## Appendix E Simulation Assay on \(\lambda_{0}(u)\)  
In this assay, we considered the case \(m=0\) of the MCA form \(\lambda_{m}(u)\) as represented by Equation (12). This is associated to constant scaling functions, \(\beta(x)=p_{0}\) and \(\alpha(x)=q_{0}\). The resulting model is set as [PERSON]'s formula of simple allometry given by Equation (1). In order to perform a simulation assay in this model, we took covariate values \(x_{i}\) for \(i=1,\ 2,\ \dots,\ N\), with \(N=500\) and such that \(x_{min}\leq x_{i}\leq x_{max}\), with \(x_{min}=1\) and \(x_{max}=10,000\). Then, we choose reference parameter values \(p_{0}=1.752\times 10^{-12}\) and \(q_{0}=2.713\) so that Equation (42) determined projected values \(y_{i}=p_{0}\exp(q_{0}lnx_{i})\). So produced data pairs \((x_{i},\ y_{i})\) are available in the supplementary files Href.txt. Next, we used the Matlab function _random_(\"_norm_\", \(\mu,\sigma\)) function in order to draw normal random numbers \(\epsilon_{k}\) for \(k=1,\ 2,\dots,\ T\) with \(T=5\). Then, Equation (43) produced replicates \(y_{k(i)}(\epsilon)=p_{0}\exp(q_{0}lnx_{i}+\epsilon_{k})\) for each reference value \(y_{i}\). In this trial, we generated \(N\times T=2500\) replicates. Resulting data can be found in the supplementary files Hrep.txt. Figure 11a shows the spread of simulated \(y_{k(i)}(\epsilon)\) replicates around the reference curve \(y_{i}=p_{0}\exp(q_{0}lnx_{i})\). A fit of the model \(y=\beta\exp(\alpha lnx)\) by direct non-linear regression to \(\big{(}x_{i},y_{k(i)}(\epsilon)\big{)}\) data resulted in estimates of \(\beta=1.231\times 10^{-12}\) and \(\alpha=2.753\ (r^{2}=0.9802)\). Figure 11b shows spread of \(y_{k(i)}(\epsilon)\) replicates around the fitted curve \((x_{i},\ \ \beta x_{i}{}^{\alpha})\). Next, we acquired \(v_{k(i)}(\epsilon)\) values from the transformation \(v_{k(i)}(\epsilon)=\log\big{(}y_{k(i)}(\epsilon)\big{)}\) and \(u_{i}=\log(x_{i})\). Then regression Equation (4) yields the linear model,  
\[v_{k(i)}(\epsilon)=\lambda_{0}(u_{i})+\epsilon_{k} \tag{45}\]  
where  
\[\lambda_{0}(u)=\ p_{0}+q_{0}u\]  
Fitting this model produced estimates \(p_{0}=-27.08\) and \(q_{0}=2.714\), \(\big{(}r^{2}=0.9985\big{)}\). Figure 11a displays the spread of \(v_{k(i)}(\epsilon)\) replicates about estimated mean response \(\lambda_{0}(u_{i})=p_{0}+q_{0}\ u_{i}\). Then, this was retransformed to arithmetical space to produce the estimated mean function \(E_{10}\ (y|x)\exp(p_{0}+q_{0}\ x(u))\delta(\epsilon)\) (cf. Equation (6)). For this try \(\delta(\epsilon)=\exp\big{(}\sigma^{2}/2\big{)}\) as corresponds to \(\epsilon\sim N(0,\sigma)\). This assumption was corroborated by applying an Anderson-Darling test (Function adtest.m in Matlab). Figure 11b allows comparison of retransformed \(\big{(}x_{i},\ E_{10}(y_{i}|x_{i})\big{)}\) and directly fitted \((x_{i},\ \beta x_{i}^{\alpha})\) curves. Shown spread suggest reliable agreement between observed values and retransforming mean response \(E_{\lambda 0}\big{(}y|x\big{)}\). No biased results can be attributed to using a logtransformation procedure in this study case. Table 12 presents model performance metrics. We can be aware of remarkableagreement of these surrogates. As a result, it can be inferred that the log transformation step in the TAMA approach dos not induce a curvature deformation in geometrical space. In order to assess error structure effects we considered the case of non-lognormally distributed replicates, and we maintained the reference curve \(y_{i}=p_{0}\exp(q_{0}nx_{i})\) as above, but according to the procedure around Equation (44) we acquired values \(\xi_{k}=random(^{\prime}exp^{\prime},\mu)*random(^{\prime}logistic^{\prime},\mu,\sigma)\), that is, \(\xi_{k}\) expresses as a product of exponentially (\(\mu=0.1\)) and logistically (\(\mu=0,\ \sigma=0.1\)) distributed random numbers. Then, we formed replicates \(y_{k(i)}(\xi)=p_{0}\exp(q_{0}lnx_{i}+\xi_{k})\). In this assay, we maintained \(N\times T=2500\) replicates. Supplemental file Hnonrep.txt includes acquired data.