Constrained Cube Lattices for Multidimensional Database Mining

In multidimensional database mining, constrained multidimensional patterns differ from the well-known frequent patterns from both conceptual and log­ical points of view because of a common structure and the ability to support various types of constraints. Classical data mining techniques are based on the power set lattice of binary attribute values and, even adapted, are not suitable when addressing the discovery of constrained multidimensional patterns. In this chapter, the authors propose a foundation for various multidimensional database mining problems by introducing a new algebraic structure called cube lattice, which characterizes the search space to be explored. This chapter takes into consideration monotone and/or anti-monotone constraints enforced when mining multidimensional patterns. The authors propose condensed representations of the constrained cube lattice, which is a convex space, and present a generalized levelwise algorithm for computing them. Additionally, the authors consider the formalization of existing data cubes, and the discovery of frequent multidimensional patterns, while introducing a perfect concise representation from which any solution provided with its conjunction, disjunction and negation frequencies. Finally, emphasis on advantages of the cube lattice when compared to the power set lattice of binary attributes in multidimensional database mining are placed.

Constrained Cube Lattices for Multidimensional Database Mining

In multidimensional database mining, constrained multidimensional patterns differ from the well-known frequent patterns from both conceptual and log­ical points of view because of a common structure and the ability to support various types of constraints. Classical data mining techniques are based on the power set lattice of binary attribute values and, even adapted, are not suitable when addressing the discovery of constrained multidimen­sional patterns. In this paper, the authors propose a foundation for various multidimensional database mining problems by introducing a new algebraic struc­ture called cube lattice, which characterizes the search space to be explored. This paper takes into consideration monotone and/or anti-monotone constraints en­forced when mining multidimensional patterns. The authors propose condensed representations of the constrained cube lattice, which is a convex space, and present a generalized levelwise algorithm for computing them. Additionally, the authors consider the formalization of existing data cubes, and the discovery of frequent multidimensional patterns, while introducing a perfect concise representation from which any solution provided with its conjunction, disjunction and negation frequencies. Fi­nally, emphasis on advantages of the cube lattice when compared to the power set lattice of binary attributes in multidimensional database mining are placed.

The mode of formation of Neumann bands. Part II.–The evidence that the bands are twins

1. Any discussion of the significance of Neumann bands must involve the geometrical relationships between them and the matrix. The orientation of the cube lattice of a meteorite having been determined by X-ray or other methods, a section parallel to a cube face can be cut, polished and etched, and the angles which the traces of the Neumann bands make with the sides or the diagonals of the cube face can be measured. If the bands are parallel to the twelve planes of the icositetrahedron {112}, their traces will lie in the directions shown in fig. 5, PVUT representing the orientation of the cube face. The spatial relationships of the planes producing such {112} traces can be visualised by means of fig. 6, an isometric projection of a cube, in which the planes producing the traces shown in fig. 5 are indicated by their traces on three mutually perpendicular planes, e. g ., PVUT (010), PTQX (001) and PXWV (100).