In this paper we present our approach for extending the OLAP model to include treatment of value uncertainty as part of a multidimensional model inhabited by flexible data and non-rigid hierarchical structures of organisation. A new multidimensional-cubic model named as the IF-Cube is introduced which is able to operate over data with imprecision either in the facts or in the dimensional hierarchies.
In this paper we introduce the semantics of the Intuitionistic Fuzzy cubic
representation in contrast to the basic multidimensional-cubic structures. The basic
cubic operators are extended and enhanced with the aid of [
Since the emergence of the OLAP technology [
Nowadays, information and knowledge-based systems need to manage imprecision
in the data and more flexible structures are needed to represent the analysis domain.
New models have appeared to manage incomplete datacube [
Each element of an Intuitionistic fuzzy [
As opposed to the classical definition of a fuzzy set given by A′ = {< x, µA′(x) > |x
∈ X} where µA(x) ∈ [
A = {< x, µA(x),vA(x) > |x ∈ X}
where: µA : X → [
Obviously, each fuzzy set may be represented by the following Intuitionistic fuzzy set A={<x, µA′ (x), (x), 1− µA′ (x)>|x ∈ X}
For each intuitionistic fuzzy set in X, we will call πA (x) = 1 − µA(x) − vA(x) an intuitionistic fuzzy index (or a hesitation margin) of x ∈ A which expresses a lack of knowledge of whether x belongs to A or not. For each x ∈ A 0<πA (x)<1.
The IF-Cube is an abstract structure that serves as the foundation for the multidimensional data cube model. 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, A dimension di = (l ≤ O, l┴, l┬) dom(di) where l = li i=1...n. li is a set of values and li ∩ lj = {}, ≤ O is a partial order between the elements of l.
To identify the level l of a dimension, as part of a hierarchy we use dl.
l┴: base level l┬: top level
for each pair of levels li and lj we have the relation
μij : li × lj [
H is an object type history that corresponds to a cubic structure( l, F, O, H′ ) which allows us to trace back the evolution of a cubic structure after performing a set of operators i.e. aggregation.
Selection (Σ): The selection operator selects a set of fact-instances from a cubic structure that satisfy a predicate (θ). A predicate (θ) involves a set of atomic predicates (θ1, …, θn ) associated with the aid of logical operators p ( i.e. ∧, ∨, etc.) . The set of possible facts (cubic instances) that satisfy the θ should carry a degree of membership μ and non-membership ν expressed as
F = {<x, min(μF(x), μ(θ (x))), max(νF(x), ν(θ (x))))> | x∈ X } 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) Mathematical notation: ∑θ (Ci ) = Co
Cubic Product ( ⊗ ): This is a binary operator Ci1 ⊗ Ci2 . It is used to relate two cubes Ci1 and Ci2 assuming that D1 ⊆ D2 and O1 , O2 are reconcilable partial orders. Thus, l1, l2 could lead to lo being a ragged hierarchy.
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 X F2
Fo ={<<x, y>, min(μf1(x), μf2(y)), max(νf1(x), νf2(y),)>|<x, y>∈ X×Y} Mathematical notation: Ci1 ⊗ Ci2 = Co
Join (Θ): It can be expressed using Cubic Product operator. Ci1 = (D1, l1, F1, O1, H1)) and Ci2 = (D2 ,l2, F2, O2, H2) are candidates to join if D1 ∩ D2 ≠ 0, Input: Ci1 = (D1, l1, F1, O1, H1) and Ci2 = (D2,l2, F2, O2, H2) Output: Co= (Do, lo, Fo, Oo, Ho) Mathematical notation: Ci1 Θ Ci2 = σp(Ci1 ⊗ Ci2)
Union (∪): The union operator is a binary operator that finds the union of two cubes. Ci1 and Ci2 have to be union compatible. The operator also coalesces the valueequivalent facts using the minimum membership and maximum non-membership. 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(μF1 (x), μF2(x)), min(ν F1(x),ν F2(x)) > | x ∈ X } Mathematical notation: Ci1 ∪ Ci2 = Co
Difference (-):. The difference operator removes the portion of the cube Ci1 that is common to both 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, min(μF1(x),μF2(x)), max(νF1(x),νF2(x)) > | x ∈ X } Mathematical notation: Ci1 – Ci2 = Co
Aggregation (A): An aggregation operator A is a function A(G) where G = {<x, μF(x) , νF(x)>| x∈ X } where x=<att1, …,attn> is an ordered tuple belonging to a given universe X, {att1, …, attn} is the set of attributes of the elements of X, μF(x) and νF(x) are the degree of membership and non-membership of x. The result is a bag of the type {<x′, μF(x′ ) , νF(x′)>| x′∈ X }. To this extent, the bag is a group of elements that can be duplicated and each one has a degree of μ and ν.
Input: Ci = (D, l, F, O, H) and the function A(G) Output: Co = (D, lo, Fo , Oo , Ho)
The definition of the extended group operators allows us to define the extended group operators Roll up (Δ), and Roll Down (Ω).
Roll up (Δ): The result of applying Roll up over dimension di at level dlr using the aggregation operator A over a datacube Ci = (Di ,li ,Fi , O , Hi ) is another datacube Co = (Do ,lo ,Fo , O , Ho ).
Input: Ci = (Di ,li ,Fi , O , Hi ) Output: Co = (Do ,lo ,Fo , O , Ho ) ω is the initial state of the cube An object of type history is a recursive structure H = (l, D, A, H’) is the state of the cube after performing an operation on the cube
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 IFSUM aggregation operator.
Roll Down (Ω): This operator performs the opposite function of the Roll Up operator. It is used to roll down from the higher levels of the hierarchy with a greater degree of generalization, to the leaves with the greater degree of precision. The result of applying Roll Down over a datacube Ci = (D, l, F, O, H) having H=( l’, D’, A’, H’ ) is another datacube Co= (D’, l’, F’, O, H’).
Input: Ci=(D, l, F, O, H) Output: Co=(D’,l’, F’, O, H’) where F’ set of fact instances defined by operator A.
To this extent, the Roll Down operative makes use of the recursive history structure previously created after performing the Roll Up operator.
The definition of aggregation operator points to the need of defining the IF
extensions for traditional group operators [
IFSUM : The IFsum aggregate, like its standard counterpart, is only defined for numeric domains. Given a fact F defined on the schema X (att1, …,attn), let attn-1 defined on the domain U={u1 , …, un ). The fact F consists of fact instances Fi with 1 ≤ i ≤ m. The fact instances Fi are assumed to take Intuitionistic Fuzzy values for the attribute attn-1 for i = 1 to m we have Fi[attn-1] = {<μi(uki), νi(uki)>/ uki | 1 ≤ ki ≤ n } . The IFsum of the attribute attn-1 of the fact table F is defined by:
IFSUM((attn-1)(F)) = {<u>/ y | (( u= min im=1 (μi(uki), νi(uki)) ∧ (y = ∑ki=k1uki ) (∀ k1, …km : 1 ≤ k1, …km ≤ n))} km
IFAVG : The IFAVG aggregate, like its standard counterpart, is only defined for numeric domains. This aggregate makes use of the IFSUM that was discussed previously and the standard COU?T. The IFAVG can be defined as:
IFAVG((attn-1)(F) = IFSUM((attn-1)(F)) / COU?T((attn-1)(F))
IFMAX : The IFMAX aggregate, like its standard counterpart, is only defined for numeric domains. The IFsum of the attribute attn-1 of the fact table F is defined by:
IFMAX((attn-1)(F)) = {<u>y|((u= min im=1 (μi(uki),νi(uki))∧ (y= maxim=1 (μi(uki),νi(uki)))(∀k1,…km :1≤k1,…km≤ n))}
IFMI) : The IFMI? aggregate, like its standard counterpart, is only defined for numeric domains. Given a fact F defined on the schema X (att1, …,attn), let attn-1 defined on the domain U={u1 , …, un ). The fact F consists of fact instances fi with 1 ≤ i ≤ m. The fact instances fi are assumed to take Intuitionistic Fuzzy values for the attribute attn-1 for i = 1 to m we have fi[attn-1] = {<μi(uki), νi(uki)>/ uki | 1 ≤ ki ≤ n } . The IFsum of the attribute attn-1 of the fact table F is defined by:
IFMI?((attn-1)(F)) = {<u>/ y|(( u= minim=1 (μi(uki),νi(uki))∧(y= minim=1 (μi(uki),νi(uki)))(∀k1,…km :1≤ k1,…km≤ n))}
We can observe that the IFMI? is extended in the same manner as IFMAX aggregate except for replacing the symbol max in the IFMAX definition with min.
Once we have defined our Intuitionistic Fuzzy multidimensional model and have defined the IF cubic-algebra, the concept of knowledge based OLAP is introduced.
Let us consider the Intuitionistic fuzzy set M defined as: {Milk<0.8,0.1>, WholeMilk<0.7,0.1>, Condensed-Milk<0.4,0.3>}} which is presented in “Fig.1”. Then the next step is to calculate the <μ, ν> values for “Pasteurized milk”, “Whole Pasteurized milk ” and “Condensed whole milk.” • If the hierarchical IF structure expresses preferences in a query, the choice of the maximum values for μ and minimum value ν from the pairs of values <μ, ν> from the parent elements to the sub elements allows us not to exclude any possible answer (high possibility necessity degrees). 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 IF represents an ill-known concept, the choice of the maximum value for μ and minimum value ν allows us to preserve all the possible values, but it also makes the answer less specific. In a way, it also participates in enlarging the query, as a less specific datum may share more common values with the query (the possibility degree of matching can thus be higher, although the necessity degree can decrease).
Pasteurizedmilk
Milk <0.8,0.1> Wholemilk <0.7,0.1>
Condensedmilk <0.4,0.3>
Pasteurizedmilk <0.8,0.1>
Condensedmilk <0.4,0.3> Milk <0.8,0.1> Wholemilk <0.7,0.1> Wholepasteurizedmilk
Condensedwholemilk
Wholepasteurizedmilk <0.8,0.1>
Condensedwholemilk <0.7,0.1> “Fig.2” is a fully weighted Hierarchy after applying the maximum values for μ and minimum value ν from the pairs of values <μ, ν> from the parent elements to the sub elements, i.e. from (whole-milk, condensed-milk) to (condensed-whole-milk), from (milk) to (pasteurized milk), and from (whole-milk, pasteurized milk) to (pasteurized-wholemilk).
The complete study of the hierarchical IF requires the formal definition of the IF hierarchical closure. We will further need to formally define the containment of an IF hierarchical set to another.
In this paper we have presented a new multidimensional-cubic model named as the IF-Cube. The main contribution of this new model is that is able to operate over data with imprecision in the facts and the summarisation hierarchies. Classical models imposed a rigid structure that made the models present difficulties when merging information from different but still reconcilable sources. We introduce the automatic recommendation of analysis according to the context of users’ explorations in order to guide the decision making with the aid of Intuitionistic fuzzy set over a universe that has a hierarchical structure and the corresponding hierarchies.
There is finally a need to formally define the closure of Intuitionistic fuzzy set over a universe that has a hierarchical structure as well the containment between different versions of these sets.