## 3 Results
### MCA Identified on Simulated Data
This for simulated pairs \((u_{i},\ v_{k(i)})\) produced an \(r^{2}=0.9963\) fit with resultant estimated parameter vector  
\[\mathbf{P}\mathbf{Q}=\left( \begin{array}{c}p_{0}\\ p_{1}\\ p_{2}\\ q_{0}\\ q_{1}\\ q_{2}\end{array} \right)=\left( \begin{array}{c}-7.261\\ 0.0004503,\\ -1.212\times 10^{-6}\\ -0.1102,\\ 0.0001862\\ 1.149\times 10^{-7}\end{array} \right) \tag{47}\]  
Figure 2a displays spread of \(v_{k(i)}\) replicates about fitted mean response \(\lambda_{2}(u)\) compared with TAMA's counterpart \(\lambda_{0}(u)\). We can ascertain remarkable account of curvature by \(\lambda_{2}(u_{i})\) and noticeable bias by the TAMA protocol on this data. Figure 2b compares retransformed mean functions \(E_{\lambda 2}(y\big{|}x)\) and corresponding TAMA for one \(E_{\lambda 0}(y\big{|}x)\) and we can ascertain biased projections by the latter. Figure 3a displays the spread of \(v_{k(i)}\) replicates about mean response \(\lambda_{2}(u)\) compared with its \(\Omega_{TSK}(u)\) counterpart. We can ascertain a dependable account of curvature by both surrogates Figure 3b compares retransformed mean functions \(E_{\lambda 2}(y\big{|}x)\) and \(E_{\Omega TSK}(y\big{|}x)\). Figure 3c presents mean response acquired by fitting the protocol H\({}_{TSK}(x)\) in direct arithmetical scales. Panel (d) is the comparison of \(E_{\Omega TSK}(y\big{|}x)\) to \(E_{PQ2}(y\big{|}x)\), the directly acquired projection function. This is gotten by replacing fitted parameter vector \(PQ\) into addressed MCA form in arithmetical space. We can ascertain remarkable reproducibility features by the \(\Omega_{TSK}(u)\), with the approach fitted in geometrical scales. Table 1 presents performance metrics for considered proxies. The failure of a TAMA approach to display suitable complexity explains poor performance in all included fitting statistics. Although, the Huxley by DNLR fit suggest fair reproducibility by \(\rho\) and \(R^{2}\) indices it is outperformed by MCA alternates in the Akaike information criterion (AIC), SEE and MPE statistics. Moreover, deviations of a \(\kappa(x)\) index from the line \(y=1\) shown in Figure 1b adds criterion that sustain selection of the MCA model. This confirms our judgement that consistency of the Huxley-DNLR approach as suggested by visual inspection of plots is only apparent. This could explain an assertion that clinging to DNLR could impair detecting inherent complexity. Then, suitable examination for this data must rely on a MCA paradigm. This example also shows how a TSK slant could offer reliability on both scales of allometric examination. Figure 2: Fit of MCA on simulated data \(\big{(}u_{i},v_{\delta(i)}(e)\big{)}\). Regression Equation (46) was fitted to simulated \(\big{(}u_{i},v_{k(i)}\big{)}\) data pairs. Panel (a) shows dispersion of \(\mathbf{v}_{k(i)}\) about the mean response function \(\lambda_{2}(u)\). This plot also shows the traditional analysis method of allometry’s (TAMA) fitted mean response \(\lambda_{0}(u)\). We can ascertain notorious bias by this last approach. Panel (b) shows corresponding retransformation results. Compared to significant reproducibility of mean response \(E_{\lambda 2}(y\big{|}x)\) TAMA’s counterpart \(E_{\lambda 0}(y\big{|}x)\) entails significant bias. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Method** & **AIC** & \(\rho\) & \(R^{2}\) & **SEE** & **MPE** \\ \hline \(E_{\lambda 0}(y|x)\) & \(-13\),\(244.48\) & \(0.8236\) & \(0.7588\) & \(0.0171\) & \(1.9240\) \\ \(E_{\lambda 2}(y|x)\) & \(-19\),\(317.76\) & \(0.9892\) & \(0.9788\) & \(0.0051\) & \(0.5706\) \\ Huudey-DNLR & \(-18\),\(704.07\) & \(0.9866\) & \(0.9728\) & \(0.0057\) & \(0.6456\) \\ EHTSK \((y|x):(q=3)\) & \(-19\),\(168.14\) & \(0.9886\) & \(0.9775\) & \(0.0052\) & \(0.5878\) \\ E\({}_{\Omega\text{TSK}}\left(y|x\right):(q=5)\) & \(-19\),\(318.23\) & \(0.9893\) & \(0.9789\) & \(0.0051\) & \(0.5703\) \\ E\({}_{\text{rm}}\left(y|x\right):(m=5)\) & \(-18\),\(749.20\) & \(0.9866\) & \(0.9734\) & \(0.0057\) & \(0.6393\) \\ \hline \hline \end{tabular}
\end{table}