## 4 Bootstrap Scheme for Uncertainty Quantification  
Generally, without repeated experiments, there is only one single estimate of flutter derivatives by using the methods mentioned above or many others. However, due to the effects of flow turbulence, measurement noise, and mathematical model errors, uncertainties in the identified flutter derivatives is inevitable, especially in the free-vibration tests. In effect, the estimation of flutter derivatives shows variations in multiple experiments [19]. If one wants to accurately characterize the uncertainties associated with flutter derivatives, a large amount of repeated tests should be performed, which is not economical and sometimes even impossible. In order to solve this problem, nonparametric resampling schemes, such as bootstrap, can be adopted in these cases. Bootstrap is a nonparametric statistical method that relies on random sampling with replacement. It can provide valid statistics (mean, variance, confidence intervals, etc.) without the distribution assumption (such as the normal distribution assumption) or sufficiently large samples [27]. Its core ideas and basic steps are as follows:  
* Repeated random sampling technique with replacement is used to extract a certain number of samples from the original sample data;
* The estimated parameter of interest \(\hat{\theta}^{*}\) is calculated based on the extracted samples;
* The steps above are repeated a large number of times, say \(N\), to obtain \(N\) estimates of \(\hat{\theta}^{*}\);
* The sample statistics (mean, variance, confidence intervals, etc.) of the \(N\) samples (\(\hat{\theta}^{*}_{1},\,\hat{\theta}^{*}_{2},\,\,\ldots,\,\,\hat{\theta}^{*}_{N}\)) are calculated, so as to quantify the uncertainty of the estimated parameters. Based on the above description, the standard deviation \(\hat{\sigma}^{*}_{\theta}\) of the samples (\(\hat{\theta}^{*}_{1}\), \(\hat{\theta}^{*}_{2}\),..., \(\hat{\theta}^{*}_{N}\)) is given as follows:  
\[\hat{\sigma}^{*}_{\theta}=\sqrt{\frac{1}{N-1}\sum_{n=1}^{N}\left(\hat{\theta}^ {*}_{n}-\overline{\theta}^{*}\right)^{2}} \tag{22}\]  
where \(\overline{\theta}^{*}\) is the bootstrap mean, which is calculated as:  
\[\overline{\theta}^{*}=\frac{1}{N}\sum_{n=1}^{N}\hat{\theta}^{*}_{n} \tag{23}\]  
Moreover, in addition to the bootstrap mean and standard deviation, the probability density function of \(\theta\) can also be estimated based on the bootstrap ensemble (\(\hat{\theta}^{*}_{1}\), \(\hat{\theta}^{*}_{2}\),..., \(\hat{\theta}^{*}_{N}\)). The percentile confidence interval at level \(1-2p\) is approximately given as:  
\[\left[\hat{\theta}_{low},\ \hat{\theta}_{up}\right]=\left[\theta^{*}_{p}, \theta^{*}_{1-p}\right] \tag{24}\]  
where \(\theta^{*}_{p}\) and \(\theta^{*}_{1-p}\) are respectively the (\(p\)-100)th and ((\(1-p\))\(\cdot\)100)th empirical percentile values. In the system identification process, the bootstrap can be utilized by fully making use of existing information to quantify the uncertainty [31; 32; 33; 34]. In this study, combining with the identification method introduced above, a bootstrap scheme is proposed to quantify the uncertainty of the flutter derivatives. Figure 3 shows the proposed scheme. Firstly, repeated free-vibration tests are performed \(L\) times for each angle of attack and wind velocity, and \(L\) data sets of free-decay time histories are obtained as the sample population. Secondly, a bootstrap sample of \(M\) free-decay time histories is generated by randomly drawing \(M\) samples from the sample population. For a practical implementation of this process, firstly number the \(L\) data sets, and then generate \(M\) random integers that is not smaller than one and not greater than \(L\). Finally take out \(M\) samples according to these numbers. \(N\) bootstrap samples are obtained by \(N\) times of random sampling with replacement.