# Structural Modal Time Domain Identification Method Based on the Bayesian Uncertain Quantification
## 3 Bayesian Modal Identification Based on Modal State Space Prediction Models
### Uncertainty Analysis of Modal Parameters  
Noise in the measurement data and errors in the theoretical prediction model can introduce uncertainty in the identified modal parameters. Utilizing the given measurements and modeling assumptions, the Bayesian approach estimates the uncertainty of the modal parameters by the posterior variance. From Equation (25), the posterior covariance matrix of the modal parameters is equal to the inverse of the Hessian matrix. Therefore, how to solve the Hessian matrix quickly and accurately is a key issue. For explicit functions, the derivative with respect to the independent variable can be analytically solved; for implicit functions, the finite difference method can be used to calculate the derivative value at the MPV of the parameter. The second derivative of the NLLF with respect to \(\sigma_{\epsilon}^{2}\) can be given by the following:  
\[\frac{\partial^{2}L(\theta)}{\partial\left(\sigma_{\epsilon}^{2}\right)^{2}}= \frac{2f\big{(}\overline{\theta}\big{)}-\sigma_{\epsilon}^{2}nN_{s}}{2\left( \sigma_{\epsilon}^{2}\right)^{3}} \tag{40}\]  
Substituting Equation (28) into the above equation yields the following:  
\[\left.\frac{\partial^{2}L(\theta)}{\partial\left(\sigma_{\epsilon}^{2}\right) ^{2}}\right|_{\theta=\theta^{*}}=\frac{nN_{s}}{2\left[\left(\sigma_{\epsilon} ^{2}\right)^{*}\right]^{2}} \tag{41}\]  
Let \(b\) denote \(f_{i}\), \(\zeta_{i}\), or \(\mathbf{\Phi}\). The second derivative of the function \(L(\theta)\) with respect to \(b\) and \(\sigma_{\epsilon}^{2}\) is as follows:  
\[\left.\frac{\partial^{2}L(\theta)}{\partial b\partial(\sigma_{\epsilon}^{2}) }\right|_{\theta=\theta^{*}}=0 \tag{42}\]  
Since \(\overline{\theta}^{*}\) makes the function \(f\big{(}\overline{\theta}\big{)}\) take an extreme value, i.e., \(\left(\partial f\big{(}\overline{\theta}\big{)}/\partial b\right)|_{b=b^{*}}=0\), there is the following:  
\[\left.\frac{\partial^{2}L(\theta)}{\partial b\partial(\sigma_{\epsilon}^{2}) }\right|_{\theta=\theta^{*}}=0 \tag{43}\]  
The second derivative of the NLLF with respect to \(b\) can be given by the following:  
\[\frac{\partial^{2}L(\theta)}{\partial b\partial b}=\frac{1}{2\sigma_{\epsilon }^{2}}\frac{\partial^{2}f\big{(}\overline{\theta}\big{)}}{\partial b\partial b} \tag{44}\]  
Equation (44) needs to be solved for the derivative function of the function \(f\big{(}\overline{\theta}\big{)}\) with respect to \(f_{i}\), \(\zeta_{i}\), and \(\mathbf{\Phi}\). Let \(l_{i}\) and \(l_{j}\) only denote \(f_{i}\) or \(\zeta_{i}\), \(i,j=1,\dots,m\). The second derivative function of the function \(f\big{(}\overline{\theta}\big{)}\) with respect to \(l_{i}\) and \(l_{j}\) is as follows:  
\[\left.\frac{\partial^{2}f\big{(}\overline{\theta}\big{)}}{ \partial l_{i}\partial l_{j}}\right\}=\left\{ \begin{aligned} \mathbf{\Phi}^{\mathsf{T}}\Bigg{\{}\sum\limits_{k=1}^{N_{s}}\left[\frac{\partial^{ 2}\eta_{i,k}^{\sim}}{\partial l_{i}\partial l_{i}}\Big{(}\mathbf{Q}_{i}+ \mathbf{Q}_{i}^{\mathsf{T}}\Big{)}+2\Bigg{(}\frac{\partial\eta_{i,k}^{\sim}}{ \partial l_{i}}\Bigg{)}^{2}\mathbf{W}_{ii}\right]\otimes\mathbf{I}_{n}\Bigg{\}} \mathbf{\Phi}-2\mathbf{b}_{i}^{\mathsf{T}}\Bigg{(}\sum\limits_{k}^{\infty}\frac{ \partial^{2}\eta_{i,k}^{\sim}}{\partial l_{i}\partial l_{i}}\tilde{\mathbf{y}}_{ k}\Bigg{)},& i=j\\ \mathbf{\Phi}^{\mathsf{T}}\Bigg{[}\left(\sum\limits_{k=1}^{N_{s}} \frac{\partial\eta_{i,k}^{\sim}}{\partial l_{i}}\frac{\partial\eta_{i,k}^{ \sim}}{\partial l_{j}}\mathbf{W}_{ii}\right)\otimes\mathbf{I}_{n}\Bigg{]}\mathbf{ \Phi},& i\
eq j\end{aligned}  \tag{45}\]  
where the \(i\)th column of the matrix \(\mathbf{Q}_{i}\in R^{m\times m}\) is \(\widetilde{\mathbf{n}}_{k}\) and the rest of the columns are zero vectors; the \((i,i)\)th element of the matrix \(\mathbf{W}_{ii}\in R^{m\times m}\) is equal to \(1\) and the rest of the elements are zero; and the \((i,j)\)th and \((j,i)\)th elements of the matrix \(\mathbf{W}_{ij}\in R^{m\times m}\) are equal to \(1\) and the rest of the elements are zero.