Estimating Parameter Uncertainty in Binding-Energy Models by the Frequency-Domain Bootstrap

We propose using the frequency-domain bootstrap (FDB) to estimate errors of modeling parameters when the modeling error is itself a major source of uncertainty. Unlike the usual bootstrap or the simple ${\chi }^2$ analysis, the FDB can take into account correlations between errors. It is also very fast compared to the Gaussian process Bayesian estimate as often implemented for computer model calibration. The method is illustrated with a simple example, the liquid drop model of nuclear binding energies. We find that the FDB gives a more conservative estimate of the uncertainty in liquid drop parameters than the ${\chi }^2$ method, and is in fair accord with more empirical estimates. For the nuclear physics application, there are no apparent obstacles to apply the method to the more accurate and detailed models based on density-functional theory.

Figure 1

Errors in the liquid-drop fit to nuclear binding energies. Blue, $r_0<-{\sigma }_r$; green, $-{\sigma }_r<r_0<0$; orange, $0<r_0<{\sigma }_r$; red, $r_0>{\sigma }_r$.

Figure 2

Distribution of residuals as a function of $A$. See text for explanation.

Figure 3

Distribution of $|\tilde{r}(n)|$ for the FDB.

Figure 4

Sample $r_A$ sets for the liquid-drop model obtained by the FDB method.

Figure 5

Distribution of $a_v$ obtained by the FDB method.