Error propagation in Amplitude Analysis - Revision history

Senderovich at 18:30, 18 December 2011

2011-12-18T18:30:13Z

2011-12-17T17:09:02Z

2011-11-23T18:43:55Z

2011-11-22T20:21:09Z

2011-11-22T20:20:34Z

2011-11-22T19:19:16Z

2011-11-22T19:17:56Z

2011-11-22T19:17:08Z

2011-11-22T19:16:13Z

2011-11-22T19:15:58Z

@@ Line 1: / Line 1: @@
-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.
 I=\frac{1}{N_{gen}}\sum_i^N{
    \sum_{\gamma,\delta}{\rho_{\gamma\delta}

@@ Line 58: / Line 58: @@
    \right|^2
+}
 </math>
 :<math>
 = \sum_{\alpha,\beta,\alpha',\beta'}^n{
        u_\alpha u_\beta^* u_{\alpha'}^* u_{\beta'}
-      \left[ \sum_{\gamma,\delta,\gamma',\delta'}{ \rho_{\gamma\delta}\rho_{\gamma'\delta'}
+       \sum_{\gamma,\delta,\gamma',\delta'}{ \rho_{\gamma\delta}\rho_{\gamma'\delta'} \left[
          \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)
        }}
        \right]
 </math>

@@ Line 43: / Line 43: @@
 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.
@@ Line 62: / Line 61: @@
 :<math>
 = \sum_{\alpha,\beta,\alpha',\beta'}^n{
-       u_\alpha u_\beta^* u_{\alpha'} u_{\beta'}^*
+       u_\alpha u_\beta^* u_{\alpha'}^* u_{\beta'}
        \left[ \sum_{\gamma,\delta,\gamma',\delta'}{ \rho_{\gamma\delta}\rho_{\gamma'\delta'}
          \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]

@@ Line 74: / Line 74: @@
 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''.
 <math>
 I = \sum_{\alpha\beta}^n{

@@ Line 76: / Line 76: @@
 <math>
 I = \sum_{\alpha\beta}^n{
-     \left(\begin{array}{lr}a_\alpha & b_\beta\end{array}\right)
+     \left(\begin{array}{lr}a_\alpha & b_\alpha\end{array}\right)
      \left(\begin{array}{lr}\Re(I_{\alpha\beta}) &-\Im(I_{\alpha\beta})
            \\               \Im(I_{\alpha\beta}) & \Re(I_{\alpha\beta})
            \end{array}\right)
-     \left(\begin{array}{c}a_\alpha \\ b_\beta\end{array}\right)
+     \left(\begin{array}{c}a_\beta \\ b_\beta\end{array}\right)
+}
 </math>

@@ Line 44: / Line 44: @@
 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.
-<b>A subtle point easy to miss at this stage is that the errors on the <font face="Times">''I''</font><sub>&alpha;&beta;</sub> are correlated because they are computed on the same MC sample.  Therefore, while the <font face="Times">''I''</font><sub>&alpha;&beta;</sub> is rank 2 in the partial wave index, its error is rank 4.</b>
+<b>A subtle point easy to miss at this stage is that the errors on the <font face="Times">''I''</font><sub>&alpha;&beta;</sub> are correlated because they are computed on the same MC sample.  Therefore, while the <font face="Times">''I''</font><sub>&alpha;&beta;</sub> is rank 2 in the partial wave index, its covariance matrix is rank 4.</b>
 <math>

@@ Line 105: / Line 105: @@
 </math>
-where ''J'' and ''C'' are both ymmetric matrices, and ''G'' is
+where ''J'' and ''C'' are both symmetric matrices, and ''G'' is
 defined as