# Transformation-Based Fuzzy Rule Interpolation Using Interval Type-2 Fuzzy Sets
## 2 Background
### Type-2 Fuzzy Sets  
A type-2 fuzzy set is characterised by a fuzzy membership function whose membership grade for each element is a fuzzy number in \([0,1]\). The formal definition is provided below.  
**Definition 1**.: _[_40_]_ _A type-2 fuzzy set, denoted \(\tilde{A}\), is characterised by a type-2 membership function \(\mu_{\tilde{A}}(x,u)\), where \(x\in X\) and \(u\in J_{x}\subseteq[0,1]\), i.e.,_  
\[\tilde{A}=\{((x,u),\mu_{\tilde{A}}(x,u))|\forall x\in X,\forall u\in J_{x} \subseteq[0,1]\} \tag{14}\]  
_in which \(0\leq\mu_{\tilde{A}}(x,u)\leq 1\). \(\tilde{A}\) can also be expressed as:_  
\[\tilde{A}=\int_{x\in X}\int_{u\in J_{x}}\mu_{\tilde{A}}(x,u)/(x,u)\;\;J_{x} \subseteq[0,1] \tag{15}\]  
_where \(\int\int\) denotes union over all admissible \(x\) and \(u\)._  
**Definition 2**.: _[_32_]_ _When all \(\mu_{\tilde{A}}(x,u)=1\) then \(\tilde{A}\) is an interval type-2 fuzzy set, which is expressed as:_  
\[\tilde{A}=\int_{x\in X}\int_{u\in J_{x}}1/(x,u)\;\;J_{x}\subseteq[0,1] \tag{16}\]Definition 3: [32] Uncertainty in the primary memberships of an interval type-2 fuzzy set, \(\tilde{A}\), consists of a bounded region that is called the footprint of uncertainty (FOU). It is the union of all primary memberships, i.e.,  
\[FOU(\tilde{A})=\bigcup_{x\in X}J_{x} \tag{17}\]  
Definition 4: [32] The upper membership function (UMF), denoted \(\tilde{A}^{U}\), and the lower membership function (LMF), denoted \(\tilde{A}^{L}\), of a type-2 fuzzy set \(\tilde{A}\) are two type-1 membership functions that bound the FOU. The UMF is associated with the upper bound of \(FOU(\tilde{A})\) and is denoted \(\mu_{\tilde{A}^{U}}(x)\), \(\forall x\in X\), and the LMF is associated with the lower bound of \(FOU(\tilde{A})\) and is denoted \(\mu_{\tilde{A}^{L}}(x)\), \(\forall x\in X\), i.e.,  
\[\mu_{\tilde{A}^{U}}(x) =\overline{FOU(\tilde{A})}\quad\forall x\in X \tag{18}\] \[\mu_{\tilde{A}^{L}}(x) =\overline{FOU(\tilde{A})}\quad\forall x\in X\]  
As indicated above, an interval type-2 fuzzy set \(\tilde{A}\) can be represented by the LMF \(\tilde{A}^{L}\) and the UMF \(\tilde{A}^{U}\), i.e., \(\tilde{A}=<\tilde{A}^{L}\), \(\tilde{A}^{U}>\). In particular, when triangular membership functions are used, such an interval type-2 fuzzy set can be illustrated as shown in Figure 3, where \(\tilde{A}^{L}=(\tilde{a}^{L}_{0},\tilde{a}^{L}_{1},\tilde{a}^{L}_{2};H_{\tilde {A}^{L}})\), \(\tilde{A}^{U}=(\tilde{a}^{U}_{0},\tilde{a}^{U}_{1},\tilde{a}^{U}_{2};H_{\tilde {A}^{U}})\), \((\tilde{a}^{U}_{0},\tilde{a}^{L}_{1},\tilde{a}^{L}_{2})\) and \((\tilde{a}^{U}_{0},\tilde{a}^{U}_{1},\tilde{a}^{U}_{2})\) denote the three key points of the LMF and those of the UMF, respectively, and \(H_{\tilde{A}^{L}}\) and \(H_{\tilde{A}^{U}}\) denote the maximum membership values of \(\tilde{A}^{L}\) and \(\tilde{A}^{U}\), with \(\tilde{a}^{U}_{0}\leq\tilde{a}^{U}_{0}\), \(\tilde{a}^{L}_{2}\leq\tilde{a}^{U}_{2}\), \(0<H_{\tilde{A}^{L}}\leq H_{\tilde{A}^{U}}=1\). Clearly, the closer the shapes of \(\tilde{A}^{L}\) and \(\tilde{A}^{U}\) are, the less uncertain the information contained within \(\tilde{A}\) is. When \(\tilde{A}^{L}\) coincides with \(\tilde{A}^{U}\), the interval type-2 fuzzy set degenerates to a type-1 fuzzy set.