## 3 Results
### MCA Identified on Real Data
#### 3.2.1 MCA Proxies Identified on the Aboveground Tree Biomass (ABG) and H Data
\begin{table}
\begin{tabular}{c c c c c c} \hline
**Method** & **AIC** & \(\rho\) & \(R^{2}\) & **SEE** & **MPE** \\ \hline \(E_{\lambda 2}(y\big{|}u)\) & 40.52 & 0.9154 & 0.8440 & 0.3538 & 7.3563 \\ \(E_{\lambda 1}(\
abla|u)\) & 44.34 & 0.8982 & 0.8152 & 0.3759 & 7.8154 \\ \(E_{\lambda 10}(\
abla|u)\) & 48.48 & 0.8760 & 0.7793 & 0.4012 & 8.3428 \\ \(E_{\Omega BL}(v|u)(m=2)\) & 38.59 & 0.9086 & 0.8333 & 0.3570 & 7.4226 \\ \(E_{\Omega ML}(v|u)\) & 39.45 & 0.9182 & 0.8476 & 0.3497 & 7.2714 \\ \(E_{\Omega TSK}(v|u)\) (\(q=2)\) & 50.97 & 0.9014 & 0.8206 & 0.3893 & 8.0954 \\ \(E_{\Omega TSK}(v|u)\) (\(q=6\)) & 36.46 & 0.9300 & 0.8691 & 0.3325 & 6.9144 \\ \(E_{\pi\pi}(v|u)\) (\(m=3\)) & 43.92 & 0.8992 & 0.8168 & 0.3742 & 7.7803 \\ \(E_{\pi\pi}(v|u)\) (\(m=5\)) & 40.68 & 0.9151 & 0.8435 & 0.3544 & 7.3688 \\ \hline \end{tabular}
\end{table} 
Table 3: Performance metrics for geometrical space models fitted on the [PERSON] et al. [6] ABG-H data set. We include proxies based on \(\lambda_{m}(u):(m=0,\ 1,\ 2),\)\(\Omega_{\mathbf{BL}}(u)\) : (biphasic)) \(\pi_{m}(u):(m=3,5),\)\(\Omega_{\mathrm{ML}}(u)\) (biphasic) and \(\Omega_{\mathrm{TSK}}(u)\) (\(q=2,\ 6\)) fitted on the Ramirez-Ramirez [6] data. 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 7: Plots of retransformed mean response functions identified for the [PERSON] et al. [6] data. Panel (**a**) presents dispersion of observed values response values about mean response function \(E_{\lambda 1}(y\big{|}x)\). Panel (**b**) displays corresponding spread about \(E_{\Omega BL}(y\big{|}x)\). Panel (**c**) shows this around \(E_{\pi 3}(y\big{|}x)\)  and panel (**d**) associates to named scattering about \(E_{\Omega ML}(y\big{|}x)\)  panel (**a**), broken line \(\Omega_{\mathbf{BL}}(u)\) (panel (**b**)), polynomial \(\pi_{3}(u)\) (panel (**c**)) and mixture lines \(\Omega_{\mathrm{ML}}(u)\) (panel (**d**)). \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Method** & **AIC** & \(\rho\) & \(R^{2}\) & **SEE** & **MPE** \\ \hline \(E_{\lambda 2}(y\big{|}x)\) & 226.66 & 0.8789 & 0.7738 & 2.6757 & 13.3503 \\ \(E_{\lambda 1}(y\big{|}x)\) & 243.72 & 0.8194 & 0.6425 & 3.2830 & 16.3803 \\ \(E_{\lambda 0}(y\big{|}x)\) & 229.39 & 0.8103 & 0.7144 & 2.8666 & 14.3030 \\ \(E_{\Omega BL}(y\big{|}x):(m=2)\) & 237.92 & 0.8415 & 0.6849 & 3.0822 & 15.3784 \\ \(E_{\Omega ML}(y\big{|}x)\) & 230.57 & 0.8728 & 0.7538 & 2.7918 & 13.9294 \\ \(E_{\Omega TSK}\)\((y\big{|}x):(q=2)\) & 248.45 & 0.8313 & 0.6670 & 3.3309 & 16.6194 \\ \(E_{\Omega TSK}\)\((y\big{|}x):(q=6)\) & 227.11 & 0.8914 & 0.7906 & 2.6413 & 13.1788 \\ \(E_{\pi\pi}\)\((y\big{|}x):(m=3)\) & 244.46 & 0.8181 & 0.6367 & 3.3093 & 16.5116 \\ \(E_{\pi\pi}\)\((y\big{|}x):(m=5)\) & 225.81 & 0.8800 & 0.7780 & 2.6510 & 13.2270 \\ \(E_{\text{HTSK}}\)\((y\big{|}x):(q=2)\) & 230.95 & 0.8716 & 0.7724 & 2.7539 & 13.7407 \\ \(E_{\text{HTSK}}\)\((y\big{|}x):(q=7)\) & 188.94 & 0.9521 & 0.9087 & 1.7444 & 8.7036 \\ Huxley-DNLR & 224.49 & 0.8532 & 0.7432 & 2.7181 & 13.5620 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Performance metrics for arithmetical space proxies identified from the [PERSON] et al. [6] ABG-H data set. We include retransformed forms of \(\lambda_{m}(u):(m=0,\ 1,\ 2)\) and \(\Omega_{\text{BL}}(u):(\text{iphasic})) \(\pi_{m}(u):(m=3,\ 5)\), \(\Omega_{ML}(u)\) (biphasic) and \(\Omega_{TSK}(u)\) (\(q=2,\ 6\)) and compare to [PERSON]’s formula of simple allometry and directly fitted \(H_{TSK}(x):(q=2,\ q=7)\). AIC stands for [PERSON]’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.