Uncertainty decomposition method and its application to the liquid drop model

A method is suggested to decompose the statistical and systematic uncertainties from the residues between the calculation of a theoretical model and the observed data. The residues and the parameters of the model can be obtained through the standard statistical fitting procedures. The present work concentrates on the decomposition of the total uncertainty, of which the distribution corresponds to that of the residues. The distribution of the total uncertainty is considered as two normal distributions, statistical and systematic uncertainties. The standard deviation of the statistical part, ${\sigma }_{\mathrm{stat}}$, is estimated through random parameters distributed around their best fitted values. The two normal distributions are obtained by minimizing the moments of the distribution of the residues with the fixed ${\sigma }_{\mathrm{stat}}$. The method is applied to the liquid drop model (LD). The statistical and systematic uncertainties are decomposed from the residues of the nuclear binding energies with and without the consideration of the shell effect in LD. The estimated distributions of the statistical and systematic uncertainties can well describe that of the residues. The normal assumption of the distribution of the statistical and systematic uncertainties is examined through various approaches. The comparison between the distributions of the specific nuclei and those of the statistical and systematic uncertainties are consistent with the physical considerations, although the latter two can be obtained without the knowledge of these considerations. Such as, the LD are more suitable to describe the heavy nuclei. The light and heavy nuclei are indeed distributed mostly inside the distributions of the systematic and statistical uncertainties, respectively. A similar situation is also found for the nuclei close to and far from shell. The present method is also performed for nuclei around the stability line. The results are used to investigate all measured nuclei, which show the usefulness of the uncertainty decomposition method in the exploration of the unmeasured nuclei.

Figure 1

$▵p=|p_3^{1/3}\left(e\right)-p_3^{1/3}\left(m_{\mathrm{stat}}\right)|+|p_4^{1/4}\left(e\right)-p_4^{1/4}\left(m_{\mathrm{stat}}\right)|,▵p_3=|p_3^{1/3}\left(e\right)-p_3^{1/3}\left(m_{\mathrm{stat}}\right)|$, and $▵p_4=|p_4^{1/4}\left(e\right)-p_4^{1/4}\left(m_{\mathrm{stat}}\right)|$ as the function of $m_{\mathrm{stat}}$, taking AME2012LD8 as an example.

Figure 2

The cumulative counts of the residues of the binding energies and the corresponding estimated uncertainties in AME2012LD6 and AME2012LD8.

Figure 3

The estimated normal statistical distribution and the simulated distribution of Eq. (8) from AME2012LD8, when the latter ignoring terms with $\sigma ={\sigma }_{a_i}{B}_{i,Z,A}<0.01,0.1$, and $0.5$ MeV.

Figure 4

The distribution of the residues of the binding energies of the light and heavy nuclei, nuclei close to and far from shell (only for AME2012LD6), all nuclei, and corresponding estimated uncertainties of AME2012LD8 and AME2012LD6.

Figure 5

The distribution of the residues of the binding energies on the chart of nuclide of the AME2012LD8 and AME2012LD6.

Figure 6

The distribution of the residues of the neutron separation energies on the chart of nuclide of the AME2012LD8.

Figure 7

The distribution of the residues of the binding energies on the chart of nuclide of the LD7 fitted to the nuclei around the stability line in AME2012. Only the nuclei around stability line are shown in the upper panel, while the others are together shown in the lower panel.

Figure 8

The distribution of the residues of the binding energies of the nuclei around stability line and all nuclei, and the corresponding estimated uncertainties of the LD7 fitted to the nuclei around stability in AME2012.