## 4 Training of MIMO-NARX Meta-Models
### Uncertainties in Ground Motion and Structural Properties  
The dynamic responses are affected by the system uncertainties in ground motion and structural properties. By varying ground motions and structural parameters, structural responses can be obtained and used as training samples. The stochastic ground motion model [37] is adopted in this study to generate ground motion uncertainties. Each ground motion is determined by six parameters \(\left(T_{a},D_{5-95},t_{mid},\omega_{mid},\omega^{\prime},\zeta_{f}\right)\) and white noise. In this study, \(\omega_{mid}\) and \(\zeta_{f}\), which control the filter frequency and damping ratio, respectively, are selected as uncertain parameters of ground motion while the other parameters (\(\vec{I}_{a}\), \(D_{5-95}\), \(t_{mid}\), \(\omega^{\prime}\)) are set as (0.0314 s\(\cdot\)g, 11.23 s, 7.85 s, \(-\)0.04 Hz/s). It is worth noting that the white noise remains unchanged to generate seismic acceleration time histories. A typical acceleration time history is shown in Figure 3.  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Floor** & \(\mathbf{k_{bw}}\) (N/mm) & \(\mathbf{\alpha_{bw}}\) & \(\mathbf{A_{bw}}\) & \(\gamma_{bw}\) & \(\beta_{bw}\) & \(\mathbf{n_{bw}}\) \\ \hline
1 & \(2.9793\times 10^{5}\) & 0.01 & 1 & 0.02 & 0.02 & 0.01 \\
2 & \(7.3787\times 10^{5}\) & 0.01 & 1 & 0.1 & 0.01 \\
3 & \(6.9252\times 10^{5}\) & 0.01 & 1 & 0.1 & 0.01 \\
4 & \(6.0177\times 10^{5}\) & 0.01 & 1 & 0.1 & 0.01 \\
5 & \(5.2397\times 10^{5}\) & 0.01 & 1 & 0.08 & 0.08 & 0.01 \\
6 & \(4.4108\times 10^{5}\) & 0.01 & 1 & 0.06 & 0.06 & 0.01 \\
7 & \(3.6986\times 10^{5}\) & 0.01 & 1 & 0.06 & 0.06 & 0.01 \\
8 & \(3.4706\times 10^{5}\) & 0.01 & 1 & 0.08 & 0.08 & 0.01 \\
9 & \(3.2748\times 10^{5}\) & 0.01 & 1 & 0.1 & 0.01 & 0.01 \\ \hline \hline \end{tabular}
\end{table}
Table 1: Bouc–Wen model parameters.  
Figure 2: Hysteresis behavior of Bouc–Wen models (a typical sample).  
A total of \(n_{s}=200\) parameter sets of (\(\omega_{mid},\zeta_{f}\)) are generated following the distribution in Figure 4a,b, which leads to 200 stochastic ground motions for MIMO-NARX training. Figures 4c and 5 present the distribution of peak ground accelerations and the mean response spectra of these 200 acceleration time histories, respectively.  
The concentrated masses of each story are considered structural uncertainties, which reflects the uncertainty of live load or temporary load changes in the building. The masses of floor 1, floor 2-8, and floor 9 are assumed to follow the Gaussian distribution as shown in Figure 6.  
The two ground motion parameters along with the three structural parameters are considered uncertain input variables in this study and are listed in Table 2. The uncertain variables of the dynamic system can be generated randomly from the distribution in Table 2  
Figure 4: Uncertainties of ground motions (200 samples). (**a**) Filter frequency \(\omega_{mid}\); (**b**) Filter damping ratio \(\zeta_{f}\); (**c**) Peak ground acceleration.  
Figure 5: Response spectra of ground motions with damping ratio 0.05 (200 samples).  
Figure 3: Time history of stochastic ground motion (a typical sample).  
for dynamic time history analysis. The sample data used for training the MIMO-NARX model are extracted from the current and past state values of structural responses.