OLAP Cube - Mathematical Definition

Mathematical Definition

In database theory, an OLAP cube is an abstract representation of a projection of an RDBMS relation. Given a relation of order N, consider a projection that subtends X, Y, and Z as the key and W as the residual attribute. Characterizing this as a function,

f : (X,Y,Z) → W,

the attributes X, Y, and Z correspond to the axes of the cube, while the W value into which each ( X, Y, Z ) triple maps corresponds to the data element that populates each cell of the cube.

Insofar as two-dimensional output devices cannot readily characterize four dimensions, it is more practical to project "slices" of the data cube (we say project in the classic vector analytic sense of dimensional reduction, not in the SQL sense, although the two are conceptually similar),

g : (X,Y) → W

which may suppress a primary key, but still have some semantic significance, perhaps a slice of the triadic functional representation for a given Z value of interest.

The motivation behind OLAP displays harks back to the cross-tabbed report paradigm of 1980s DBMS. The resulting spreadsheet-style display, where values of X populate row $1; values of Y populate column $A; and values of g : ( X, Y ) → W populate the individual cells "southeast of" $B2, so to speak, $B2 itself included.