## 4 Discussion  
### Error Results for THMA (Error 1)  
The following observations are derived from the information in plots and tables:  
* As shown in Figures 3 and A3, Figures A4 and A5, Tables 2 and 3, the median \(x^{50}\) for all cases, variables and stories is very close to 0, and the percentiles \(x^{10}\) and \(x^{90}\) are approximately contained within the indicative EAR [\(-0.15\); \(+0.25\)] except at roof, where a tendency for under-prediction (\(x^{10}\) below \(-0.15\)) exists for all variables except \(u_{\rm Ler,proof}\); for this variable the prediction is excellent, as expected from the fact that \begin{table}
\begin{tabular}{c c c c c c c c c c c c c c c c} \hline \hline  & & \multicolumn{8}{c}{**First Story**} & \multicolumn{8}{c}{**Average Across Stories**} & \multicolumn{8}{c}{**Roof**} \\ \cline{3-14}  & & **min** & \(x^{10}\) & \(x^{50}\) & \(x^{90}\) & **max** & **min** & \(x^{10}\) & \(x^{50}\) & \(x^{90}\) & **max** & **min** & \(x^{10}\) & \(x^{50}\) & **max** \\ \hline \multirow{5}{*}{\(\sigma\)} & \(u_{3}\) & 0.02 & 0.01 & 0.01 & 0.01 & 0.16 & 0.02 & 0.00 & 0.00 & 0.00 & 0.02 & 0.03 & 0.01 & 0.00 & 0.01 & 0.00 \\  & \(\
u_{5}\) & 0.02 & 0.01 & 0.01 & 0.02 & 0.14 & 0.01 & 0.00 & 0.00 & 0.00 & 0.01 & 0.03 & 0.00 & 0.01 & 0.01 & 0.01 \\  & \(\
u_{3}\) & 0.03 & 0.01 & 0.01 & 0.01 & 0.04 & 0.02 & 0.01 & 0.01 & 0.07 & 0.09 & 0.04 & 0.02 & 0.05 & 0.15 & 0.38 \\  & \(\Delta_{3}\) & 0.02 & 0.01 & 0.01 & 0.01 & 0.16 & 0.01 & 0.01 & 0.00 & 0.02 & 0.02 & 0.03 & 0.04 & 0.01 & 0.06 & 0.15 \\  & \(\
abla_{3}\) & 0.02 & 0.01 & 0.01 & 0.02 & 0.14 & 0.01 & 0.01 & 0.02 & 0.04 & 0.05 & 0.02 & 0.04 & 0.04 & 0.08 & 0.27 \\  & \(V_{53}\) & 0.01 & 0.02 & 0.02 & 0.02 & 0.41 & 0.04 & 0.01 & 0.00 & 0.02 & 0.07 & 0.03 & 0.03 & 0.01 & 0.07 & 0.27 \\  & \(V_{\rm DS}\) & 0.02 & 0.01 & 0.01 & 0.02 & 0.13 & 0.03 & 0.00 & 0.02 & 0.04 & 0.05 & 0.10 & 0.03 & 0.04 & 0.08 & 0.26 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: \(Cont.\)  
\begin{table}
\begin{tabular}{c c c c c c c c c c c c c c c c} \hline \hline  & & \multicolumn{8}{c}{**Average Across Stories**} & \multicolumn{8}{c}{**Roof**} \\ \cline{3-14}  & & **min** & \(x^{10}\) & \(x^{50}\) & \(x^{90}\) & **max** & **min** & \(x^{10}\) & \(x^{50}\) & \(x^{90}\) & **max** & **min** & \(x^{10}\) & \(x^{50}\) & \(x^{90}\) & **max** \\ \hline \multirow{5}{*}{\(\mu\)} & \(u_{1}\) & \(-  
the roof displacement is the key variable in the modal method. Indeed, the relative velocity error \(v_{1,\text{error},\text{root}}\) is also very small. However, this is not the case for \(a_{1,\text{error},\text{root}}\). * In general, the dispersion for all variables (except \(u_{1}\) and \(v_{1}\)) is smaller in central stories of the frame and increases at the top and bottom (Figure 3); the 3-story frame is an exception where \(x^{90}\) and max generally diminish at the top. * The dispersion \(\sigma\) across cases included in Tables 2 and 3 is very small for all statistics except max, indicating a similar behavior across cases. * When the average across stories is examined (Figures A3-A5, Tables 2 and 3), THMA offers a reasonable accuracy for all variables analyzed. The behavior is particularly good for \(\Delta_{1,\text{error},\text{average}}\), and for the static variables \(V_{\text{SI},\text{error},\text{average}}\) and \(V_{\text{D1},\text{error},\text{average}}\). The worst behavior is observed for \(a_{1,\text{error},\text{average}}\), with a tendency to under-prediction. In all cases, a reduced number of extreme values give errors well outside the indicative EAR. Extreme errors are more pronounced for shear-type systems except for \(V_{\text{SI},\text{error},\text{average}}\). * When the first story error is examined, larger extreme errors are found, particularly for the maximum in all variables except \(a_{1,\text{error},\text{first}}\). The over-conservative error reaches particularly high values (up to 6.66 for B5-1) for \(V_{\text{SI},\text{error},\text{first}}\). For other variables, approximately up to 3. Remarkable minimum extreme errors are also found.