Error estimation

Error estimation in any physics field can sometimes be difficult and is often poorly understood.  It is common for errors to be underestimated because of a failure to correlate error contributions from all significant (strongly coupled) fit parameters, and fallacies about error propagation abound.

==Diagonal errors versus correlated errors==

The ''diagonal'' error is the error one obtains by varying only a single parameter "x" above and below its minimum [[chi-squared]] value to find the intersection of the chi-squared vs. x curve with the line defined by chi-squared = minimum chi-squared + 1.  The diagonal error is generally much smaller than the full correlated error, and the latter is the error that should be quoted.

The ''correlated error'' is defined by the lower and upper limits of the parameter being measured along the intersection of the [[reduced chi-squared]] surface with the plane defined by <math>\stackrel{\sim}{\chi}^2 = \stackrel{\sim}{\chi}^2_{\rm min} + \frac{\stackrel{\sim}{\chi}^2_{\rm min}}{N}</math>, where <math>N</math> is the number of [[degrees of freedom]] in the fit and <math>\stackrel{\sim}{\chi}^2</math> is the "reduced <math>\chi^2</math> value.  Ideally, this contour of intersection is elliptical, but this is not always the case.

==Proper treatment of correlated errors in the general case==

Assuming that the errors in each measured parameter obey approximately a normal distribution, the correct derivation of the correlated error in each parameter is described in detail by Cline and Lesser<ref>D. Cline and P.M.S. Lesser, "Error estimation in non-linear least squares analysis of data," NIM '''82''' (1970) 291-293.</ref>.  This paper compares an incorrect procedure sometimes used with the correct treatment.

==Correlated errors calculated by Gosia's OP,ERRO function==


==Correlated errors using gosia2==

Gosia2 fits parameters (matrix elements) for the nucleus being studied (the "investigated nucleus") by normalization of the Coulomb excitation gamma-ray yields of the investigated nucleus to those of the collision partner.  However, it only correlates errors among the measured matrix element(s) of each collision partner individually, i.e., it does not do a complete correlated error calculation including both the beam nucleus and target nucleus free parameters.  Hence, using the present version of gosia2, to get a complete correlated error analysis, some brute-force hand-calculations are necessary.  Generally, gosia2 is used to measure only a few parameters for both collision partners, since the original Gosia can be used to self-normalize for large data sets.  This makes it feasible to calculate the correlated errors by hand using gosia2 (see below).

===Brute force total correlated error estimation in gosia2 problems===

The following procedure can be used to calculate the correlated error in one matrix element <math>\langle f\| ML\| i\rangle</math>.  It is assumed that all sensitive parameters have been fit to the experimental data for both nuclei using gosia2 prior to starting this procedure.  This procedure can be iterated over all sensitive matrix elements to get the correlated error in each.

# Set appropriate fit limits for all sensitive fit parameters (matrix elements) except the one whose correlated error is to be measured.
# Set the matrix element <math>\langle f\| ML\| i\rangle=x</math> whose correlated error is to be measured at approximately its best value, and set it as a "fixed" matrix element.  This can be done by setting the last two fields for this matrix element to "1", e.g.<pre>1, 2, 0.25, 1, 1</pre>
# Fit all matrix elements using the procedure 
## Calculate the corrected yields using OP,CORR.
## Minimize the reduced-<math>\chi^2\equiv\stackrel{\sim}{\chi}^2</math> value using OP,MINI.  (The reduced-<math>\chi^2</math> value is reported separately for the projectile and target nuclei in the two gosia2 output files.)
# Convert the two <math>\stackrel{\sim}{\chi}^2</math> values to <math>\chi^2</math> by multiplying each by the number of yield data points in the corresponding yield file for that nucleus.
# Sum the two <math>\chi^2</math> values to get <math>\chi^2_{\rm total}</math>.
# Iterate this procedure, stepping <math>x=\langle f\| ML\| i\rangle</math> through values below and above the best value from the gosia2 fit and recording the points <math>x,\chi^2</math>.

Using the procedure above, a plot of <math>\chi^2</math> vs. <math>x</math> can be generated, which should look similar to the sketch in Figure 1.  [[File:Gosia2_correlated_error_sketch.png|thumb|right|Figure 1: A sketch of a plot used to estimate correlated errors.  A parabolic fit to the chi-squared points is shown in black.  The green line indicates the new chi-squared minimum from the correlated error calculation, while the dashed red line indicates the min chi^2 + 1 criterion.  The vertical lines indicate the correlated error on the matrix element.]]  This may give a slightly improved best value of the matrix element as well.

The correlated error for this matrix element <math>\langle f\| ML\| i\rangle</math> is then given by matrix element values where a fit to the points (a parabolic fit is usually sufficient) intersects the line defined by <math>\chi^2=\chi^2_{\rm min} + 1</math>.  As in the sections above, this criterion is derived in <ref>D. Cline and P.M.S. Lesser, "Error estimation in non-linear least squares analysis of data," NIM '''82''' (1970) 291-293.</ref>

==References==


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