228 S. Chauhan and S.I. Ahmed Fig. 22.5 Bootstrap and Monte Carlo means FRFs and Confidence Intervals along with Havg Figure 22.5, also shows bootstrap CIs and mean value compared with those obtained from Monte Carlo simulations along with averaged FRF (Havg). This plot shows that bootstrap mean value is expectedly closer to the Havg. which is based on the same sample that is used for bootstrapping. The CIs from two methods don’t really overlap, but their spread is similar. It should be noted that these plots are developed for block size of 1024 and 100 averages. 22.4.1 Effect of Number of Averages Few terminologies and symbols are introduced before the start of this section to facilitate easier understanding. !theo , theo– Theoretical modal frequency and damping, !BS; BS–Mean ofNboot bootstrapped estimates of modal frequency and damping, !MC; MC–Mean ofNboot Monte-Carlo estimates of modal frequency and damping, and !est , est—Modal frequency and damping estimated on the basis of averaged Havg, following the general modal parameter estimation procedure. A general understanding related to averaging is that it reduces random error, hence more the number of averages, closer one gets to the correct estimates. In statistical terms, number of averages is equivalent to size of the sample used to characterize the population. The bigger the sample size, the closer one is to overall population statistics. If bias errors are not present, this also means that statistics calculated on the basis of bigger samples are bound to be more accurate and precise in comparison to comparatively smaller sample. This section is devoted to understanding these aspects in relation to suggested bootstrapping procedure. Note that a block size of 1024 is used for this study. Figures 22.6 and 22.7 show a bar plot comparing bootstrap standard errors with those from Monte Carlo simulation with increasing number of averages, for modal frequency and damping estimates. The Bootstrap Standard Error of a statistic is the standard deviation of the bootstrap distribution of that statistic [11]. Standard error represents how far the sample mean is likely to be from population mean. For uncertainty quantification purposes, it provides an idea about how certain one can be about the estimated statistic. A low standard error signifies that the estimate does not have much dispersion and is close to the true population statistic. As expected, these plots show that standard error reduces as the number of averages increases. This can be attributed to the growing sample size. As the sample size increases, it is expected to be more and more close
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