## 3 Results
### MCA Identified on Simulated Data  
For the aim of performing MCA simulation studies, we adapted a response range as determined by leaf area values in the eelgrass data set reported by [PERSON] et al. [92]. This way, we considered covariate values \(x_{i}\) for \(i=1,\ 2\dots 500\) such that \(0<x_{i}\leq x_{max}\) where \(x_{max}=10,000\). Then, reference values \(y_{i}\) are produced by Equation (42), with \(\beta(x_{i})=\exp(P_{m}(x_{i}))\) and \(\alpha(x_{i})=Q_{m}(x_{i})\) being \(P_{m}(x_{i})\) and \(Q_{m}(x_{i})\) as given by Equations (9) and (10) respectively. Results for the case \(\lambda_{0}(u)\ (m=0)\) addressing [PERSON]'s model \(y_{i}=\exp(p_{0})\exp(q_{0}lnx_{i})\) are presented in Appendix E. This assay included the aims of (1) demonstrating that curvature in geometrical space could not be considered as a consequence of a logtransformation itself (2) explaining the skills of the \(\Omega_{TSK}(u)\) and \(\mathrm{H}(x)\) formulations to adapt complexity as necessary in interpreting inherent allometric pattern and (3) verifying the dependability of the curvature index \(\kappa(x)\) criterion of Equation (38). Indeed, as portrayed in Figure A5c,d for this simulation assay deviations of \(\kappa(x)\) about the line \(y=1\) are practically vanishing for all values of \(x\). In order to explore full MCA, without loss of generality, we circumscribed our study to the case \(m=2\) of Equation (12). In particular, we set reference parameter vector \(\mathbf{P}\mathbf{Q}\). \[\mathbf{P}\mathbf{Q}=\left( \begin{array}{c}p_{0}\\ p_{1}\\ p_{2}\\ q_{0}\\ q_{1}\\ q_{2}\end{array} \right)=\left( \begin{array}{c}-7.247\\ 0.001128\\ -1.037\times 10^{-6}\\ -0.1189\\ 8.74\times 10^{-5}\\ 9.848\times 10^{-8}\end{array} \right) \tag{45}\]  
Acquired reference data pairs \((x_{i},~{}y_{i})\) are provided in the Com2 ref.txt supplementary file. We used \(\mu=0\) and \(\sigma=0.1\) in Equation (43) in order to generate normally distributed random numbers \(\epsilon_{k}\), \(k=1,~{}2,\dots,~{}5\). Then, Equation (43) produced \(y_{k(i)}\) replicates for each reference value \(y_{i}\). Resulting data can be obtained from the supplementary file Com2 rep.txt. Moreover, fitting [PERSON]'s power function model (Equation (1)) to \(\left(x_{i},y_{k(i)}\right)\) pairs produced estimated parameter values \(\beta=1.113\text{e}~{}-12\) and \(=2.765~{}(r^{2}=0.9729)\). Figure 1a presents the distribution of \(y_{k(i)}\) values around the reference curve, and also compared to the fitted Huxley curve. This seems to provide a suitable approximation to the reference one. Nevertheless, we can be aware of deviations attributed to curvature induced by the conceived MCA form. Indeed, deviations from the line \(y=1\) by curvature index \(\kappa(x)\), in Figure 1b suggest the reliability of [PERSON]'s model for this data being only putative. Next, we acquired \(v_{k(i)}\) values from the log-transformation \(v_{k(i)}~{}=\log\Bigl{(}y_{k(i)}\Bigr{)}\) and \(u_{i}=\log(x_{i})\). According to Equation (11), choosing the case \(m=0\) in Equation (12) brings about a TAMA protocol. This produced estimates \(p_{0}=-14.92\) and \(q_{0}=1.313\left(r^{2}=0.829\right)\). We then considered the \(m=2\) case of the MCA of Equation (12). According to Equation (11), the regression model to be considered becomes:  
\[v_{k(i)}~{}=\lambda_{2}(u_{i})+\epsilon \tag{46}\]  
with  
\[\lambda_{2}(u)=(p_{0}+p_{1}\exp(u)+p_{2}\exp(2u))+(q_{0}+q_{1}\exp(u)+q_{2} \exp(2u))u.\]  
Figure 1: [PERSON]’s model of simple allometry fitted on multiple-parameter complex allometry (MCA) data simulated based on a lognormal distribution Panel (**a**) displays spread of \(y_{k(i)}(\epsilon)\) replicates around the reference curve \(y=\exp(P_{2}(x))\exp(Q_{2}(x)lnx)\) compared to the fitted [PERSON]’s mean response. Panel (**b**) shows the plot of curvature index \(\kappa(x)\). We can ascertain that deviations from linearity in geometrical space can be expected for this data.