## 4 Property-Based Ontology Representation
### Property Relevance  
The investigation of attribute relevance is a standard method in data mining, especially in the classification domain, where the correlation between the attribute and the target variable determines the importance of the attribute [44]. The relevance ordering of the attributes can be used, among others, in data reduction or data representation. If we consider the FCA approach, we can see that in FCA every attribute has the same importance, and all possible formal concepts are contained in the generated concept sets. In practical applications, only a small subset of concepts is used and recognized. The selected concepts have a larger practical importance, however. This simplified concept model is represented in the semantic models constructed for the application programs; thus, the generated schema should express the relevance value of the involved concepts. Applying this idea to the constructed ontologies, we can introduce an importance measure for the attributes and properties.  
In FSA, for every attribute \(a\) there exists a concept \(c\), where  
\[A(c)=a^{\prime\prime}\text{ i.e., }a\Rightarrow A(c).\]  
This concept meets the condition  
\[a\in A(c^{\prime})\Rightarrow c^{\prime}\subseteq c.\]  
This concept is the unique supremum of attribute \(a\). In ontology or other schema modeling, this concept \(c\) may be missing. In this case, attribute \(a\) may be less important than the attributes with a unique supremum in the ontology.  
Based on this consideration, for a property \(p\) in the ontology we can define the relevance factor \(r(p)\) as  
\[r(p)=\frac{\sum_{x\in a}\sum_{c\in D_{a}}d(\top,c)}{|\{a:p\in r\}|}\]  
Symbol \(p\in a\) means that the property part of \(a\) is equal to \(p\). This measure shows how far the supremum concepts of \(a\) are from the optimal position. In the best case, the supremum of every attributes related to \(p\) is the child node of the top element. In this model, a lower \(r(p)\) value means a higher importance.  
Here, we take the properties defined in the reduced pizza ontology (Section 4.2). Based on the generated property-oriented ontology representation, we obtain the following relevance values:  
* _hasCountryOfOrigin_, \(r=8\)
* \(a_{6}:\)_hasCountryOfOrigin value America_, \(r=8\)
* _hasBase_, \(r=2\)
* \(a_{2}:\)_hasBase some PizzaBase_, \(r=2\)
* _hasTopping_, \(r=3.3\)
* _a_\(1:\)_hasTopping some FruitTopping_, \(r=2\)
* \(a_{3}:\)_hasTopping some CheeseTopping_, \(r=3\)
* \(a_{4}:\)_hasTopping some \(\top\)_, \(r=3\)
* \(a_{5}:\)_hasTopping some MetaTopping_, \(r=3\)
* \(a_{7}:\)_hasTopping some MozzarellaTopping_, \(r=3\)
* \(a_{8}:\)_hasTopping some PeperoniSausageTopping_, \(r=3\)
* \(a_{9}:\)_hasTopping some TomatoTopping_, \(r=3\)
* \(a_{10}:\)_hasTopping some HotGreenPerperTopping_, \(r=4\)
* \(a_{11}:\)_hasTopping some JalapentoPepperTopping_, \(r=4\)
* \(a_{12}:\)_hasTopping some OnionTopping_, \(r=4\)
* \(a_{13}:\)_hasTopping some PrawsTopping_, \(r=4\)
* \(a_{14}:\)_hasTopping some TobascoPepperSoucTopping_, \(r=4\)
* \(a_{15}:\)_hasTopping some SpicyTopping_, \(r=3\)
Based on calculations, the property _hasBase_ has the largest importance in the presented example ontology. The property _hasTopping_ has a relatively high importance as well, while _hasCountryOfOrigin_ is a property with low importance.