## 5 Conclusions
Writing--original draft, H.E.-H.; Writing--review and editing, H.E.-H.; C.L.-R., E.V.-D. and J.R.C.-R. This research received no external funding. Two anonymous reviewers provided important comments that greatly improved our final presentation. The authors declare no conflict of interest. Appendix A Identification of the Model of Complex Allometry by Approximation of \(\beta(x)\) and \(\alpha(x)\) through Polynomial Forms  
The proposed MCA formula takes on the form of the generalized power function of Equation (2) namely:  
\[y=\beta(x)x^{\alpha(x)}\]  
with \(x\) varying in an interval \([c,d]\subset R\), and also assuming that both \(\beta(x)\) and \(\alpha(x)\) are continuous functions on that domain. Additionally, we undertake that \(\beta(x)\) maps \([c,d]\) into \(R^{+}\). Complexity of the general settings of the model of Equation (2) poses significant difficulties when using direct nonlinear regression as an identification device. However, we can provide an alternative fitting protocol in geometrical space that overcomes these technical complications. In order to advance this, we set \(\beta(x)=\ \exp(f(x))\) with \(f(x)\) continuous in \([c,d]\) so we can express Equation (2) in the equivalent form:  
\[y=e^{f(x)}x^{\alpha(x)} \tag{1}\]Now, since \(f(x)\) is continuous on a real interval, then, the Weierstrass approximation theorem assures that for any \(\varepsilon>0\), there exists a \(r\) th-order polynomial \(P_{r}(x)\) with \(r\) depending on \(\varepsilon\) and set by:  
\[P_{r}(x)=\ p_{0}+\sum\
olimits_{1}^{r}p_{k}x^{k} \tag{10}\]  
where \(p_{i}\) for \(0,\ 1,\ldots,r\) are coefficients, and such that, \(\left|P_{r}(x)-f(x)\right|<\varepsilon\), for all \(x\ \varepsilon[c,d]\). Moreover for that value of \(\varepsilon\) we can also choose a \(s\)_th_-order polynomial \(Q_{s}(x)\) given by:  
\[Q_{s}(x)=\ q_{0}+\sum\
olimits_{1}^{s}q_{k}x^{k}, \tag{11}\]  
such that we also have, \(\left|Q_{s}(x)-\alpha(x)\right|<\varepsilon\). Let \(m=\max(r,s)\). Then, if \(m=r\) we set \(q_{k}=0\), for \(s+1\leq k\leq m\). Conversely, if \(m=s\), we set \(p_{k}=0\) for \(r+1\leq k\leq m\). This way we can consider an homogeneous degree \(m\) for polynomials \(P_{r}(x)\) and \(Q_{s}(x)\) in Equations (10) and (11) that will ensure tolerances \(\left|P_{m}(x)-f(x)\right|<\varepsilon\), and \(\left|\ Q_{m}(x)-\alpha(x)\right|<\varepsilon\). Therefore, Equation (10) can be written in the form:  
\[y=e^{P_{m}(x)}\ a^{Q_{m}(\ x)}\ e^{\varepsilon(x)} \tag{12}\]  
where \(\varepsilon(x)\) is a remainder that, according to the Weierstrass approximation theorem, can become negligible by a suitable choosing of \(m\). Assuming that this is satisfied we can take \(\varepsilon(x)\) as a random variable \(\varepsilon\) which allows to consider the regression model:  
\[w=e^{P_{m}(x)}\ a^{Q_{m}(\ x)}\ e^{\varepsilon} \tag{13}\]  
Then, \(\varepsilon^{c}\) can be interpreted as a multiplicative random error term. The transformation \(v=lny\) and \(u=lnx\) carries Equation (13) into:  
\[v=P_{m}(\varepsilon^{u})+Q_{m}(\varepsilon^{u})u+\varepsilon \tag{14}\]  
with \(\varepsilon\) interpreted as an additive error term. Using Equations (10) and (11) to set forms \(P_{m}(x)\) and \(Q_{m}(\ x)\), Equation (14) yields,  
\[v=\sum\
olimits_{0}^{m}p_{k}\varepsilon^{ku}+\sum\
olimits_{0}^{m}q_{k}u \varepsilon^{ku}+\varepsilon \tag{15}\]  
Therefore, we can express the regression model of Equation (14) in the form:  
\[v=\ \lambda_{m}(u)+\varepsilon \tag{16}\]  
with  
\[\lambda_{m}(u)=\ \sum\
olimits_{0}^{m}(p_{k}+q_{k}u)e^{ku}\]  
that allows to obtain estimators of the parameters \(p_{k}\) and \(q_{k}\) defining the \(P_{m}(x)\) and \(Q_{m}(x)\) that approximate \(\beta(x)\) and \(\alpha(x)\) in Equation (13). Equation (16) provides a non-linear regression protocol involving \(u\) and powers of \(e^{u}\). A characterization of the log transformed allometric response \(v\) as a polynomial of the linked covariate \(u\) corresponds to a complex allometry model with a varying exponent in arithmetical space.