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).