## Appendix C The Takagi-Sugeno-Kang (TSK) Fuzzy Model  
Ying [133] demonstrated that the general output of a TSK fuzzy can uniformly approximate any continuous function to arbitrarily high precision. In the present allometric examination settings, a TSK a fuzzy model is intended to provide a device for the identification of the mean response \(z(u)\) associating to the regression model of Equation (4) involving the logtransformed response \(v\) expressed as a nonlinear function of a descriptor \(u\), namely:  
\[v=\ln\!\left(\beta(\epsilon^{u})x^{\alpha(\epsilon^{u})}\right)\] (A28)  
where \(v\) and \(u\) take values on domains \(V\) and U one to one. In order to elaborate on the construction of a TSK fuzzy model, we firstly need to bring up an abstract structure known as a general fuzzy inference system. For that aim, we introduce a set \(L_{U}\) containing a number \(q\) of linguistic terms \(\Phi_{k}(u)\) characterizing the input variable \(u\), formally:  
\[L_{U}=\{\Phi_{k}(u)\big{|}k=1,\ 2,\ldots,q\ \}\] (A29)Each linguistic term \(\Phi_{k}(u)\) is associated to a membership function \(\mu\Phi_{k}(u)\) that setting a mapping \(\mu\Phi_{k}(u):U\rightarrow[0,1]\) characterizes \(\Phi_{k}(u)\) as the fuzzy set:  
\[\Phi_{k}(u)=\{\ \mu\Phi_{k}\left(u_{1}\right)/u_{1},\ \mu\Phi_{k}\left(u_{2} \right)/u_{2},\ldots,\ \mu\Phi_{k}\left(u_{n}\ \right)/u_{n}\ \} \tag{10}\]  
where \(u_{1},u_{2},\ldots,\ u_{n}\) stand for the values that \(u\) takes on. We use the symbol \(\mu_{U}\) to denote the collection of membership functions describing the variable \(u\), that is,  
\[\mu_{U}=\{\mu\Phi_{k}(u)\big{|}k=1,\ 2,\ldots,q\ \} \tag{11}\]  
Likewise, the pair \((L_{U},\mu_{U})\) will stand for what is known as a fuzzy partition of the input domain \(U\). Respectively, for \(j=1,\ 2,\ldots,r\) we designate linguistic terms \(\Psi_{j}(v)\) for the output variable \(v\) so we can consider a set \(L_{v}\) of linguistic terms, i.e.,  
\[L_{V}\ =\left\{\Psi_{j}(v)\ \big{|}\ j=1,\ 2,\ldots,r\right\} \tag{12}\]  
Similarly, for each linguistic term \(\Psi_{j}(v)\), we associate a membership function \(\mu\Psi_{j}(v)\), such that the mapping \(\mu\Psi_{j}(v)\ :V\rightarrow[0,1]\) establishes the fuzzy set:  
\[\Psi_{j}\left(v\right)=\left\{\mu\Psi_{j}(v_{1})/v_{1},\mu\Psi_{j}(v_{2})/v_{ 2},\ldots,\mu\Psi_{j}(v_{m})/v_{m}\right\} \tag{13}\]  
where \(v_{1},\ldots,v_{m}\) denote the values that \(v\) acquires. Concurrently, we have the collection \(\mu_{V}\) of membership functions tied to \(V\),  
\[\mu_{V}=\left\{\mu\Psi_{j}\left(v\right)\big{|}j=1,\ 2,\ldots,r\ \right\} \tag{14}\]  
and concomitantly, we may also say that the pair \((L_{V}\,\mu_{V})\) sets a fuzzy partition for the output domain \(V\). Additionally, for \(i=1,\ 2,\ldots,q\), in \(L_{U}\) and \(L_{V}\) we advance a correspondence \(i\rightarrow\Phi_{k(i)}(u)\), and \(i\rightarrow\Psi_{j(i)}(v)\), so we can contemplate antecedent conjunctions of the form,  
\[P^{i}(u):\left[u\text{ is }\Phi_{k(i)}(u)\right] \tag{15}\]  
and a consequent  
\[Q^{i}(v):\left[v\text{ is }\Psi_{j(i)}(v)\right] \tag{16}\]  
backing inferential rules \(R^{i}\) :  
\[R^{i}:\left\{ \begin{array}{rcl}if&:&P^{i}(u))\\ then&:&Q^{i}(v)\end{array} \right\} \tag{17}\]  
We may then envisage a general fuzzy inference system F as a construct that includes an application F: \(U\to V\) characterized by the sets of fuzzy partitions \((L_{U},\mu_{U})\) and \((L_{V},\mu_{V})\), a set of inference rules \(R=\text{U}_{1}^{p}\big{[}R^{i}\big{]}\) and a defuzzification operator \(D\) that associates to the fuzzy set \(Q^{i}(v)\) in Equation (16) a crisp value \(v\) in \(V\). In a TSK fuzzy inference system representation, we consider decision rules \(R^{i}\) having an antecedent \(P^{i}(u)\) of the form given by Equation (15) but with the consequent \(Q^{i}(v)\) in expression (16) taking a crisp functional form \(v=f^{i}(u)\). That is, in the TSK fuzzy inference system we may envision rules having the form  
\[R^{i}:\left\{ \begin{array}{rcl}if&:&u\text{ is }\Phi_{k(i)}(u)\\ then&:&v=f^{i}(u)\end{array}