## 4 Discussion
### Identification of Significant Parameters  
The error results presented indicate that either method can render un-conservative or over-conservative results for certain cases (combinations of frame and added damping system). It is of interest to link the error to specific values of parameters describing the system. Such identification could be used to impose additional conditions to damped systems analyzed with classical modal procedures to keep the error below a certain threshold. In order to identify the relevant parameters, the variability of the THMA error (error 1) statistics min, \(x^{10}\), \(x^{90}\) and max with respect to the parameters considered in the study is analyzed through their coefficient of variation \(c_{v}\) calculated as follows: for a certain case the error results are sorted according to a certain parameter, thus obtaining 4 independent subsets, each one associated with a specific value of the parameter; statistics (min, \(x^{10}\), \(x^{90}\) and max) of each sub-set are calculated; the sample standard deviation \(\sigma\) and mean \(\mu\) across the four values for each statistic are derived and \(c_{v}\) is found as \(\sigma/\mu\). An example for case B0-\(\infty\) and \(\Delta_{1,\rm error}\) is shown in Table 4; in this table, the second column on the left indicates that the largest variation of the min of \(\Delta_{1,\rm error,first}\) is due to \(\alpha_{k}\) (0.63, highlighted in bold), and the second largest variation is due to \(\alpha_{d}\) (0.36, highlighted in italics). \begin{table}
\begin{tabular}{c c c c c c c c c c c c c c c c} \hline \hline  & \multicolumn{4}{c}{**First**} & \multicolumn{4}{c}{**Average**} & \multicolumn{4}{c}{**Central**} & \multicolumn{4}{c}{**Roof**} \\ \cline{2-13}  & **min** & \(x^{10}\) & \(x^{90}\) & **max** & **min** & \(x^{10}\) & \(x^{90}\) & **max** & **min** & \(x^{10}\) & \(x^{90}\) & **max** & **min** & \(x^{10}\) & \(x^{90}\) & **max** \\ \hline \(T_{1}\) & 0.09 & 0.22 & 0.44 & 0.37 & 0.08 & 0.07 & 0.47 & 0.31 & 0.07 & 0.13 & 0.59 & 0.38 & 0.11 & 0.25 & 0.25 & 0.25 \\ \(\xi_{a,1}\) & 0.26 & 0.39 & 0.50 & 0.39 & 0.22 & 0.29 & 0.96 & 0.67 & **0.30** & 0.31 & 0.20 & 0.53 & _0.17_ & _0.47_ & 0.65 & 0.66 \\ \(N\) & 0.05 & 0.20 & 0.11 & 0.27 & 0.06 & 0.22 & 0.55 & 0.21 & 0.09 & 0.20 & 0.63 & 0.71 & 0.02 & 0.01 & 0.48 & 0.61 \\ \(\alpha_{k}\) & **0.63** & _0.84_ & 0.08 & 0.12 & 0.02 & 0.19 & 0.48 & **0.78** & 0.02 & **0.59** & 0.39 & _0.77_ & 0.02 & 0.01 & _0.90_ & **1.00** \\ \(\beta_{k}\) & 0.07 & 0.32 & 0.11 & 0.09 & 0.04 & 0.18 & 0.26 & 0.19 & 0.03 & 0.18 & 0.41 & 0.16 & 0.02 & 0.02 & 0.32 & 0.45 \\ \(a_{d}\) & _0.36_ & **1.08** & **1.36** & **1.40** & **0.32** & **0.75** & **1.08** & 0.32 & _0.17_ & 0.39 & **1.42** & **1.44** & **1.01** & **1.39** & **1.17** & _0.77_ \\ \(\beta_{d}\) & 0.00 & 0.08 & _0.83_ & _0.82_ & 0.13 & _0.49_ & 0.10 & 0.00 & 0.11 & _0.56_ & _0.89_ & 0.20 & 0.05 & 0.18 & 0.13 & 0.00 \\ \hline \hline \end{tabular} Maximum and second maximum \(c_{v}\) for each column identified in **bold** and _italics_, respectively.  
\end{table} 
Table 4: Coefficient of variation \(c_{v}\) for \(\Delta_{1,\rm error}\) and case B0-\(\infty\). This procedure is repeated for each shear-type case (B0-\(\infty\), B3-\(\infty\), B5-\(\infty\), C0-\(\infty\), C3-\(\infty\), C5-\(\infty\)); the most and second most influential parameter for each case are identified, and the overall count across cases for \(\Delta_{1,\text{error}}\) is summarized in Table 5. In the total count the average has been left out to avoid double-counting. This table shows a remarkable consistency of the error dependency from certain parameters between cases, but the relevant parameters are not the same across stories and variable statistics. For instance, the min error at first story (\(\Delta_{1,\text{min,first}}\)) is mostly dependent on \(\alpha_{d}\) and \(\alpha_{k}\); however, in all other stories (\(\Delta_{1,\text{min,average}}\), \(\Delta_{1,\text{min,central}}\), \(\Delta_{1,\text{min,roof}}\)) it is dependent on \(\alpha_{d}\) and \(\xi_{a,1}\).