## 3 Numerical Studies  
To investigate the effectiveness of the SCEM-UA algorithm for parameter estimation and uncertainty assessment of structural models, numerical simulations of MDOF structural system models are carried out. The two-dimensional shear frame type structure shown in Figure 2 is analyzed. The dynamic equation of the structure motion can be described as  
Figure 1: The flowchart of the algorithm. \[\mathbf{M\ddot{x}}(t)+\mathbf{C}\mathbf{x}(t)+\mathbf{K}\mathbf{x}(t)=\mathbf{F}(t) \tag{2}\]  
where \(\mathbf{M}\), \(\mathbf{C}\), and \(\mathbf{K}\) represent the mass, damping and stiffness matrices, respectively. \(\mathbf{F}\) is the input force vector, and \(\mathbf{x}\) is the displacement vector. The mass of each floor is concentrated and the damping is assumed as the Rayleigh damping matrix \(\mathbf{C}\). The damping ratio, \(\zeta_{\mathrm{r}}\), of the first two modes of vibration. Therefore, the n-DOF structure system can be described by the parameter set \(\theta\) = (\(m_{1}\), \(m_{2}\),..., \(m_{n}\), \(k_{1}\), \(k_{2}\),..., \(k_{n}\), \(\zeta_{1}\), \(\zeta_{2}\)). The structure is excited by known forces, \(\mathbf{F}(t)\) and the acceleration response of the structure, \(\ddot{\mathbf{x}}(t)\), is measured at some given point. Let \(\mathbf{\hat{x}}_{j}(t_{i})\) also for \(j\) = 1, 2,..., \(ny\), and \(i\) = 1, 2,..., T denotes the value of the \(j\)th response of the identified system at the \(i\)th time step. All available output records can be written as  
\[\mathbf{\hat{y}}=\left(\mathbf{\hat{x}}_{1}(t),\mathbf{\hat{x}}_{2}(t),\dots \mathbf{\hat{x}}_{ny}(t)\right)_{t=t_{1},-t_{\mathrm{T}}}^{\mathrm{T}}. \tag{3}\]  
First, the authors consider the parameter estimation for an 8-DOF structural model and the structural properties are shown in Table 1 with no prior knowledge about stiffness and damping. The parameters of stiffness and damping are all constant over time and the structural response to a given excitation is carried out by fourth-order Runge-Kutta method. The structure is excited by the Niigata earthquake (Japan, 2000). This paper elucidates the uncertainty assessment for parameter estimations and how their limited available observations with error are handled. The input and output (I/O) data are polluted by Gaussian, zero mean, and white noise. Root mean square (RMS) value of the noise is a certain percentage of the real time histories. The \"full output\" indicates that measurement data are available at all floors, whereas, the \"partial output\" indicates that data are available at Floors 1, 3, 5, and 7. The SCEM-UA algorithm was implemented with q = 10 complexes including 30 points in each complex, indicating a population size s = 300. The prior information consists of lower and upper bounds on each parameter distribution as shown in Table 2; thus, a feasible parameter space can be defined and a uniform prior distribution is imposed on this rectangle. Three scenarios are tested: Case1: 0% noise + Full Outputs; Case2: 10% noise + Full Outputs, and Case3: 10% noise + Partial Outputs. \begin{table}
\begin{tabular}{c c} \hline \hline
**Mass (kg)** & \\ \hline \(m_{1}\)\(\sim\)\(m_{7}\) & 4.95 \\ \(m_{8}\) & 4.51 \\ Stiffness (kN/m) & \\ \(k_{1}\) & \(5.53\times 10^{3}\) \\ \(k_{2}\)\(\sim\)\(k_{8}\) & \(2.72\times 10^{3}\) \\ Damping ratio & \\ \(\zeta_{1}\) & \(1\times 10^{-2}\) \\ \(\zeta_{2}\) & \(3\times 10^{-2}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Structure properties of the 8-DOF system. Figure 2: n-DOF structure model. The \(\sqrt{SR}\) defined by [PERSON] [33] was utilized to evaluate the convergence of the MCMC algorithm. When the factor is less than 1.2, which indicates the Markov chain is converged to a stationary distribution, the evolution stops. The evolution of the Gelman-Rubin factor for the parameters \(k_{1}\)-\(k_{8}\), \(\zeta_{1}\) and \(\zeta_{2}\) of the three cases using the SCEM-UA algorithms is shown in Figure 3a-c, respectively.