Sébastien Nedjar

Emerging cubes for trends analysis in OLAP databases

In various approaches, data cubes are pre-computed in order to efficiently answer Olap queries. Such cubes are also successfully used for multidimensional analysis of data streams. In this paper, we address the issue of performing cube comparisons in order to exhibit trend reversals between two cubes. Mining such trend changes provides users with a novel and specially interesting knowledge. For capturing the latter, we introduce the concept of emerging cube. Moreover, we provide a condensed representation of emerging cubes which avoids to compute two underlying cubes. Finally, we study an algorithmic way to achieve our representation using cube maximals and cube transversals.

Closed Cube Lattices

In this paper we propose a lattice-based approach intended for summa- rizing the Data Cubes. With this intention, we introduce a novel concept: the cube closure over the cube lattice (multidimensional search space) of a categorical data- base relation. We introduce the cube connection, show that it is a Galois connection and derive a closure operator over the cube lattice. We introduce the concept of Closed Cube lattice which is a cover for Data Cube and show that it is isomorphic to, on one hand the Galois (concept) lattice and, on the other hand the Quotient Cube. Proposed by Lakshmanan et al., the Quotient Cube is a succinct summary of a Data Cube preserving the Rollup/Drilldown semantics. We show that the Quo- tient Cube, provided with a closure-based characterization, can be derived from the Closed Cube. Thus these two structures have a similar expression power but the Closed Cube is smaller. Finally, we perform some experiments in order to measure the benefit of our approach.

Emerging Cubes: Borders, size estimations and lossless reductions

Discovering trend reversals between two data cubes provides users with a novel and interesting knowledge when the real world context fluctuates: What is new? Which trends appear or emerge? Which tendencies are immersing or disappear? With the concept of Emerging Cube, we capture such trend reversals by enforcing an emergence constraint. We resume the classical borders for the Emerging Cube and introduce a new one which optimizes both storage space and computation time, provides a simple characterization of the size of Emerging Cubes, as well as classification and cube navigation tools. We soundly state the connection between the classical and proposed borders by using cube transversals. Knowing the size of Emerging Cubes without computing them is of great interest in particular for adjusting at best the underlying emergence constraint. We address this issue by studying an upper bound and characterizing the exact size of Emerging Cubes. We propose two strategies for quickly estimate their size: one based on analytical estimation, without database access, and one based on probabilistic counting using the proposed borders as the input of the near-optimal algorithm HyperLogLog. Due to the efficiency of the estimation algorithm various iterations can be performed to calibrate at best the emergence constraint. Moreover, we propose reduced and lossless representations of the Emerging Cube by using the concept of cube closure. Finally, we perform experiments for different data distributions in order to measure on one hand the size of the introduced condensed and concise representations and on the other hand the performance (accuracy and computation time) of the proposed estimation method.

Extracting semantics in OLAP databases using emerging cubes

Data cubes capture general trends aggregated from multidimensional data from a categorical relation. When provided with two relations, interesting knowledge can be exhibited by comparing the two underlying data cubes. Trend reversals or particular phenomena irrelevant in one data cube may indeed clearly appear in the other data cube. In order to capture such trend reversals, we have proposed the concept of Emerging Cube. In this article, we emphasize on two new approaches for computing Emerging Cubes. Both are devised to be integrated within standard Olap systems, since they do not require any additional nor complex data structures. Our rst approach is based on Sql. We propose three queries with dierent aims. The most ecient query uses a particular data structure merging the two input relations to achieve a single data cube computation. This query works ne even when voluminous data are processed. Our second approach is algorithmic and aims to improve eciency and scalability while preserving integration capability. The E-Idea algorithm works a la Buc and takes the specic features of Emerging Cubes into account. EIdea is automaton-based and adapts its behavior to the current execution context. Our proposals are validated by various experiments where we measure query response time. Comparative experiments show that E-Idea’s response time is proportional to the size of the Emerging Cube. Experiments also demonstrate that extracting Emerging Cubes can be computed in practice, in a time compatible with user expectations.

Convex cube: towards a unified structure for multidimensional databases

In various approaches, data cubes are pre-computed in order to efficiently answer Olap queries. Such cubes are also successfully used for multidimensional analysis of data streams. The notion of data cube has been explored in various ways: iceberg cubes, range cubes, differential cubes or emerging cubes. In this paper, we introduce the concept of convex cube which captures all the tuples satisfying a monotone and/or antimonotone constraint combination. It can be represented in a very compact way in order to optimize both computation time and required storage space. The convex cube is not an additional structure appended to the list of cube variants but we propose it as a unifying structure that we use to characterize, in a simple, sound and homogeneous way, the other quoted types of cubes.

The agree concept lattice for multidimensional database analysis

In this paper we propose the characterization of two new structures, the Agree Concept Lattice and the Quotient Agree Lattice of a database relation. Both of them are of great interest for multidimensional database analysis. They provide a formal framework which makes it possible to improve computation time, reduce representation and easily navigate through the Hasse diagram. These structures are generic, apply to various database analysis problems and combine formal concept analysis and database theory. They make use of the concepts of agree set and database partition. Agree set and partition are associated to define the Agree Concept of a database relation. The set of all the Agree Concepts is organized within the Agree Concept Lattice. The Quotient Agree Lattice is along the lines of both the TITANIC framework and the quotient cube. We also briefly present three application fields of the proposed structures. The first two ones are classical since they concern on the one hand the discovery of functional and approximate dependencies for database design and tuning and on the other hand the data cube computation and representation. The latter field has been recently investigated. The underlying issue is to retrieve the most relevant objects according to the user expectations: the SKYLINE. The multidimensional generalization of the SKYLINE has been proposed through the SKYCUBE. The proposed structures smartly solve the problem of partial materialization of SKYCUBE with reconstruction guarantee.

Reduced representations of Emerging Cubes for OLAP database mining

In this paper, we investigate reduced representations for the Emerging Cube. We use the borders, classical in data mining, for the Emerging Cube. These borders can support classification tasks to know whether a trend is emerging or not. However, the borders do not make possible to retrieve the measure values. This is why we introduce two new and reduced representations without measure loss: the L-Emerging Closed Cube and Emerging Quotient Cube. We state the relationship between the introduced representations. Experiments performed on various data sets are intended to measure the size of the three reduced representations.

Constrained Closed and Quotient Cubes

In this chapter, we investigate reduced representations for the Constrained Cube. We use the borders, classical in data mining, for the Constrained Cube. These borders are the boundaries of the solution space and can support classification tasks. However, the borders do not make possible to retrieve the measure values and therefore cannot be used to answer Olap queries. This is why we introduce two new and reduced representations without measure loss: the Constrained Closed Cube and Constrained Quotient Cube. The former representation is based on the concept of cube closure. It is one of the smallest possible representations of cubes. Provided with the Constrained Closed Cube and thus by storing the minimal information, it is possible to answer efficiently queries which can be answered from the Constrained Cube itself. The latter representation is supported by the structure of the Quotient Cube which was proposed to summarize data cubes. The Quotient Cube is revisited in order to provide it with a closure-based semantics and thus adapt it to the context of the Constrained Cube. The resulting Constrained Quotient Cube is less reduced than the Constrained Closed Cube but it preserves the “specialization / generalization” property of the Quotient Cube which makes it possible to navigate within the Constrained Cube. We also state the relationship between the two introduced representations and the one based on the borders. Experiments performed on various data sets are intended to measure the size of the three representations. As expected in the most common situations (real data), the space reduction for each representation is significant comparatively to the size of the Constrained Cube. Thus depending on the user future needs, each of the proposed representations supplies a significant space reduction for a specific use: Borders for Olap classification, Constrained Closed Cube for Olap querying and Constrained Quotient Cube for cube navigation.

Upper Borders for Emerging Cubes

The emerging cube computed from two relations r 1 , r 2 of categorical attributes gather the tuples for which the measure value strongly increases from r 1 to r 2 . In this paper, we are interested in borders for emerging cubes which optimize both storage space and computation time. Such borders also provide classification and cube navigation tools. Firstly we study the condensed representation through the classical borders Lower / Upper, then we propose the borders Upper*/ Uppermore reduced than the previous ones. We soundly state the connexion between the two representations by using cube transversals. Finally, we perform experiments about the size of the introduced representations. The results are convincing and reinforce our idea that the proposed borders are relevant candidates to be the smallest condensed representation of emerging cubes and thus can be really interesting for trend analysis in Olap databases.

Constrained closed datacubes

This paper focuses on borders and lossless representations for Constrained Datacubes of database relations, which can represent many-valued contexts. The final goal is to optimize both storage space and computation time. First we study the succinct representation through the borders Lower / Upper and Upper