coulomb excitation data analysis codes; gosia 2007 - Physics and ...

The sensitivity maps provide useful information that can be used during fittingtodefine the currentsets of correlated matrix elements to be varied, to check to what extent the different sets of experimentaldata are independent and, finally, to ascertain which matrix elements included in the initial setup haveasignificant influence on the processes analyzed. The data obtained during the compilation of the yieldsensitivity map also are used to generate the correlation matrix, selecting the subsets of matrix elementsthat are strongly correlated for the error estimation (see Section 4.6). It should be noted, however, that thecompilation of the sensitivity maps is time-consuming, thus it should be requested only periodically whenneeded. The sensitivity maps also are helpful when planning an experiment, in which case a set of simulatedγ-ray yields can be used as experimental data for a dummy minimization run (simulated yields can be createdby GOSIA using OP,POIN - see V.18). A comparison of the sensitivity parameters for different (existing orplanned) experiments give an indication to what extent the additional sets of data provide qualitatively newinformation.4.6 Estimation of Errors of the Fitted matrix ElementsThe most commonly used methods of estimating the errors of fitted parameters (in our case the matrixelements) resulting from a least-squares minimization, are derived using the assumptions that do not necessarilyapply to Coulomb excitation analyses. Usually, the errors of the fitted parameters are evaluatedeither, using the curvature matrix, or by requesting an increase of the least-squares statistic dependent onthe number of degrees of freedom and the value of this statistic at the minimum. An approach based on thesecond-order approximation is not applicable because this approximation is in general not valid even in thevicinity of the fitted set of matrix elements, not to mention the problems resulting from the so-called “nuisanceparameters“, i.e. parameters that can be introduced formally, but have no influence on the observedprocesses. The inclusion of such parameters prohibits a straightforward inversion of the second-derivativematrix. Moreover, the curvature matrix approximation assumes that the fit is perfect, i.e. the gradient at thesolution point vanishes, thus only the second-order term describes the behavior of the least-squares statistic.Practically, however, we have to assume that a fitting procedure must be stopped at some point even thougha number of the matrix elements that have a weak influence on the Coulomb excitation process, are far fromtheir best values. This is not important from the point of view of extracting the information contained inthe experimental data but can considerably disturb the error estimation of the significant dependences whenthe second-order approximation is used.The unavoidable presence of the nuisance parameters also prohibits the error estimation procedures basedon the assumption that the least-squares statistic should obey the χ 2 distribution with a given number ofdegrees of freedom. It should be noted that the concept of the number of degrees of freedom is implicitlybased on the assumption that all the parameters are of about equal significance, which is obviously not truein the case of Coulomb excitation since the sensitivity of the observed γ yields to the various matrix elementsdiffers by the orders of magnitude. This is illustrated by noticing that to describe a given experiment onecan arbitrarily add any number of unobserved levels connected by any number of the matrix elements havingno influence on either the Coulomb excitation or the γ deexcitation, thus arbitrarily changing the numberof parameters and, consequently, the number of degrees of freedom. Finally, any procedure based on thenumber of degrees of freedom will result in ascribing the errors of the fitted matrix elements according tothe subjective feeling as to what is a valid parameter of the theory and what is not. The error estimationformalism used in GOSIA, described in subsequent sections, has been derived without employing either thelocal quadratic approximation or the concept of the degrees of freedom to provide a reliable (however ingeneral non-analytic) method of calculating the errors of the fitted parameters.4.6.1 Derivation of the Error Estimation MethodLet us assume that a set of observables, Y k , measured with known and fixed experimental errors, σ k ,isused to determine the values of a set of parameters, x i . We also assume that the observables are relatedto the parameters by an error-free functional dependence, i.e. that if the experiments were error-free eachexperimental point Y k could be exactly reproduced as:Y k = f k (¯x) (4.57)52