## 4 Training of MIMO-NARX Meta-Models
### MIMO-NARX Model Training
The full NARX model consists of potential NARX terms including seismic acceleration from 0 to \(l\) steps forward \(\ddot{x}_{g}(t-l\Delta t)\), velocity responses of the reduced coordinates from 1 to \(l\) steps forward \(\dot{q}_{j}(t-l\Delta t)\), one step ahead absolute value of velocity responses of the reduced coordinates \(\left|\dot{q}_{j}(t-\Delta t)\right|\), \(\left|\dot{q}_{j}(t-\Delta t)\right|\dot{q}_{j}(t-l\Delta t)\), \(\left|\dot{q}_{j}(t-\Delta t)\right|\ddot{x}_{g}(t-l\Delta t)\) and 1, with \(l=1\), 2, 3, 4, \(j=1\), 2, 3, 4, which leads to \(5+4\times 4+4\) + \(4\times 4\times 4+4\times 5+1=110\) terms. It should be noticed that the absolute value terms directly use velocity responses of the reduced coordinates rather than velocity responses in the original coordinates \(\boldsymbol{\Phi}_{n}^{m}\dot{\boldsymbol{q}}(t-\Delta t)\) used in previous studies [14]. This results in a smaller  
Figure 8: Procedure for MIMO-NARX training. Figure 9: Structure of MIMO-NARX full model. number of terms to avoid redundant terms, which brings the benefits of faster computation and almost similar accuracy. The final MIMO-NARX model can be expressed as follows:  
\[ \begin{array}{ll}\dot{q}_{1}(t)=y_{1,1}\ddot{x}_{g}(t)&+y_{1,2}\ddot{x}_{g}(t- \Delta t)+y_{1,3}\ddot{x}_{g}(t-2\Delta t)+y_{1,4}\ddot{x}_{g}(t-3\Delta t)\\ &+y_{1,5}\ddot{x}_{g}(t-4\Delta t)+\cdots+y_{1,55}|\ddot{q}_{2}(t-\Delta t)| \ddot{x}_{g}(t-4\Delta t)\\ &+y_{1,56}|\ddot{q}_{3}(t-\Delta t)|\ddot{x}_{g}(t-4\Delta t)+y_{1,57}|\ddot{q} _{4}(t-\Delta t)|\ddot{x}_{g}(t-4\Delta t)\\ \end{array}  \tag{31}\]  
\[ \begin{array}{ll}\dot{q}_{2}(t)=y_{2,1}\ddot{x}_{g}(t)&+y_{2,2}\ddot{x}_{g}(t -\Delta t)+y_{2,3}\ddot{x}_{g}(t-2\Delta t)+y_{2,4}\ddot{x}_{g}(t-3\Delta t)\\ &+y_{2,5}\ddot{x}_{g}(t-4\Delta t)+\cdots+y_{2,60}|\ddot{q}_{2}(t-\Delta t)| \ddot{x}_{g}(t-4\Delta t)\\ &+y_{2,61}|\ddot{q}_{3}(t-\Delta t)|\ddot{x}_{g}(t-4\Delta t)+y_{2,62}|\ddot{q} _{4}(t-\Delta t)|\ddot{x}_{g}(t-4\Delta t)\\ \end{array}  \tag{32}\]  
\[ \begin{array}{ll}\dot{q}_{3}(t)=y_{3,1}\ddot{x}_{g}(t)&+y_{3,2}\ddot{x}_{g}(t -4\Delta t)+y_{3,3}\dot{q}_{2}(t-3\Delta t)+y_{3,4}\dot{q}_{2}(t-4\Delta t)\\ &+y_{3,5}\ddot{q}_{3}(t-\Delta t)+\cdots+y_{3,27}|\ddot{q}_{2}(t-\Delta t)| \ddot{x}_{g}(t)\\ &+y_{3,28}|\dot{q}_{3}(t-\Delta t)|\ddot{x}_{g}(t)+y_{3,29}|\ddot{q}_{4}(t- \Delta t)|\ddot{x}_{g}(t)\\ \end{array}  \tag{33}\]  
\[ \begin{array}{ll}\dot{q}_{4}(t)=y_{4,1}\ddot{x}_{g}(t)&+y_{4,2}\ddot{x}_{g}( t-\Delta t)+y_{4,3}\ddot{x}_{g}(t-2\Delta t)+y_{4,4}\ddot{x}_{g}(t-3\Delta t)\\ &+y_{4,5}\ddot{x}_{g}(t-4\Delta t)+\cdots+y_{4,47}|\ddot{q}_{2}(t-\Delta t)| \ddot{x}_{g}(t-4\Delta t)\\ &+y_{4,48}|\dot{q}_{3}(t-\Delta t)|\ddot{x}_{g}(t-4\Delta t)+y_{4,49}|\ddot{q} _{4}(t-\Delta t)|\ddot{x}_{g}(t-4\Delta t)\\ \end{array}  \tag{34}\]  
where \(y_{i,j}\) represents the NARX coefficient for the \(j\)th NARX term of the \(i\)th output. It can be observed that there are 57, 62, 29, and 49 terms for the four outputs \(\dot{q}_{1}(t)\), \(\dot{q}_{2}(t)\), \(\dot{q}_{3}(t)\), and \(\dot{q}_{4}(t)\), respectively. Higher order of DOF does not necessarily lead to more terms for the MIMO-NARX model.