# Transformation-Based Fuzzy Rule Interpolation Using Interval Type-2 Fuzzy Sets
## 3 Proposed Interval Type-2 Transformation-Based Interpolation
Calculate representative values: The lower and upper representative values \(\text{Rep}(\tilde{A}^{K})_{x}\) and \(\text{Rep}(\tilde{A}^{K})_{y}\) of a given interval type-2 fuzzy set \(\tilde{A}\) are calculated first using Equation (19). The shape diversity factors \(f_{\tilde{A}}^{K}\) and weight factors \(w_{\tilde{A}}^{K}\) are computed according to Equations (20) and (21), respectively. The overall Rep \(\text{Rep}(\tilde{A})\) is then obtained by Equation (22), \(K=L,U\). The calculations for all of the antecedent variables of all rules and their counterparts in the observation follow the same procedure. Choose closest \(N\) rules: The distances between the observation and all rules in the rule base are calculated using Equations (2) and (3). The \(N\) (\(N\geq 2\)) rules that have minimal distances are then chosen as the \(N\) closest rules to perform interpolation. 3. Construct intermediate rule The normalised weight \(w^{\prime}_{\bar{A}_{ij}}\) of the \(j\)-th antecedent of the \(i\)-th chosen rule, which is calculated by Equation (4), together with the parameter \(\delta_{\bar{A}_{j}}\), which is calculated by Equation (6), are used in Equation (5) to obtain the value of each antecedent variable \(x_{j}\) within the intermediate rule \(\bar{A}^{\prime}_{j}\), \(i\in\{1,\ldots,N\}\), \(j\in\{1,\ldots,M\}\). From this, two parameters \(w^{\prime}_{\bar{B}_{i}}\) and \(\delta_{\bar{B}}\) are computed using Equation (8) and are then used to construct \(\bar{B}^{\prime}\) from Equation (7), resulting in the intermediate rule \(\bar{A}^{\prime}_{1}\wedge\cdots\bar{A}^{\prime}_{j}\wedge\cdots\bar{A}^{ \prime}_{M}\Rightarrow\bar{B}^{\prime}\). 4. Perform scale, move and height transformations: In conjunction with the given \(\bar{A}^{*}_{j}\) for each antecedent variable \(x_{j}\), the rates \(s^{K}_{j}\), \(m^{K}_{j}\) and \(h_{j}\), \(K\in\{L,U\}\), can then be calculated using Equations (9), (11) and (23). Due to the extra uncertainty encountered in the membership functions, a further transformation on the height of the LMF is needed (because the LMFs of different interval type-2 fuzzy sets may have different heights), while the height of the UMF remains the same owing to its normality. This additional transformation is introduced to transform the heights of \(\bar{A}^{\prime L}_{j}\) to those of \(\bar{A}^{*L}_{j}\), with the height rate \(h\) being calculated by: \[h_{j}=\frac{H^{*L}_{\bar{A}_{j}}}{H^{\prime L}_{\bar{A}_{j}}}\] (23) where \(0<H^{*L}_{\bar{A}_{j}}\leq H^{*U}_{\bar{A}_{j}}=1\) and \(0<H^{\prime L}_{\bar{A}_{j}}\leq H^{\prime U}_{\bar{A}_{j}}=1\), as defined previously. This constraint applies to the interpolated conclusion, as well. That is, if the height of \(\bar{B}^{*L}\) is greater than the height of \(\bar{B}^{*L}\) after the height transformation, then \(H^{*L}_{\bar{B}}=H^{*U}_{\bar{B}}\). 5. Derive interpolated conclusion: The second intermediate term \(\bar{B}^{\prime\prime}\) and the interpolated result \(\bar{B}^{*}\) can then be estimated by the combined \(s^{K}_{\bar{B}}\), \(m^{K}_{\bar{B}}\) and \(h_{\bar{B}}\), \(K\in\{L,U\}\). Here, \(s^{K}_{\bar{B}}\) and \(m^{K}_{\bar{B}}\) are computed following Equations (9)-(13), respectively, and \(h_{\bar{B}}\) is computed according to Equation (23) such that: \[h_{\bar{B}}=\frac{1}{M}\sum_{j=1}^{M}h_{j}\] (24)
6. Implement modified procedure: To obtain intuitive interpolated conclusions for interval type-2 fuzzy sets, the relative location between the LA and UA of an interval type-2 fuzzy set should be considered [39]. For this purpose, \(B^{\prime\prime}\) is modified into \(B^{\prime\prime}_{c}\) to maintain the relative location both before and after the scale transformation.