## 3 Results
### MCA Identified on Real Data
#### 3.2.2 MCA Proxies Identified on the _Zostera Marina_ Leaf Area-Leaf Biomass Data
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Method** & **AIC** & \(\mathbf{\rho}\) & \(\mathbf{R^{2}}\) & **SEE** & **MPE** \\ \hline \(E_{\lambda 2}(y|x)\) & \(-7104.84\) & \(0.9737\) & \(0.9430\) & \(0.0018\) & \(2.1340\) \\ \(E_{\lambda 1}(y|x)\) & \(-6474.24\) & \(0.9388\) & \(0.8632\) & \(0.0028\) & \(5.3011\) \\ \(E_{\lambda 0}(y|x)\) & \(-6837.92\) & \(0.9490\) & \(0.9167\) & \(0.0022\) & \(2.5723\) \\ \(E_{\text{DBL}}(y|x):(m=2)\) & \(-7008.52\) & \(0.9712\) & \(0.9345\) & \(0.0019\) & \(2.2837\) \\ \(E_{\text{fML}}(y|x)\) & \(-7092.99\) & \(0.9739\) & \(0.9421\) & \(0.0018\) & \(2.1515\) \\ \(E_{\text{GTSK}}(y|x):(q=2)\) & \(-7116.99\) & \(0.9747\) & \(0.9443\) & \(0.0018\) & \(2.1133\) \\ \(E_{\text{GTSK}}(y|x):(q=5)\) & \(-7199.19\) & \(0.9770\) & \(0.9502\) & \(0.0017\) & \(1.9968\) \\ \(E_{\text{rm}}\) (\(y|x):(m=3)\) & \(-7060.11\) & \(0.9729\) & \(0.9390\) & \(0.0019\) & \(2.2039\) \\ \(E_{\text{rm}}\) (\(y|x):(m=5)\) & \(-7096.27\) & \(0.9738\) & \(0.9423\) & \(0.0018\) & \(2.1466\) \\ \(E_{\text{HTSK}}(y|x):(q=2)\) & \(-7405.16\) & \(0.9809\) & \(0.9625\) & \(0.0015\) & \(1.7324\) \\ \(E_{\text{HTSK}}(y|x):(q=3)\) & \(-7412.54\) & \(0.9811\) & \(0.9629\) & \(0.0014\) & \(1.7236\) \\ Huxley & \(-7414.89\) & \(0.9808\) & \(0.9624\) & \(0.0015\) & \(1.7279\) \\ \hline \hline \end{tabular}
\end{table} 
Table 6: Performance metrics for MCA proxies for observed eelgrass leaf biomass. We include retransformed forms of \(\lambda_{m}(u):(m=0,\ 1,\ 2)\) and \(,\Omega_{BL}(u):(\text{biphasic})\)\(\pi_{m}(u):(m=3,\ 5)\), \(\Omega_{ML}(u)\) (biphasic) and \(\Omega_{TSK}(u)\)\((q=2,\ 5)\) compared to [PERSON]’s formula of simple allometry and directly fitted \(H_{TSK}(x):(q=2,q=3)\). AIC stands for [PERSON]’s information criterion, \(\mathbf{\rho}\) symbolizes [PERSON]’s concordance correlation coefficient, \(R^{2}\) denotes determination coefficient, SEE standard error of estimation and MPE mean prediction error. Figure 12: Plots of retransformed \(\Omega_{TSK}(u)\) and direct \(H_{TSK}(x)\) proxies identified on eelgrass data. Panel (**a**) shows spread about \(E_{\OmegaTSK}(y|x)\). Panel (**b**) depicts dispersion about fitted biphasic form \(H_{TSK}(x)\), \((q=2)\). Panel (**c**) compares \(H_{TSK}(x)\), \((q=2)\), to the power function by [PERSON]’s model of simple allometry fitted on direct scales. Panel (**d**) displays the plot of curvature index \(\kappa(x)\). This plot exhibits curvature effects more pronounced for the smaller leaves. cases be corroborated by performance metrics presented in the tables. The criterion for identifying break points based in MCA forms \(\lambda_{m}(u)\) and \(\pi_{m}(u)\) fitted in geometrical space provides reliable estimations as compared with procedures directly aimed to this endeavor and presently exemplified by the \(\Omega_{TSK}(u)\), \(\Omega_{BL}(u)\), and \(\Omega_{ML}(u)\) approaches. Moreover, as shown by Figure 3d, the suggested MCA form also allows CF-free retransformation step that grants reliable reproducibility. Performance of the curvature index \(\kappa(x)\) enhances our perception of the present MCA arrangement as an effective scheme for allometric examination in the general settings. Besides the aforementioned empirical advantages, we expect that addressing MCA could overcome the controversy around using DNLR or log-transformation methods in identification tasks.