Multidimensional Algebra and Its Graphical Representation

1. Multidimensional Algebra and Its Graphical Representation Lukáš STRYKA1, Tomáš HRUŠKA1, Michal MÁČEL2 {stryka, hruska}@fit.vutbr.cz1, macel@vema.cz2 Department of Information Systems Faculty of Information Technology Brno University of Technology1 Vema a.s., Brno2

2. Outlines • Motivation • Multidimensional algebra • Multidimensional algebra expressions • Dynamic tables • Conclusion

3. Motivation • Decision support problems • Wide using of data warehousing and OLAP analysis • Solution for web based intranets required

4. Multidiensional Databases • There is no universally accepted multidimenaional algebra. • There are few papers with multidimensional modeling and conceptual level description. • We use a terminology based on UML core metaclasses extension described in YAM (Yet Another Multidimensional Model): An Extenssion of UML writenby A.Abello et.al.. • Multidimensional databases are based on Data Cube

5. Multidiensional Databases Definitions: A Dimension (subclass of UML Classifier) contains Levels (subclass of UML Class) representing different granularities (or levels of class of UML Attribute). Fact (Subclass of UML classifier) contains Cells (subclass of UML Class), which contain Measures (subclass of UML Attribute). One Cell represents those individual cells of the same granularity that show data regarding the same Fact A Cube is an injective function from an n-dimensional finite space (defined by the Cartesian product of n functionally independent Levels {L1,…, Ln}, to the set on instances of a Cell (Cc).

6. Data cube – basic operations • ChangeBase operation • reallocates exactly the same instances of a Cell in a new n-dimensional space with exactly the same number of points • Drill-across operation • relates instances of different Facts (joining two or more Facts with same ganularity) • Dice operation • allows to choose the subset of points of interest out of the whole n-dimensional space • Projection operation • just selects a subset of Measures from those available in the Cube. It is equivalent to projection from relational algebra

7. Data cube – basic operations • Roll-up operation • groups cells in the Cube based on an aggregation hierarchy • Union operation • Can be applied on cubes defined on the same domain (n-dimensional space) • i.e. recover the cells removed by means of Dice • Drill-down operation • can applied only if we used the Roll-up operation before and we didn’t loose the correspondence between cells. • Pivoting operation • represents a special case of changeBase operation. It is realized by multidimensional space reorganizing (B  A instead of A  B) based on Levels order changing

8. Present approaches for expressions of multidimensional algebra • However, data cubes and OLAP analysis results are not limited to three dimensions; the visualization of these multidimensional cubes comprises a problem from spatial or geometric point of view. • With respect to web based application we distinguish two main types of these methods: • tabular based methods • Graphical methods – graphs, schemas, etc.

9. time seller (difficult to dispay other dimensions) product

10. Tabular representation of n-dimensional space • Cross-tables (Cross-tabs) • 2D tables representing intersection of two dimensions • supplemented by subtotals and totals • results of symmetric aggregation. • Pros: method is well-arranged presentation of data in 2D table that provides easy orientation to user. • Weakness: ability of only 2 dimensions displaying; provide only limited facilities for multidimensional algebra operations

11. Tabular representation of n-dimensional space • Pivot table • multidimensional view on data with escalated legibility • includes 4 basic parts: Page, Column, Row, Data. • enables specific rotation so rows and columns exchanging, combination and hierarchical structure of rows and columns. • Pros: ability of displaying more than two dimensions; easy displaying of results follows from storing the data in a table • Weakness: orientation deteriorating with higher number of dimensions.

12. Pivot table – source data

13. Pivot table – initial table column page row data

14. Pivot table – pivoting

15. Pivot table – roll-up drill-down

16. Graphical representation of n-dimensional space • more favourite for their information capability • basic classification is to: • 2D or 3D graphs • additional classification is by character of data: • continuous or discrete • special graphs for specific tasks (i.e. spatial graphs connected to spatial databases, economic graphs, etc.).

17. Graphical representation of n-dimensional space • VRML • <?php • header ("Content-type: model/vrml"); • echo "#VRML V2.0 utf8\n"; • ?> • Shape { • geometry Box { • <?php • $size = 2; • echo " size $size $size $size \n"; • ?> • } • }

18. Dynamic tables • developed in the VEMA Brno a.s. • alternative way to display n-dimensional space as a 2D tabular expression • consist of 4 basic parts: Dimensions, Data, Filters and Statistics

19. Data area Roll up Pivotting control Dynamic tables Drill down slice Dimensions area Codebook filter Expression filter Column filters statistics

20. Dynamic tables – pivoting (by changing columns order)

21. Dynamic tables – roll-up and drill-down(by hiding columns for higher aggregation, and vice versa)

22. Dynamic tables slice & dice(by showing and hiding of columns or rows)

23. Conclusions • We discussed a web-based OLAP and its multidimensional data cube algebra. • We focused on visualization methods and possibilities of their application in WOLAP. We classify the methods to tabular and graphical ones. • At the end, we introduced a new tabular method for multidimensional data displaying called Dynamic tables. This method provides all basic operations of multidimensional algebra. The Dynamic tables forming is very simple and its usage is very suitable for web based analyzing tools.

24. Thank you for your patience

25. References • Abello A., Samos J., and Salto F.: YAM (Yet Another Multidimensional Model): An Extenssion of UML. In Int. Database Engineering and Applications Symposium. IEEE,2002. • Please mail hruska@fit.vutbr.cz or stryka@fit.vutbr.cz for more literature.