For example May 2005 could be summarized into Second Quarter 2005 which in turn would be summarized in the Year 2005. Similarly the cities could be summarized into regions, countries and then global regions; products could be summarized into larger categories; and cost headings could be grouped into types of expenditure. Conversely the analyst could start at a highly summarized level, such as the total difference between the actual results and the budget, and drill down into the cube to discover which locations, products and periods had produced this difference.
Slice: A slice is a subset of a multi-dimensional array corresponding to a single value for one or more members of the dimensions not in the subset.
Dice: The dice operation is a slice on more than two dimensions of a data cube (or more than two consecutive slices).
Drill Down/Up: Drilling down or up is a specific analytical technique whereby the user navigates among levels of data ranging from the most summarized (up) to the most detailed (down).
Roll-up: A roll-up involves computing all of the data relationships for one or more dimensions. To do this, a computational relationship or formula might be defined.
Pivot: To change the dimensional orientation of a report or page display.
Linking cubes is a method of overcoming sparsity. Sparsity arises when not every cell in the cube is filled with data and so valuable processing time is taken by effectively adding up zeros. For example revenues may be available for each customer and product but cost data may not be available with this amount of analysis. Instead of creating a sparse cube, it is sometimes better to create another separate, but linked, cube in which a sub-set of the data can be analyzed into great detail. The linking ensures that the data in the cubes remain consistent.
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,
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 clearly conceptually homologous), perhaps
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. One may wish for a spreadsheet-style display, where—to appropriate the Microsoft Excel paradigm—values of X populate row $1; values of Y populate column $A; and values of W : (X, Y ) → W populate the individual cells "southeast of" $B2, so to speak, $B2 itself included. While one can certainly use the DML (Data Manipulation Language) of traditional SQL to display (X, Y, W ) triples, this output format is not nearly as convenient as the cross-tabbed alternative: certainly, the former requires one to hunt linearly for a given (X, Y ) pair in order to determine the corresponding W value, while the latter enables one to more conveniently scan for the intersection of the proper X column with the proper Y row.
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