# Intuitionistic and Interval-Valued Fuzzy Set Representations for Data Mining
## 4 Generalization Involving Interval-Valued and Intuitionistic Fuzzy Information
### Generalization with Ordinary Hierarchies
#### 4.1.4 IFS Data
First, if  
\[\text{m(a}^{\prime}\text{,c}_{j}^{\prime})\in\text{I}_{1}\left(\text{a}^{ \prime}\right)\text{, then }\text{I}^{\text{e}}_{1}\left(\text{c}_{j}^{\prime}\right)=\text{I}_{1} \left(\text{a}^{\prime}\right) \tag{32}\]  
Otherwise, we use an application-derived threshold, 0 < h < 1. Therefore, if m(a\({}^{\prime}\text{,c}_{j}^{\prime}\)) is within this threshold of the upper or lower bounds of the interval, we modify the interval by the membership. Therefore, if  
\[\text{ml(a}^{\prime})-\text{m(a}^{\prime}\text{,c}_{j}^{\prime})<\text{h, then }\text{ml*(a}^{\prime})=\text{m(a}^{\prime}\text{,c}_{j}^{\prime})\]  
or if  
\[\text{m(a}^{\prime}\text{,c}_{j}^{\prime})-\text{mu(a}^{\prime})<\text{h, then }\text{mu*(a}^{\prime})=\text{m(a}^{\prime}\text{,c}_{j}^{\prime}) \tag{33}\]  
where we allow the respective interval bound to be modified. Finally, if a\({}^{\prime}\) generalizes to multiple values in the hierarchy, we must use the above approach for multiple fuzzy membership values in the generalized tuples. However, since this is a generalization to different concepts, there are multiple tuples that cannot be merged. Recall that our objective in attribute generalization is to reduce the number of tuples in a given relation; therefore, in general, only one of these tuples should be maintained. We need to establish criteria to select which tuple to keep. We can proceed by comparing the intervals using our previous interval measure IW. Let the multiple concepts be indexed \(c_{1}\)... \(c_{n}\). Then, for \(a^{\prime}\) with \(\mathbf{I}_{k=1,\ n}\) (\(a^{\prime}\)) in the m tuples, we have the intervals \(\mathbf{I}^{*}_{1}\) (\(c_{1}\)),... \(\mathbf{I}^{*}_{m}\) (\(c_{n}\)\({}^{\prime}\)), as above. Then, our criteria is  
\[\underset{j=1}{\overset{n}{\text{min}}}(\text{IW}\ (I\ *\ (c_{j})) \tag{34}\]  
Let us illustrate this with an example for a data value \(a^{\prime}\), \(\mathbf{I}\) (\(a^{\prime}\)) = [lb = 0.2, ub = 0.6], for two tuples where \(a^{\prime}\) generalizes to two concepts, \(c_{1}\) and \(c_{2}\). Let the memberships be  
\[m\ (a^{\prime},c_{1})=0.3;\ \ \ m(a^{\prime},c_{2})=0.7 \tag{35}\]  
First, \(m\ (a^{\prime},c_{1})\in\mathbf{I}\) (\(a^{\prime}\)), so \(\mathbf{I}^{*}_{1}\) (\(c_{1}\)\({}^{\prime}\)) = \(\mathbf{I}\) (\(a^{\prime}\)). If we let the threshold \(h=0.15\), then we can extend the upper bound for \(c_{2}\)\({}^{\prime}\), so \(\mathbf{I}^{*}_{1}\) (\(c_{2}\)\({}^{\prime}\)) = [lb = 0.2, ub = 0.7]. \[\text{IW}(\mathbf{I}^{*}_{1}\ (c_{1}\)\({}^{\prime}))=0.4;\ \ \ \text{IW}(\mathbf{I}^{*}_{1}\ (c_{2}\)\({}^{\prime}))=0.5 \tag{36}\]  
Now, to compare, we use the interval width. Then, with this evaluation, we retain the tuple \(t\) (\(a^{\prime}\), \(\mathbf{I}^{*}_{1}\) (\(c_{1}\)\({}^{\prime}\))) as the generalization result. If the membership is not within the threshold, then the semantics of the relationship between \(a^{\prime}\) and \(c_{i}\)\({}^{\prime}\) is such that the tuple with the data value \(a^{\prime}\) cannot be generalized in a consistent manner.