functional representation

OLAP cube

An OLAP cube is a data structure that allows fast analysis of data. The arrangement of data into cubes overcomes a limitation of relational databases. Relational databases are not well suited for near instantaneous analysis and display of large amounts of data. Instead, they are better suited for creating records from a series of transactions known as OLTP or On-Line Transaction Processing. Although many report-writing tools exist for relational databases, these are slow when the whole database must be summarized.


OLAP cubes can be thought of as extensions to the two-dimensional array of a spreadsheet. For example a company might wish to analyze some financial data by product, by time-period, by city, by type of revenue and cost, and by comparing actual data with a budget. These additional methods of analyzing the data are known as dimensions. Because there can be more than three dimensions in an OLAP system the term hypercube is sometimes used.


The OLAP cube consists of numeric facts called measures which are categorized by dimensions. The cube metadata is typically created from a star schema or snowflake schema of tables in a relational database. Measures are derived from the records in the fact table and dimensions are derived from the dimension tables.


A financial analyst might want to view or "pivot" the data in various ways, such as displaying all the cities down the page and all the products across a page. This could be for a specified period, version and type of expenditure. Having seen the data in this particular way the analyst might then immediately wish to view it in another way. The cube could effectively be re-oriented so that the data displayed now has periods across the page and type of cost down the page. Because this re-orientation involves re-summarizing very large amounts of data, this new view of the data has to be generated efficiently to avoid wasting the analyst's time, i.e within seconds, rather than the hours a relational database and conventional report-writer might have taken.


Each of the elements of a dimension could be summarized using a hierarchy. The hierarchy is a series of parent-child relationships, typically where a parent member represents the consolidation of the members which are its children. Parent members can be further aggregated as the children of another parent.

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.

OLAP operations

The analyst can understand the meaning contained in the databases using multi-dimensional analysis. By aligning the data content with the analyst's mental model, the chances of confusion and erroneous interpretations are reduced. The analyst can navigate through the database and screen for a particular subset of the data, changing the data's orientations and defining analytical calculations. The user-initiated process of navigating by calling for page displays interactively, through the specification of slices via rotations and drill down/up is sometimes called "slice and dice". Common operations include slice and dice, drill down, roll up, and pivot.

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 and sparsity

The commercial OLAP products have different methods of creating the cubes and hypercubes and of linking cubes and hypercubes (see Types of OLAP in the article on OLAP.)

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.

Variance in products

The data in cubes may be updated at times, perhaps by different people. Techniques are therefore often needed to lock parts of the cube while one of the users is writing to it and to recalculate the cube's totals. Other facilities may allow an alert that shows previously calculated totals are no longer valid after the new data has been added, but some products only calculate the totals when they are needed.

Technical 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,

W : (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 clearly conceptually homologous), perhaps

W : (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. 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.

See also


External references

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