Extracting semantics from data cubes using cube transversals and closures

Abstract

In this paper we propose a lattice-based approach intended for extracting semantics from datacubes: borders of version spaces for supervised classification, closed cube lattice to summarize the semantics of datacubes w.r.t. COUNT, SUM, and covering graph of the quotient cube as a visualization tool of minimal multidimensional associations. With this intention, we introduce two novel concepts: the cube transversals and the cube closures over the cube lattice of a categorical database relation. We propose a levelwise merging algorithm for mining minimal cube transversals with a single database scan. We introduce the cube connection, show that it is a Galois connection and derive a closure operator over the cube lattice. Using cube transversals and closures, we define a new characterization of boundary sets which provide a condensed representation of version spaces used to enhance supervised classification. The algorithm designed for computing such borders improves the complexity of previous proposals. We also introduce the concept of closed cube lattice and show that it is isomorph to on one hand the Galois lattice and on the other hand the quotient cube w.r.t. COUNT, SUM. Proposed in [16], the quotient cube is a succinct summary of a datacube preserving the Rollup/Drilldown semantics. We show that the quotient cube w.r.t. COUNT, SUM and the closed cube lattice have a similar expression power but the latter has the smallest possible size. Finally we focus on the multidimensional association issue and introduce the covering graph of the quotient cube which provides the user with a visualization tool of minimal multidimensional associations.

Searching Data Cube for Submerging and Emerging Cuboids

“…The computing time and storage space can be optimized based on equivalence relations defined on aggregate functions [16] [17] or on the concept of closed itemsets in frequent itemset mining [18] or by reducing redundancies between tuples in cuboids, using tuple references [13] [23]. In these approaches, the computation is usually organized on the complete lattice of subschemes of the fact table dimension scheme.…”

A Data Cube Representation for Efficient Querying and Updating

“…The computing time and storage space can be optimized based on equivalence relations defined on aggregate functions [11] [16] or on the concept of closed itemsets in frequent itemset mining [15] or by reducing redundancies between tuples in cuboids, using tuple references [2] [21]. In these approaches, the computation is usually organized on the complete lattice of subschemes of the fact table dimension scheme.…”

Constrained Closed and Quotient Cubes

“…Their main idea is to propose a structure which avoids to store the intrinsic redundancies existing within a data cube. They propose respectively the Closed Cube [Casali et al, 2009a, Casali et al, 2003b and the Quotient Cube [Lakshmanan et al, 2002]. The former is one of the most reduced representations of cubes which can be used for OLAP queries.…”

“…The first one, called Constrained Closed Cube, is the most reduced representation which encompasses constrained tuples (tuples respecting a constraint or a combination of constraints). It is based on the concept of cube closure [Casali et al, 2003b]. From this representation, it is possible to retrieve the measure values associated to the various trends thus any OLAP query can be answered.…”