Imprecise Data and Knowledge Based OLAP

1. Imprecise Data and Knowledge Based OLAP Harrow School of Computer Science Data & Knowledge Management Group University of WestminsterWatford Road, Northwick Park, HA1 3TPLondon, UK Ermir Rogova, Panagiotis Chountas, KrassimirAtanassov CLBME Bulgarian Academy of Sciences Sofia, Bulgaria

2. Overview • Imprecision and OLAP • IF-Cube • IF-OLAP Operators • Knowledge Based OLAP • Summary

3. Imprecision and OLAP • Current models are mainly concerned with the querying of imprecision at the fact table. • Use of inflexible hierarchies makes it difficult to reconcile data coming from different sources. • Knowledge has to be incorporated as part of the OLAP operators in order to enhance the results.

4. The need for the IF Cube • To accommodate imprecise data using IFS measures • To accommodate precise information expressed in the form of concepts i.e. Whole milk, whole pasteurised milk, etc. • In the latter case, the information stored in the cube is precise, but the concept is not, i.e. Whole pasteurised milk satisfies requests for whole milk as well (to some extent)

5. The IF Cube A cube is an abstract structure that serves as the foundation for the multidimensional data cube model. A cube C is defined as a five-tuple (D, l, F, O, H) where: • D is a set of dimensions • l is a set of levels l1,…, ln, • F is a set of facts instances • O is a partial order between the elements of l. • H is an object type history which allows to trace the evolution of the cubic structure

6. The need for IF OLAP operators • Standard OLAP operators cannot cope with imprecise facts • Standard OLAP operators cannot cope with concept-based information i.e. How do you detect and SUM up reconcilable concepts? i.e. whole semi-skimmed and skimmed milk?

7. IF OLAP operators Selection (Σ): • The selectionoperator selects a set of fact-instances from a cubic structure that satisfy a predicate (  ). • Input: Ci = (D, l, F, O, H) and the predicate θ • Output: Co=(D, l, Fo , O, H)where • FoF and Fo={f | (f F)(f satisfies θ) • Σ(amount>1000  (μ>0.4  ν<0.3)  year=2004)(Sales)=CResult

8. IF OLAP operators Cubic Product (): • This is a binary operator Ci1  Ci2 . It is used to relate two cubes Ci1 and Ci2 Input: Ci1 = (D1, 11, F1, O1, H1) and Ci2 = (D2,l2, F2, O2, H2) Output: Co= (Do, lo, Fo, Oo, Ho) where Do= D1D2 , lo= l1l2, Oo= O1O2 Ho= H1H2, Fo= F1 X F2, = {<<x, y>, min(μf1(x), μf2(y)), max(νf1(x), νf2(y),)>|<x, y> XY}

9. IF OLAP operators Union (): • The union operator is a binary operator that finds the union of two cubes. Ci1and Ci2 have to be union compatible. • Input: Ci1 = (D1, l1, F1, O1, H1) and Ci2 = (D2,l2, F2, O2, H2) • Output:Co= (Do, lo, Fo, Oo, Ho) where Do=D1=D2, lo=l1=l2, Oo=O1=O2, Ho=H1=H2, • Fo= F1 F2 = { < x, max(F1 (x), F2(x)), min(F1(x),F2(x)) > | x  X }

10. History • Why do we need to keep track of the history? • The structured history of the datacube allows us to keep all the information when applying Roll up and get it all back when Roll Down is performed. To be able to apply the operation of Roll Up we need to make use of the IFSSUMaggregation operator • H is an object type history that corresponds to a cubic structure ( l, D, A, H’ )which allows us to trace back the evolution of a cubic structure after performing a set of operators i.e. aggregation.

11. Roll up (  ): The result of applying Roll up over dimension di at level dlr using the aggregation operatorover a datacube Ci is another datacube (Co ) Input: Ci = (Di ,li ,Fi , O , Hi ) Output: Co = (Do ,lo ,Fo , O , Ho ) An object of type history is the initial state of the cube is a recursive structure H = ( l, D, A, H’ )is the state of the cube after performing an operation on the cube

12. Group Operators • Will the result over which the aggregate is performed be either crisp or Intuitionistic Fuzzy? • What is the meaning of the result after the IF aggregation is performed? • Using the standard definitions for the group operators (SUM, AVG, MIN and MAX), we provide their IFS extensions and meaning.

13. Group Operators IFSSUM : Example: IFSUM((Amount)(ProdID)) {<.8,.1>/10}+{(<.4,.2>/11),(<.3,.2>/12)}+{(<.5,.3>/13),(<.5,.1>/12)} ={(<.3, .3>/34), (<.4, .2>/33), (<.3, .3>/35)} IFAVG : The IFAVG aggregate makes use of the IFSUMthat was discussed previously and the standard COUNT.

14. IFSMAX : Group Operators Example: IFMAX((Amount)(ProdID)) {<.8,.1>/10},{(<.4,.2>/11),(<.3,.2>/12)},{(<.5,.3>/13),(<.5, 0.1>/120)} ={(<.3,.3>/13),(<.3,.2>/12)}

15. Knowledge Based OLAPKNOLAP • Concepts are used to describe how the data is organized in the data sources. • These definitions are used to rewrite queries conditions and to combine OLAP features in this process. • This supports the analysis according to the context of users’ explorations in order to guide the decision making, feature inexistent in current analytical tools.

16. The KNOLAP environment Milk <0.8, 0.1> Skim milk <0.8, 0.1> Whole milk <1.0, 0.0> Half skim milk <0.8, 0.1> <0.8, 0.1> Sweetened milk <0.8, 0.1> Pasteurized milk Condensed milk <0.4, 0.3> <1.0, 0.0> Condensed whole milk <0.4, 0.3> Sweetened condensed milk <1.0, 0.0> Whole pasteurized milk

17. KNOLAP (cont.) • If the hierarchical IFS structure expresses preferences in a query, the choice of the maximum allows us not to exclude any possible answer. In real cases, the lack of answers to a query generally makes this choice preferable, because it consists of widening the query answer rather than restricting it. • If the hierarchical IFS set represents an ill-known concept, the choice of the maximum allows us to preserve all the possible values, but it also makes the answer less specific.

18. Conclusions-Future Work • Flexible hierarchies and data • Extend querying language • Extend the OLAP modelling construct (definition of facts – cube) • Intuitionistic Fuzzy OLAP • Extending the OLAP operators • KNOLAP • Imprecise hierarchical Intuitionistic fuzzy concepts • Involvement of the domain knowledge in answering OLAP queries • Automatic navigation/summarisation paths • Implementation

19. Thank you for your attention Harrow School of Computer Science Data & Knowledge Management Group University of WestminsterWatford Road, Northwick Park, HA1 3TPLondon, UK CLBME Bulgarian Academy of Sciences Sofia, Bulgaria