Data Warehouse Concepts - Data Warehouse Systems: Design and Implementation - page 55

corresponding category. All categories are grouped under a member called
all , which is the only member of the distinguished level All .Thismemberis
used for obtaining the aggregation of measures for the whole hierarchy, that
is, for obtaining the total sales for all products.

All all

Beverages ... Seafood

Category

Chai Chang .. . Ikura Konbu . ..

Product

Fig. 3.3 Members of a hierarchy Product → Category

In real-world applications, there exist many kinds of hierarchies. For
example, the hierarchy depicted in Fig. 3.3 is balanced , since there is
the same number of levels from each individual product to the root of
the hierarchy. In Chaps. 4 and 5 , we shall study these and other kinds of
hierarchies in detail, covering both their conceptual representation and their
implementation in current data warehouse and OLAP systems.

3.1.2 Measures

Each measure in a cube is associated with an aggregation function that
combines several measure values into a single one. Aggregation of measures
takes place when one changes the level of detail at which data in a cube are
visualized. This is performed by traversing the hierarchies of the dimensions.
For example, if we use the Customer hierarchy in Fig. 3.2 for changing the
granularity of the data cube in Fig. 3.1 from City to Country , then the sales
figures for all customers in the same country will be aggregated using, for
example, the SUM operation. Similarly, total sales figures will result in a cube
containing one cell with the total sum of the quantities of all products, that
is, this corresponds to visualizing the cube at the All level of all dimension
hierarchies.
Summarizability refers to the correct aggregation of cube measures
along dimension hierarchies, in order to obtain consistent aggregation results.
To ensure summarizability, a set of conditions may hold. Below, we list some
of these conditions

￿ Disjointness of instances: The grouping of instances in a level with
respect to their parent in the next level must result in disjoint subsets.
For example, in the hierarchy of Fig. 3.3 , a product cannot belong to two