# Structural Modal Time Domain Identification Method Based on the Bayesian Uncertain Quantification
## 3 Bayesian Modal Identification Based on Modal State Space Prediction Models
### Time Domain Bayesian Theoretical Framework
For the prediction error sequence \(\mathcal{D}_{\epsilon}=\{\epsilon_{k}\in R^{n}:k=1,\ldots,N_{\mathrm{s}}\}\), \(\epsilon_{k}\) is a zero mean Gaussian white noise stochastic process with independent components, and the joint probability density function of \(\epsilon_{k}\) can be given by the probability distribution theory of multivariate random variables [28]:  
\[p\Big{(}\mathcal{D}_{\epsilon}\Big{|}\sigma_{\epsilon}^{2}\Big{)}=\Big{(}2\pi \sigma_{\epsilon}^{2}\Big{)}^{-\frac{nN_{\mathrm{s}}}{2}}\exp\Bigg{\{}-\frac{ 1}{2\sigma_{\epsilon}^{2}}\sum_{k=1}^{N_{\mathrm{s}}}\left(\epsilon_{k} \right)^{\mathrm{T}}\epsilon_{k}\Bigg{\}} \tag{20}\]  
The likelihood function \(p(\mathcal{D}|\theta)\) is obtained by substituting \(\epsilon_{k}=\hat{\mathbf{y}}_{k}-\mathbf{y}_{k}(\overline{\theta})\) from Equation (16) into the above equation:  
\[p(\mathcal{D}|\theta)=\Big{(}2\pi\sigma_{\epsilon}^{2}\Big{)}^{-\frac{nN_{ \mathrm{s}}}{2}}\exp\Bigg{\{}-\frac{1}{2\sigma_{\epsilon}^{2}}\sum_{k=1}^{N_{ \mathrm{s}}}\left[\hat{\mathbf{y}}_{k}-\mathbf{y}_{k}(\overline{\theta}) \right]^{\mathrm{T}}\left[\hat{\mathbf{y}}_{k}-\mathbf{y}_{k}(\overline{ \theta})\right]\Bigg{\}} \tag{21}\]  
Taking the negative logarithm of Equation (21), according to the \(\ln\) algorithm, the NLLF can be obtained as follows:  
\[L(\theta)=-ln(p(\mathcal{D}|\theta))=\frac{nN_{\mathrm{s}}}{2}\ln(2\pi)+\frac{ nN_{\mathrm{s}}}{2}\ln\Big{(}\sigma_{\epsilon}^{2}\Big{)}+\frac{1}{2\sigma_{ \epsilon}^{2}}\sum_{k=1}^{N_{\mathrm{s}}}\left[\hat{\mathbf{y}}_{k}-\mathbf{ y}_{k}(\overline{\theta})\right]^{\mathrm{T}}\left[\hat{\mathbf{y}}_{k}- \mathbf{y}_{k}(\overline{\theta})\right] \tag{22}\]  
In the Bayesian approach, the most probable value (MPV) \(\theta^{*}\) maximizes the PDF, and, by the orthogonal relation \(p(\theta|\mathcal{D})\propto e^{-L(\theta)}\), \(\theta^{*}\) minimizes the NLLF, i.e.,  
\[\theta^{*}=\underset{\theta}{\mathrm{argmax}}(p(\theta|\mathcal{D}))=\underset {\theta}{\mathrm{argmin}}(L(\theta)) \tag{23}\]  
Therefore, \(\theta^{*}\) can be calculated by solving for the minimum value of \(L(\theta)\) with an optimization algorithm. The MPV in the Bayesian approach is consistent with the results of the maximum likelihood estimation (MLE) in traditional statistical theory due to the assumption that the prior PDF \(p(\theta)\) obeys a uniform distribution [29]. However, a distinct advantage of the Bayesian approach is that it can give the posterior covariance matrix of the parameters, thus quantifying the uncertainty of the identification results. When the length of the measured data is large enough, according to [PERSON]'s method of asymptotic approximation, the posterior PDF \(p(\mathcal{D}|\theta)\) can be well approximated by a Gaussian PDF [30]:  
\[p(\theta|\mathcal{D})\propto\exp\left[-\frac{1}{2}(\theta-\theta^{*})^{ \mathrm{T}}(\mathbf{C}^{*})^{-1}(\theta-\theta^{*})\right] \tag{24}\]  
where \(\theta^{*}\) and \(\mathbf{C}^{*}\) are the mean and covariance matrices of parameters \(\theta\), respectively. The covariance matrix \(\mathbf{C}^{*}\) is equal to the inverse of the Hessian matrix of the NLLF:  
\[\mathbf{C}^{*}=[\mathbf{H}_{L}(\theta^{*})]^{-1} \tag{25}\]  
where \(\mathbf{H}_{L}(\theta^{*})\) denotes the Hessian matrix of \(L(\theta)\) at \(\theta^{*}\) with elements of \(\left[\mathbf{H}_{L}(\theta^{*})\right]_{ij}=\frac{\partial^{2}L(\theta)}{ \partial\theta_{i}\partial\theta_{j}}|_{\theta=\theta^{*}}\).