## Materials and Methods
### Multiple-Parameter Complex Allometry (MCA) Identification Protocols
#### 2.2.2 Identification of \(z(u)\) Involving Polynomial Forms  
The Weierstrass approximation theorem [112] offers an approach that lodges \(z(u)\) involving terms \(ln\beta(x)\) and \(\alpha(x)\) that express through polynomials. Certainly, as we explain in Appendix A, for a proper integer \(m\) we can consider polynomials \(P_{m}(x)\) for \(ln(\beta(x))\) and \(Q_{m}(x)\) for \(\alpha(x)\) such that the MCA of Equation (2) can be written in the form:  
\[y=e^{P_{m}(x)}\ x^{Q_{m}(\ x)}\ e^{\epsilon(x)} \tag{8}\]  
where the function \(\exp(\epsilon(x))\) stands for involved approximation error. We further assume that the \(mth\)-degree polynomials \(P_{m}(x)\) and \(Q_{m}(x)\) define through,  
\[P_{m}(x)=p_{0}+\sum\
olimits_{1}^{m}p_{k}x^{k} \tag{9}\]  
\[Q_{m}(x)=q_{0}+\sum\
olimits_{1}^{m}q_{k}x^{k}, \tag{10}\]where \(p_{k}\) and \(q_{k}\) for \(k=0,\ 1,\ \ldots,m\) are coefficients. According to the [PERSON] approximation theorem for \(m\) large enough \(\epsilon(x)\) will display negligible values so we can consider that Equation (8) associates to the regression model:  
\[v=\lambda_{m}(u)+\epsilon \tag{11}\]  
where  
\[\lambda_{m}(u)=\sum\
olimits_{0}^{m}(p_{k}+q_{k}u\ )e^{ku} \tag{12}\]  
Then, \(\lambda_{m}(u)\) can be interpreted as a mixture of linear models \(f^{k}(u)=p_{k}+q_{k}u\) weighted by exponential powers \(e^{ku}\). Through, we will use the symbol \(PQ\) to symbolize the vector of parameters characterizing \(\lambda_{m}(u)\). It turns out that by setting \(m=0\) Equation (11) returns the linear regression model of the TAMA protocol. The resulting mean response function is denoted by means of the symbol \(E_{\lambda m}(v|u)\) and turns out to be \(E_{\lambda m}(v|u)=\lambda_{m}(u)\). Moreover, by defining auxiliary variables \(w_{k}(u)=e^{ku}\) and \(g_{k}(u)=ue^{ku}\) Equation (11) becomes a multiple linear regression model. According to our notation convention, the corresponding mean response function in arithmetical space is denoted by means of \(E_{\lambda m}(y|x)\). It becomes:  
\[E_{\lambda m}(y|x)=e^{\lambda_{m}(x(u))}\delta(\epsilon) \tag{13}\]  
As given by Equation (12) the \(\lambda_{m}(u)\) function depends on both \(u\) and \(exp(ku)\), but, as it is explained in Appendix A, it is possible to offer an equivalent form expressed as a \((2m+1)th\)-degree polynomial of \(u\) plus a remainder, namely  
\[\lambda_{m}(u)=\pi_{m}(u)+\mathrm{R}_{m}(u) \tag{14}\]  
where  
\[\pi_{m}(u)=\sum\
olimits_{n=0}^{2m+1}c_{n}u^{n} \tag{15}\]  
where Equation (A15) estates how the \(c_{n}\) coefficients relate to those conforming \(P_{m}(x)\) and \(Q_{m}(x)\). Moreover, by the Weierstrass approximation theorem for \(m\) large enough we could expect \(\mathrm{R}_{m}(u)\) becoming negligible, so the regression Equation (11) can be also written in the form:  
\[v=\pi_{m}(u)+\epsilon \tag{16}\]  
with,  
\[\pi_{m}(u)=\sum\
olimits_{k=0}^{2m+1}c_{k}u^{k}. \tag{17}\]  
Corresponding mean response function \(E_{\pi m}(y|x)\) in arithmetical space becomes:  
\[P_{m}(x)=p_{0}+\sum\
olimits_{1}^{m}p_{k}x^{k} \tag{9}\]0  
Identification of \(z(u)\) by means of an operating-regime-based modeling approach:  
In order to place MCA into an ORBM scheme, we set \(U=\{u|u_{min}\leq u\leq u_{max}\}\). Then, we contemplate a collection of disjoint intervals \(I_{k}\), with \(k=1,\ 2,\ldots,n\), given by  
\[I_{k}= \begin{cases}[u_{min}\ b_{1})&for\ k=1\\ \{b_{k-1},\ b_{k}\}&k=2,\ 3\ldots n-1.\\ [b_{n-1},\ u_{max}]&k=n\end{cases}  \tag{19}\]  
This way, the \(I_{k}^{\prime}\)s compose a partition \(\mathrm{U}_{1}^{n+1}[I_{k}]\) for \(U\). Now, for \(k=1,\ 2,\ldots,\ n\), we take weight functions \(\mathfrak{s}^{k}(u)\) defined through,  
\[\mathfrak{s}^{k}(u)= \begin{cases}1&u\in I_{k}\\ 0&\text{elsewhere}\end{cases} .