# Hybrid Learning for General Type-2 TSK Fuzzy Logic Systems
## 3 Description of the Hybrid Learning Technique for GT2 TSK FLS  
The proposed approach, being hybrid in nature, is composed of a sequence of multiple steps, each using different algorithms in order to achieve the final model which is shown in this section. The hybrid learning technique requires a set of meaningful centers \(x\) for the antecedents of the rule base, these can be obtained via any method, such as clustering algorithms; for this paper, a fuzzy c-means (FCM) clustering algorithm [26] was used. Via these cluster centers, subsets belonging to each cluster center are selected through Euclidean distance, where the nearest data point to each cluster center is a member of its subset. After all subsets are found, cluster coverages can be calculated, i.e., the standard deviation \(\sigma\), obtained through Equation (14), where \(\sigma_{q,k}\) is the standard deviation of the \(q\)-th rule and \(k\)-th input, \(x_{i}\) is each datum from the subsets previously obtained, \(x_{q,k}^{c}\) is the cluster center for the \(q\)-th rule and \(k\)-th input, and \(n\) is the cardinality of the subset. \[\sigma_{q,k}=\frac{\sum_{i=1}^{n}\left(x_{i}-x_{q,k}^{c}\right)^{2}}{n-1} \tag{14}\]  
Up to now, a Type-1 Gaussian membership function can be formed with \(\{\sigma,x\}\). However, the sought end product is a GT2 Gaussian primary membership function with uncertain mean and Gaussian secondary membership functions, which requires the parameters: \(\{\sigma,m_{1},m_{2},\rho\}\). So far, we have calculated \(\sigma\) and partially \(m_{1}\) and \(m_{2}\), which are based on \(x\). The following process obtains the remaining required parameters \(\{m_{1},m_{2},\rho\}\) to form GT2 FSs of the antecedents. To obtain \(m_{1}\) and \(m_{2}\), the principle of justifiable granularity is used as a means to heuristically measure uncertainty via the difference of the intervals \(a\) and \(b\). This is carried out by extending each information granule to its highest coverage by using the user criterion value of \(\alpha_{0}\) for each side of the information granule's interval, as described by Equations (12) and (13). When both intervals \(a\) and \(b\) are obtained, their difference will define the amount of uncertainty which will be used to calculate parameters \(\{m_{1},m_{2}\}\), as shown in Equation (15), where \(\tau\) is a measure of uncertainty for the \(q\)-th rule and \(k\)-th input. \[\tau_{q,k}=\left|a_{q,k}-b_{q,k}\right| \tag{15}\]  
The obtained value of \(\tau_{q,k}\) is used by Equations (16) and (17) to offset the centers \(x_{q,k}^{c}\) of the Gaussian primary membership function by adding uncertainty in the mean, thus obtaining \(\{m_{1},m_{2}\}\). \[m_{1}^{q,k}=x_{q,k}^{c}-\tau_{q,k} \tag{16}\]  
\[m_{2}^{q,k}=x_{q,k}^{c}+\tau_{q,k} \tag{17}\]  
For practical reasons, the final missing value of \(\{\rho\}\) is set to zero \(\rho=0\) for all membership functions, as it was found that it has no effect on classification performance if other values are set; some experimentation to demonstrate this is shown in Section 4. This ends the parameter acquisition phase for all antecedents in the GT2 TSK FLS. All IT2 linear TSK consequents are calculated in a two-step process. First, a Least Square Estimator (LSE) algorithm [27; 28] is used twice; as the Gaussian primary membership function with uncertain mean is parameterized by a left and right T1 Gaussian membership function on the support FOU, the LSE is applied as if two T1 TSK FLSs existed, using the following sets of parameters: for the left \(\left\{\sigma_{q,k},m_{1}^{q,k}\right\}\) and right side \(\left\{\sigma_{q,k},m_{2}^{q,k}\right\}\). When all TSK coefficients \(\varphi\) are obtained, the average of both sets of parameters is used, as shown in Equation (18), where \(\varphi_{l}\) and \(\varphi_{r}\) are the coefficient sets for the left and right side respectively. This set \(C\) represents all \(c_{q,k}\) coefficients of all IT2 TSK linear equations.