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Error propagation in Amplitude Analysis (view source)
Revision as of 03:21, 22 November 2011
, 03:21, 22 November 2011no edit summary
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\right) + \frac{\partial}{\partial u_l}\left(+ \sum_{\gamma',\delta'}{\rho_{\gamma'\delta'}+ \right)
−}
+
+
+
+
= \sum_{\gamma,\delta}{\rho_{\gamma\delta}
= \sum_{\gamma,\delta}{\rho_{\gamma\delta}
\sum_{\alpha,\beta}^n{
\sum_{\alpha,\beta}^n{
− u_\alpha u_\beta^* J_{\alpha \beta}
+ u_\alpha u_\beta^* J^{\gamma\delta}_{\alpha \beta}
}
}
}
}
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When considering the uncertainty on the overall integral, both the errors on ''u'' parameters
When considering the uncertainty on the overall integral, both the errors on ''u'' parameters
−and those from the finite MC set of events will contribute. A single detected event (i) can be viewed as one sample of a Poisson process, having therefore an uncertainty of
+and those from the finite MC set of events will contribute. A single detected event (i) can be viewed as one sample in a process of independent events, having therefore a count uncertainty of
σ<sub>i</sub>=1. An integral over such events is then a weighted sum of such samples,
σ<sub>i</sub>=1. An integral over such events is then a weighted sum of such samples,
−having therefore a contribution to the variance:
+having resulting in a contribution to the variance:
<math>
<math>
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<math>
<math>
\sigma_{fit}^2=
\sigma_{fit}^2=
−\sum_{k,l}^n{ \sigma_{u_k}\sigma_{u_l}
+\left| \sum_k^n{ \sigma_{u_k}
\frac{\partial}{\partial u_k}\left(
\frac{\partial}{\partial u_k}\left(
− \sum_{\gamma,\delta}{\rho_{\gamma\delta}
+ \sum_{\gamma,\delta}{ \rho_{\gamma\delta}
− \sum_{\alpha,\beta}^n{
+ \sum_{\alpha,\beta}^n{u_\alpha u_\beta^* J_{\alpha\beta}}
− u_\alpha u_\beta^* J_{\alpha\beta}
+ }
+ \right)
+}
+\right|^2
+</math>
+::<math>
+=\left( \sum_k^n{ \sigma_{u_k}
+ \sum_{\gamma,\delta}{ \rho_{\gamma\delta}
+ \sum_{\alpha,\beta}^n{\delta_{k\alpha} u_\beta^* J^{\gamma\delta}_{\alpha\beta}}
+ }
+}\right)
+\left( \sum_{k'}^n{ \sigma^*_{u_{k'}}
+ \sum_{\gamma',\delta'}{ \rho_{\gamma'\delta'}
+ \sum_{\alpha',\beta'}^n{\delta_{k'\alpha'} u_{\beta'} J^{\gamma'\delta'*}_{\alpha'\beta'}}
+ }
+}\right)
+</math>
+::<math>
+= \sum_{\gamma,\delta,\gamma',\delta'}{
+ \rho_{\gamma\delta} \rho_{\gamma'\delta'}
+ \sum_{\alpha,\beta,\alpha',\beta'}^n{
+ \left(\sigma_{u_\alpha}\sigma^*_{u_{\alpha'}}\right)
+ \left(u_\beta^* J^{\gamma\delta}_{\alpha\beta}\right)
+ \left(u_{\beta'} J^{\gamma'\delta'*}_{\alpha'\beta'} \right)
}
}
}
}
−</math>
−::<math>
−= \sum_{\gamma,\delta,\gamma',\delta'}{
− \sum_{\alpha',\beta'}^n{
+ \rho_{\gamma\delta} \rho_{\gamma'\delta'}
− u_{\alpha'} u_{\beta'}^* J_{\alpha'\beta'}
+ \sum_{\alpha,\alpha'}^n{
+ \left(\sigma_{u_\alpha}\sigma^*_{u_{\alpha'}}\right)
+ G^{\gamma\delta*}_\alpha G_{\alpha'}^{\gamma\delta}
}
}
}
}
−</math>
</math>
−The product of σ terms in the summation are the error matrix derived from the fit.
+The product of σ terms in the summation is represented by the error matrix derived from the fit. ''G'' was defined as
+<math>
+G_\alpha^{\gamma\delta}=\sum_\beta{ u_\beta J_{\alpha\beta}^{\gamma\delta}}
+</math>
+The overall uncertainty in the integral ''I'' defined in the beginning comes out to:
+<math>
+\sigma_I=\sqrt{\sigma^2_{MC} + \sigma^2_{fit}}
+</math>