# Intuitionistic and Interval-Valued Fuzzy Set Representations for Data Mining
## 4 Generalization Involving Interval-Valued and Intuitionistic Fuzzy Information
### Generalization with Ordinary Hierarchies
#### 4.1.1 IVF Data  
For interval-valued fuzzy data, we use the following notation, \(\mathbf{I}(\mathrm{a^{\prime}})\), for a data value's membership \(\mathrm{m}(\mathrm{a^{\prime}})\):  
\[\mathrm{m}(\mathrm{a^{\prime}})=\mathbf{I}(\mathrm{a^{\prime}})=[\ \mathrm{m} \mathrm{l}(\mathrm{a^{\prime}}),\ \mathrm{m}\mathrm{u}(\mathrm{a^{\prime}})]. \tag{17}\]  
Then, consider the two tuples  
\[\mathrm{t_{1}}\ (\mathrm{a^{\prime}},\mathbf{I}_{1}(\mathrm{a^{\prime}}))\ \mathrm{and}\ \mathrm{t_{2}}\ ( \mathrm{b^{\prime}},\mathbf{I}_{2}\ (\mathrm{b^{\prime}})) \tag{18}\]  
Here, the data values \(\mathrm{a^{\prime}}\) and \(\mathrm{b^{\prime}}\) generalize to one concept in \(\mathrm{CH_{i}}\), labeled as \(\mathrm{c^{\prime}}\). This produces two tuples for which the membership values of the data are not affected by the generalization when using a crisp hierarchy:  
\[\mathrm{t_{1}}\ (\mathrm{c^{\prime}},\ \mathbf{I}_{1}(\mathrm{a^{\prime}}))\ \mathrm{and}\ \mathrm{t_{2}}\ ( \mathrm{c^{\prime}},\ \mathbf{I}_{2}(\mathrm{b^{\prime}})) \tag{19}\]  
Now, to assess the similarity of the tuples, \(\mathrm{Sim}\ (\mathrm{t_{1}},\ \mathrm{t_{2}})\), we must consider the relationships possible between the membership intervals in the tuples. There are several cases to consider for the generalization of interval-valued fuzzy data:  
1. Data intervals are contained: \(\mathbf{I_{j}}\subseteq\mathbf{I_{k}}\).  
2. Data interval overlap: \(\mathbf{I_{j}}\cap\mathbf{I_{k}}\
eq\emptyset\).  
3. The intervals are disjoint, so tuples cannot be combined: \(\mathbf{I_{j}}\cap\mathbf{I_{k}}=\emptyset\).  
For the first two cases, we must combine the membership intervals of the generalized tuples that are being merged. The first approach uses an arithmetic averaging of interval bounds, \(\mathrm{F_{avg}}\):  
\[\mathrm{F_{avg}}\ (\mathbf{I}\ (\mathrm{a^{\prime  
\[=[\min(\text{m}\text{l}(\text{a}^{\prime}),\,\text{m}\text{l}(\text{b}^{\prime})),\, \max(\text{mu}(\text{a}^{\prime}),\,\text{mu}(\text{b}^{\prime}))]\]  
Here is an example of three tuples that have been generalized:  
\[\text{t}_{1}\;(\text{c}^{\prime},[0.2,\,0.6]);\,\text{t}_{2}\;(\text{c}^{\prime},[0.3,\,0.5]);\,\text{t}_{3}\;(\text{c}^{\prime},[0.4,\,0.8]) \tag{20}\]  
Using Favg, we can obtain the merged tuple t\({}^{\prime}\) (c\({}^{\prime},[0.3,\,0.635]\)), and for Fmd, there is the tuple t\({}^{\prime}\) (c\({}^{\prime}\), [0.2, 0.8]).  
To compare these results, we can use the interval width IW (a\({}^{\prime}\)) = 1 ml(a\({}^{\prime}\)) - mu(a\({}^{\prime}\)). Note that 0 \(<\) IW (a\({}^{\prime}\)) \(<\) 1, where a greater interval width implies more uncertainty in the interval. Then, the interval width of the merged tuples for Favg is IW (c\({}^{\prime}\)) = 0.335, and for Fmd, IW (c\({}^{\prime}\)) = 0.6. Therefore, Favg has preserved more information in the merged tuples. We also note that the average IW for the three original tuples is 0.333, so the information content has essentially been preserved.