Bootstrapping Analysis
To examine the reliability of the above calibrations, we conducted a
(nonparametric) bootstrapping analysis.126,127 As
shown previously with Mössbauer isomer shifts as an exemplary data set,
bootstrapping increases the robustness of statistical measures such as
fit parameters and relative performances of density
functionals.24 Here, we applied Bayesian
bootstrapping,128 which yields smoother results than
its original variant.129 The results of the
bootstrapping analysis applied to the B3LYP contact density are shown in
Figure 5, details are given in the Supporting Information.
The ensembles of regression lines (blue) were obtained by bootstrapping
samples from the data set and regressing each sample. The mean over each
regression ensemble is marked as a black line and used to make
predictions for the isomer shift. The transparent red bands represent
1.96 times the prediction uncertainty (assuming a normal distribution),
i.e. it is estimated that 95% of the population is located inside the
bands. The results shown in Figure 5 (left) were obtained by
bootstrapping all data points except 9 and 10 (not
shown) as discussed above. 100% of the data lies within the uncertainty
band; since each of the 18 remaining data points makes up
>5% of the data set, we do not consider this finding a
violation of the statistical hypothesis (95% confidence).