# Hybrid Learning for General Type-2 TSK Fuzzy Logic Systems
## 2 Background
### General Type-2 Membership Function Parameterization  
The proposed hybrid learning technique depends on the parameterization of a GT2 FS in the form of a Gaussian primary membership function with uncertain mean and Gaussian secondary membership  
Figure 1: Generic general type-2 fuzzy set (GT2 FS) as shown from the primary function’s perspective.  
Figure 2: Generic GT2 FS as shown from an isometric view.  
functions. This GT2 membership function requires four parameters: \(\{\sigma,m_{1},m_{2},\rho\}\), where \(\sigma\) is the standard deviation of the primary membership function, \(m_{1}\) and \(m_{2}\) are the left and right centers of the Gaussian membership function with uncertain mean, and \(\rho\) is a fraction of uncertainty which affects the support of the secondary membership function. Here, for the sake of simplification, the primary membership function is best represented by the support footprint of uncertainty (FOU) of the primary membership function in the form of an IT2 membership function, as shown in Figure 3. Based on the parameterized structure of the support FOU, the hybrid learning technique performs two type-1 TSK optimizations, as if optimizing two distinct type-1 TSK FLSs.  
The parameterization of the GT2 membership function is as follows. First, the support of the GT2 membership function is created by Equations (4)-(7), using \(\{x,u,\sigma,m_{1},m_{2},\rho\}\), where \(x\in X\) on the Universe of Discourse \(X\), and \(u\in U\) such that \(u\in J_{x}\subseteq[0,\;1]\). Creating an IT2 MF with \(\overline{\mu}(x)\) and \(\underline{\mu}(x)\), for the upper and lower membership function respectively, as shown in Figure 3.  
\[\mu_{1}(x)=exp\left[-\frac{1}{2}\left(\frac{x-m_{1}}{\sigma}\right)^{2}\right] \tag{4}\]  
\[\mu_{2}(x)=exp\left[-\frac{1}{2}\left(\frac{x-m_{2}}{\sigma}\right)^{2}\right] \tag{5}\]  
\[\overline{\mu}(x)=\left\{\begin{array}{l}\mu_{1}(x)\;x<m_{1}\\ 1\;m_{1}\leq x\leq m_{2}\\ \mu_{2}(x)\;x>m_{2}\end{array}\right. \tag{6}\]  
\[\underline{\mu}(x)=\left\{\begin{array}{l}\mu_{2}(x)\;x\leq\frac{m_{1}+m_{2 }}{2}\\ \mu_{1}(x)\;x>\frac{m_{1}+m_{2}}{2}\end{array}\right. \tag{7}\]  
Afterwards, all parameters required to form the individual secondary membership functions must be calculated, as shown in Equations (8)-(10), where \(p_{x}\) and \(\sigma_{u}\) are the center and standard deviation of the secondary Gaussian membership function, and \(\varepsilon\) is a very small number, e.g., 0.000001.  
\[p_{x}=exp\left[-\frac{1}{2}\left(\frac{x-m}{\sigma}\right)^{2}\right];\;\text {where }m=\frac{m_{1}+m_{2}}{2} \tag{8}\]  
\[\delta=\overline{\mu}(x)-\underline{\mu}(x) \tag{9}\]  
Figure 3: Support of the primary membership function of the used GT2 FS.  
\[\sigma_{u}=(1-\rho)\frac{\delta}{2\sqrt{3}}+\varepsilon \tag{10}\]  
Finally, each secondary membership function can be calculated by Equation (11), such that \(\widetilde{\mu}(x,u)\) is the secondary function on \(x\). Therefore, forming a complete GT2 membership function would be achieved by calculating for all \(x\in X\).  
\[\widetilde{\mu}(x,u)=exp\left[-\frac{1}{2}\left(\frac{u-p_{x}}{\sigma_{u}} \right)^{2}\right] \tag{11}\]