# 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  
The structure of the intuitionistic fuzzy data and concept hierarchies has complications that are not found in the generalization of the interval-valued fuzzy data. Therefore, in this section, we describe some representations for IFS data that can potentially facilitate attribute generalization techniques. The issue is that positive and negative memberships that are found in IFS data may have different semantics for each membership, which is not what we encounter with the interval approach. Next, we consider how to generalize tuples in which the attributes for the data have fuzzy intuitionistic memberships. We must consider how the crisp concept hierarchies for such attributes can be specified for this case. We propose to derive separate tuples in which the positive and negative memberships are separated before generalization. This can allow a simpler process for merging tuples after generalization. Now, we can describe a complex hierarchy following the simple concept hierarchy with respect to partitions, as in Section 3. Therefore, at each level k, we have a defining many-to-many relationship:  
\[\Gamma_{k}:\text{P(A)}\times\text{P(A)}\rightarrow\text{[0, 1]} \tag{29}\]  
Such a relationship entails many-to-many equivalence classes of concepts at each level. However, these sets do not form a formal set partitioning of a domain as for crisp equivalence classes in simple concept hierarchies. This means that, instead of a partitioning of the domain P(A), there is set decomposition. At each level, this is \(\text{Sd}_{k\,11}\), \(\text{Sd}_{k\,12}\),..., and, in general,  
\[\text{Sd}_{k\,1i}\cap\text{Sd}_{k\,1j}\
eq\emptyset \tag{30}\]  
This means that there may not be a unique concept at level r to which a value at level r \(-\)1 generalizes, as seen in Figure 2, which then allows fuzzy concept hierarchies. Lastly, in this section, we describe the extensions for the generalization of interval-valued fuzzy data representations using fuzzy hierarchies. The generalization of crisp and fuzzy data for fuzzy hierarchies has been previously considered [12; 33; 35]. For crisp and fuzzy data, the only factor is the modification of the count kept when similar tuples are merged after generalization. Now, consider the relationship of the membership found in the fuzzy hierarchy concepts and the interval-valued fuzzy data. As shown in Figure 2, a data value may generalize to exactly one concept c\({}_{i}\) in CH\({}_{C}\), implicitly with a membership degree of 1, m(a, c\({}_{j}\)) = 1. In general, a data value can generalize to two or more concepts with associated membership values. Let us consider a specific data value a\({}^{\prime}\) which generalizes to two concepts, denoted c\({}_{j}\)and c\({}_{k}\). There are then memberships m(a\({}^{\prime}\), c\({}_{j}^{\prime}\)) and m(a\({}^{\prime}\), c\({}_{k}^{\prime}\)), where m(a\({}^{\prime}\), c\({}_{j}^{\prime}\)) + m(a\({}^{\prime}\), c\({}_{k}^{\prime}\)) = 1. In particular, we must consider the relationship to the interval-valued fuzzy membership of the data being generalized and how this changes the interval after generalization. First, we analyze where the data value generalizes to the first one of the values in the hierarchy, i.e., the value a\({}^{\prime}\) generalizes to c\({}_{j}\)with membership m(a\({}^{\prime}\), c\({}_{j}^{\prime}\)). \[\text{t}_{1}\left(\text{a}^{\prime},\text{I}_{1}\left(\text{a}^{\prime} \right)\right)\rightarrow\text{t}_{1}\left(\text{c}_{j}^{\prime},\text{I}^{ \text{e}}_{1}\left(\text{c}_{j}^{\prime}\right)\right) \tag{31}\]  
where I\({}^{\text{e}}_{1}\) = \(F\left(\text{m(a}^{\prime}\text{,c}_{j}^{\prime}\right)\), \(\text{I}_{1}\left(\text{a}^{\prime}\right)\right)\). How does this function, \(F\), produce the new interval-valued membership I*? We must consider three possibilities.