A Model for Fuzzy Multidimensional Spaces Claudia González2, Raimundo Mirisola2, Leonid Tineo2, and Angélica Urrutia1 1
Universidad Católica del Maule, Departamento de Computación, Talca, Chile
[email protected] 2 Universidad Simón Bolívar, Departamento de Computación, Apartado 89000, Caracas 1080-A, Venezuela {claudia,raimundo,leonid}@ldc.usb.ve
Abstract. Fuzzy data representation and manipulation is needed in Data Warehouse due to imprecision or uncertainty from different sources. Nevertheless, this problem has not been wide explored. We propose a fuzzy multidimensional model, cube and operators with a notion of fuzzy hierarchy based on fuzzy functional dependencies. Keywords: Data Warehouse, Fuzzy Databases, Multidimensional Model.
1 Introduction Different aspects of multidimensional databases have taken the attention of previous works. Formal definitions of cube, operators and representation in multidimensional databases and relational databases have been presented in [Error! Reference source not found.]. On the other hand, several efforts have been made in order to add fuzziness in databases. In particular, [Error! Reference source not found.] introduced a model for fuzzy data. Fuzzy OLAP to support the qualitative analysis in data warehousing has been presented in [Error! Reference source not found.]. Based on the use of fuzzy data model [Error! Reference source not found.], we extend here the model proposed in [Error! Reference source not found.].
2 Fuzzy Space, Cube and Operations We take the concept of Generalized Fuzzy Domain form the GREFRED Model [Error! Reference source not found.]. It includes traditional (crisp) data but also special values (Unknown, Undefined, Null) and fuzzy numbers. Comparison operators are generalized in this model by means of the Possibility Measure. A fuzzy multidimensional space is defined by dimensions, dimensional levels with associated domains that might be generalized fuzzy ones. A value in a dimensional level can have ancestors or descendants. We allow fuzzy dependency functions, obtaining fuzzy ancestor-descendant relations. The dependency degree is defined as a possibility measure. A base cube Cb is defined by , being Db the R. Meersman, Z. Tari, P. Herrero et al. (Eds.): OTM 2007 Ws, Part I, LNCS 4805, pp. 21–22, 2007. © Springer-Verlag Berlin Heidelberg 2007
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dimensions, Lb the dimensional levels lb1 … lbm with fuzzy generalized domains Dom(lbi) and Rb a data set of rows in Dom(lb1)×…×Dom(lbm). A cube C is a substructure , being D⊆Db, L the projection of Lb in D and R the projection of Rb in D. Operation of on fuzzy cubes are. ⎯ Level Climbing rolls up the associated levels to a dimension set belonging to the cube dimensions. This operator is extended by means of the fuzzy ancestor relation. ⎯ Function Application transforming values, applied functions are automatically extended to fuzzy generalized domain. ⎯ Projection: this operation is the trivial extension of traditional one giving fuzzy data in corresponding places. ⎯ Packing makes a pack of those rows that have equal dimensional value, and the measure associated to each pack is the set of measures associated to the rows of the pack. In our extension the equality is defined by a possibility measure. Its process is a possibilistc reasoning that involves a user given consistence level. ⎯ Navigation rolls up a cube to a specific dimension, packing the result and applying an aggregated function to allow fuzzy data and its operators. ⎯ Dicing performs a selection in both, the cube and the base cube. In the cube filters the elements that satisfy a given fuzzy condition with a threshold, in the base cube considers also ancestors. ⎯ Slicing allows rolling up the cube to a specific dimension, cut out the dimension D and apply it a parking and an aggregated function, according to corresponding fuzzy semantics.
3 Conclusions and Future Works In this work Multidimensional space was extended to include fuzzy dimensions defined over functional dependencies. Adequate version for ancestor and descendants were provided. Also fuzzy cube and its operators were presented. In future works we hope extend SQL 2003 with presented model and implement it in a RDBMS. Acknowledgments. This work was supported in part by Venezuelan Governmental Foundation for Science, Innovation and Technology FONACIT Grant G-2005000278. Main support for whole live comes form Jesus Christ, my personal Lord (Leonid).
References 1. Galindo, J., Urrutia, A., Piattini, M.: Representation of Fuzzy Knowledge in Relational Databases. In: Galindo, F., Takizawa, M., Traunmüller, R. (eds.) DEXA 2004. LNCS, vol. 3180, pp. 917–921. Springer, Heidelberg (2004) 2. Vassiliadis, P.: Modeling Multidimensional Databases, Cubes and Cube Operations. Statistical and Scientific Database Management, 53–62 (1998) 3. Pavan Kumar, K.V.N.N., Radha Krishna, P., De Kumar, S.: Fuzzy OLAP cube for qualitative analysis. Intelligent Sensing and Information Processing, 290–295 (2005)
