Classification trees in a box extent lattice

Abstract

Abstract. In this paper we show that, during an elementary extension of a context, each of the classification trees of the newly created box extent lattice can be obtained by modifying the classification trees of the box extent lattice of the original, smaller context. We also devise an algorithm which, starting from a classification tree of the box extent lattice of the smaller context (H, M, I ∩ H × M ), gives a classification tree of the extended context (G, M, I) which contains the new elements inserted. The efficiency of the method is given by the fact that it is sufficient to know the original context, the classification tree of the box extent lattice and its box extents while the knowledge of a new box extension of the extended context mesh elements is not required (except for one, which is the new element box extension). (G, M, I) where G and M are sets and I ⊆ G × M is a binary relation. The elements of G and M are called objects and attributes of the context, respectively. The relation gIm means that the object g has the attribute m. A small context can be easily represented by a cross table, i.e., by a rectangular table with rows headed by the object names and the columns by the attribute names. A cross in the intersection of row g and column m, means that object g has attribute m. For all sets A ⊆ G and B ⊆ M , we define A = {m ∈ M | g I m for all g ∈ A}, B = {g ∈ G | g I m for all m ∈ B}. Preliminaries: Box lattice, extent lattice A context ([2]) is a triple

An incremental method for the construction of the box extents of a context

“…The notion of the box extents of a context and the box elements of a concept lattice are related to the application of Formal Concept Analysis [1] in data classification by defining partitions of a universe putting its elements with similar attributes into the same class, see [4,6] and [2]. This task transparently appears in some clustering problems originating in database analysis, as well as in the engineering discipline Group Technology, see, e.g., [9] or [8]. This discipline exploits the similarities between technological objects and divides them into (relatively) homogeneous groups in order to optimize the manufacturing processes.…”