Elsevier

Discrete Applied Mathematics

Volume 156, Issue 8, 15 April 2008, Pages 1271-1294
Discrete Applied Mathematics

Spatial classification

https://doi.org/10.1016/j.dam.2007.04.031Get rights and content
Under an Elsevier user license
open archive

Abstract

The aim of a spatial classification is to position the units on a spatial network and to give simultaneously a set of structured classes of these units “compatible” with the network. We introduce the basic needed definitions: compatibility between a classification structure and a tessellation, (m,k)-networks as a case of tessellation, convex, maximal and connected subsets in such networks, spatial pyramids and spatial hierarchies. As like Robinsonian dissimilarities induced by indexed pyramids generalize ultrametrics induced by indexed hierarchies we show that a new kind of dissimilarity called “Yadidean” induced by spatial pyramids generalize Robinsonian dissimilarities. We focus on spatial pyramids where each class is a convex for a grid, and we show that there are several one-to-one correspondences with different kinds of Yadidean dissimilarities. These new results produce also, as a special case, several one-to-one correspondences between spatial hierarchies (resp. standard indexed pyramids) and Yadidean ultrametrics (resp. Robinsonian) dissimilarities. Qualities of spatial pyramids and their supremum under a given dissimilarity are considered. We give a constructive algorithm for convex spatial pyramids illustrated by an example. We show finally by a simple example that spatial pyramids on symbolic data can produce a geometrical representation of conceptual lattices of “symbolic objects”.

Keywords

Pyramidal clustering
Spatial classification
Symbolic data analysis
Conceptual Lattices
Kohonen mapping

Cited by (0)