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Harrow School of Computer Science Harrow School of Computer Science
Data & Knowledge Management GroupData & Knowledge Management Group
University of WestminsterUniversity of WestminsterWatford Road, Northwick Park, HA1 3TPWatford Road, Northwick Park, HA1 3TP
London, UKLondon, UK
Ermir Rogova, Panagiotis Chountas, Krassimir AtanassovErmir Rogova, Panagiotis Chountas, Krassimir Atanassov
CLBME CLBME
Bulgarian Academy of SciencesBulgarian Academy of Sciences
Sofia, BulgariaSofia, Bulgaria
Imprecision and OLAP
IF-Cube
IF-OLAP Operators
Knowledge Based OLAP
Summary
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.
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)
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 dimensionsl 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
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?
Selection (Σ):
◦ The selection operator 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
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> XY}
CSales’ CDiscounts = CResult
ProdID StoreID Amount <μ, ν> ProdID StoreID Discount <μ, ν> P1 S1 10 .7, .2 P2 S1 2 .5, .5 P2 S2 15 .5, .5 P3 S3 5 .3, .3
= S.ProdID S.StoreID S.Amount D.ProdID D.StoreID D.Discount <μ, ν>
P1 S1 10 P2 S1 2 .5, .5 P1 S1 10 P3 S3 5 .3, .3 P2 S2 15 P2 S1 2 .5, .5 P2 S2 15 P3 S3 5 .3, .5
Union (): The union operator is a binary operator that finds
the union of two cubes. Ci1 and 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 }
CSales_North CSales_South = CResult
ProdID StoreID Amount <μ, ν> ProdID StoreID Amount <μ, ν>
P1 S1 10 .7, .2 P1 S1 10 .5, .5 P2 S2 15 .5, .5 P3 S3 5 .3, .3
=
S.ProdID S.StoreID S.Amount <μ, ν>
P1 S1 10 .7, .2 P2 S2 15 .5, .5 P3 S3 5 .3, .3
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 IFSSUM aggregation 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.
The result of applying Roll up over dimension di at level dlr using the aggregation operator over 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
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.
IFSUM((attn-1)(F)) =
{<u>/ y | (( u= mi 1min (μi(uki), νi(uki)) (y =
km
kki kiu1
) ( k1, …km : 1 k1, …km n))}
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 IFSUM that was discussed previously and the standard COUNT.
IFSSUM :
IFMAX((attn-1)(F)) = {<u>y|((u= m
i 1min (μi(uki),νi(uki)) (y= m
i 1max (μi(uki),νi(uki)))(k1,…km :1k1,…km n))}
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)}
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.
Milk<0.8, 0.1>
Whole milk<1.0, 0.0>
Condensed whole milk
Skim milk
Pasteurized milk
Half skim milk
Sweetened milk
Condensed milk<0.4, 0.3>
Whole pasteurized milkSweetened condensed milk
<0.8, 0.1>
<0.8, 0.1>
<0.8, 0.1>
<0.4, 0.3>
<1.0, 0.0>
<0.8, 0.1>
<1.0, 0.0>
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.
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
Harrow School of Computer Science Harrow School of Computer Science
Data & Knowledge Management GroupData & Knowledge Management Group
University of WestminsterUniversity of WestminsterWatford Road, Northwick Park, HA1 3TPWatford Road, Northwick Park, HA1 3TP
London, UKLondon, UK
CLBME CLBME
Bulgarian Academy of SciencesBulgarian Academy of Sciences
Sofia, BulgariaSofia, Bulgaria