0 Hierarchies in Multidimensional Databases

Hierarchies play a fundamental role in knowledge representation and reasoning. They have been considered as the structures created by abstraction processes. According to Smith and Smith (1977), an abstraction process is an instinctively known human activity, and abstraction processes and their properties are generally used for multilevel object representation in information systems. An abstraction can be understood as a selection of a set of attributes, objects, or actions from a much larger set of attributes, objects, or actions according to certain criteria. Repeating this selection several times, that is, continuing to choose from each subset of objects, another subset of objects with even more abstract properties, we create other levels of (semantic) details of objects. The complete structure created by the abstraction process is a hierarchy and the type of hierarchy depends on the operation used for the abstraction process and the relations. As for the relations, the best known in the literature are classification, generalization, association (or grouping), and aggregation. Their main characteristics are briefly listed in the following. Classification is a simple form of data abstraction in which an object type is defined as a set of instances. It introduces an instance-of relationship between an object type in a schema and its instances in the database (Brodie, Mylopoulos, & Schmidt, 1984). Generalization is a form of abstraction in which similar objects are related to a higher level generic object. It forms a new concept by leaving out the properties of an existing concept. With such an abstraction, the similar constituent objects are specializations of the generic objects. At the level of the generic object, the similarities of the specializations are emphasized, while the differences are suppressed (Brodie, et al., 1984). This introduces an is-a relationship between objects. This relation covers a wide range of categories that are used in other frameworks, such as inheritance, implication, and inclusion. It is the most frequent relation resulting from subdividing concepts, called taxonomies in lexical semantics. The inverse of the generalization relation, called specialization, forms a new concept by adding properties to an existing concept (Borgida, Mylopoulos, & Wong, 1984). A particular type of generalization hierarchy, named filter hierarchy, is defined by the so-called filtering operation. This operation applies a filter function to a set of objects on one level and generates a subset of these objects on a higher level. The main difference from the generalization hierarchy is that the objects that do not pass the filter will be suppressed at the higher level (Timpf, 1999). Association or grouping is a form of abstraction in which a relationship between member objects is considered as a higher level set of objects. With this relationship, the details of member objects are suppressed and properties of the set object are emphasized. This introduces the member-of relationship between a member object and a set of objects (Brodie, 1981). Aggregation is a form of abstraction in which a relationship between objects is considered as a higher level aggregate object (Brodie et al., 1984). Each instance of an aggregate object can be decomposed into instances of the component objects. This introduces a part-of relationship between objects. The type of hierarchy constructed by this abstraction is called an aggregation hierarchy. Like data warehousing and OLAP (online analytical processing), the above-mentioned aggregation hierarchies are widely used to support data aggregation (Lenz & Shoshani, 1997). In a simple form, such a hierarchy shows the relationships between domains of values. Each operation on a hierarchy can be viewed as a mapping from one domain to a smaller domain. In the OLAP environment, hierarchies are used to conceptualize the process of generalizing data as a transformation of values from one domain to values of another smaller or bigger domain by means of drill-down or roll-up operators. In the next sections, the roles of aggregation hierarchies in analysis dimensions of a data cube will be analyzed.