## 3 Results
### MCA Identified on Real Data
#### 3.2.2 MCA Proxies Identified on the _Zostera Marina_ Leaf Area-Leaf Biomass Data
Panel (c) compares the fitted function \(H_{TSK}(x)\) to power function produced by [PERSON]'s model of simple allometry. Panel (d) presents the dynamics of curvature index \(\kappa(x)\). Deviation of this index from the line \(y=1\) validate curvature detected in geometrical space for this data. This illustrates that manifestation of curvature is explained by complexity inherent to the allometric response-covariate association in direct scales. Table 6 presents 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. Figure 10: Fit of the \(\Omega_{TSK}(u)\) proxy on the eelgrass data. Panel (**a**) shows firing strengths \(\delta^{1}(u)\) and \(\delta^{2}(u)\) for the biphasic characterization of \(\Omega_{TSK}(u)\). Panel (**b**) exhibits dispersion of \(v\) about fitted TSK mean response \(E_{\Omega TSK}(v|u)\) vertical segment signposts estimated break point. Panel (**c**) shows residual dispersion about the zero line. Panel (**d**) displays normal QQ plot. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Method** & **AIC** & \(\rho\) & \(R^{2}\) & **SEE** & **MPE** \\ \hline \(E_{\lambda 2}(v|u)\) & 883.75 & 0.9485 & 0.9021 & 0.4433 & \(-0.5635\) \\ \(E_{\lambda 1}(v|u)\) & 937.92 & 0.9440 & 0.8939 & 0.4608 & \(-0.5857\) \\ \(E_{\lambda 0}(v|u)\) & 1182.97 & 0.9192 & 0.8505 & 0.5465 & \(-0.6945\) \\ \(E_{\Omega BL}(v|u)(m=2)\) & 858.81 & 0.9501 & 0.9049 & 0.4363 & \(-0.5546\) \\ \(E_{\Omega ML}(v|u)\) & 852.50 & 0.9510 & 0.9062 & 0.4338 & \(-0.5515\) \\ \(E_{\Omega TSK}(v|u)\) (\(q=2\)) & 863.76 & 0.9503 & 0.9053 & 0.4366 & \(-0.5550\) \\ \(E_{\Omega TSK}(v|u)\) (\(q=5\)) & 848.63 & 0.9514 & 0.9073 & 0.4321 & \(-0.5492\) \\ \(E_{\tau m}(v|u)\) (\(m=3\)) & 894.59 & 0.9474 & 0.9001 & 0.4472 & \(-0.5685\) \\ \(E_{\tau m}(v|u)\) (\(m=5\)) & 871.19 & 0.9495 & 0.9038 & 0.4394 & \(-0.5586\) \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Comparison of reproducibility strengths of MCA protocols fitted in geometrical space for the _Zostera marina_ data. We include \(\lambda_{m}(u):(m=0,\ 1,\ 2)\) and, \(\Omega_{BL}(u)\) : (biphasic) \(\pi_{m}(u):(m=3,\ 5)\), \(\Omega_{ML}(u)\) (biphasic) and \(\Omega_{TSK}(u)\) (\(q=2,\ 5\)) and \(\pi_{m}(u)(m=3,\ 5)\). AIC stands for Akaike’s information criterion, \(\rho\) symbolizes [PERSON]’s concordance correlation coefficient, \(R^{2}\) denotes determination coefficient, SEE standard error of estimation and MPE mean prediction error. Figure 11: Plots of retransformed MCA protocols fitted on logtransformed eelgrass data. Panel (**a**) \(E_{\lambda 2B}(y|x)\) function obtained by retransformation of interpolation lines \(E_{\lambda 2B}(v|u)\) (Figure 9a) Panel (**b**) spread about \(E_{\Omega BL}(y|x)\). Panel (**c**) presents dispersion around mean function \(E_{\pi 5}(y|x)\) produced by the \(\pi_{5}(u)\) polynomial. Panel (**d**) displays spread about mean funcion \(E_{\Omega ML}(y|x)\) deriving from retransformation of the mixture lines proxy \(\Omega_{ML}(u)\). These plots suggest remarkable consistency of the retransformed MCA output. The present results obtained on real and simulated data suggest adequacy of the MCA scheme proposed as an analytical tool for efficient allometric projection of plant biomass units. Consistency submitted by plots of dispersion about identified mean response functions in direct scales can in all