The Skyline concept has been introduced in order to exhibit the best objects according to all the criterion combinations and makes it possible to analyze the relationships between Skyline objects. Like the data cube, the Skycube groups all the multidimensional Skylines and it is so voluminous that reduction approaches are a necessity. In this paper, we define an approach which partially materializes the Skycube. The underlying idea is to discard from the representation the Skycuboids which can be computed again easily. To meet this reduction objective, we characterize a formal framework: the Agree Concept Lattice. It provides a formal framework which makes it possible to improve computation time, reduce representation and easily navigate through the Hasse diagram in order to focus on the most relevant skycuboids. This structure is generic, applies to various database analysis problems and combines both formal concept analysis and database theory. It makes use of the concepts of agree set and database partition. They 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. From this structure, we derive the Skyline concept lattice which is one of its constrained instances for efficient multidimensional Skyline analysis. The strong points of our approach are: (i) it is attribute oriented; (ii) it provides a boundary for the number of lattice nodes; (iii) it facilitates the navigation within the Skycuboids.

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.

Constrained multidimensional patterns differ from the well-known frequent patterns from a conceptual and logical points of view because they are provided with a common structure and support various types of constraints. Classical data mining techniques are based on the power set lattice of binary attributes and, even extended, are not suitable when addressing the discovery of constrained multidimensional patterns. In this paper we propose a foundation for various multidimensional data mining problems by introducing a new algebraic structure called cube lattice which characterizes the search space to be explored. We take into consideration monotone and/or antimonotone constraints enforced when mining multidimensional patterns. In addition, we propose condensed representations of the constrained cube lattice which is a convex space. Finally, we place emphasis on advantages of the cube lattice when compared to the power set lattice of binary attributes used for multidimensional data mining.