## 3 Results
### MCA Identified on Real Data
#### 3.2.2 MCA Proxies Identified on the _Zostera Marina_ Leaf Area-Leaf Biomass Data  
Figure 9 shows plots of MCA protocols fitted on the _Zostera marina_ data set. Panel (a) displays dispersion of response \(v\) about mean functions \(E_{\lambda 2}(v|u)\). Shown \(E_{\lambda 2 BI}(v|u)\) lines stand for the biphasic approximation to \(E_{\lambda 2}(v|u)\) acquired by using the criterion in Appendix A. The position of the determined break point is signposted by a vertical segment. Panel (b) portrays dispersion of \(v\) values about \(E_{\Omega BL}(v|u)\). Panel (c) portrays \(v\) dispersion about the polynomial mean response \(E_{\pi 5}(v|u)\). Panel (d) shows corresponding spread about the mixture lines mean response \(E_{\Omega ML}(v|u)\). Vertical segments in plots of panels (b), (c) and (d) associate to identified break points. As explained above, the break point for \(\pi_{5}(u)\) was also estimated by the empirical interpolation criterion described in Appendix A. Resulting estimations of this threshold trough are consistent with reported by [PERSON] et al. [92]. In all cases scattering of \(v\) values about MCA fitted mean response functions suggest steady agreement. Figure 10 presents plots resulting from a \(\Omega_{TSK}(u)\) protocol fitted on present eelgrass data. Panel (a) shows firing strengths \(\delta^{1}(u)\) and \(\delta^{2}(u)\) for a biphasic (\(q=2\)) characterization. This plot displays a breakpoint \(b_{1}\) as determined by intersection of \(\delta^{1}(u)\) and \(\delta^{2}(u)\) curves. Panel (b) depicts dispersion of log-transformed response \(v\) about mean response \(E_{\Omega TSK}(v|u)\). The vertical segment shown indicates placing of estimated break point \(b_{1}\). Panel (c) and panel (d) display residual and normal QQ plots one to one. Residual dispersion about the zero line is fair, and suggests heteroscedasticity. The QQ plot shows large plateau where residuals track a normal distribution pattern. Figure 10a provides further corroboration of adequacy of firing strength intersection criteria for break point detection derived from  
Figure 9: MCA protocols fitted in geometrical space for the _Zostera marina_ data set. Panel (**a**) dispersion of response \(v\) about mean functions \(E_{\lambda 2}(v|u)\). The break point signposted by vertical segment is determined by the interpolation lines criterion presented in Appendix A. Panel (**b**) shows corresponding spreading about \(E_{\Omega BL}(v|u)\). Panel (**c**) shows deviations of \(v\) values around polynomial mean response \(E_{\pi 5}(v|u)\). Panel (**d**) is for \(v\) spread about \(E_{ML}(v|u)\). Plots suggest consistency of MCA fits in all cases. the \(\Omega_{TSK}(u)\) approach. This estimate of \(b_{1}\) agrees with corresponding estimations by MCA methods described above. Figure 11 displays plots of retransformed \(\lambda_{2}(u)\), \(\Omega_{BL}(u),\pi_{5}(u)\) and \(\Omega_{ML}(u)\) protocols. Panel (a) shows spread about \(E_{\lambda 2 BI}(y|x)\) function obtained by retransformation of interpolation lines \(E_{\lambda 2 BI}(y|u)\) for \(\lambda_{2}(u)\) shown in Figure 9a. Panel (b) depicts spread about mean function \(E_{\Omega BL}(y|x)\) associated with the broken line method \(\Omega_{\mathbf{BL}}(u)\). Panel (d) presents dispersion about retransformed mean function of polynomial model \(\pi_{5}(u)\). Panel (d) portrays dispersion about \(E_{\Gamma ML}(y|x)\) deriving from the mixture lines procedures \(\Omega_{ML}(u)\). Table 5 presents model comparison metrics for MCA surrogates fitted in geometrical space. We can learn that, among considered methods, an interpolation mode of the \(\Omega_{TSK}(u)\) approach brings about better performance. Figure 12 presents plots of spread about retransformed mean response \(E_{\Omega TSK}(y|x)\), associating to the fitted biphasic \(\Omega_{TSK}(u)\) (panel (a)). Panel (b) displays spread about curves produced by a biphasic form of \(H_{TSK}(x)\) fitted on direct scales.