## 4 Discussion
### Correlation of Error Results with Existing Non-Proportionality Indices  
To investigate the relationship between the NPIs described in Section 2.2 and Appendix A with the error results described in Section 3, values of the indices are calculated for each system in the study; for mode-dependent indices, the average value across modes is taken. Index \(I_{\text{{PF1}}}\) is left out because of its numerical instability. Indices \(I_{A1}\) and \(I_{\text{{LKZ4}}}\) are very similar, showing a sample Pearson correlation coefficient \(\rho(I_{\text{{LKZ4}}};I_{A1})=0.98\) (calculated including all systems analyzed in this study), for this reason the discussion and conclusionsfor \(I_{A1}\) can be extended to \(I_{LK24}\). Two conditions indicate a good correlation of error and index: 1) the dispersion of errors (measured by the difference between extremes) must increase steadily with increasing index value; 2) for index values close to 0 the error should be close to 0; limiting the index value will then result in an effective error limitation. Figure 7 presents a scatter plot of \(\Delta_{1,\text{error},\text{average}}\), calculated for case B3-\(\infty\), versus all NPIs considered in the study (except \(I_{P1}\)); each sub-plot contains 13,312 points, each point corresponding to a different system. The plot shows clearly that only four indices (\(I_{P55}\), \(I_{TL1}\), \(I_{LK24}\) and \(I_{A1}\)) fulfil both conditions. Plots for other cases, variables and stories, not shown for brevity, confirm this observation. The scatter plot for each index, variable, story, and case is enveloped by a minimum monotonically increasing curve and a maximum monotonically decreasing curve, hereinafter referred to as max and min envelopes, respectively. Plot of these envelopes comprising all cases considered in the study are provided in Figure 8 (index \(I_{P55}\)), Figure 9 (index \(I_{TL1}\)), and Figure 10 (index \(I_{A1}\)). \(I_{LK24}\) is omitted because the corresponding plots are virtually identical to those for \(I_{A1}\). These plots establish a direct relationship between a maximum or minimum value of error for any variable and the relevant NPI at every case. The following conclusions are derived:  
* The relationship between THMA error for a certain variable at a certain story and the same NPI follows similar trends for all cases in the study, regardless of the index considered. The curves corresponding to different cases are, in all cases, closely squeezed together. * The differences between cases are mostly due to model type: shear-type cases (B0-\(\infty\), B3-\(\infty\), B5-\(\infty\), C0-\(\infty\), C3-\(\infty\), C5-\(\infty\)) present more extreme errors and generally correspond to the outermost curve; contrariwise, the extreme case of frame-type system (B3-0.10) presents the innermost curve of the collection. The only exception is \(V_{\text{S1,average}}\) where for all indices the min un-conservative error corresponds to B3-0.10. * Inherent damping and ground type exert a very mild influence; for the same \(\lambda\), the curves corresponding to different inherent damping ratios are practically identical, as are the curves corresponding to different ground types. An exception happens at \(V_{\text{D1,error},\text{average}}\) with indices \(I_{P55}\) and \(I_{A1}\), where the 0% case presents a steeper increase of conservative error. This trend is not detected in index \(I_{TL1}\). * The absolute value of un-conservative error is, in general, much smaller than the conservative error. However, for some specific variables and stories, similar orders of magnitude are obtained for both extreme errors. Figure 7: Scatter plots of \(\Delta_{1,\text{error},\text{average}}\) for case B3-\(\infty\) versus different non-proportionality indices. Figure 8: THMA error (type 1) max and min envelope for all cases versus index \(I_{PS5}\). Figure 9: THMA error (type 1) max and min envelope for all cases versus index \(I_{TLL1}\).