# Transformation-Based Fuzzy Rule Interpolation Using Interval Type-2 Fuzzy Sets
## 3 Proposed Interval Type-2 Transformation-Based Interpolation  
Suppose that an interval type-2 fuzzy set \(\tilde{A}\) as defined in Figure 3 has the following six distinct coordinates: \((\tilde{a}^{L}_{0},0)\), \((\tilde{a}^{L}_{1},H^{L}_{\tilde{A}})\), \((\tilde{a}^{L}_{2},0)\), \((\tilde{a}^{U}_{0},0)\), \((\tilde{a}^{U}_{1},H^{U}_{\tilde{A}})\) and \((\tilde{a}^{U}_{2},0)\). The lower and upper Reps \(\text{Rep}(\tilde{A}^{L})\) and \(\text{Rep}(\tilde{A}^{U})\) of \(\tilde{A}\) can then be computed according to Equation (1), such that:  
\[\left\{ \begin{aligned} \text{Rep}(\tilde{A}^{K})_{x}& =\frac{1}{3}(\tilde{a}^{K}_{0}+\tilde{a}^{K}_{1}+\tilde{a}^{K}_{2 })\\ \text{Rep}(\tilde{A}^{K})_{y}&=\frac{1}{3}H^{K}_{ \tilde{A}}\end{aligned} \right. \tag{19}\]  
where \(x\) and \(y\) denote a certain variable and the corresponding membership function, respectively, \(K\in\{L,U\}\). Note that in the existing T-FRI, \(\text{Rep}(\tilde{A})_{y}\) is a constant, so only the \(x\) coordinate value is therefore considered there. However, this is no longer the case in this work due to the introduction of additional  
Figure 3: Lower membership function \(\tilde{A}^{L}\) and upper membership function \(\tilde{A}^{U}\) of a triangular interval type-2 fuzzy set \(\tilde{A}\). uncertainty, and both values of the \(x\) and \(y\) coordinates need to be considered. The calculation for \(\text{Rep}(\tilde{A})_{y}\) follows the same mechanism as the one used to compute \(\text{Rep}(\tilde{A})_{x}\), to maintain consistency. In order to distinguish different shapes of interval type-2 fuzzy sets, the shape diversity factor \(f\) needs to be introduced. In this work, its specification follows the conventional definition of the statistical standard deviation (although this may be defined differently). **Definition 5**. : _The lower and upper shape diversity factors \(f_{\tilde{A}}^{L}\) and \(f_{\tilde{A}}^{U}\) are defined by:_  
\[f_{\tilde{A}}^{K}=\sqrt{\frac{\sum\limits_{i=0}^{2}(\tilde{a}_{i}^{K}-Rep( \tilde{A}^{K})_{x})^{2}}{3}},\ \ K=L,U \tag{20}\]  
A small shape diversity factor implies that the four key points of \(\tilde{A}^{L}\) (\(\tilde{A}^{U}\)) tend to be close to those of the lower (upper) Rep. That is, the smaller the shape diversity factor, the smaller the area of the LMF (UMF). Mimicking the development of T-FRI where a Rep is used to succinctly depict a given type-1 fuzzy set, while extending T-FRI to dealing with interval type-2 fuzzy sets, a single overall Rep of a given interval type-2 fuzzy set is also introduced. For this, the weight factor \(w\) of the LMF (UMF) is first defined as follows. **Definition 6**. : _The lower and upper weight factors \(w_{\tilde{A}}^{L}\) and \(w_{\tilde{A}}^{U}\) are defined as the weights of the shape diversity factors, in terms of the areas of the LMF and UMF, such that:_  
\[w_{\tilde{A}}^{K}=\frac{f_{\tilde{A}}^{K}}{f_{\tilde{A}}^{L}+f_{\tilde{A}}^{U }},\ \ K=L,U \tag{21}\]  
_where \(f_{\tilde{A}}^{L}+f_{\tilde{A}}^{U}\
eq 0\). If however, \(f_{\tilde{A}}^{L}+f_{\tilde{A}}^{U}=0\), i.e., \(f_{\tilde{A}}^{L}=0\) and \(f_{\tilde{A}}^{U}=0\), the interval type-2 fuzzy set degenerates to a singleton value, \(w_{\tilde{A}}^{L}=w_{\tilde{A}}^{U}=1/2\)._  
**Definition 7**. : _The overall representative value \(\text{Rep}(\tilde{A})\) of a given interval type-2 fuzzy set \(\tilde{A}\) is defined by:_  
\[\text{Rep}(\tilde{A})=\sum\limits_{K\in\{L,U\}}(w_{\tilde{A}}^{K}\sum\limits _{e\in\{x,y\}}\text{Rep}(\tilde{A}^{K})_{e}) \tag{22}\]  
_where the lower (upper) shape diversity factor is regarded as the weight of the lower (upper) Rep of the LMF (UMF). This is necessary, as otherwise, the same value for Rep would be derived from different shapes of interval type-2 fuzzy sets._  
Given the above definitions, again, by mimicking the approach of the original T-FRI, the extended algorithm for deriving the interpolated conclusion with multiple multi-antecedent rules can be established and is summarised below. 1.