## Materials and Methods
### Multiple-Parameter Complex Allometry (MCA) Identification Protocols
#### 2.2.2 Identification of \(z(u)\) Involving Polynomial Forms
\tag{20}\]Then, chosen \(\delta^{k}(u)\) functions satisfy the normalizing condition,  
\[\sum\
olimits_{1}^{n}\delta^{k}(u)=1. \tag{21}\]  
Now, for \(k=1,\ 2,\ \ldots,\ n\) we contemplate parameters \(\alpha_{k}\) and \(ln\beta_{k}\), and also assume that \(\beta(e^{u})\) and \(\alpha\ (e^{u})\) setting \(z(u)\) in Equation (5) are defined through:  
\[\beta(e^{u})=e^{\sum\
olimits_{1}^{n}\delta^{k}(u)ln\beta_{k}} \tag{22}\]  
\[\alpha\ (e^{u})=\sum\
olimits_{1}^{n}\delta^{k}(u)\alpha_{k} \tag{23}\]  
This way regression Equation (4) hosts a characterization:  
\[v=\ \Omega_{\text{BL}}(u)+\epsilon \tag{24}\]  
where \(\Omega_{\text{BL}}(u)\) is the arranged ORBM form of \(z(u)\) that becomes,  
\[\Omega_{\text{BL}}(u)=\sum\
olimits_{1}^{n}\delta^{k}(u\ )f^{k}(u) \tag{25}\]  
with  
\[f^{k}(u)=ln\beta_{k}+\alpha_{k}u. \tag{26}\]  
In a PLA arrangement, variability of the log-transformed response \(v\), is interpreted through a collection of linear sub models defined on domains conceived as disjoint subintervals composing covariate range \(U\). The local linear models switch on thresholds also called break points. This way a PLA assemblage provides interpolation features for the identification of curvature in geometrical space. But, beyond empirical gains, the parameters composing the local linear models admit an interpretation as allometric exponents according to [PERSON]'s original formulation. Then, Equation (19) through Equation (26), readily bring about a PLA arrangement. Certainly, for \(k=1,\ 2,\ldots,\ \ n-1\) a threshold \(b_{k}\) interprets as a break point for transition from a \(k\)th allometric phase associated to a linear sub model \(f^{k}(u)\) to one \((k+1)\)th for local model \(f^{k+1}(u)\). Moreover, \(\Omega_{\text{BL}}(u)\) as given by Equation (25) entails a mixture of linear models \(f^{k}(u)\) weighted by factors \(\delta^{k}(u)\). Along with this, Equation (24) becomes a mixture regression model [104; 113; 114]. For the piecewise linear setting of Equation (25) identification can be achieved by means of the segmented package in R [115]. Alternatively, we can conceive the regression model:  
\[v=\ \Omega_{\text{ML}}(u)+\ \epsilon \tag{27}\]  
where  
\[\Omega_{\text{ML}}(u)=\ \sum\
olimits_{1}^{n}\delta^{k}(u\ )f^{k}(u) \tag{28}\]  
but, this time, we let \(\delta^{k}(u)\) vary continuously over the whole domain \(U\), with range \(0\leq\delta^{k}(u)\leq 1\) and also satisfying the normalization condition of Equation (21). Then, Equation (28) entails a mixture of weighted linear models [114; 116; 117; 118; 119]. For instance, for a biphasic characterization \((n=2)\) of \(\Omega_{\text{ML}}(u)\) we may take \(\delta^{1}(u)\) as a normal survival function. Consequently, Equation (21) sets \(\delta^{2}(u)=1-\delta^{1}(u)\). Details of the related fitting procedure are provided in Equations (A24) through (A27) in Appendix B. Nevertheless, this slant poses significant technical difficulties for the case \(n>2\). Moreover, a Takagi-Sugeno-Kang (TSK) fuzzy model [120; 121] offers a versatile PLA identification procedure that allows consideration of multiple interpolation sub models \((n>2)\). Associated regression model expresses here by:  
\[v=\ \Omega_{TSK}(u)+\ \epsilon \tag{29}\]with \(\Omega_{TSK}(u)\) taking the ORBM form:  
\[\Omega_{TSK}(u)\ =\ \sum\
olimits_{1}^{n}\delta^{k}(u\ )f^{k}(u) \tag{30}\]  
with weights \(\delta^{k}(u)\) being continuous functions acquired through fuzzy clustering techniques, and \(f^{k}(u)\) linear sub models having the form set by Equation (26) and to be identified through recursive least squares techniques. Appendix C explains details on putting forward \(\Omega_{TSK}(u)\) as given by Equation (30) This is achieved by adapting Equation (A42) for the present allometric settings. A first step involves acquiring a fuzzy partition of covariate domain \(U\). This characterizes the set \(L_{U}\) specified by Equation (A29), composing linguistic terms \(\Phi_{1}(u),\dots,\ \Phi_{q}(u)\) that create a fuzzy partition of \(U\).