-\langle x \rangle}}{E\lrb{\langle x \rangle}}=
|
|
\DSF{1}{2\lambda}+\DSF{1}{2\lambda^2}-\DSF{1-\EE^{-2\lambda}}{4\lambda^3}
|
|
\]
|
|
The relative error on $\langle x \rangle$ is
|
|
\[
|
|
\epsilon = \DSF{\sigma_x}{\langle x \rangle} =
|
|
\DSF{\lambda^{1/2}}{\sqrt{2} \lambda}=\DSF{1}{\sqrt{2\lambda}}
|
|
\]
|
|
therefore
|
|
\[
|
|
\DSF{E\lrb{\langle x \rangle_{\mathrm{w(2)}}-\langle x \rangle}}{E\lrb{\langle x \rangle}}=
|
|
O(\epsilon^2)
|
|
\]
|
|
Therefore, the expectation value of the error (relative) involved in taking
|
|
the M-P weighted average instead of the simple average is negligible.
|
|
|
|
|
|
|
|
\subsubsection{Numerical comparison of averages}
|
|
|
|
In the next table numerical results are displayed. An exact random-Poisson generator has been used to generate Poisson deviates of given average value
|
|
$\lambda$, with $\lambda=1,10,100,\ldots,1000000$. For each value $\lambda$
|
|
$N=10^8$ deviates have been generated. Then averages have been taken for each value $\lambda$ and compared with the true value.
|
|
For each value $\lambda$ - in order to have a scale for comparison -
|
|
we evaluate the expected absolute s.d. of averages as $\xi_\lambda=\sqrt{\lambda/N}$, and the relative s.d. of averages as $\epsilon_\lambda=\sqrt{\lambda/N}/\lambda=1/\sqrt{N\lambda}$. Then - for each averaging method - we evaluate the error $E_\lambda$ (average minus $\lambda$),
|
|
the relative error $e_\lambda=E_\lambda/\lambda$, and finally the comparison criterion $e_\lambda/\epsilon_\lambda$ (bold). The comparison criterion is expected to be close to 1 in absolute value. Values much larger than one mean that we are introducing a systematic error.
|
|
|
|
\footnotesize
|
|
|
|
\begin{tabular}{l|llll}
|
|
\footnotesize
|
|
% # trials N = 100000000
|
|
\ &&&&\ \\
|
|
& \multicolumn{4}{l}{$\lambda =$ 1. ; $\xi_\lambda = $0.0001 ; $\epsilon_\lambda$ = 0.0001}\\
|
|
& ${\langle x \rangle_{\!\mathrm{w(1)}}}$ & ${\langle x \rangle_{\!\mathrm{w(2)}}}$ & $\langle x \rangle^*$ & $\langle x \rangle$\\
|
|
Averages & 1.303772380383934 & 0.9999155361216990 & 1.581941754994651 & 0.9999283300000000 \\
|
|
$E_\lambda$ & 0.3037723803839338 & -0.8446387830096658E-04 & 0.5819417549946508 & -0.7166999999996815E-04\\
|
|
$e_\lambda$ & 0.3037723803839338 & -0.8446387830096658E-04 & 0.5819417549946508 & -0.7166999999996815E-04\\
|
|
$e_\lambda/\epsilon_\lambda$ &{\textbf{ 3037.723803839338 }}&{\textbf{ -0.8446387830096658 }}&{\textbf{ 5819.417549946508 }}&{\textbf{ -0.7166999999996815 }} \\
|
|
\ &&&&\ \\
|
|
& \multicolumn{4}{l}{$\lambda =$ 10.000000000000002 ; $\xi_\lambda = $0.00031622776601683794 ; $\epsilon_\lambda$ = 0.00003162277660168379}\\
|
|
& ${\langle x \rangle_{\!\mathrm{w(1)}}}$ & ${\langle x \rangle_{\!\mathrm{w(2)}}}$ & $\langle x \rangle^*$ & $\langle x \rangle$\\
|
|
Averages & 8.848248847530357 & 10.00025732384808 & 10.00052232372917 & 10.00006800000000 \\
|
|
$E_\lambda$ & -1.151751152469645 & 0.2573238480785278E-03 & 0.5223237291644978E-03 & 0.6799999999884676E-04\\
|
|
$e_\lambda$ & -0.1151751152469645 & 0.2573238480785278E-04 & 0.5223237291644977E-04 & 0.6799999999884675E-05\\
|
|
$e_\lambda/\epsilon_\lambda$ &{\textbf{ -3642.156939527943 }}&{\textbf{ 0.8137294562072904 }}&{\textbf{ 1.651732660112730 }}&{\textbf{ 0.2150348808878029 }} \\
|
|
\ &&&&\ \\
|
|
& \multicolumn{4}{l}{$\lambda =$ 100.00000000000004 ; $\xi_\lambda = $0.0010000000000000002 ; $\epsilon_\lambda$ = 0.000009999999999999997}\\
|
|
& ${\langle x \rangle_{\!\mathrm{w(1)}}}$ & ${\langle x \rangle_{\!\mathrm{w(2)}}}$ & $\langle x \rangle^*$ & $\langle x \rangle$\\
|
|
Averages & 98.98978896904168 & 100.0001037814804 & 100.0002153600000 & 100.0002153600000 \\
|
|
$E_\lambda$ & -1.010211030958359 & 0.1037814803765968E-03 & 0.2153599999559219E-03 & 0.2153599999559219E-03\\
|
|
$e_\lambda$ & -0.1010211030958359E-01 & 0.1037814803765968E-05 & 0.2153599999559218E-05 & 0.2153599999559218E-05\\
|
|
$e_\lambda/\epsilon_\lambda$ &{\textbf{ -1010.211030958359 }}&{\textbf{ 0.1037814803765968 }}&{\textbf{ 0.2153599999559219 }}&{\textbf{ 0.2153599999559219 }} \\
|
|
\ &&&&\ \\
|
|
& \multicolumn{4}{l}{$\lambda =$ 1000.0000000000007 ; $\xi_\lambda = $0.0031622776601683803 ; $\epsilon_\lambda$ = 0.000003162277660168378}\\
|
|
& ${\langle x \rangle_{\!\mathrm{w(1)}}}$ & ${\langle x \rangle_{\!\mathrm{w(2)}}}$ & $\langle x \rangle^*$ & $\langle x \rangle$\\
|
|
Averages & 999.0029754507847 & 1000.003978305674 & 1000.003836760000 & 1000.003836760000 \\
|
|
$E_\lambda$ & -0.9970245492160075 & 0.3978305673513205E-02 & 0.3836759999330752E-02 & 0.3836759999330752E-02\\
|
|
$e_\lambda$ & -0.9970245492160069E-03 & 0.3978305673513202E-05 & 0.3836759999330750E-05 & 0.3836759999330750E-05\\
|
|
$e_\lambda/\epsilon_\lambda$ &{\textbf{ -315.2868458625229 }}&{\textbf{ 1.258050715667192 }}&{\textbf{ 1.213290043331128 }}&{\textbf{ 1.213290043331128 }} \\
|
|
\ &&&&\ \\
|
|
& \multicolumn{4}{l}{$\lambda =$ 10000.00000000001 ; $\xi_\lambda = $0.010000000000000005 ; $\epsilon_\lambda$ = 9.999999999999995E-7}\\
|
|
& ${\langle x \rangle_{\!\mathrm{w(1)}}}$ & ${\langle x \rangle_{\!\mathrm{w(2)}}}$ & $\langle x \rangle^*$ & $\langle x \rangle$\\
|
|
Averages & 9998.995728116572 & 9999.995828163173 & 9999.995919900000 & 9999.995919900000 \\
|
|
$E_\lambda$ & -1.004271883437468 & -0.4171836835666909E-02 &-0.4080100008650334E-02 &-0.4080100008650334E-02\\
|
|
$e_\lambda$ & -0.1004271883437467E-03 & -0.4171836835666905E-06 & -0.4080100008650330E-06 &-0.4080100008650330E-06\\
|
|
$e_\lambda/\epsilon_\lambda$ &{\textbf{ -100.4271883437468 }}&{\textbf{ -0.4171836835666907 }}&{\textbf{ -0.4080100008650331 }}&{\textbf{ -0.4080100008650331 }} \\
|
|
\ &&&&\ \\
|
|
& \multicolumn{4}{l}{$\lambda =$ 100000.0000000002 ; $\xi_\lambda = $0.031622776601683826 ; $\epsilon_\lambda$ = 3.162277660168376E-7}\\
|
|
& ${\langle x \rangle_{\!\mathrm{w(1)}}}$ & ${\langle x \rangle_{\!\mathrm{w(2)}}}$ & $\langle x \rangle^*$ & $\langle x \rangle$\\
|
|
Averages & 99999.01275394148 & 100000.0127639189 & 100000.0125627100 & 100000.0125627100 \\
|
|
$E_\lambda$ & -0.9872460587212117 & 0.1276391866849735E-01 & 0.1256270980229601E-01 & 0.1256270980229601E-01\\
|
|
$e_\lambda$ & -0.9872460587212097E-05 & 0.1276391866849733E-06 & 0.1256270980229599E-06 & 0.1256270980229599E-06\\
|
|
$e_\lambda/\epsilon_\lambda$ &{\textbf{ -31.21946156583365 }}&{\textbf{ 0.4036305486159527 }}&{\textbf{ 0.3972677655897895 }}&{\textbf{ 0.3972677655897895 }} \\
|
|
\ &&&&\ \\
|
|
& \multicolumn{4}{l}{$\lambda =$ 1000000.0000000013 ; $\xi_\lambda = $0.10000000000000006 ; $\epsilon_\lambda$ = 9.999999999999993E-8}\\
|
|
& ${\langle x \rangle_{\!\mathrm{w(1)}}}$ & ${\langle x \rangle_{\!\mathrm{w(2)}}}$ & $\langle x \rangle^*$ & $\langle x \rangle$\\
|
|
Averages & 999999.1188353101 & 1000000.118835812 & 1000000.118809340 & 1000000.118809340 \\
|
|
$E_\lambda$ & -0.8811646911781281 & 0.1188358106883243 & 0.1188093387754634 & 0.1188093387754634 \\
|
|
$e_\lambda$ & -0.8811646911781270E-06 & 0.1188358106883241E-06 & 0.1188093387754633E-06 & 0.1188093387754633E-06\\
|
|
$e_\lambda/\epsilon_\lambda$ &{\textbf{ -8.811646911781276 }}&{\textbf{ 1.188358106883242 }}&{\textbf{ 1.188093387754633 }}&{\textbf{ 1.188093387754633}} \\
|
|
\end{tabular}
|
|
|
|
\normalsize
|
|
|
|
As it is visible from the table:
|
|
\begin{itemize}
|
|
\item[1.\quad]{${\langle x \rangle_{\!\mathrm{w(1)}}}\ :$ the weighted average using straight Poisson statistics is consistenty bad at all values of $\lambda$, that means at all counting levels;}
|
|
\item[2.\quad]{${\langle x \rangle^*}\ $: the normal average skipping zero count data is bad for $\lambda<100$, that means at low counting levels (of course at higher counting levels zeroes are not happening);}
|
|
\item[3.\quad]{${\langle x \rangle}\ \text{and}\ {\langle x \rangle_{\!\mathrm{w(2)}}}$: the normal average including zero count data and the Mighell-Poisson weighted average
|
|
are consistently and equivalently good at all counting levels.}
|
|
\end{itemize}
|
|
Therefore there is no bias when using the Mighell-Poisson weighted method to average data w.r.t. the usual average. The former, however,
|
|
has already accomplished the passage to normal statistics, therefore all operations on data that are not simple averaging can be done in
|
|
the framework of normal statistics, where everything is known and clear. In the next section, on the opposite, it is shown that
|
|
even simple operations as scaling data lead to the necessity of abandoning Poisson statistics in order to estimate correctly the standard deviations.
|
|
|
|
|
|
|
|
|
|
|
|
\subsection{Scaling Poisson variates}\label{sec:3}
|
|
|
|
If we have a count value $C_0$ that follows a Poisson distribution,
|
|
we can assume immediately that the average is $C_0$ and the s.d. is $\sqrt{C_0}$.
|
|
I.e., repeated experiments would give values $n$
|
|
distributed according to the normalized distribution
|
|
\[
|
|
P(n)=\DSF{C_0^n\EE^{-C_0}
|
|
}{
|
|
n!}
|
|
\]
|
|
This obeys
|
|
\[
|
|
\mathop{\sum}_{n=0}^{+\infty}
|
|
P(n)=1\ ;
|
|
\]
|
|
\[
|
|
\langle n\rangle=\mathop{\sum}_{n=0}^{+\infty}
|
|
nP(n)=C_0\ ;
|
|
\]
|
|
\[
|
|
\langle n^2\rangle=\mathop{\sum}_{n=0}^{+\infty}
|
|
n^2 P(n)=C_0^2+C_0\ ;
|
|
\]
|
|
\[
|
|
\sigma_{C_0}=\sqrt{\langle n^2\rangle-\langle n\rangle^2}=\sqrt{C_0}
|
|
\]
|
|
Suppose now that
|
|
our observable is
|
|
\[
|
|
X_0=\eta_0 C_0
|
|
\]
|
|
where $\eta_0$ is a known error-free scaling factor.
|
|
The distribution of $X$ is
|
|
\[
|
|
P'(X)=P(X/\eta_0)=P(n)\qquad\Biggl|\Biggr.\qquad \frac{X}{\eta_0}\equiv n\in\mathbb{Z}
|
|
%=\left.\DSF{(X_0/\eta_0)^{X/\eta_0}\EE^{-X_0/\eta_0}}{(X/\eta_0)!}\right|_{\frac{X}{\eta_0}\in\mathbb{Z}}
|
|
\]
|
|
and now,
|
|
\[
|
|
\mathop{\sum}_{n=0}^{+\infty}
|
|
P(n)=1\ ;
|
|
\]
|
|
\[
|
|
\langle X\rangle=\mathop{\sum}_{n=0}^{+\infty}
|
|
\eta_0 nP(n)=\eta_0 C_0=X_0\ ;
|
|
\]
|
|
\[
|
|
\langle X^2\rangle=\mathop{\sum}_{n=0}^{+\infty}
|
|
\eta_0^2 n^2 P(n)=\eta_0^2(C_0^2+C_0)=X_0^2+\eta_0 X_0\ ;
|
|
\]
|
|
\[
|
|
\sigma_X=\sqrt{\langle X^2\rangle-\langle X\rangle^2}=\sqrt{\eta_0 X_0}=\eta_0\sqrt{C_0}=\sqrt{\eta_0}\sqrt{X_0}
|
|
\]
|
|
Now it is no more valid that $\sigma_X=\sqrt{\langle X\rangle}=\sqrt{X_0}$, instead
|
|
\[
|
|
\sigma_X=\sqrt{\eta_0}\sqrt{X_0}=\eta_0\sqrt{C_0}=\eta_0\sigma_{C_0}
|
|
\]
|
|
that is the characteristic relationship for a normal-variate distribution.
|
|
|
|
Moreover, assume now that the scaling factor is not exctly known
|
|
but instead it is a normal-variate itself with average $\eta_0$, s.d.
|
|
$\sigma_{\eta_0}$, and distribution
|
|
\[
|
|
\widehat{P}(\eta)=\DSF{
|
|
\EE^{
|
|
-\frac{1}{2}
|
|
\lrb{
|
|
\frac{\eta-\eta_0}{\sigma_{\eta_0}}
|
|
}^2
|
|
}
|
|
}{
|
|
\sigma_{\eta_0}\sqrt{2\pi}
|
|
}
|
|
\]
|
|
Then,
|
|
\[
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|
\int_{-\infty}^{+\infty}\DD{\eta}\mathop{\sum}_{n=0}^{+\infty}
|
|
P(n)\widehat{P}(\eta)=1\ ;
|
|
\]
|
|
\[
|
|
\langle X\rangle=\int_{-\infty}^{+\infty}\DD{\eta}\mathop{\sum}_{n=0}^{+\infty}
|
|
\widehat{P}(\eta)\eta nP(n)=
|
|
\mathop{\sum}_{n=0}^{+\infty}
|
|
nP(n)
|
|
\int_{-\infty}^{+\infty}\DD{\eta} \widehat{P}(\eta)\eta
|
|
=
|
|
\eta_0 C_0=X_0\ ;
|
|
\]
|
|
\[
|
|
\langle X^2\rangle=\int_{-\infty}^{+\infty}\DD{\eta}\mathop{\sum}_{n=0}^{+\infty}
|
|
\widehat{P}(\eta)\eta^2 n^2 P(n)=
|
|
\int_{-\infty}^{+\infty}\DD{\eta}\widehat{P}(\eta)\eta^2
|
|
\mathop{\sum}_{n=0}^{+\infty}
|
|
n^2 P(n)
|
|
=
|
|
(\eta_0^2+\sigma_{\eta_0}^2)(C_0^2+C_0)\ ;
|
|
\]
|
|
\[
|
|
\DSF{\sigma_X}{\langle X\rangle}=\DSF{\sqrt{\langle X^2\rangle-\langle X\rangle^2}}{\langle X\rangle}
|
|
=\DSF{\sqrt{
|
|
\eta_0^2 C_0+\sigma_{\eta_0}^2C_0^2+\sigma_{\eta_0}^2 C_0
|
|
}}{\eta_0C_0}=
|
|
\sqrt{
|
|
\lrb{\DSF{ \sigma_{C_0}}{C_0}}^2
|
|
+\lrb{\DSF{\sigma_{\eta_0}}{\eta_0}}^2+\lrb{\DSF{\sigma_{\eta_0}}{\eta_0}\DSF{\sigma_{C_0}}{C_0}}^2
|
|
}\approx\sqrt{
|
|
\lrb{\DSF{ \sigma_{C_0}}{C_0}}^2
|
|
+\lrb{\DSF{\sigma_{\eta_0}}{\eta_0}}^2
|
|
}
|
|
%=\sqrt{\eta_0^2\sigma_{C_0}^2}
|
|
\]
|
|
where in the last we discard, as usual, the 4th order in the relative errors. Both the exact and approximated forms
|
|
are exactly the same as if both distributions were to be normal.
|
|
|
|
|
|
|
|
|
|
|
|
\subsection*{Bibliography}
|
|
|
|
[1] - B. E. Warren, {\textit{"X-Ray Diffraction"}} (Dover:1990)
|
|
|
|
[2] - A. Guinier, {\textit{"X-Ray Diffraction In Crystals, Imperfect Crystals, and Amorphous Bodies"}} (Dover:1994)
|
|
|
|
[3] - G. L. Squires, {\textit{"Introduction to the Theory of Thermal Neutron Scattering"}} (Dover:1997)
|
|
|
|
[4] - G. E. P. Box, G. C. Tiao, {\textit{"Bayesian inference in statistical analysis"}} (Wiley, NY: 1996)
|
|
|
|
[5] - E. Prince, P. T. Boggs, {\textit{"International Tables for Crystallography"}} Vol. C, ch.~8.1, pp.~678-688 (First online edition : 2006)
|
|
|
|
[6] - K. J. Mighell, {\textit{Astrophys. J.}} {\textbf{518}} (1999) p. 380--393
|