A Fuzzy Multidimensional Model for Supporting Imprecision in OLAP Miguel Delgado, Carlos Molina, Daniel S´anchez and Amparo Vila
L´azaro Rodr´ıguez-Ariza
Dpt. of Computer Science and Artificial Intelligent University of Granada Granada, Spain Telephone: +34 958 240468 Email:
[email protected]
Dpt. Financial Economy and Accounting University of Granada Granada, Spain
Abstract— The use of OLAP technology in new knowledge fields and the merge of data from different sources have made appeared new requirements for models to support this technology. What we propose in this paper is a new multidimensional model that can manage imprecision both in the dimensions and the facts. This enables the multidimensional structure to model the imprecision of the data as a result of the integration of data from different sources or even information from experts. This is done by means of fuzzy logic.
I. I NTRODUCTION Since the appearance of the OLAP technology ( [1]) different proposals have been made to give support to the special necessities of this technology. In the literature we can see two different approaches. One of this is to extend the relational model to support the structures and operations typical of OLAP. The first one following this idea can be found in [2]. From then on, more proposes have appeared ( [3]) and most of the present relational systems include extension to represent datacubes and operate over them. The other approach is to develop new models using a multidimensional view of the data. Many authors proposed model in this way ( [4–7]). In the early 70’s, the necessity of flexible models and query languages to manage the ill-defined nature of information in DSS is identified ( [8]).Nowadays, the application of the OLAP technology to other knowledge fields (e.g. medical data) and the use of semi-structured (e.g. XML) and non-structured (e.g. plain text) sources introduce new requirements to the models. Now the systems need to manage imprecision in the data and more flexible structures to represent the analysis domain. New models have appeared to manage incomplete datacube ( [9]), imprecision in the facts ( [10]) and the definition of fact using different levels in the dimensions ( [11]). Nevertheless, these models continue using rigid hierarchies that made very difficult to model some problems that can be translate into loss of information when we need to merge data from different sources with some incompatibilities in their schemata. What we propose in this paper is a new multidimensional model that is able to treat with imprecision over hierarchies and facts using fuzzy logic. The use of fuzzy hierarchies enables to define the structures of the dimension in a more intuitive way to the final user and then allow a more intuitive use of the
system. Furthermore, this allow to merge information from different sources which structures have some incompatibilities or even thought use information given by experts to improve the multidimensional schema. In the next section we first introduce the classical multidimensional model as an introduction to present our model. Then, in section III we present an example over the structure proposed to show how to apply the operations over the multidimensional structure. In section IV we present the position of the model inside the company information structure. The main conclusions and future work is presented in the last section. II. M ULTIDIMENSIONAL STRUCTURE In this section we present the new multidimensional model that we propose. We first introduce the classical models. Next we propose our multidimensional structure and the normal operations (roll-up, drill-down, dice, slice and pivot) over it. A. Classical Multidimensional Models In classical multidimensional models we can differentiate two different types of data. In one hand we have the facts that are the object of the analysis. On the other hand, the dimensions are the facts context. In the dimensions we can define hierarchies. The different levels of the dimensions allow us to access the facts at different levels of granularity. To do it, classical aggregation operators are needed (maximum, minimum, average, etc). The hierarchies defined use many-to-one relations, so one element in a level can only be grouped by a unique value of each upper level in the hierarchy. This made the final structure of a datacube to be rigid and well defined in the sense that given two values in the same level of a dimension, the set of facts related to the values have empty intersection. Over this structure the normal operations (roll-up, drill-down, dice, slice and pivot) are defined. B. Multidimensional structure Definition 1: A dimension is a tuple d = (l, ≤d , l⊥ , l ) where l = li , i = 1, ..., n so that each li is a set of values and li ∩lj = ∅ if i=j, and ≤d is a partial order relation between
Fig. 1.
Example of a hierarchy over ages
Fig. 2.
the elements of l. l⊥ and l are two elements of l so that ∀li ∈ l l⊥ ≤d li and li ≤d l . We denote level to each element li . To identify the level l of the dimension d we will use d.l. The two special levels l⊥ and l will be called base level and top level respectively. The partial order relation in a dimension is what gives the hierarchical relation between the levels. In the figure 1 you can see a definition of a hierarchy over the ages. The definition of the dimension as we have presented it would be Age = (Age, Group, legal age, All, ≤Age , Age, All) and the relation Age ≤Age Age, Group ≤Age Group, Legal age ≤Age Legal age, All ≤Age All, Age ≤Age Group, Age ≤Age Legal age, Age ≤Age All, Group ≤Age All and Legal age ≤Age All Definition 2: For each dimension d the domain is dom(d) = li In the example above the domain of the dimension Age is dom(Age) = {1, ..., 100, Y oung, Adult, Old, Y es, N o, All}. Definition 3: For each li the set Hi = {lj /lj =li ∧lj ≤d li ∧ ¬∃lk lj ≤d lk ≤d li }
(1)
We call it set of children of the level l. Using the same example, we have that the set of children of the level All is HAll = {Group, Legal age}. In all dimensions that we can define this set for the base level will be always the empty set, as you can see from the definition of set of children. Definition 4: For each li the set Pi = {lj /li =lj ∧li ≤d lj ∧ ¬∃lk lj ≤d lk ≤d li }
(2)
and we call it set of parents of the level l. Over the hierarchy we have defined, the set of parents of the level Age is PAge = {Legal age, Group}. In the case of the top level of a dimension this set will be always the empty set. Definition 5: For each pair of levels li and lj so that we have the relation µij : li × lj → [0, 1]
(3)
We call it kinship relation. The degree of inclusion of the elements of a level in the elements of their parent levels can be defined using this relation. If we use only the values 0 and 1 and we only allow an element to be include with degree 1 by a unique element of its parent levels, this relation represents a crisp hierarchy. Following the example, the relation between the levels Legal
Kinship relation between levels Group and Age
age and Age is of this type. The parent relation in this situation is 1 if x ∈ [18, 100] µLegalAge,Age (Y es, x) = 0 in other case 1 if x ∈ [1, 17] µLegalAge,Age (N o, x) = 0 in other case If we relax these condition and we allow to use values in the interval [0,1] without any other limitation, we have a fuzzy hierarchical relation. This allows represent several hierarchical relations in a more intuitive way. An example can be seen in the figure 2 where we present the group of ages according to linguistic labels. Furthermore, this fuzzy relation allows to define hierarchies in which there is imprecision in the relationship between elements in different levels. In this situation, the value in the interval shows the degree of confidence in the relation. Definition 6: For each pair of levels li and lj of the dimension d so that lj ≤d li ∧lj =li we have the relation ηij : li × lj → [0, 1] defined as if lj ∈Hli µij (a, b) ηij (a, b) = (µik (a, c) ⊗ ηkj (c, b)) in other case lk ∈Hli c∈lk
(4) where ⊗ and ⊕ are a t-norm and a t-conorm respectively or operators from the families MOM and MAM defined by Yager ( [12]), that include the t-norms and t-conorms. We call this extended kinship relation. This relation gives us information about the degree of relation between two values in different levels inside the same dimension. To obtain this value, it considers all the possible paths between the elements in the hierarchy. Each one is calculate aggregating the kinship relation between elements in two consecutive levels using a t-norm. Then the final value is the aggregation of the result of each path using a t-conorm. As an example, we will show how to calculate the value of ηAll,Age (All, 25). In this situation we have two different paths. Let see each one: • All - Legal age - Age. In the figure 3.a you can see the two ways to get to 25 from All going pass the level legal age. The result of this path is (1 ⊗ 1) ⊕ (1 ⊗ 0). • All - Group - Age. This is a situation very similar to the previous one. In the figure 3.b you can see the three different paths going through the level Group. The result of this path is (1 ⊗ 0.7) ⊕ (1 ⊗ 0.3) ⊕ (1 ⊗ 0). Now we have to aggregate these two values using a tconorm to obtain the result. If we use the maximum as t-norm
Fig. 3. Example of the calculation of the extended kinship relation. a) path All - Legal age - Age b) path All - Group - Age
and the minimum as t-conorm, the result is ((1⊗1)⊕(1⊗0))⊕ ((1 ⊗ 0.7) ⊕ (1 ⊗ 0.3) ⊕ (1 ⊗ 0)) = (1 ⊕ 0) ⊗ (0.7 ⊕ 0.3 ⊕ 0) = 1 ⊕ 0, 7 = 1. So the value of ηAll,Age (All, 25) is 1. Definition 7: We call fact to any pair (h, α) where h is a m-tuple over the domain of the attributes we want to analyze, and α ∈ [0, 1]. The management of uncertainty in the facts is carried out using a degree of certainty. This degree of certainty allows us to use values in analysis that can be interesting to the decisor but imply imprecision. Definition 8: An object of type history is the recursive structure Ω H= (5) (A, lb , F, G, H ) where Ω is the recursive clausure, F is a set of facts, lb is a set of levels (l1b , , lnb ),A is an application from lb to F , G is an aggregation operator, H is a structure of type history. The important of this structure will be clear when we talk about the operations. Definition 9: A datacube is a tuple C = (D, lb , F, H, A) so that D = (d1 , ..., dn ) is a set of dimensions, lb = (l1b , ..., lnb ) is a set of levels so that lib belongs to di , F = R∪∅ where R is the set of facts and ∅ is a special symbol, H is an object of type history, and A is an application defined as A : l1b ×...×lnb → F. If for a = (a1 , ..., an ) we have A(a) = ∅, this means that there isn’t defined a fact for this combination of values. Definition 10: We say a datacube is basic if lb = (l1⊥ , ..., ln⊥ ) and H = Ω. Now, we have all the tools to define a multidimensional schema using the model proposed. In the figure 4 you can see an example of multidimensional schema. This tries to model the analysis of the causes of customers’ complaints. The information of the company has been improved using two external sources. One of it has been the experts’ opinion about the severity of the cause of the complaint. The level Severity in the dimension Cause manages this information. The other external source has been use to introduce the classification of the provider looking at the quality. The level Quality in the dimension Product reflects this data. In the figure, the continuous lines represent crisp relation in the hierarchy. The discontinuous lines represent fuzzy hierarchical relation where one element can be related to elements in the upper level with values in the interval [0,1].
Fig. 4.
Example of multidimensional schema
The structure of the datacube related to this schema is C = (Customer, P roduct, T ime, Cause, Count ∪ ∅, Ω, A) where the dimension are defined as Customer = (Age, Legal age, Group, All, ≤Cu , Age, All), P roduct = (P roduct, P rovider, Quality, All, ≤P r , P roduct, All), T ime = (Date, M onth, Y ear, All, ≤T i , Date, All) and Cause = (Cause, Severity, All, ≤Ca , Cause, All). The other element to define is the relation A. In this situation the structure of this relation is A : Age × P roduct × Date × Cause → Count ∪ ∅. C. Operations Once we have the structure of the multidimensional model, we need the operations to analyze the data in the datacube. In this section we present the normal operations (roll-up, drilldown, slice, dice and pivot) over the structured proposed. In section III we present an example of the application of these operations over a multidimensional schema. Definition 11: An aggregation operator G is a function G(B) where B = (h, α)/(h, α) ∈ F and the result is a tuple (h , α ). The parameter that operator needs can be seen as a fuzzy bag ( [13]). In this structure there is a group of elements that can be duplicated, and each one has a degree of membership. Definition 12: For each value a belonging to di we have the set Fb /b ∈ lj ∧ µij (a, b) > 0 if li = lb li ∈Hli Fa = {h/h ∈ H ∧ ∃a1 , ..., an A(a1 , ..., an ) = h} if li = lb (6) The set Fa represents all the facts that are related to the value a. Definition 13: The result of applying roll-up over the dimension di , level lr (lr = l⊥ ), using the aggregation operator G over a datacube C = (D, lb , F, A, H) is another datacube C = (D, lb , F , A , H ) where lb = (l1b , , lr , , lnb ),
A (a1 , ..., an ) = G({(b, α ⊗ ηrb (a, c))/(b, α) ∈ Fa ∧ A(a1 , ..., an ) = (b, α)}), F is the range of A , and H = (A, lb , F, G, H). Definition 14: The result of applying drill-down over a datacube C = (D, lb , F, A, H) having H = (A , l b, F , H ) is another datacube C = (D, lb , F , A , H ). The structure history of the datacube allow us to keep all the information when applying roll − up and get it back with the drill − down operations. Definition 15: The result of applying dice with the condition β over the level lr of the dimension di in a datacube C = (D, lb , F, A, H) is another datacube C = (D , lb , F , A , Ω) where D = d1 , ..., di , ..., dn with di = (li , ≤di , lb , l ) having lj and l = lj /lb ≤d if lj = lr {v/v ∈ lj ∧ β(v)} {v/v ∈ di .lj ∧ δrj (v)} if lj ≤d lr where di .lj = {v/v ∈ di .lj ∧ δjr (v)} if lr ≤d lj δij (v) = ∃x ∈ lr β(x) ∧ ηij (x, v) > 0, A (a1 , ..., ai , ..., an ) = (h, α ⊗ µβ )/a1 ∈ d1 .l b ∧ ... ∧ an ∈ dn .lb ∧ A(a1 , ..., an ) = (h, α) where µβ = ηrb (c, ai ), and F is the range of c∈di .lr
A . Definition 16: The result of applying slice over the dimension di using the aggregation operator G in a datacube C = (D, lb , F, A, H) is another datacube C = (D , lb , F , A , Ω) = (d1 , ..., di−1 , di+1 , ..., dn ), lb = having D (lib , ..., li−1b , li+1b , ..., lnb ), A (a1 , ..., ai−1 , ai+1 , ..., an ) = G({(h, α)/∃xA(a1 , ..., ai−1 , x, ai+1 , ..., an ) = (h, α)}), and F is the range of A . Definition 17: The result of applying pivot over the dimensions di and dj in a datacube C = (D, lb , F, A, H) is another datacube C = (D , lb , F, A , Ω) where = (d1 , ..., di−1 , dj , di+1 , ..., dj−1 , di , dj+1 , ..., dn ), D = (l1b , ..., li−1b , ljb , li+1b , ..., lj−1b , lib , lj+1b , ..., lnb ), lb = and A (a1 , ..., ai−1 , ai , ai+1 , ..., aj−1 , aj , aj+1 , ..., an ) A (a1 , ..., ai−1 , aj , ai+1 , ..., aj−1 , ai , aj+1 , ..., an ) Now we have the operations to work with the structure proposed. But this structure can represent objects that are not suitable for the operations defined above. So, we have to say when a datacube is valid to work with it. Definition 18: A datacube is valid if it is basic or it has been obtained applying a finite number of operations over a basic datacube. D. User view We have presented a structure that manages imprecision by mean of fuzzy logic. We need to use aggregation operators over fuzzy bags to apply some of the operations presented. Most of the methods in the literature give as a result a fuzzy set. This situation can made the result be difficult to understand and use in a decision process. So, we propose a two layer model. One of them is the structure presented in the previous section. The other is defined over this and the main objective is to hide the complexity of the model and gives a more understandable result to de user. To do it, we propose to use a fuzzy summary operator that gives a more intuitive result but
Fig. 5.
Structure of dimensions Cause, Product and Time
keeping as much information as possible. Using this kind of operator we define the user view. Definition 19: Given a summary operator M , we define the user view of a datacube C = (D, lb , F, A, H) using M as the structure CM = (D, lb , FM , AM ) where AM (a1 , ..., an ) = M (A(a1 , ..., an )) and FM is the range of AM . We can define as many user views of a datacube as summary operators we use. So, using a datacube each user can have a user view with most intuitive view of data according to her/his preferences. As an example of this type of operator we can use the proposed in [14]. This operator proposes to use the fuzzy number that fits better, in the sense of fuzziness, to the fuzzy set or fuzzy bag result. III. E XAMPLE Once we have defined our multidimensional model, we present now an example to show how to define a datacube and how to apply the operations that we have defined over the structure. The multidimensional schema we use is the same that we presented in figure 4 and in section II-B. We need to define in a more detailed way the structure of the dimensions. The dimension Customer is the same that we have use in the rest of the paper as an example of the dimension structure. The elements of the others dimensions are shown in figure 5. We don’t define the partial order relations because they can be obtained in the multidimensional schema. Over this example we are going to apply operations using our multidimensional model and the classical one. We do this to be able to compare both focuses. If we want to know ”the amount of complaints made by adults depending on the severity of the causes and the quality of the providers” the sequence of operations applied is the following: 1) Dice over the dimension Customer, in the level Group with the condition β(x) = ”x is Adult”. 2) Slice over the dimension Time using as aggregation operator the Sum.
TABLE I No.
Date
Age
Cause
Product
No.
α
No.
Date
Age
Cause
Product
No.
α
1 2 3 4 5 6 7 8 9 10 21
2002-12-01 2002-12-12 2002-12-10 2002-12-01 2002-11-22 2002-12-05 2002-12-18 2002-12-26 2002-12-23 2003-01-03 2003-02-03
40 50 45 35 41 38 20 18 81 16 61
Elec. Prob. Slight dam. Elec. Prob. Slight dam. Breaking Missing spare Slight dam. Missing spare Slight dam. Breaking Slight dam.
Radio Discman TV Video TV Video TV Discman Radio Radio Discman
1 2 2 1 3 1 1 2 3 2 1
1 1 1 1 1 1 1 1 1 1 1
11 12 13 14 15 16 17 18 19 20
2002-12-30 2003-01-03 2002-12-30 2003-01-12 2003-01-19 2003-01-23 2003-01-20 2003-01-20 2003-02-03 2003-02-14
72 60 46 41 37 25 28 30 34 57
Elec. Prob. Breaking Slight dam. Missing spare Missing spare Breaking Elec. Prob. Slight dam. Elec. Prob. Elec. Prob.
Discman TV Video Video TV Discman Video Radio TV Radio
2 1 1 2 3 2 1 1 1 1
1 1 1 1 1 1 1 1 1 1
Example data over the multidimensional schema
3) Roll-up over the dimension Customer and level Group, dimension Cause and level Severity and dimension Product and level Quality, using the Sum operator. If we want to refine the results to know ”the average amount of complaints made by adults depending on the quality of the providers” we have to apply over the datacube result of the other operations Slice over the dimensions Customer and Product using the Average operator. The result of the operations is recollected in tables II-IV. The result of the second operation (slice over Time) isn’t presented in the tables because it doesn’t change the facts since combination of the values of the other dimensions are not repeated. To be able to apply the operations of roll-up and slice we need to use of sum and average aggregation operators. In the crisp case we use the classical one. In our multidimensional model we need operators over fuzzy bags. In the example we have used the ones proposed by Rundensteiner ( [15]) for a fuzzy relational model. We need to adapt the operators to our case. If R is one of the operators defined by Rundensteiner, we define the operator GR for our model as GR (h) = (R(h), 1). As summary operator we have used the linguistic summary proposed in [14]. As you can see, the results in both focuses are very similar. The main different is that, using our approach for OLAP we use all the information taken from external sources and we get more expressive results and understandable by the user. IV. A RCHITECTURE In this section we present where this model fits inside the company information structure. As we have said, most of the multidimensional models impose a rigid structure that have problem when we need to merge information from different sources. If we try to integrate data from extern sources the situation is even more complicate. In this position is where our model can help to improve the information structure of the company. So we propose our model as a new layer over the existing structure to merge information from different sources inside the company, even OLAP server already implanted, or extern to the company.
Fig. 6.
Position of the model inside the company information structure
V. C ONCLUSIONS AND FUTURE WORK In this paper we have presented a new multidimensional model. The main contribution of this new model is that is able to operate over data with imprecision in the facts and the dimensions. Classical models impose a rigid structure that made the models to be difficult to merge information from different sources if they have some incompatibilities in the schemata. Our model can manage these problems by means of fuzzy logic that enabled the model to carry out the integration relaxing the schemata to obtain a new one that cover the other and try to preserved as much information as possible. Furthermore, our model can manage information given by experts that most of the time is imprecise. This data can be use to improve the multidimensional schema and enabled the final user to use it in the decision process. The other advantage of the model is that it can model situation in a more natural way to the user so that he/she can access the information more intuitively. To complete the model we need to study the properties of the operations over the structure. Another line is to develop a graphical way to represent the results of the operations to get a more intuitive way to read the information obtained. To finish the decision process, we need to study the integration
TABLE II Fact
1
2
3
4
5
6
7
8
9
10
11
Fuzzy Crisp
1 X
1 X
1 X
1 X
1 X
1 X
0 -
0 -
0 -
0 -
0.15 -
Fact
12
13
14
15
16
17
18
19
20
21
Fuzzy Crisp
0.75 -
1 X
1 X
1 X
1 X
1 X
1 X
1 X
0.9 -
0.7 -
Result of applying dice over the dimension Customer, in the level Group with condition β(x) = ”x is Adult” over the datacube C. In the fuzzy case, the value is the new α for the fact. In the crisp case, an X means that the fact satisfies the condition TABLE III Cause Fuzzy Product
Crisp
Light
Serious
C
CM
C
CM
Light
Serious
Good
{1/2, 0.7/8, 0.2/17},1
(2,2,0,14.4) ”Greater than 2”
{1/4, 0.8/7, 0.75/8, 0.3/13, 0.2/17, 0.15/19},1
(4,4,0,13.8) ”Greater than 4”
2
4
Bad
{1/3, 0.7/4, 0.2/6, 0.1/18},1
(3,3,0,6) ”A bit greater than 3”
{1/1, 0.9/2, 0.8/4, 0.15/6, 0.1/20},1
(1,1,0,8.8) ”A bit greater than 1”
3
1
Result of applying roll-up over the dimension Customer and level Group, dimension Cause and level Severity and dimension Product and level Quality, using the Sum operator. We don’t show the Customer dimension because there is only one value (Adult) TABLE IV Fuzzy C
Severity
Crisp CM
Light
{1/3, 0.7/4, 0.2/6, 0.1/18, 0.7/8, 0.2/17},1
(3,3,0,6) ”A bit greater than 3”
3
Serious
{1/4, 0.15/6, 0.1/20, 0.8/7, 0.75/8, 0.3/13,0.2/17, 0.15/19},1
(4,4,0,14.2) ”Greater than 4”
4
Result of applying slice over the dimensions Customer and Product using the Average operator
process so we get a formal way to merge data from different sources, including experts knowledge. Another future line is to study the possible relation between our model and fuzzy object oriented relational models. R EFERENCES [1] E. Codd, “Providing OLAP (On-line Analytical Processing) to useranalysts: An IT mandate,” E.F. Codd and Associates, Tech. Rep., 1993. [2] J. Gray, S. Chaudhuri, A. Bosworth, A. Layman, D. Reichart, and M. Venkatrao, “Data Cube: A relational aggregation operator generalizing group-by, cross-tab, and sub-totals,” Data Mining and Konwledge Discovery, vol. 1, pp. 29–53, 1997. [3] R. Kimball, The Data Warehouse Toolkit. New York: John Wiley & Sons, 1996. [4] R. Agrawal, A. Gupta, and S. Sarawagi, “Modeling multidimensional databases,” IBM, IBM Almaden Research Center, Tech. Rep., September 1995. [5] L. Cabibbo and R. Torlone, “Querying multidimensional databases,” in Proceeding of the 6th Int. Workshop on databases programming languages (DBPL6), Estes Pork (U.S.A.), 1997. [6] ——, “A logical approach to multidimensional databases,” in Advances in Databases Technology (EDTB’98), ser. LNCS, no. 1337. Springer, 1998, pp. 183–197.
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