# Intuitionistic and Interval-Valued Fuzzy Set Representations for Data Mining
## 4 Generalization Involving Interval-Valued and Intuitionistic Fuzzy Information
### Generalization with Ordinary Hierarchies
#### 4.1.3 Evaluation of Generalization  
In order to make effective usage of generalizations for decision-making criteria, measures and metrics must be used to provide an analysis of the generalizations. We can consider measures such as granularity and the overlap of data associated with different concept hierarchies. The most granular partitioning of data occurs when all values are lumped into one set, and the finest partition is where each data value is in a separate set. A measure that is termed as coarseness or granularity was used to characterize partitioning, where the coarseness of the maximum partition was the greatest, 1, and the minimum was the finest, with 0 granularity.  
Let R\({}_{i}\) (A\({}_{1}\), A\({}_{2}\),..., A\({}_{n}\)) be a relation with attributes that have interval-valued fuzzy membership values for their data values. Depending on specific applications, there may be more than just one concept hierarchy relevant that can be used to generalize these data values. As a consequence, we would like to compare the effectiveness of generalization using m different hierarchies, \(\text{CH}_{1}\),..., \(\text{CH}_{\text{m}}\). For this, we compare the sets of generalized data values, \(\text{S}_{1}\), \(\text{S}_{2}\),..., \(\text{S}_{\text{m}}\), with the granularity measure. The formulation of granularity for crisp sets uses their cardinality, Cd, but must be extended for imprecise data, such as for fuzzy sets [15]. Here, as above, we use the information measure IW of the intervals in the generalized data for Cd.  
\[\text{Cd}(\text{S}_{\text{j}})=|\text{S}_{\text{j}}|=\sum_{\text{c}_{\text{i}} \in\text{S}_{\text{i}}}\text{Iw}(\text{c}_{\text{i}}) \tag{22}\]  
Then, the expression for the granularity **G**, extended from the form of [PERSON] (2008), is  
\[\textbf{G}(\text{S}_{1},\text{S}_{2}\ldots\text{S}_{\text{m}})=\sum_{\text{j} =1}^{\text{m}}\left(\text{Cd}(\text{S}_{\text{i}})^{2}\right)-\Delta)/Z \tag{23}\]  
where  
\[Z=(\Delta^{2}-\Delta)\text{and}\Delta=\sum_{\text{j}=1}^{\text{m}}\text{Cd}( \text{S}_{\text{j}}) \tag{24}\]  
Now, we consider the use of three potential concept hierarchies, \(\text{CH}_{1}\), \(\text{CH}_{2}\) and \(\text{CH}_{3}\), to generalize some of the data in a relation \(\text{R}_{\text{i}}\). This example corresponds to three sets of the data values in the final generalized tuples, \(\text{S}_{1}\), \(\text{S}_{2}\) and \(\text{S}_{3}\).  
\[\text{S}_{1}=\{\text{c}_{1},\text{IW}(\text{c}_{1})=0.7;\text{c}_{2},\text{IW }(\text{c}_{2})=0.6\text{ c}_{3},\text{IW}(\text{c}_{3})=0.6\}\]  
\[\text{S}_{2}=\{\text{c}_{1},\text{IW}(\text{c}_{1})=0.5;\text{c}_{2},\text{IW }(\text{c}_{2})=0.4\}\]  
\[\text{S}_{3}=\{\text{c}_{2},\text{IW}(\text{c}_{2})=0.8;\text{c}_{3},\text{IW }(\text{c}_{3})=0.7\}\]  
Therefore, the granularity of the possible choices, \(\text{S}_{1}\) and \(\text{S}_{2}\), \(\text{S}_{2}\) and \(\text{S}_{3}\), and \(\text{S}_{1}\) and \(\text{S}_{3}\) must be considered. First, we have  
\[\textbf{G}\ (\text{S}_{1},\text{S}_{2})=(\text{Cd}\ (\text{S}_{1})^{2}+\text{ Cd}\ (\text{S}_{2})^{2})-\Delta)/Z=((3.61+0.81)-2.8)/0.04=0.32 \tag{25}\]  
Similarly,  
\[\textbf{G}\ (\text{S}_{2},\text{S}_{3})=0.66/3.36=0.196;\textbf{G}\ (\text{S}_{1},\text{S}_{3})=2.46/8.16=0.30\]  
For the purposes of data mining, this provides a criterion to provide multiple generalizations that may be useful to provide alternative data representations for evaluation and interpretation. Therefore, the best choice for this example is using \(\text{CH}_{1}\), with \(\text{CH}_{2}\) corresponding to \(\text{S}_{1}\) and \(\text{S}_{2}\), although \(\text{S}_{1}\) and \(\text{S}_{3}\) derived from \(\text{CH}_{1}\) and \(\text{CH}_{3}\) are acceptable.