## 3 Results
### MCA Identified on Real Data
#### 3.2.1 MCA Proxies Identified on the Aboveground Tree Biomass (ABG) and H Data  
The [PERSON] and Ramirez et al. (2019) tree aboveground biomass and height (ABG-H) data set is available from: Ramiez-Ramirez.txt in the supplementary files section. Figure 5 presents \((u,v)=(\log(H),\log(ABG))\) plots showing dispersion of response \(v\): about mean functions \(E_{\lambda 1}(v|u)\) panel (a)), about \(E_{\Omega BL}(v|u)\) (panel (b)), \(E_{\pi 3}(v|u)\) (panel (c)) and about \(E_{\Omega ML}(v|u)\) (panel (d)). Vertical segments in plots signpost identified break points. Break points for MCA approaches \(\lambda_{1}(u)\) and \(\pi_{3}(u)\) where estimated by the empirical criterion described in Appendix A. Estimations of this threshold trough present methods seem to correspond with value estimated from Figure 2b in [PERSON] et al. (2019) by electronic scanning. The scattering of \(v\) values about MCA fitted mean response functions suggests reliable agreement in all cases. Figure 6 portrays fitted firing strengths \(\vartheta^{1}(u)\) and \(\vartheta^{2}(u)\) for fitted biphasic form \((q=2)\) of \(\Omega_{\text{TSK}}(u)\), (panel (a)). Panel (b) shows dispersion of log-transformed response values \(v\) about mean response \(E_{\Omega TSK}(v|u)\). The vertical segment shown indicates positioning of estimated break point \(b_{1}\). Panel (c) and panel (d) display residual and normal QQ (quantile-quantile) plots one to one. Plot in Figure 6a stands visual corroboration of criteria for break point identification derived from intersection of firing strength functions characterizing \(\Omega_{\text{TSK}}(u)\). Residual and QQ plots corroborate a consistent fit. Figure 5: MCA proxies fitted in geometrical space for the [PERSON] et al. (2019) data. Panel (**a**) dispersion of response \(v\) about mean functions \(E_{\lambda 1}(v|u)\). Panel (**b**) corresponding spreading about \(E_{\Omega BL}(v|u)\). Panel (**c**) shows deviations of \(v\) values around polynomial mean response \(E_{\pi 3}(v|u)\). Panel (**d**) shows scattering about \(E_{\Omega ML}(v|u)\). The plots suggest consistency of MCA fits in all cases. Figure 7 displays retransformed conventional MCA models. We provide plots of retransformed \(\lambda_{1}(u)\) (panel (a)), broken line \(\Omega_{\mathbf{BL}}(u)\) (panel (b)), polynomial \(\pi_{3}(u)\) (panel (c)), and mixture lines \(\Omega_{\text{ML}}(u)\) (panel (d)). Table 3 presents model comparison metrics for all MCA surrogates fitted in geometrical space. We can learn that among considered methods an interpolation mode of the \(\Omega_{\text{TSK}}(u)\) approach brings about better performance. Figure 6: Fit of the \(\Omega_{\text{TSK}}(u)\) proxy fitted for the [PERSON] et al. [6] data. Panel (**a**) shows firing strengths \(\delta^{1}(u)\) and \(\delta^{2}(u)\) for the biphasic characterization \((q=2)\) of \(\Omega_{\text{TSK}}(u)\) intersecting at break point. Panel (**b**) exhibits dispersion of \(v\) about fitted TSK mean response, vertical segment signposts estimated break point. Panel (**c**) shows residual dispersion about the zero line. Panel (**d**) displays normal QQ (quantile-quantile) plot. Figure 7: _Cont._  
Figure 8 shows results for retransformed biphasic mean response \(E_{\Omega TSK}(y\big{|}x)\) (panel (a)), direct scales fitted \(H_{TSK}(x)\) (panel (b)), [PERSON]'s model of simple allometry (panel (c)) and curvature index \(\kappa(x)\) (panel (d)). We can learn of the deviation of index \(\kappa(x)\) from the line \(y=1\) that explain curvature detected in geometrical space for this data is inherent to complexity of the allometric relationship response-covariate in direct scales. Table 4 presents the performance metrics of retransformed proxies compared to [PERSON]'s formula of simple allometry and characterizations of directly fitted \(H_{TSK}(x)\) : \((q=2,q=7).\) Again, the retransformed \(\Omega_{TSK}(u)\) and directly fitted \(H_{TSK}(x)\) displayed the highest reproducibility strengths.