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Error propagation in Amplitude Analysis (view source)
Revision as of 18:30, 18 December 2011
, 18:30, 18 December 2011no edit summary
The following is a review of error propagation needed to compute the errors on the normalization integrals and the intensity sum that is based on them. Consider the estimator for the intensity for a given PWA solution, based on a sum over a Monte Carlo sample with N<sub>gen</sub> phase space events generated and N reconstructed and passing all cuts. <math>
The following is a review of error propagation needed to compute the errors on the normalization integrals and the intensity sum that is based on them. Consider the estimator for the intensity for a given PWA solution, based on a sum over a Monte Carlo sample with N<sub>gen</sub> phase space events generated and N reconstructed and passing all cuts.
<math>
I=\frac{1}{N_{gen}}\sum_i^N{
I=\frac{1}{N_{gen}}\sum_i^N{
\sum_{\gamma,\delta}{\rho_{\gamma\delta}
\sum_{\gamma,\delta}{\rho_{\gamma\delta}
When considering the uncertainty on the overall integral, both the errors on the ''u'' parameters
When considering the uncertainty on the overall integral, both the errors on the ''u'' parameters
and those from the finite MC statistics will contribute. The part of the error on the intensity coming from the finite MC statistics is computed using the usual rules for error propagation.
and those from the finite MC statistics will contribute. The part of the error on the intensity coming from the finite MC statistics is computed using the usual rules for error propagation.
\right|^2
\right|^2
}
}
-
\frac{1}{N_{gen}}
\left|
\sum_{\alpha,\beta}^n{
u_\alpha u_\beta^* I_{\alpha\beta}
}
\right|^2
</math>
</math>
:<math>
:<math>
= \sum_{\alpha,\beta,\alpha',\beta'}^n{
= \sum_{\alpha,\beta,\alpha',\beta'}^n{
u_\alpha u_\beta^* u_{\alpha'} u_{\beta'}^*
u_\alpha u_\beta^* u_{\alpha'}^* u_{\beta'}
\sum_{\gamma,\delta,\gamma',\delta'}{ \rho_{\gamma\delta}\rho_{\gamma'\delta'} \left[
\frac{1}{N_{gen}^2} \sum_i^N{
\frac{1}{N_{gen}^2} \sum_i^N{
A_\alpha^{\gamma}(x_i) A_\beta^{\delta *}(x_i)
A_\alpha^{\gamma}(x_i) A_\beta^{\delta *}(x_i)
A_{\alpha'}^{\gamma'*}(x_i) A_{\beta'}^{\delta'}(x_i)
}
\right.
}
}
</math>
::::<math>
-
\left.
\frac{1}{N_{gen}^3} \left(
\sum_i^N{
A_\alpha^{\gamma}(x_i) A_\beta^{\delta *}(x_i)
}\right)
\left(\sum_i^N{
A_{\alpha'}^{\gamma'}(x_i) A_{\beta'}^{\delta' *}(x_i)
A_{\alpha'}^{\gamma'}(x_i) A_{\beta'}^{\delta' *}(x_i)
}}
}\right)
\right]
\right]
</math>
</math>
The relevant piece to pre-compute over the event set for error calculation is shown in brackets.
The relevant piece to pre-compute over the event set for error calculation is shown in brackets.
Turning our attention now to the contribution to error from the fit parameters ''u'', it is convenient to write them explicitly in terms of their real and imaginary parts as ''u = a + ib''.
Turning our attention now to the contribution to error from the fit parameters ''u'', it is convenient to write them explicitly in terms of their real and imaginary parts as ''u = a + ib''.
<math>
<math>
I = \sum_{\alpha\beta}^n{
I = \sum_{\alpha\beta}^n{